Appendix A Plane and Solid Angles

1.0 A.1 Plane Angles y = tan θ y = sin θ The angle θ between two intersecting lines is shown in 0.8 y = θ (radians) Fig. A.1. It is measured by drawing a circle centered on the vertex or point of intersection. The arc length s on that part of the circle contained between the lines measures the angle. 0.6 In daily work, the angle is marked off in degrees. y In some cases, there are advantages to measuring the an- 0.4 gle in radians. This is particularly true when trigonometric functions have to be differentiated or integrated. The angle in radians is defined by 0.2 s θ = . (A.1) r 0.0 0 20 40 60 80 Since the circumference of a circle is 2πr, the angle corre- θ sponding to a complete rotation of 360 ◦ is 2πr/r = 2π. (degrees) Other equivalences are Fig. A.2 Comparison of y = tan θ, y = θ (radians), and y = sin θ Degrees Radians 360 2π π 180 (A.2) One of the advantages of radian measure can be seen in 57.2958 1 Fig. A.2. The functions sin θ,tanθ, and θ in radians are plot- 10.01745 ted vs. angle for angles less than 80 ◦. For angles less than 15 ◦, y = θ is a good approximation to both y = tan θ (2.3 % Since the angle in radians is the ratio of two distances, it is di- error at 15 ◦) and y = sin θ (1.2 % error at 15 ◦). mensionless. Nevertheless, it is sometimes useful to specify that something is measured in radians to avoid confusion.

A.2 Solid Angles

A plane angle measures the diverging of two lines in two dimensions. Solid angles measure the diverging of a cone of lines in three dimensions. Figure A.3 shows a series of rays diverging from a point and forming a cone. The solid angle Fig. A.1 A plane angle θ is measured by the arc length s on a circle of Ω is measured by constructing a sphere of radius r centered radius r centered at the vertex of the lines defining the angle at the vertex and taking the ratio of the surface area S on the

R. K. Hobbie, B. J. Roth, Intermediate Physics for Medicine and Biology, 567 DOI 10.1007/978-3-319-12682-1, c Springer International Publishing Switzerland 2015 568 A Plane and Solid Angles

When the included angle in the cone is small, the differ- ence between the surface area of a plane tangent to the sphere and the sphere itself is small. (This is difficult to draw in three dimensions. Imagine that Fig. A.5 represents a slice through a cone; the difference in length between the circular arc and the tangent to it is small.) This approximation is often useful. A3× 5-in. card at a distance of 6 ft (72 in.) subtends a solid angle which is approximately

3 × 5 − = 2.9 × 10 3 sr. 722 Fig. A.3 A cone of rays in three dimensions It is not necessary to calculate the surface area on a sphere of 72-in. radius.

Problems

Problem 1. Convert 0.1 radians to degrees. Convert 7.5 ◦ to radians. Problem 2. Use the fact that sin θ ≈ θ ≈ tan θ to estimate the sine and tangent of 3 ◦. Look up the values in a table and Fig. A.4 The solid angle of this cone is Ω = S/r2. S is the surface see how accurate the approximation is. area on a sphere of radius r centered at the vertex Problem 3. What is the solid angle subtended by the pupil of the eye (radius = 3 mm) at a source of light 30 m away? Problem 4. Figure A.2 suggests that y = θ is a bet- ter approximation to sin θ than to tan θ and that y = θ overestimates sin θ and underestimates tan θ. Calculate (See Appendix D) or look up the Taylor expansions of sin θ and tan θ and use the first two nonzero terms in each expansion Fig. A.5 For small angles, the arc length is very nearly equal to the to verify this behavior. length of the tangent to the circle Problem 5. What is the solid angle subtended by the “cap” of a sphere from the sphere center, where the “cap” is defined using spherical coordinates (Appendix L) as the surface of sphere enclosed by the cone to r2: the sphere between θ = 0 and θ = 30 ◦. Hint: In spherical S coordinates an element of surface area on a sphere is dS = Ω = . (A.3) r2 r2 sin θdθdφ. This is shown in Fig. A.4 for a cone consisting of the planes defined by adjacent pairs of the four rays shown. The unit of solid angle is the steradian (sr). A complete sphere subtends a solid angle of 4π steradians, since the surface area of a sphere is 4πr2. Appendix B Vectors; Displacement, Velocity, and Acceleration

B.1 Vectors and Vector Addition

A displacement describes how to get from one point to an- other. A displacement has a magnitude (how far point 2 is from point 1 in Fig. B.1) and a direction (the direction one has to go from point 1 to get to point 2). The displacement of point 2 from point 1 is labeled A. Displacements can be added: displacement B from point 2 puts an object at point 3. The displacement from point 1 to point 3 is C and is the sum Fig. B.2 Vectors A and B can be added in either order of displacements A and B:

C = A + B. (B.1) 5 Sum 3 A displacement is a special example of a more general 4 quantity called a vector. One often finds a vector defined as 2 a quantity having a magnitude and a direction. However, the complete definition of a vector also includes the requirement that vectors add like displacements. The rule for adding two 1 vectors is to place the tail of the second vector at the head of the first; the sum is the vector from the tail of the first to the head of the second. Fig. B.3 Addition of several vectors A displacement is a change of position so far in such a direction. It is independent of the starting point. To know where an object is, it is necessary to specify the starting point Displacement B can first be made from point 1 to point 4, as well as its displacement from that point. followed by displacement A from 4 to 3. The sum is still C: Displacements can be added in any order. In Fig. B.2, = + = + either of the vectors A represents the same displacement. C A B B A. (B.2) The sum of several vectors can be obtained by first adding two of them, then adding the third to that sum, and so forth. This is equivalent to placing the tail of each vector at the head of the previous one, as shown in Fig. B.3. The sum then goes from the tail of the first vector to the head of the last. The negative of vector A is that vector which, added to A, yields zero: A + (−A) = 0. (B.3)

Fig. B.1 Displacement C is equivalent to displacement A followed by It has the same magnitude as A and points in the opposite displacement B: C = A + B direction.

R. K. Hobbie, B. J. Roth, Intermediate Physics for Medicine and Biology, 569 DOI 10.1007/978-3-319-12682-1, c Springer International Publishing Switzerland 2015 570 B Vectors; Displacement, Velocity, and Acceleration

Fig. B.5 Addition of components in three dimensions

Fig. B.4 Vector A has components A and A x y pointing in the x, y, and z directions. (In some books, the unit vectors are denoted by ˆi, jˆ, and kˆ instead of xˆ, yˆ, and zˆ.) With this notation, instead of A , one would always write A xˆ. Multiplying a vector A by a scalar (a number with no x x The addition of vectors is often made easier by using associated direction) multiplies the magnitude of vector A components. The sum A + B = C can be written as by that number and leaves its direction unchanged.

Axxˆ + Ayyˆ + Azzˆ + Bxxˆ + Byyˆ + Bzzˆ

= Cxxˆ + Cyyˆ + Czzˆ.

B.2 Components of Vectors Like components can be grouped to give

(Ax + Bx)xˆ + (Ay + By)yˆ + (Az + Bz)zˆ Consider a vector in a plane. If we set up two perpendicular = C xˆ + C yˆ + C zˆ. axes, we can regard vector A as being the sum of vectors x y z parallel to each of these axes. These vectors, Ax and Ay in Therefore, the magnitudes of the components can be added 1 Fig. B.4, are called the components of A along each axis .If separately: vector A makes an angle θ with the x axis and its magnitude C = A + B , is A, then the magnitudes of the components are x x x Cy = Ay + By, (B.7) = + A = A cos θ, Cz Az Bz. x (B.4) Ay = A sin θ.

2 + 2 = The sum of the squares of the components is Ax Ay 2 2 + 2 2 = 2 2 + 2 A cos θ A sin θ A (sin θ cos θ). Since, by B.3 Position, Velocity, and Acceleration Pythagoras’ theorem, this must be A2, we obtain the trigono- metric identity The position of an object at time t is defined by specifying cos2 θ + sin2 θ = 1. (B.5) its displacement from an agreed-upon origin: = + + In three dimensions, A Ax Ay Az. The magnitudes R(t) = x(t)xˆ + y(t)yˆ + z(t)zˆ. can again be related using Pythagoras’ theorem, as shown 2 = 2 + 2 in Fig. B.5. From triangle OPQ, Axy Ax Ay.From The average velocity vav(t1,t2) between times t1 and t2 is triangle OQR, defined to be − 2 = 2 + 2 = 2 + 2 + 2 R(t2) R(t1) A Axy Az Ax Ay Az. (B.6) vav(t1,t2) = . t2 − t1 In our notation, Ax means a vector pointing in the x direc- This can be written in terms of the components as tion, while Ax is the magnitude of that vector. It can become       difficult to keep the distinction straight. Therefore, it is cus- x(t2) − x(t1) y(t2) − y(t1) z(t2) − z(t1) vav = xˆ+ yˆ+ zˆ. tomary to write xˆ, yˆ, and zˆ to mean vectors of unit length t2 − t1 t2 − t1 t2 − t1 The instantaneous velocity is 1 Some texts define the component to be a scalar, the magnitude of the dR dx dy dz v(t) = = xˆ + yˆ + zˆ component defined here. dt dt dt dt Problems 571

= vx(t)xˆ + vy(t)yˆ + vz(t)zˆ. (B.8) t = 0and3s? Problem 2. The position of an object as a function of time The x component of the velocity tells how rapidly the x is R(t) = (20 + 4t)xˆ + (10 + 5t − 49t2)yˆ. Determine the component of the position is changing. instantaneous velocity and acceleration as functions of time. The acceleration is the rate of change of the velocity with Problem 3. The electric field E is a vector (see Chap. 6). −1 time. The instantaneous acceleration is E1 has a magnitude of 30 V m and is directed along the y axis. E has a magnitude of 15 V m−1 and is directed at dv dvx dvy dvz 2 a(t) = = xˆ + yˆ + zˆ. + ◦ = + dt dt dt dt an angle of 30 from the x axis. Calculate E E1 E2. Express your answer in two ways: give the magnitude and direction of E, and give Ex and Ey.

Problems

Problem 1. At t = 0, the position of an object is given by R = 10xˆ + 5yˆ, where R is in meters. At t = 3 s, the position is R = 16xˆ − 10yˆ. What was the average velocity between Appendix C Properties of Exponents and Logarithms

m = = In the expression a , a is called the base and m is called the log10(100). Similarly, 3 log10(1000),4 log10(10, 000), exponent. Since a2 = a × a, a3 = a × a × a, and and so forth. The most useful property of logarithms can be derived by m = × × ×···× a (a a a a), letting m times y = am, it is easy to show that z = an, aman = (a × a × a ×···×a)(a × a × a ×···×a), + m times n times w = am n,

+ so that aman = am n. (C.1) = m loga y, If m>n, the same technique can be used to show that = m n loga z, a − = am n. (C.2) n + = a m n loga w. If m = n, this gives Then, since am+n = aman,

m a − w = yz, 1 = = am m = a0, am = = + loga(yz) loga w loga y loga z. (C.6) This result can be used to show that a0 = 1. (C.3) log(ym) = log(y × y × y ×···×y) The rules also work for m

R. K. Hobbie, B. J. Roth, Intermediate Physics for Medicine and Biology, 573 DOI 10.1007/978-3-319-12682-1, c Springer International Publishing Switzerland 2015 Appendix D Taylor’s Series

Consider the function y(x) shown in Fig. D.1.Thevalueof y(x) y the function at x1, y1 = y(x1), is known. We wish to estimate y(x1 + x). The simplest estimate, labeled approximation 0 in Fig. D.1, is to assume that y does not change: y(x1 + ≈ x) y(x1). A better estimate can be obtained if we as- Approximation 2 sume that y changes everywhere at the same rate it does at x1. Approximation 1 is (dy/dx)| Δx x1  2 2 Δ 2  + (1/2)(d y/dx )|x ( x) dy  1 y(x1 + x) ≈ y(x1) +  x. dx x1

The derivative is evaluated at point x1. An even better estimate is shown in Fig. D.2. Instead of fitting the curve by the straight line that has the proper first Δ derivative at x1, we fit it by a parabola that matches both the x1 x1 + x x first and second derivatives. The approximation is   Fig. D.2 The second-order approximation fits y(x) with a parabola  2  dy  1 d y  2 y(x1 + x) ≈ y(x1) +  x +  (x) . dx 2 dx2 x1 x1 That this is the best approximation can be derived in the following way. Suppose the desired approximation is more n n general and uses terms up to (x) = (x − x1) : y(x) y 2 n yapprox = A0 +A1(x−x1)+A2(x−x1) +···+An(x−x1) . Actual (D.1) The constants A0,A1,...,An are determined by making the value of yapprox and its first n derivatives agree with the value of y and its first n derivatives at x = x1. When x = x1,all terms with x − x1 in yapprox vanish, so that

Approximation 1 yapprox(x1) = A0. Δ (dy/dx)|x x 1 The first derivative of yapprox is y1 Approximation 0 d(yapprox) = A + 2A (x − x ) dx 1 2 1 2 n−1 Δ + 3A3(x − x1) +···+nAn(x − x1) . x1 x1 + x x

Fig. D.1 The zeroth-order and first-order approximations to y(x)

R. K. Hobbie, B. J. Roth, Intermediate Physics for Medicine and Biology, 575 DOI 10.1007/978-3-319-12682-1, c Springer International Publishing Switzerland 2015 576 DTaylor’sSeries

Table D.1 y = e2x and its derivatives 20 y = e2x Function or derivative Value at x1 = 0 2 3 = 2x y = 1 + 2x + 2x + 4x /3 y e 1 15 y = 1 + 2x + 2x2 dy = 2e2x 2 dx y = 1 + 2x d2y 10 = 4e2x 4 dx2 y d3y = 8e2x 8 5 dx3 y = 1 Table D.2 Values of y and successive approximations 0 xy= e2x 1 + 2x 1+2x +2x2 1 + 2x +2x2 + 4 x3 3 -5 −20.0183 −3.05.0 −5.67 −1.50.0498 −2.02.5 −2.0 -2 -1 0 1 2 3 4 5 −10.1353 −1.01.0 −0.33 x −0.40.4493 0.2000 0.5200 0.4347 − 0.20.6703 0.6000 0.6800 0.6693 Fig. D.3 The function y = e2x with Taylor’s series expansions about − 0.10.8187 0.8000 0.8200 0.8187 x = 0ofdegree0,1,2,and3 01.0000 1.0000 1.0000 1.0000 0.11.2214 1.2000 1.2200 1.2213 0.21.4918 1.4000 1.4800 1.4907 2x 3.0 y = e 0.42.2255 1.8000 2.1200 2.2053 2 3 1.07.389 3.0000 5.0000 6.33 y = 1 + 2x + 2x + 4x /3 2.054.60 5.013.023.67 2.5 y = 1 + 2x + 2x2 The second derivative is 2.0 y − 2A + 3 × 2A (x − x ) +···+n(n − 1)A (x − x )n 2, 2 3 1 n 1 1.5 y = 1 + 2x and the nth derivative is y = 1 1.0 n(n − 1)(n − 2) ···2An = n!An. 0.5 Evaluating these at x = x1 gives   d(yapprox) 0.0  = A1, -0.5 0.0 0.5 1.0 dx x x1

 = 2  Fig. D.4 An enlargement of Fig. D.3 near x 0 d (yapprox)  = 2 × 1 × A , dx2  2 x1  3  d (yapprox) larger region about x1 as more terms are added. The function  = 3 × 2 × 1 × A , 2x dx3  3 being approximated is y = e . The derivatives are given in x1 Table D.1. The expansion is made about x = 0.  1 n  Finally, the Taylor’s series expansion for y = ex about d (yapprox)  = n!An. = x x dxn x 0 is often useful. Since all derivatives of e are e ,the x1 value of y and each derivative at x = 0 is 1. The series is Combining these expressions for An with Eq. D.1, we get  N n  ∞ m 1 d y  n x 1 2 1 3 x y(x1 + x) ≈ y(x1) +  (x) . (D.2) e = 1 + x + x + x +···= . (D.3) n! dxn 2! 3! m! n=1 x1 m=0 Tables D.1 and D.2 and Figs. D.3 and D.4 show how the Taylor’s series approximation gets better over a larger and (Note that 0!=1 by definition.) Problems 577

Problems Hint: first make a Taylor series expansion of sin x and then divide by x. Problem 1. Make a Taylor’s series expansion of y = a + Problem 5. Derive a Taylor’s series of y = 1/(1 − x) bx + cx2 about x = 0. Show that the expansion exactly about x = 0. Plot y(x) vs x including approximation 0, reproduces the function. approximation 1, and approximation 2 as in Figs. D.1 and Problem 2. Repeat the previous problem, making the expan- D.2. sion about x = 1. Problem 6. Derive a Taylor’s series of y(x) = ln(1 + x) Problem 3. (a) Make a Taylor’s series expansion of the co- about x = 0. Plot y(x) vs x, including approximation 0, sine function about x = 0. Remember that d(sin x)/dx = approximation 1, and approximation 2, as in Figs. D.1 and cos x and d(cos x)/dx =−sin x. D.2. (b) Make a Taylor’s series expansion of the sine function. Problem 4. The “sinc” function is defined as sin x/x.Make a Taylor’s series expansion of the sinc function about x = 0. Appendix E Some Integrals of Sines and Cosines

The average of a function of x with period T is defined to be 1

 (a) (b) 1 x +T f  = f(x)dx. (E.1) T x 0 π The sine function is plotted in Fig. E.1a. The integral over 0 2 a period is zero, and its average value is zero. The area above the axis is equal to the area below the axis. Figure E.1bshows a plot of sin2 x. Since sin x varies between −1 and +1, sin2 x -1 varies between 0 (when sin x = 0) and +1 (when sin x = ± 1 Fig. E.2 Plot of one period of a y = sin x sin 2x; b y = sin x cos x 1). Its average value, from inspection of Fig. E.1bis 2 .If you do not want to trust the drawing to convince yourself of this, recall the identity sin2 θ + cos2 θ = 1. Since the sine function and the cosine function look the same, but are just These could be used to show that the average value of sin x shifted along the axis, their squares must also look similar. or cos x is zero. Then Eq. E.2 could be used to show that the Therefore, sin2 θ and cos2 θ must have the same average. But average of sin2 x is 1 . if their sum is always 1, the sum of their averages must be 1. 2 The integral of sin2 x over a period is its average value If the two averages are the same, then each must be 1 . 2 times the length of the period: These same results could have been obtained analytically by using the trigonometric identity T T T sin2 xdx= cos2 xdx= . (E.4) 2 1 1 sin x = − cos 2x. (E.2) 0 0 2 2 2 The integrals of sin x and cos x are We will also encounter integrals like

1 sin ax dx =− cos ax, T a = (E.3) sin mx sin nx dx, m n, 1 0 cos ax dx = sin ax. a

T cos mx cos nx dx, m = n, (E.5) 1 1 0 Av = 1/2

T Av = 0 cos mx sin nx dx, m = n, m = n. 0 0 0 All these integrals are zero. This can be shown using in- tegral tables. Or, you can see why the integrals vanish by -1 -1 considering the specific examples plotted in Fig. E.2. Each integrand has equal positive and negative contributions to the Fig. E.1 a Plot of y = sin x. b Plot of y = sin2 x total integral. R. K. Hobbie, B. J. Roth, Intermediate Physics for Medicine and Biology, 579 DOI 10.1007/978-3-319-12682-1, c Springer International Publishing Switzerland 2015 580 E Some Integrals of Sines and Cosines

Problems Problem 2. Use the trigonometric relationship 1 Problem 1. Plot the following functions over the range 0 sin A sin B = [cos(A − B) − cos(A + B)], to 2π as in Fig. E.2, and show by inspection that the neg- 2 1 ative and positive areas cancel, giving an integral of zero: cos A cos B = [cos(A − B) + cos(A + B)], cos θ sin 2θ,sin2θ sin 3θ, cos θ cos 2θ, and cos 2θ sin 2θ. 2 1 sin A cos B = [sin(A − B) + sin(A + B)], 2 to verify that all of the integrals in Eq. E.5 are zero. Appendix F Linear Differential Equations with Constant Coefficients

The equation is called the characteristic equation of this differential equa- dy tion. It can be written in a much more compact form using + by = a (F.1) dt summation notation: is called a linear differential equation because each term in- volves only y or its derivatives (not y(dy/dt) or (dy/dt)2, N n = etc.). A more general equation of this kind has the form bns 0, (F.5) n=0 dN y dN−1y dy + b − +···+b + b y = f(t). N N 1 N−1 1 0 (F.2) dt dt dt with bN = 1. + = The highest derivative is the Nth derivative, so the equation For Eq. F.1, the characteristic equation is s b 0or =− = is of order N. It has been written in standard form by di- s b, and a solution to the homogeneous equation is y Ae−bt . viding through by any bN that was there originally, so that the coefficient of the highest term is one. If all the bsare If the characteristic equation is a polynomial, it can have = snt constants, this is a linear differential equation with constant up to N roots. For each distinct root sn, y Ane is a so- coefficients. The right-hand side may be a function of the lution to the differential equation. (The question of solutions independent variable t,butnot of y.Iff(t)= 0, it is a homo- when there are not N distinct roots will be taken up below.) geneous equation; if f(t)is not zero, it is an inhomogeneous This is still not the solution to the inhomogeneous equation. 1 equation. However, one can prove that the most general solution to Consider first the homogeneous equation the inhomogeneous equation is the sum of the homogeneous solution, dN y dN−1y dy + b − +···+b + b y = . N N N 1 N−1 1 0 0 (F.3)  dt dt dt snt y = Ane , The exponential est (where s is a constant) has the property n=1 that d(est)/dt = sest, d2(est)/dt2 = s2est, dn(est)/dtn = and any solution to the inhomogeneous equation. The val- snest. The function y = Aest satisfies Eq. F.3 for any value ues of the arbitrary constants A are picked to satisfy some of A and certain values of s. The equation becomes n other conditions that are imposed on the problem. If we can

N st N−1 st st st guess the solution to the inhomogeneous equation, that is A s e + b − s e +···+b se + b e = 0, N 1 1 0 fine. However we get it, we need only one such solution to the inhomogeneous equation. We will not prove this as- N N−1 st sertion, but we will apply it to the first- and second-order A s + bN−1s +···+b1s + b0 e = 0. equations and see how it works. This equation is satisfied if the polynomial in parentheses is equal to zero. The equation

N N−1 s + b − s +···+b s + b = 0(F.4) N 1 1 0 1 See any calculus text.

R. K. Hobbie, B. J. Roth, Intermediate Physics for Medicine and Biology, 581 DOI 10.1007/978-3-319-12682-1, c Springer International Publishing Switzerland 2015 582 F Linear Differential Equations with Constant Coefficients

F.1 First-Order Equation cos(ωt + φ). The rows are labeled on the left by which term of the differential equation they came from. The homogeneous equation corresponding to Eq. F.1 has solution y = Ae−bt . There is one solution to the inhomoge- Term Coefficients + + neous equation that is particularly easy to write down: when sin(ωt φ) cos(ωt φ) y is constant, with the value y = a/b, the time derivative d2y/dt2 α2 − ω2 −2αω vanishes and the inhomogeneous equation is satisfied. The b1(dy/dt) −b1αb1ω most general solution is therefore of the form b0yb0 0

− a y = Ae bt + . The only way that the equation can be satisfied for all times is b if the coefficient of the sin(ωt + φ) term and the coefficient of the cos(ωt + φ) term separately are equal to zero. This If the initial condition is y(0) = 0, then A can be determined means that we have two equations that must be satisfied (call from 0 = Ae−b0 + a/b. Since e0 = 1, this gives A =−a/b. b = ω2): Therefore, 0 0 = a − 2αω b1ω, y = 1 − e bt . (F.6) b 2 − 2 − + 2 = α ω b1α ω0 0. A physical example of this is given in Sect. 2.8. From the first equation 2α = b1, while from this and the 2 − 2 − 2 + 2 = 2 = 2 − 2 second, α ω 2α ω0 0, or ω ω0 α . Thus, the solution to the equation

d2y dy F.2 Second-Order Equation + 2α + ω2y = 0(F.9) dt2 dt 0 The second-order equation is − y = Ae αt sin(ωt + φ) (F.10a) d2y dy + b1 + b0y = 0(F.7)where dt2 dt 2 = 2 − 2 ω ω0 α ,α<ω0. (F.10b) has a characteristic equation s2 + b s + b = 0 with roots 1 0 Solution F.10 is a decaying exponential multiplied by a - sinusoidally varying term. The initial amplitude A and the −b ± b2 − b 1 1 4 0 phase angle φ are arbitrary and are determined by other con- s = . (F.8) 2 ditions in the problem. The constant α is called the damping. Parameter ω is the undamped frequency, the frequency of This equation may have zero, one, or two solutions. 0 oscillation when α = 0. ω is the damped frequency. If it has two solutions s1 and s2, then the general solution When the damping becomes so large that α = ω0, then the of the homogeneous equation is y = A es1t + A es2t . 1 2 solution given above does not work. In that case, the solution If b2 − 4b is negative, there is no solution to the equation 1 0 is given by for a real value of s. However, a solution of the form y = −αt + −αt Ae sin(ωt φ) will satisfy the equation. This can be seen y = (A + Bt)e ,α= ω0. (F.11) by direct substitution. Differentiating this twice shows that This case is called critical damping and represents the case dy − − =−αAe αt sin(ωt + φ) + ωAe αt cos(ωt + φ), in which y returns to zero most rapidly and without multiple dt oscillations. The solution can be verified by substitution. If α>ω0, then the solution is the sum of the two 2 exponentials that satisfy Eq. F.8: d y = 2 −αt + α Ae sin(ωt φ) − − dt2 y = Ae at + Be bt , (F.12a) − − − 2αωAe αt cos(ωt + φ) − ω2Ae αt sin(ωt + φ). where - If these derivatives are substituted in Eq. F.7, one gets the = + 2 − 2 a α α ω0, (F.12b) following results. The terms are written in two columns. One column contains the coefficients of terms with sin(ωt + φ), - = − 2 − 2 and the other column contains the coefficients of terms with b α α ω0. (F.12c) F.2 Second-Order Equation 583

Table F.1 Solutions of the harmonic oscillator equation d2y dy + 2α + ω2y = 0 dt2 dt 0 Case Criterion Solution −αt Underdamped α<ω0 y = Ae sin(ωt + φ) 2 = 2 − 2 ω ω0 α −αt Critically α = ω0 y = (A + Bt)e Fig. F.1 Different starting points on the sine wave give different damped combinations of the initial position and the initial velocity −at −bt Overdamped α>ω0 y = Ae + Be = + 2 − 2 1/2 a α (α ω0) = − 2 − 2 1/2 b α (α ω0) When α = 0, the equation is

d2y + ω2y = 0. (F.13) dt2 0 The solution may be written either as

y = C sin(ω0t + φ) (F.14a) C sin(ω0t + φ) = C[sin ω0t cos φ + cos ω0t sin φ]. Compar- or as ison with Eq. F.14b shows that B = C cos φ, A = C sin φ. Squaring and adding these gives C2 = A2 + B2, while y = A cos(ω0t) + B sin(ω0t). (F.14b) dividing one by the other shows that tan φ = A/B. The simplest physical example of this equation is a mass Changing the initial phase angle changes the relative val- on a spring. There will be an equilibrium position of the mass ues of the initial position and velocity. This can be seen from = (y 0) at which there is no net force on the mass. If the mass the three plots of Fig. F.1, which show phase angles 0, π/4, is displaced toward either positive or negative y, a force back and π/2. When φ = 0, the initial position is zero, while the toward the origin results. The force is proportional to the dis- initial velocity has its maximum value. When φ = π/4, the =− placement and is given by F ky. The proportionality initial position has a positive value, and so does the initial constant k is called the spring constant. Newton’s second velocity. When φ = π/2, the initial position has its max- F = ma m(d2 2 =− 2 = law, ,is y/dt ) ky or, defining ω0 k/m, imum value and the initial velocity is zero. The values of A and B are determined from the initial position and veloc- d2y + ω2y = 0. ity. At t = 0, Eq. F.14b and its derivative give y(0) = A, dt2 0 dy/dt(0) = ω0B. This (as well as the equation with α = 0) is a second-order The term in the differential equation equal to 2α(dy/dt) differential equation. Integrating it twice introduces two corresponds to a drag force acting on the mass and damping constants of integration: C and φ,orA and B.Theyare the motion. Increasing the damping coefficient α increases usually found from two initial conditions. For the mass the rate at which the oscillatory behavior decays. Figure F.2 on the spring, they are often the initial velocity and initial shows plots of y and dy/dt for different values of α. position of the mass. The second-order equation we have just studied is called The equivalence of the two solutions can be demonstrated the harmonic oscillator equation. Its solution is summarized by using Eqs. F.14a and a trigonometric identity to write in Table F.1.

Underdamped Critically damped Overdamped

t t t

Fig. F.2 Plot of y(t) (solid line) and dy/dt (dashed line) for different values of α 584 F Linear Differential Equations with Constant Coefficients

Problems (a) for critical damping, α = ω0, (b) for no damping, and Problem 1. From Eq. F.14 with ω0 = 10, find A, B, C, and (c) for overdamping, α = 2ω0. φ for the following cases: Problem 4. Show using numerical examples or physical ar- (a) y(0) = 5, (dy/dt)(0) = 0. guments that the overdamped and critically damped solutions (b) y(0) = 5, (dy/dt)(0) = 5. can cross the y = 0 axis at most once. Draw a plot of one (c) y(0) = 0, (dy/dt)(0) = 50. such case. (d) What values of A, B, and C would be needed to have Problem 5. Start with Eq. F.9. Add the function f(t) = the same φ as in case (b) and the same amplitude as in sin ω1t to the right-hand side so you have an inhomogeneous case (a)? equation. Search for a solution to the inhomogeneous equa- Problem 2. Verify Eq. F.11 in the critically damped case. tion (sometimes called a particular solution) by guessing that Problem 3. Find the general solution of the equation y(t) = A sin ω1t +B cos ω1t. Put this back in the differential % equation and find values of A and B that satisfy the equation. d2y dy 0,t≤ 0 + 2α + ω2y = For what values of ω1 will A and B be largest? This is an ex- 2 0 2 ≥ dt dt ω0y0,t 0 ample of resonance: when the system is driven at its natural frequency, the response is largest. subject to the initial conditions y(0) = 0, (dy/dt)(0) = 0 Appendix G The Mean and Standard Deviation

In many measurements in physics or biology there may be The table and the histogram give all the information that several possible outcomes to the measurement. Different val- there is to know about the experiment unless the result de- ues are obtained when the measurement is repeated. For pends on some variable that was not recorded, such as the age example, the measurement might be the number of red cells of the student or where the student was sitting during the test. in a certain small volume of blood, whether a person is right In many cases the frequency distribution gives more in- handed or left handed, the number of radioactive disintegra- formation than we need. It is convenient to invent some tions of a certain sample during a 5-min interval, or the scores quantities that will answer the questions: Around what val- on a test. ues do the results cluster? How wide is the distribution of Table G.1 gives the scores on an examination admin- results? Many different quantities have been invented for an- istered to 30 people. These results are also plotted as a swering these questions. Some are easier to calculate or have histogram in Fig. G.1. more useful properties than others. The mean or average shows where the distribution is cen- Table G.1 Quiz scores tered. It is familiar to everyone: add up all the scores and Student No. Score Student No. Score divide by the number of students. For the data given above, 1801671 the mean is x = 77.8. 2681783 It is often convenient to group the data by the value ob- 3901888 tained, along with the frequency of that value. The data of 4721975 5652069 Table G.1 are grouped this way in Table G.2. The mean is 6812150 calculated as  7852281  8932394 1 i fixi x = fixi =  , 9762473 N i fi 10 86 25 79 i 11 80 26 82 where the sum is over the different values of the test scores 12 88 27 78 13 81 28 84 that occur. For the example in Table G.2, the sums are = = = = 14 72 29 74 i fi 30, i fixi 2335, so x 2335/30 77.8. 15 67 30 70 If a large number of trials are made, fi/N can be called the probability pi of getting result xi. Then 6  _ x = x p . (G.1) 5 x i i σσ i 4  3 Note that pi = 1. 2 The average of some function of x is  1 g(x) = g(xi)pi. (G.2) Frequency of score, f 0 i 0 20 40 60 80 100 Score, x For example,  2 2 x = (xi) pi. Fig. G.1 Histogram of the quiz scores in Table G.1 i

R. K. Hobbie, B. J. Roth, Intermediate Physics for Medicine and Biology, 585 DOI 10.1007/978-3-319-12682-1, c Springer International Publishing Switzerland 2015 586 G The Mean and Standard Deviation

Table G.2 Quiz scores grouped by score gives Eq. G.4. In summary, Score Score xi Frequency of - number i score, f f x i i i = 2 1501 50 σ (x) , 2651 65 (G.4) 3671 67 σ 2 = (x)2 = (x − x)2 = x2 − x2. 4681 68 5691 69 This equation is true as long as the p s are accurately known. 6701 70 i 7711 71 If the pis have only been estimated from N experimental 8 72 2 144 observations, the best estimate of σ 2 is N/(N − 1) times the 9731 73 value calculated from Eq. G.4. 10 74 1 74 For the data of Fig. G.1, σ = 9.4. This width is shown 11 75 1 75 12 76 1 76 along with the mean at the top of the figure. 13 78 1 78 14 79 1 79 15 80 2 160 16 81 3 243 17 82 1 82 18 83 1 83 Problems 19 84 1 84 20 85 1 85 Problem 1. Calculate the variance and standard deviation 21 86 1 86 for the data in Table G.2. 22 88 2 176 23 90 1 90 Problem 2. Use the data in Table G.1 to calculate the mean 2 2 24 93 1 93 of the squares and then verify that σ = x2 − x . 25 94 1 94 Problem 3. Another way to characterize the distribution of values is the mode: the most common value recorded, or the one with the highest probability. Find the mode of the data in The width of the distribution is often characterized by the Table G.1. dispersion or variance:  Problem 4. Still another way to describe a distribution is 2 2 2 the median: line up all the values in order from the smallest (x) = (x − x) = pi(xi − x) . (G.3) i to largest, and find the middle value. (If you have an even number of values, average the middle two.) Find the median This is also sometimes called the mean square variation: the of the data in Table G.1. mean of the square of the variation of x from the mean. A Problem 5. Find the mean and standard deviation of the fol- measure of the width is the square root of this, which is called lowing data: 14, 8, 12, 13, 7, 7, 11 and 9. You do not have the standard deviation σ. The need for taking the square root enough data to know the probabilities accurately, so use the is easy to see since x may have units associated with it. If x factor N/(N − 1) to calculate the variance. is in meters, then the variance has the units of square meters. Problem 6. Imagine that the data in Table G.1 represent The width of the distribution in x must be in meters. 30 measurements of some quantity. The measurements con- A very useful result is tain errors, which explain why the values are not all the same. One property of the mean and standard deviation is − 2 = 2 − 2 (x x) x x . that approximately two-third of the measurements should fall within the range x ± σ . (This is true for a Gaussian To prove this, note that (x − x)2 = x2 − 2x x + x2.The i i i distribution of data and is approximately true for many oth- variance is then    ers.) Check whether this is approximately true for the data in 2 = 2 − + 2 Table G.1. (x) pixi 2 xi xpi pix . i i i Problem 7. Suppose that you make a set of measurements that have mean x and standard deviation σ . You now repeat The first sum is the definition of x2. The second sum has a this set of measurements N times, so that you have N mean number x in every term. It can be factored in front of the sum, values. These mean values will have a distribution that is − − 2 to make the second term 2x xipi, which is just 2(x) . narrower than the distribution of the values in a single set 2 2 The last term is (x) pi = (x) . Combining all three sums of measurements. The√standard deviation of the mean is de- noted by σmean = σ/ N. Calculate the standard deviation of the mean for the data in Table G.1. Appendix H The Binomial Probability Distribution

Consider an experiment with two mutually exclusive out- The remaining factor, pn(1 − p)N−n, is the probability of comes, which is repeated N times, with each repetition being taking n tries in a row and having success and taking N − n independent of every other. One of the outcomes is labeled tries in a row and having failure. “success”; the other is called “failure.” The experiment could The binomial distribution applies if each “try” is indepen- be throwing a die with success being a three, flipping a coin dent of every other try. Such processes are called Bernoulli with success being a head, or placing a particle in a box with processes (and the binomial distribution is often called the success being that the particle is located in a subvolume v. Bernoulli distribution). In contrast, if the probability of an In a single try, call the probability of success p and the outcome depends on the results of the previous try, the ran- probability of failure q. Since one outcome must occur and dom process is called a Markov process. Although such both cannot occur at the same time, processes are important, they are more difficult to deal with and are not discussed here. p + q = 1. (H.1) Some examples of the use of the binomial distribution are given in Chap. 3. As another example, consider the prob- Suppose that the experiment is repeated N times. The prob- lem of performing several laboratory tests on a patient. In ability of n successes out of N tries is given by the binomial the 1970s it became common to use automated machines probability distribution, which is stated here without proof.1 for blood-chemistry evaluations of patients; such machines We can call the probability P(n; N), since it is a function automatically performed (and reported) 6, 12, 20, or more of n and depends on the parameter N. Strictly speaking, it tests on one small sample of a patient’s blood serum, for less depends on two parameters, N and p: P(n; N,p).Itis2 cost than doing just one or two of the tests. But this meant that the physician got a large number of results—many more   N! − than would have been asked for if the tests were done one at P(n; N) = P(n; N,p) = pn(1 − p)N n. n!(N − n)! a time. When such test batteries were first done, physicians (H.2) were surprised to find that patients had many more abnormal The factor N!/[n!(N −n)!] counts the number of different tests than they expected. This was in part because some tests ways that one can get n successful outcomes; the probabil- were not known to be abnormal in certain diseases, because ity of each of these ways is pn(1 − p)N−n. In the example no one had ever looked at them in that disease. But there still of three particles in Sect. 3.1, there are three ways to have was a problem that some tests were abnormal in patients who one particle in the left-hand side. The particle can be either appeared to be perfectly healthy. particle a or particle b or particle c. The factor gives directly We can understand why by considering the following ide-   alized situation. Suppose that we do N independent tests, N! 3! 3 × 2 × 1 6 and suppose that in healthy people, the probability that each = = = = 3. n!(N − n)! 1!2! (1)(2 × 1) 2 test is abnormal is p. (In our vocabulary, having an abnor- mal test is “success”!). The probability of not having the test abnormal is q = 1 − p. In a perfect test, p wouldbe0for healthy people and would be 1 in sick people; however, very 1 A detailed proof can be found in many places. See, for example, F. few tests are that discriminating. The definition of normal Reif (1964). Statistical Physics.BerkeleyPhysicsCourse,Vol.5.New York, McGraw-Hill, p. 67. vs abnormal involves a compromise between false positives 2 N! is N factorial and is N(N − 1)(N − 2) ···1. By definition, 0!=1. (abnormal test results in healthy people) and false negatives (normal test results in sick people). Good reviews of this

R. K. Hobbie, B. J. Roth, Intermediate Physics for Medicine and Biology, 587 DOI 10.1007/978-3-319-12682-1, c Springer International Publishing Switzerland 2015 588 H The Binomial Probability Distribution

1.0 N N N!n − 34 Clinically normal patients n = nP (n; N) = pn(1 − p)N n. n!(N − n)! 0.8 Binomial distribution n=0 n=0 P(n,12) if p = 0.05 The first term of each sum is for n = 0. Since each term is multiplied by n, the first term vanishes, and the limits of the 0.6 sum can be rewritten as N N!n − pn(1 − p)N n. 0.4 n!(N − n)! n=1 Fraction of patients To evaluate this sum, we use a trick. Let m = n−1 and M = 0.2 N − 1. Then we can rewrite various parts of this expression as follows: n 1 1 = = , 0.0 n! (n − 1)! m! 0 1 2 3 n = m Number of abnormal tests p pp , N!=(N)(N − 1)!, Fig. H.1 Measurement of the probability that a clinically normal pa- tient having a battery of 12 tests done has n abnormal tests (solid line) (N − n)!=[N − 1 − (n − 1)]!=(M − m)!. and a calculation based on the binomial distribution (dashed line). The = = = = The limits of summation are n 1orm 0, and n N or calculation assumes that p 0.05 and that all 12 tests are independent. = Several of the tests in this battery are not independent, but the general m M. With these substitutions features are reproduced M M! − n = Np pm(1 − p)M m. m!(M − m)! m=0 problem have been written by Murphy and Abbey3 and by This sum is exactly the sum of a binomial distribution over Feinstein.4 In many cases, p is about 0.05. Now suppose that all possible values of m and is equal to one. We have the p is the same for all the tests and that the tests are indepen- result that, for a binomial distribution, dent. Neither of these assumptions is very good, but they will n = Np. (H.3) show what the basic problem is. Then, the probability for all of the N tests to be normal in a healthy patient is given by This says that the average number of successes is the total the binomial probability distribution: number of tries times the probability of a success on each try. If 100 particles are placed in a box and we look at half N! the box so that p = 1 , the average number of particles in P( ; N,p) = p0qN = qN . 2 0 ! ! × 1 = 0 N that half is 100 2 50. If we put 500 particles in the box 1 and look at 10 of the box, the average number of particles If p = 0.05, then q = 0.95, and P(0; N,p) = 0.95N . Typ- in the volume is also 50. If we have 100,000 particles and ical values are P(0; 12) = 0.54 and P(0; 20) = 0.36. If the v/V = p = 1/2000, the average number is still 50. assumptions about p and independence are right, then only For the binomial distribution, the variance σ 2 can be ex- 36 % of healthy patients will have all their tests normal if 20 pressed in terms of N and p using Eq. G.4. The average of tests are done. n2 is Figure H.1 shows a plot of the number of patients in a se-  N N! − ries who were clinically normal but who had abnormal tests. n2 = P(n; N)n2 = n2pn(1 − p)N n. n!(N − n)! The data have the general features predicted by this simple n n=0 model. 2 = − + We can derive simple expressions to give the mean and The trick to evaluate this is to write n n(n 1) n. With standard deviation if the probability distribution is binomial. this substitution we get two sums: Themeanvalueofn is defined to be N N! − n2 = n(n − 1)pnqN n n!(N − n)! n=0 3 E. A. Murphy and H. Abbey (1967). The normal range—a common N ! N n n N−n misuse. J Chronic Dis 20: 79. + p q . n!(N − n)! 4 A. R. Feinstein (1975). Clinical biostatistics XXVII. The derange- n=0 ments of the normal range. Clin Pharmacol Therap 15: 528. Problems 589

0.6 Problems

0.4 Problem 1. Calculate the probability of throwing 0, 1,...,9 1/2 heads out of a total of nine throws of a coin. Problem 2. Assume that males and females are born with

[p(1-p)] 0.2 equal probability. What is the probability that a couple will have four children, all of whom are girls? The couple has had three girls. What is the probability that they will have a 0.0 0.0 0.2 0.4 0.6 0.8 1.0 fourth girl? Why are these probabilities so different? Problem 3. The Mayo Clinic reported that a single stool p specimen in a patient known to have an intestinal parasite   Fig. H.2 Plot of p(1 − p) 1/2 yields positive results only about 90 % of the time (R. B. Thomson, R. A. Haas, and J. H. Thompson, Jr. (1984). In- testinal parasites: The necessity of examining multiple stool Mayo Clin Proc 59 The second sum is n = Np. The first sum is rewritten by specimens. : 641–642). What is the prob- noticing that the terms for n = 0 and n = 1 both vanish. Let ability of a false negative if two specimens are examined? m = n − 2 and M = N − 2: Three? Problem 4. The Minneapolis Tribune on October 31, 1974, M ! listed the following incidence rates for cancer in the Twin M 2 m M−m n2 = Np + N(N − 1) p p q Cities greater metropolitan area, which at that time had a to- m!(M − m)! m=0 tal population of 1.4 million. These rates are compared to = Np + N(N − 1)p2 = Np + N 2p2 − Np2. those in nine other areas of the country whose total popula- tion is 15 million. Assume that each study was for 1 year. Are Therefore, the differences statistically significant? Show calculations to support your answer. How would your answer differ if the 2 2 (n)2 = n2 − n = Np − Np = Np(1 − p) = Npq. study were for several years?

For the binomial distribution, then, Type of cancer Incidence per 100,000 per year $ $ Twin Cities Other = = σ Npq nq. Colon 35.630.9 Lung (women) 34.240.0 The standard deviation for the binomial distribution for Lung (men) 63.672.0 fixed p goes as N 1/2. For fixed N, it is proportional to √ Breast (women) 81.373.8 p(1 − p), which is plotted in Fig. H.2. The maximum Prostate (men) 69.960.8 value of σ occurs when p = q = 1 .Ifp is very small, 2 Overall 313.8 300.0 the event happens rarely; if p is close to 1, the event nearly always happens. In either case, the variation is reduced. On Problem 5. The probability that a patient with cystic fibro- the other hand, if N becomes large while p becomes small in sis gets a bad lung illness is 0.5 % per day. With treatment, such a way√ as to keep n fixed, then σ increases to a maximum which is time consuming and not pleasant, the daily probabil- value of n. This variation of σ with N and p is demon- ity is ten times less.5 Show that the probability of not having strated in Fig. H.3. Figure H.3a–c shows how σ changes as an illness in a year is 16 % without treatment and 83 % with N is held fixed and p is varied. For N = 100, p is 0.05, 0.5, treatment. and 0.95. Both the mean and σ change. Comparing Fig. H.3b with H.3d shows two different cases where n = 50. When p is very small because N is very large in Fig. H.3d, σ is larger 5 than in Fig. H.3b. These numbers are from W. Warwick, MD, private communication. See also A. Gawande, The bell curve. The New Yorker, December 6, 2004, pp. 82–91. 590 H The Binomial Probability Distribution

0.20 (a) N = 100 (c)

p = 0.05 0.15 p = 0.95 _ _ n = Np = 5 n = Np = 95

σ2 2 P(n) 0.10 = Np(1-p) σ = Np(1 - p) = 100(.05)(.95) = 100(0.95)(0.05) = 4.75 = 4.75

σ 0.05 = 2.18 σ = 2.18

0.00 0 20 40 60 80 100 n -3 100x10 (b)

80 N = 100 p = 0.5 60 _ P(n) n = 50 40 σ = 5 20

0 0 20 40 60 80 100 n 100x10-3

80 (d)

60 N = 1,000,000 P(n) p = 0.00005 _ 40 n = 50 σ = 7.07 20

0 0 20 40 60 80 100 n

Fig. H.3 Examples of the variation of σ with N and p.(a), (b), and (c) show variations of σ with p when N is held fixed. The maximum value of σ occurs when p = 0.5. Note that (a)and(c) are both in the top panel. Comparison of (b)and(d) shows the variation of σ as p and N change together in such a way that n remains equal to 50 Appendix I The Gaussian Probability Distribution

Appendix H considered a process that had two mutually 3 exclusive outcomes and was repeated N times, with the y=ln(m) probability of “success” on one try being p. If each try is in- 2 dependent, then the probability of n occurrences of success y in N tries is 1 N! − P(n; N,p) = pn(1 − p)N n. (I.1) n!(N − n)! 0 2 4 6 8 10 12 14 This probability distribution depends on two parameters N m and p. We have seen two other parameters, the mean, which roughly locates the center of the distribution, and the stan- Fig. I.1 Plot of y = ln m used to derive Stirling’s approximation dard deviation, which measures its width. These parameters, n and σ , are related to N and p by the equations Table I.1 Accuracy of Stirling’s approximation nn! ln(n!)nln n − n Error % Error n = Np, 5 120 4.7875 3.047 1.74 36 10 3.6 × 106 15.104 13.026 2.08 14 σ 2 = Np(1 − p). 20 2.4 × 1018 42.336 39.915 2.42 6 100 9.3 × 10157 363.74 360.51 3.23 0.8 It is possible to write the binomial distribution formula in terms of the new parameters instead of N and p. At best, however, it is cumbersome, because of the need to evaluate To derive it, write ln(n!) as so many factorial functions. We will now develop an approx- n imation that is valid when N is large and which allows the ln(n!) = ln 1 + ln 2 +···+ln n = ln m. probability to be calculated more easily. m=1 The procedure is to take the log of the probability, y = ln(P ) and expand it in a Taylor’s series (Appendix D) about The sum is the same as the total area of the rectangles in some point. Since there is a value of n for which P has a Fig. I.1, where the height of each rectangle is ln m and the maximum and since the logarithmic function is monotonic, width of the base is one. The area of all the rectangles is y has a maximum for the same value of n. We will expand approximately the area under the smooth curve, which is a plot of ln m. The area is approximately about that point; call it n0. Then the form of y is    2  n dy  1 d y  2 ln mdm= [m ln m − m]n = n ln n − n + 1. y = y(n0) +  (n − n0) +  (n − n0) +··· . 1 dn 2 dn2 1 n0 n0 Since y is a maximum at n , the first derivative vanishes and This completes the proof of Eq. I.2. Table I.1 shows values of 0 ! it is necessary to keep the quadratic term in the expansion. n and Stirling’s approximation for various values of n.The To take the logarithm of Eq. I.1, we need a way to handle approximation is not too bad for n>100. the factorials. There is a very useful approximation to the We can now return to the task of deriving the binomial factorial, called Stirling’s approximation: distribution. Taking logarithms of Eq. I.1, we get

ln(n!) ≈ n ln n − n. (I.2) y = ln P = ln(N!) − ln(n!) − ln(N − n)! +n ln p + (N − n) ln(1 − p). R. K. Hobbie, B. J. Roth, Intermediate Physics for Medicine and Biology, 591 DOI 10.1007/978-3-319-12682-1, c Springer International Publishing Switzerland 2015 592 I The Gaussian Probability Distribution

With Stirling’s approximation, this becomes

y = N ln N − n ln n − N ln(N − n) + n ln(N − n) + n ln p + (N − n) ln(1 − p). (I.3)

The derivative with respect to n is dy =−ln n + ln(N − n) + ln p − ln(1 − p). dn Fig. I.2 Evaluating the normalization constant The second derivative is

d2y 1 1 =− − . dn2 n N − n

The point of expansion n0 is found by making the first derivative vanish: (N − n)p 0 = ln . n(1 − p)

Since ln 1 = 0, this is equivalent to (N − n0)p = n0(1 − p) Fig. I.3 The allowed values of x are closely spaced in this case or n0 = Np. The maximum of y occurs when n is equal to the mean. At n = n0, the value of the second derivative is Therefore the normalization integral is (letting x = n − n) d2y 1 1 1 =− − =− . ∞ $ dn2 Np N(1 − p) Np(1 − p) − 2 2 e x /2σ dx = 2πσ 2. = −∞ It is still necessary to evaluate y0 y(n0).Ifwetryto √ = 2 do this by substitution of n n0 in Eq. I.3, we get zero. The The normalization constant is C0 = 1/ 2πσ , so that the reason is that the Stirling approximation we used is too crude Gaussian or normal probability distribution is for this purpose. (There are additional terms in Stirling’s ap- 1 − − 2 2 proximation that make it more accurate.) The easiest way to P(n)= √ e (n n) /2σ . (I.4) 2 find y(n0) is to call it y0 for now and determine it from the re- 2πσ quirement that the probability be normalized. Therefore, we It is possible, as in the case of the random-walk problem, have that the measured quantity x is proportional to n with a very 1 y = y − (n − Np)2 small proportionality constant, x = kn, so that the values 0 − 2Np(1 p) of x appear to form a continuum. As shown in Fig. I.3,the so that, in this approximation, number of different values of n (each with about the same value of P(n)) in the interval dx is proportional to dx.The − − 2 [ − ] P(n)= ey = ey0 e (n Np) / 2Np(1 p) . easiest way to write down the Gaussian distribution in the continuous case is to recognize that the mean is x = kn, and = y0 = − = 2 With Np n, e C0, and Np(1 p) σ ,thisis 2 = − 2 = 2 − 2 = 2 2 the standard deviation is σx (x x) x x k n 2 2 2 2 − − 2 2 −k n = k σ .ThetermP(x)dxis given by P(n)times the P(n)= C e (n n) /2σ . 0 number of different values of n in dx. This number is dx/k. Therefore, To evaluate C0, note that the sum of P(n) for all n is the area of all the rectangles in Fig. I.2.Thisareais dx 1 − − 2 2 P(x)dx = P(n) = dx √ e (x/k x/k) /2σ approximately the area under the smooth curve, so that k k 2πσ 2 ∞ 1 − − 2 2 −(n−n)2/2σ 2 = √ (x x) /2σx 1 = C0 e dn. dx e . (I.5) −∞ 2πσx

It is shown in Appendix K that half of this integral is To recapitulate: the binomial distribution in the case of # large N can be approximated by Eq. I.4, the Gaussian ∞ − 2 1 π or normal distribution, or Eq. I.5 for continuous variables. dx e bx = . 0 2 b The original parameters N and p are replaced in these approximations by n (or x) and σ . Problems 593

Problems Expand Table I.1 to include entries using this approximation. Problem 2. Let y = (x − x)/σ. Express the Gaussian prob- Problem 1. An improved approximation to Stirling’s for- ability distribution as a function of y. Calculate the mean and mula1 is standard deviation of this distribution. ln(2πn) ln n!≈n ln n − n + 2

1 For more about Stirling’s formula, see N. D. Mermin (1994) Stirling’s formula! Am J Phys 52: 362–365. Appendix J The Poisson Distribution

Appendix H discussed the binomial probability distribution. If these factors are multiplied out, the first term is N x,fol- If an experiment is repeated N times, and has two possi- lowed by terms containing N x−1, N x−2, ...,downtoN 1. ble outcomes, with “success” occurring with probability p But there is also a factor N x in the denominator of the in each try, the probability of getting that outcome x times in expression for P , which, combined with this gives N tries is − − 1 + (something)N 1 + (something)N 2 +··· . N! − P(x; N,p) = px(1 − p)N x. x!(N − x)! As long as N is very large, all terms but the first can be The distribution of possible values of x is characterized by a neglected. With these substitutions, Eq. J.1 takes the form mean value x = Np and a variance σ 2 = Np(1 − p).Itis   possible to specify x and σ 2 instead of N and p to define the −x 1 x −x x distribution. P(x)= x e 1 − . (J.2) x! N Appendix I showed that it is easier to work with the Gaus- sian or normal distribution when N is large. It is specified in The values of x for which P(x) is not zero are near x, terms of the parameters x and σ 2 instead of N and p: which is much less than N. Therefore, the last term, which − x 1 − − 2 2 is really 1/(1 p) , can be approximated by one, while P(x; x,σ2) = e (x x) /2σ . (2πσ 2)1/2 such a term raised to the Nth power had to be approximated by e−x. If this is difficult to understand, consider the fol- The Poisson distribution is an approximation to the bino- lowing numerical example. Let N=10,000 and p = 0.001, mial distribution that is valid for large N and for small p so x = 10. The two terms we are considering are (1 − (when N gets large and p gets small in such a way that their 10/10, 000)10,000 = 4.517×10−5, which is approximated by product remains finite). To derive it, rewrite the binomial e−10 = 4.54 × 10−5, and terms like (1 − 10/10, 000)−10 = probability in terms of p = x/N: 1.001, which are approximated by 1. With these approximations, the probability is P(x) = N! − − P(x)= (x/N)x(1 − x/N)N x [(x)x/x!]e x or, calling x = m, x!(N − x)!     N −x x N! 1 x x m − = xx 1 − 1 − . P(x)= e m. (J.3) x!(N − x)! N x N N x! (J.1) This is the Poisson distribution and is an approximation to It is necessary next to consider the behavior of some of the binomial distribution for large N and small p, such that these factors as N becomes very large. The factor (1−x/N)N the mean x = m = Np is defined (that is, it does not go to approaches e−x as N →∞, by definition (see p. 34). The infinity or zero as N gets large and p gets small). factor N!/(N − x)! can be written out as This probability, when summed over all values of x, should be unity. This is easily verified. Write N(N − 1)(N − 2) ···1 = N(N− )(N− ) ···(N−x+ ). − − − ··· 1 2 1 (N x)(N x 1) 1 ∞ ∞ x − m P(x)= e m . x! x=0 x=0

R. K. Hobbie, B. J. Roth, Intermediate Physics for Medicine and Biology, 595 DOI 10.1007/978-3-319-12682-1, c Springer International Publishing Switzerland 2015 596 J The Poisson Distribution

Table J.1 Comparison of the binomial, Gaussian, and Poisson 1.0 distributions N! Binomial P(x; N,p) = px (1 − p)N−x 0.8 x!(N − x)! x = m = Np P(0) σ 2 = Np(1 − p) = m(1 − p) 0.6 P 1 2 2 Gaussian P(x; m, σ) = e−(x−m) /2σ 0.4 P(1) (2πσ 2)1/2 P(2) P(3) mx 0.2 Poisson P(x; m) = e−m x! = m Np 0.0 σ 2 = m 0 1 2 3 4 5 6

m = Νλt

m −m m = But the sum on the right is the series for e and e e 1. Fig. J.1 Plot of P(0) through P(3) vs. Nλt The same trick can be used to verify that the mean is m:

∞ ∞ x ∞ x m − m − xP(x) = x e m = x e m. proportional to the length of the time interval: p = λdt. x! x! x=0 x=0 x=1 The average number of decays if many time intervals are examined is The index of summation can be changed from x to y = x−1: m = Np = Nλdt. ∞ ∞ ∞ y (y + 1) − m − xP(x) = myme m = m e m = m. The probability of x decays in time dt is (y + 1)! y! x=0 y=0 y=0 x (Nλdt) − P(x)= e Nλdt. One can show that the variance for the Poisson distribution x! 2 = − 2 = is σ (x m) m. As dt → 0, the exponential approaches one, and Table J.1 compares the binomial, Gaussian, and Poisson distributions. The principal difference between the binomial (Nλdt)x P(x)→ . and Gaussian distributions is that the latter is valid for large x! N and is expressed in terms of the mean and standard devi- The overwhelming probability for dt → 0 is for there to be ation instead of N and p. Since the Poisson distribution is no decays: P(0) ≈ (Nλdt)0/0!=1. The probability of a valid for very small p, there is only one parameter left, and single decay is P(1) = Ndt; the probability of two decays σ 2 = m rather than m(1 − p). during dt is (Nλdt)2/2, and so forth. The Poisson distribution can be used to answer questions If time interval t is finite, it is still possible for the Poisson like the following: criterion to be satisfied, as long as p = λt is small. Then the 1. How many red cells are there in a small square in a hemo- probability of no decays is cytometer? The number of cells N is large; the probability p of each cell falling in a particular square is small. The − − P(0) = e m = e Nλt. variable x is the number of cells per square. 2. How many gas molecules are found in a small volume of The probability of one decay is gas in a large container? The number of tries is the total − number of molecules. The probability that an individual P(1) = (Nλt)e Nλt. molecule is in the smaller volume is p = V/V0, where V is the small volume and V0 is the volume of the entire This probability increases linearly with t at first and then de- box. creases as the exponential term begins to decay. The reason 3. How many radioactive nuclei (or excited atoms) decay for the lowered probability of one decay is that it is now more (or emit light) during a time dt? The probability of decay probable for two or more decays to take place in this longer during time dt is proportional to how long dt is: p = time interval. As t increases, it is more probable that there λdt. The number of tries is the N nuclei that might decay are two decays than one or none; for still longer times, even during that time. more decays become more probable. The probability that n The last example is worth considering in greater detail. decays occur in time t is P(n). Figure J.1 shows plots of The probability p that each nucleus decays in time dt is P(0), P(1), P(2), and P(3),vsm = Nλt. Problems 597

Problems Number of packets Number of times observed reaching the end plate Problem 1. In the USA 400,000 people were killed or in- 018 jured one year in automobile accidents. The total population 144 was 200,000,000. If the probability of being killed or injured 255 is independent of time, what is the probability that you will 336 escape unharmed from 70 years of driving? 425 Problem 2. Large proteins consist of a number of smaller 512 subunits that are stuck together. Suppose that an error is made 65 − in inserting an amino acid once in every 105 tries; p = 10 5. 72 If a chain has length 1000, what is the probability of making 81 a chain with no mistakes? If the chain length is 105? 90 Problem 3. The muscle end plate has an electrical response whenever the nerve connected to it is stimulated. I. A. Boyd Analyze these data in terms of a Poisson distribution. and A. R. Martin (The end plate potential in mammalian muscle. J Physiol 132: 74–91 (1956)) found that the electri- cal response could be interpreted as resulting from the release of packets of acetylcholine by the nerve. In terms of this model, they obtained the following data: Appendix K 2 Integrals Involving e−ax

2 Integrals involving e−ax appear in the Gaussian distribution. The desired integral is, therefore, The integral # ∞ ∞ π − 2 −ax2 I = e ax dx I = e dx = . (K.1) −∞ −∞ a can also be written with y as the dummy variable: This integral is one of a sequence of integrals of the ∞ − 2 general form I = e ay dy. −∞ ∞ n −ax2 In = x e dx. These can be multiplied together to get 0 ∞ ∞ − 2 − 2 From Eq. K.1, we see that I 2 = dxdy e ax e ay −∞ −∞ # I 1 π I0 = = . (K.2) ∞ ∞ 2 2 a − 2+ 2 = dxdy e a(x y ). −∞ −∞ The next integral in the sequence can be integrated di- = 2 A point in the xy plane can also be specified by the polar rectly with the substitution u ax : coordinates r and θ (Fig. K.1). The element of area dxdy is ∞ ∞ −ax2 1 −u 1 replaced by the element rdrdθ: I1 = xe dx = e du = . (K.3) 0 2a 0 2a 2π ∞ ∞ 2 = −ar2 = π −ar2 I dθ rdre 2 rdre . A value for I2 can be obtained by integrating by parts: 0 0 0

2 ∞ To continue, make the substitution u = ar , so that du = 2 −ax2 I2 = x e dx. 2ardr. Then 0

∞ π   π 2 1 −u −u ∞ 2 2 I = 2π e du = −e = . u = x dv = xe−ax dx =−( / a)d(e−ax ) 2a a 0 a Let  and  1 2 . 0 Since udv = uv − vdu,

∞ −ax2 − 2 xe 1 − 2 x3e ax dx =− + e ax dx. 0 2a 2a

This expression is evaluated at the limits 0 and ∞.Theterm −ax2 xe vanishes at both limits. The second term is I0/2a. Therefore, # 1 π I = . 2 2 × 2a a This process can be repeated to get other integrals in the Fig. K.1 An element of area in polar coordinates sequence. The even members build on I0; the odd members

R. K. Hobbie, B. J. Roth, Intermediate Physics for Medicine and Biology, 599 DOI 10.1007/978-3-319-12682-1, c Springer International Publishing Switzerland 2015 2 600 K Integrals Involving e−ax build on I1. General expressions can be written. Note that 2n Therefore, and 2n + 1 are used below to assure even and odd exponents: ∞ − n! Γ(n+ 1) # xne axdx = = . ∞ n+1 n+1 (K.6) − 2 1 × 3 × 5 × (2n − 1) π 0 a a x2ne ax dx = , (K.4) 2n+1an a 0 The gamma function Γ(n) = (n − 1)! if n is an integer. Unlike n!, it is also defined for noninteger values. Although ∞ + − 2 n! we have not shown it, Eq. K.6 is correct for noninteger values x2n 1e ax dx = ,(a>). n+1 0 (K.5) 0 2a of n as well, as long as a>0 and n>−1. The integrals in Appendix I are of the form ∞ − 2 2 e x /2σ dx. −∞ Problems 2 This√ integral is 2I0 with a = 1/(2σ ). Therefore, the integral is 2πσ 2. Problem 1. Use integration by parts to evaluate Integrals of the form ∞ 3 −ax2 ∞ I3 = x e dx. − J = xne axdx, 0 0 Compare this result with Eq. K.5. ∞ −ax2 can be transformed to the forms above with the substitution Problem 2. Show that −∞ xe dx = 0. Note the lower y = x1/2, x = y2, dx = 2ydy. Then limit is −∞, not 0. There is a hard way and an easy way to ∞ ∞ show this. Try to find the easy way. − 2 + − 2 J = y2ne ay 2ydy= 2 y2n 1e ay dy. 0 0 Appendix L Spherical and Cylindrical Coordinates

It is possible to use coordinate systems other than the rect- calculation1 shows that the divergence is angular (or Cartesian) (x,y,z): In spherical coordinates 1 ∂ 1 ∂ (Fig. L.1), the coordinates are radius r and angles θ and φ: div J =∇·J = (r2J ) + (sin θJ ) r2 ∂r r r sin θ ∂θ θ 1 ∂ x = r sin θ cos φ, + (J ). (L.3) r sin θ ∂φ φ y = r sin θ sin φ, (L.1) The gradient, which appears in the three-dimensional dif- z = r cos θ. fusion equation (Fick’s first law), can also be written in spherical coordinates. The components are In Cartesian coordinates a volume element is defined by surfaces on which x is constant (at x and x + dx), y is ∂C (∇C) = , constant, and z is constant. The volume element is a cube r ∂r with edges dx, dy, and dz. In spherical coordinates, the cube has faces defined by surfaces of constant r, constant θ, and 1 ∂C (∇C) = , (L.4) constant φ (Fig. L.2). A volume element is then θ r ∂θ

2 1 ∂C dV = (dr)(r dθ)(r sin θdφ)= r sin θdθdφdr. (L.2) (∇C) = . φ r sin θ ∂φ To calculate the divergence of vector J, resolve it into Figure L.2 also shows that the element of area on the sur- = 2 components Jr , Jθ , and Jφ, as shown in Fig. L.2. These face of the sphere is (r dθ)(r sin θdφ) r sin θdθdφ.The components are parallel to the vectors defined by small element of solid angle is therefore displacements in the r, θ, and φ directions. A detailed dΩ = sin θdθdφ.

This is easily integrated to show that the surface area of a sphere is 4πr2 or that the solid angle is 4π sr.

π 2π π S = r2 sin θdθ dφ = 2πr2 sin θdθ 0 0 0 = π 2 − π = π 2 2 r [ cos θ]0 4 r . Similar results can be written down in cylindrical coordi- nates (r,φ,z), shown in Fig. L.3. Table L.1 shows the divergence, gradient, and curl in rect- angular, cylindrical, and spherical coordinates, along with the Laplacian operator ∇2.

1 H. M. Schey (2005). Div, Grad, Curl, and All That. 4th. ed. New York, Fig. L.1 Spherical coordinates. Norton.

R. K. Hobbie, B. J. Roth, Intermediate Physics for Medicine and Biology, 601 DOI 10.1007/978-3-319-12682-1, c Springer International Publishing Switzerland 2015 602 L Spherical and Cylindrical Coordinates

Fig. L.2 The volume element and element of surface area in spherical coordinates

Fig. L.3 A cylindrical coordinate system L Spherical and Cylindrical Coordinates 603

Table L.1 The vector operators in rectangular, cylindrical, and spherical coordinates Rectangular x,y,z Cylindrical r, φ, z Spherical r, θ, φ Gradient ∂C ∂C ∂C (∇C) = (∇C) = (∇C) = x ∂x r ∂r r ∂r ∂C 1 ∂C 1 ∂C (∇C) = (∇C) = (∇C) = y ∂y φ r ∂φ θ r ∂θ ∂C ∂C 1 ∂C (∇C) = (∇C) = (∇C) = z ∂z z ∂z φ r sin θ ∂φ

Laplacian       ∂2C ∂2C ∂2C 1 ∂ ∂C 1 ∂2C ∂2C 1 ∂ ∂C 1 ∂ ∂C ∇2C = + + ∇2C = r + + ∇2C = r2 + sin θ ∂x2 ∂y2 ∂z2 r ∂r ∂r r2 ∂φ2 ∂z2 r2 ∂r ∂r r2 sin θ ∂θ ∂θ 2 + 1 ∂ C r2 sin2 θ ∂φ2 Divergence ∂j ∂j ∂j 1 ∂(rj ) 1 ∂j ∂j 1 ∂(r2j ) 1 ∂(sin θj ) 1 ∂j ∇·j = x + y + z ∇·j = r + φ + z ∇·j = r + θ + φ ∂x ∂y ∂z r ∂r r ∂φ ∂z r2 ∂r r sin θ ∂θ r sin θ ∂φ

Curl ∂j ∂j 1 ∂j ∂j 1 ∂(sin θj ) ∂(j ) (∇×j) = z − y (∇×j) = z − φ (∇×j) = × φ − θ x ∂y ∂z r r ∂φ ∂z r r sin θ ∂θ ∂φ

∂j ∂j ∂j ∂j 1 ∂j sin θ∂(rj ) (∇×j) = x − z (∇×j) = r − z (∇×j) = × r − φ y ∂z ∂x φ ∂z ∂r θ r sin θ ∂φ ∂r

∂j ∂j 1 ∂(rj ) 1 ∂j 1 ∂(rj ) ∂j (∇×j) = y − x (∇×j) = φ − r (∇×j) = θ − r z ∂x ∂y z r ∂r r ∂φ ∂φ r ∂r ∂θ Appendix M Joint Probability Distributions

In both physics and medicine, the question often arises of what is the probability that x has a certain value xi while y has the value yj . This is called a joint probability. Joint prob- ability can be extended to several variables. This appendix derives some properties of joint probabilities for discrete and continuous variables.

M.1 Discrete Variables

Consider two variables. For simplicity assume that each can have only two values. The first might be the patient’s health with values healthy and sick; the other might be the results of some laboratory test, with results normal and abnormal. Table M.1 shows the values of the two variables for a sample Fig. M.1 The results of measuring two continuous variables simulta- of 100 patients. The joint probability that a patient is healthy neously. Each experimental result is shown as a point and has a normal test result is P(x = 0,y = 0) = 0.6; the probability that a patient is sick and has an abnormal = test is P(1, 1) 0.15. The probability of a false positive independent of y, Px(x), and so forth. Then test is P(0, 1) = 0.20; the probability of a false negative is  = P(1, 0) = 0.05. Px(x) y P(x,y) The probability that a patient is healthy regardless of the  (M.1) Py(y) = P(x,y). test result is obtained by a summing over all possible test x outcomes: P(x = 0) = P(0, 0)+P(0, 1) = 0.6+0.2 = 0.8. Since any measurement must give some value for x and y, In a more general case, we can call the joint proba- we can write bility P(x,y), the probability that x has a certain value    = = 1 x Px(x) x y P(x,y),    (M.2) = = 1 y Py(y) y x P(x,y).

Table M.1 The results of measurements on 100 patients showing whether they are healthy or sick and whether a laboratory test was normal or abnormal M.2 Continuous Variables Healthy (x = 0) Sick (x = 1) Normal test (y = 0) 60 5 When a variable can take on a continuous range of values, Abnormal test (y = 1) 20 15 it is quite unlikely that the variable will have precisely the

R. K. Hobbie, B. J. Roth, Intermediate Physics for Medicine and Biology, 605 DOI 10.1007/978-3-319-12682-1, c Springer International Publishing Switzerland 2015 606 M Joint Probability Distributions value x. Instead, there is a probability that it is in the interval (x, dx), meaning that it is between x and x + dx. For small values of dx, the probability that the value is in the interval is proportional to the width of the interval. We will call it px(x)dx. The extension to joint probability in two dimen- sions is p(x,y)dxdy. This is the probability that x is in the interval (x, dx) and y is in the interval (y, dy). Figure M.1 shows each outcome of a joint measurement as a dot in the xy plane. The probability that x is in (x, dx) regardless of the value of y is  

px(x)dx = p(x,y)dy dx. (M.3)

It is proportional to the total number of dots in the vertical strip in Fig. M.1. Normalization requires that

1 = px(x)dx = dx dy p(x,y). (M.4) Fig. M.2 Perspective drawing of p(x,y) The first strip could be taken horizontally:

1 = py(y)dy = dy dx p(x,y).

Figure M.2 shows a perspective drawing of p(x,y).The volume of the shaded column is p(x,y)dxdy.Thevolume of the slice is px(x)dx. The entire volume under the surface is equal to 1. Appendix N Partial Derivatives

When a function depends on several variables, we may want Suppose now that we allow small changes in both r and h. to know how the value of the function changes when one or The difference in volume is more of the variables is changed. For example, the volume of = + + − a cylinder is V V(r r, h h) V(r,h). V = πr2h. We can add and subtract the term V(r,h+ h):

How does V change when r is changed while the height of V = V(r+ r, h + h) − V(r,h+ h) the cylinder is kept fixed? + V(r,h+ h) − V(r,h) V(r+ r) = π(r + r)2h = π(r2 + 2rr + r2)h. V(r+ r, h + h) − V(r,h+ h) = r r Subtracting the original volume, we have V(r,h+ h) − V(r,h) + h. h V = π(2rr + r2)h. In the limit as r and h → 0, the first term is   In the limit of small r,thisis ∂V r, ∂r dV = 2πhrdr. h evaluated at h+h. If the derivatives are continuous at (r, h), This is the same answer we would have gotten if h had the derivative evaluated at (r, h + h) is negligibly differ- been regarded as a constant. The partial derivative of V with ent from the derivative evaluated at (r, h). Therefore, we can respect to r is defined to be write         ∂V ∂V ∂V V(r+ r, h) − V(r,h) dV = dr + dh. = lim = 2πrh. ∂r h ∂h r ∂r r→0 r h This result is true for several variables. For a function w(x,y,z) The subscript h in the partial derivative symbol means that h ,       is held fixed during the differentiation. Sometimes it is omit- = ∂w + ∂w + ∂w ted; when it is not there, it is understood that all variables dw dx dy dz. (N.1) ∂x y,z ∂y x,z ∂z x,y except the one following the ∂ are held fixed. If the cylinder radius is held fixed while the height is The derivatives are evaluated as though the variables varied, we can write being held fixed were ordinary constants. If w = 3x2yz4,   2 ∂w V = V(r,h+ h) − V(r,h)= πr h. = 6xyz4, ∂x y,z   The partial derivative is ∂w = 3x2z4,     ∂y ∂V V(r,h+ h) − V(r,h)  x,z = lim = πr2. → ∂w ∂h r h 0 h = 12x2yz3. ∂z x,y

R. K. Hobbie, B. J. Roth, Intermediate Physics for Medicine and Biology, 607 DOI 10.1007/978-3-319-12682-1, c Springer International Publishing Switzerland 2015 608 N Partial Derivatives

It is also possible to take higher derivatives, such as ∂2w/∂x2 or ∂2w/∂x∂y. One important result is that the or- der of differentiation is unimportant, if the function, its first derivatives, and the derivatives in question are continuous at the point where they are evaluated. Without filling in all the details of a rigorous proof, we will simply note that   ∂w w(x + x, y) − w(x, y) f = = lim ∂x x→0 x   ∂w w(x, y + y) − w(x, y) g = = lim . ∂y y→0 y The mixed partials are   ∂2w ∂f f(x,y+ y) − f(x,y) = = lim ∂y∂x ∂y y→0 y   w(x + x, y + y) − w(x, y + y) − w(x + x, y) + w(x, y) = lim x→0 x y y→0

2 ∂ w = ∂g ∂x∂y ∂x   w(x + x, y + y) − w(x + x, y) − w(x, y + y) + w(x, y) = lim . y→0 x y x→0 The right side of each of these equations is the same, except for the order of the terms. Thus, ∂ ∂w ∂ ∂w = . ∂x ∂y ∂y ∂x

Problems

Problem 1. If w = 12x3y + z, find the three partial derivatives ∂w/∂x, ∂w/∂y, and ∂w/∂z. Problem 2. If V = xyz and x = 5, y = 6, z = 2, find dV when dx = 0.01, dy = 0.02, and dz = 0.03. Make a geometrical interpretation of each term. Appendix O Some Fundamental Constants and Conversion Factors

The values of the fundamental constants are from the 2010 least-squares adjustment, available at http://www.nist.gov/pml/data/index.cfm Symbol Constant Value SI units c Velocity of light in vacuum 2.997925 × 108 ms−1 e Elementary charge 1.602177 × 10−19 C F Faraday constant 9.64853 × 104 Cmol−1 g Standard acceleration of free fall 9.80665 m s−2 h Planck’s constant 6.626070 × 10−34 Js  Planck’s constant (reduced) 1.054572 × 10−34 Js 6.582119 × 10−16 eV s −23 −1 kB Boltzmann’s constant 1.380649 × 10 JK 8.617343 × 10−5 eV K−1 −31 me Electron rest mass 9.109383 × 10 kg 2 −14 mec Electron rest energy 8.187105 × 10 J 5.10999 × 105 eV −27 mp Proton rest mass 1.672622 × 10 kg 23 −1 NA Avogadro’s number 6.022141 × 10 mol −15 re Classical electron radius 2.817940 × 10 m R Gas constant 8.31446 J mol−1 K−1 u Mass unit (12C standard) 1.660539 × 10−27 kg uc2 Mass unit (energy units) 9.31494 × 108 eV −12 2 −1 −2 0 Electrical permittivity of free space 8.85419 × 10 C N m 9 2 −2 1/4π 0 8.98755 × 10 Nm C −8 −2 −4 σSB Stefan Boltzmann constant 5.67037 × 10 Wm K −12 λC Compton wavelength of electron 2.42631 × 10 m −24 −1 μB Bohr magneton 9.274010 × 10 JT −7 −1 μ0 Magnetic permeability 4π × 10 TmA of space ≈ 12.566 × 10−7 −27 −1 μN Nuclear magneton 5.050784 × 10 JT

R. K. Hobbie, B. J. Roth, Intermediate Physics for Medicine and Biology, 609 DOI 10.1007/978-3-319-12682-1, c Springer International Publishing Switzerland 2015 610 O Some Fundamental Constants and Conversion Factors

Some of the more useful conversion factors for converting from older units to SI units are listed. (Taken from Standard for Metric Practice, ASTM E 380-76, Copyright 1976 by the American Society for Testing and Materials, Philadelphia) To convert from To Multiply by Angstrom Meter 1.000000 × 10−10 Atmosphere (standard) Pascal 1.013250 × 105 Bar Pascal 1.000000 × 105 Barn Meter2 1.000000 × 10−28 Calorie (thermochemical) Joule 4.184000 Centimeter of mercury (0 ◦C) Pascal 1.33322 × 103 Centimeter of water (4 ◦C) Pascal 9.80638 × 101 Centipoise Pascal second 1.000000 × 10−3 Curie Becquerel 3.700000 × 1010 Dyne Newton 1.000000 × 10−5 Electron volt Joule 1.60218 × 10−19 Erg Joule 1.000000 × 10−7 Fermi (femtometer) Meter 1.000000 × 10−15 Gauss Tesla 1.000000 × 10−4 Liter Meter3 1.000000 × 10−3 Mho Siemens 1.000000 Millimeter of mercury Pascal 1.33322 × 102 Poise Pascal second 1.000000 × 10−1 Roentgen Coulomb per kilogram 2.58 × 10−4 Torr Pascal 1.33322 × 102 Index

A scan, 373 Agre, P., 118 Anode, 202 AAPM, 405, 406, 421, 469, 507 Ahlen, S. P., 442, 444, 448 Anode-break excitation, 181, 210 AAPM Report 96, 492 Ahrens, E. T., 557 Anomalous rectification, 264 Abbey, H., 588 Air Antidiuretic hormone, 294 Abduct, 7 acoustic impedance, 367 Antineutrino, 509 Abductor muscle, 7 attenuation, 371 Antiscatter grid, 474 Aberration density and specific heat, 65 Antonini, E., 81 chromatic, 412 speed of sound, 366 Anumonwo, J. B., 172 spherical, 412 Air bladder, 27 Aorta, 21, 479 Ablation, 404 Alberts, B., 481 Apoptosis, 480 Able,K.P.,229 Albumin, 136 Appa Y., 422 Able,M.A.,229 Aldosterone, 294 Aquaporins, 118 Abraham, R., 277 Algae, 229 Aquaspirillum magnetotacticum, 228 Abramowitz, M., 209, 263, 359, 361, 362 Aliasing, 314, 329 Aqueous, 411 Absolute temperature, 61, 62 in an image, 350 Arakane K., 423 Absorbed dose, 452 Allen, A. P., 51 Armato, S. G., 470 Absorption coefficient, 387 Allen, R. D., 108 Armstrong, B. K., 402 Absorption edge, 428 Almond, P. R., 488 Armstrong, C. M., 394, 421 Acceleration, 571 α particle, 448, 506, 525 Arqueros, F., 450, 515, 532 Accommodation, 412 Alternans, 298 Arterial spin labeling, 556 Acetabulum, 8 Altes, T. A., 559 Arteriole, 21 Acetylcholine, 110, 143, 194, 597 Alveoli, 1, 20, 82, 110, 270 Artificial insemination, 77 ACHD, 399 Alzheimer’s disease, 226 Artificial kidney, 126 Achilles tendon, 6 American Association of Physicists in Asano H., 423 Acoustic impedance, 366 Medicine, see AAPM Ascites, 122 Acoustic shadow, 378 Ampere (unit), 151 Ashcroft, F., 250 Actin, 85 Ampere’s law, 216, 225, 227 Astigmatism, 412 Actinometry, 405 and Biot-Savart law, 232 Astumian, R. D., 256, 259, 260, 265, 337, 338 Action potential, 141 Amplitude attenuation coefficient, 370 Ataxia-tangliectasia, 483 foot, 181 Ampullae of Lorenzini, 256, 266 Atherosclerosis, 399 Gaussian approximation, 190 Anaplasia, 401 Atkins, P. W., 75, 83 propagating, 166 Anderka, M., 313 Atmosphere, 46 space-clamped, 165 Anderson K. E., 422 pressure variation, 64, 80 Activating function, 209 Anderson, H. L., 379 Atmosphere (pressure unit), 14 Active transport, 519 Anderson, J. R., 533 Atomic deexcitation, 434 Activity, 81, 506, 511 Anesthetic, 169 Atomic energy levels, 425 cumulated, 511, 512 Aneurysm, 27 Atomic number, 503 Activity vector, 188 Angiography, 399, 476 Atoms per unit volume, 388 Acton, F. S., 340 Angioplasty, 524 ATP, 3, 71, 555 Adair, R. K., 256, 257, 260, 337 Angioscopy, 399 Atrioventricular node, see see AV node Adenosine triphosphate (ATP), 3 Angstrom, 3 Atrium, 193 Adiabatic approximation, 440 Angular Frequency, 308 Attenuation Adiabatic process, 58 Angular momentum, 536 of sound wave, 370 ADP, 72, 555 Angular wave number, 346 water, 371 Adrenal gland, 294 Anisotropy, 28, 200, 201 Attenuation coefficient, 387 Afterloading, 523 Annihilation radiation, 438, 510 amplitude, 370

R. K. Hobbie, B. J. Roth, Intermediate Physics for Medicine and Biology, 611 DOI 10.1007/978-3-319-12682-1, c Springer International Publishing Switzerland 2015 612 Index

effective, 391 Balloon angioplasty, 524 Bessel function, 263, 359, 361, 372 intensity, 370 Bambynek, W., 435 modified, 208 linear, 433 Banavar, J. R., 45, 51 Beta decay, 506, 508 mass, 433 Bandettini, P. A., 557 spectrum, 509 Attenunator Bar (unit), 13 Bethe, H., 442 ladder, 179 Barach, J. P., 223, 229, 232 Bethe–Bloch formula, 444 Attix, F. H., 430, 431, 438, 444, 446, 450, Barium, 472 Bevington, P. R., 307 453, 455, 462, 464, 469, 489 Barium fluorobromide, 470 Beyer, R. T., 371, 379 Attractor, 276, 284 Barker, A. T., 226 Bianchi, A. M., 328, 329 Auditory evoked response, 328 Barlow, H. B., 415, 421 Bidomain model, 200, 201, 205, 208, 233 Auger electron, 435, 449, 507, 524 Barn (unit), 428 Bieber, M. T., 113, 115 cascade, 436, 461, 507 Barnes, F. S., 257, 266 Biersack, J. P., 439, 442, 445, 446, 448 Augmented limb leads, 197 Barold, S. S., 203 Bifocal lenses, 412 Autocorrelation function, 318, 543 Barr M. L., 422 Bifurcation, 282 of exponential pulse, 325 Barr, G., 91, 114 diagram, 285 and energy spectrum, 325 Barr, R. C., 161, 167 Bilderback D., 115 and power spectrum, 320 Barrett, H. H., 345, 353, 362 Bilinear interpolation, 360 of noise, 334 Barrett, J. N., 251 Bilirubin, 400 of sine wave, 319 Bart K., 115 Binding energy of square wave, 319 Bartels, L. W., 379 electronic, 507 AV node, 194, 203 Barth, R. F., 489 nuclear, 505 Average, 585 Bartlett, A. A., 42, 51, 289 per nucleon, 505 ensemble, 56 Barysch M. J., 421 Binkert,C.A.,379 time, 57 Barysch, M. J., 403 Binomial probability distribution, 55, 106, Average reference recording, 205, 211 Basal cell, 401, 403 587 Avogadro’s number, 64, 89, 433 Basal cell carcinoma (BCC), 402, 403, 486 Bioheat equation, 404 definition in SI units, 388 Basal metabolic rate, 65 Biomagnetism, 213 Axel, L., 546, 555 Base, 573 Biot-Savart law, 216 Axelrod, D., 101, 114, 115 Basford, J., 27, 30 and Ampere’s law, 232 Axial vector, 6 Basilar membrane, 370 and magnetic field around an axon, 219 Axon, 2, 141 Basin of attraction, 281 current in, 218 cable model, 156 Bass, M., 422, 423 Bipolar electrode, 203 electric field, 154 Basser, P. J., 110, 115, 136, 558 Birch, R., 463 membrane capacitance, 155, 169 Bastian, J., 256 Bird, R. B., 94 membrane capacitance and conductance, Battocletti, J. H., 555 Birds, 229 157 Baylor, D. A., 415, 422 Birefringence, 393 membrane equivalent circuit, 162 Bazylinski, D. A., 228 Births, 341 membrane time constant, 156 Beam hardening, 471 spontaneous, 313, 314 myelinated, 142, 167 Bean, C. P., 91, 115, 129, 133, 134, 177 Bistable systems, 298 potassium gate, 164 Beaurepaire, E., 423 Blackbody, 395 potassium Nernst potential, 162 Beck,R.E.,134 Blackbody radiation potential outside, 185 Becklund, O. A., 348, 362 and heat loss, 398 sodium conductance, 164 Becquerel (unit), 495, 506, 511 vs. frequency, 397 sodium gate, 165 Beer’s law, 388 vs. wavelength, 396 sodium Nernst potential, 162 Bees, 229, 235 Blackman, R. B., 314 space-clamped, 161, 165 Begon, M., 281 Blackman-Tukey method, 329 surface charge density, 155 BEIR, 494, 495, 525, 526, 532 Blagoev, K. B., 3 unmyelinated, 142, 167 Belousov–Zhabotinsky reaction, 290 Blair, D. F., 101, 115 voltage-clamped, 161 Bénard, H., 296 Blakemore, N., 228 Axoplasm, 142 Bender, M. A., 531 Blakemore,R.P.,228 Ayotte, P., 495 Benedek, G. B., 11, 15, 30, 95, 105, 115 Bland, E. F., 39, 51 Bennison, L. J., 48 Bloch equations, 539 Bequerel second, 506 Bloch, F., 442, 539 B scan, 373, 378 Berg, H. C., 85, 101, 115 Blood flow Background Berg, M. J., 48, 51 pulmonary artery, 316 natural, 492 Berg, W. A., 379 Blood pressure, 269 Backscatter Berger, M. J., 515, 532 Blood-brain barrier, 123, 519 factor, 455 Bergmanson, J. P. G., 403, 421 Blount, W. P., 11 Backx, P. H., 194 Berliner,L.J.,545 Blue (color), 415 Bacteria, 2 Berlinger, W. G., 51 Blume, S., 399, 422 magnetotactic, 228, 235 Berne, B. J., 393, 422 Boas, D. A., 422 orientation in a magnetic field, 235 Bernoulli equation, 19, 29 Boccara, A. C., 423 Bacteriophage, 2, 527 Bernoulli process, 587 Bockris, J. O’M., 247 Badeer, H. S., 19, 31 Bernstein, M. A., 553 Bodurka, J., 557 Bagavathiappan, S., 422 Berry, E., 422, 423 Bohr, N., 383, 442 Bainton, C. R., 276 Berwick M., 422 Boice, J. D., 490 Index 613

Bolch, W. E., 511, 532 Cable equation, 156, 159 prokaryotic, 2 BOLD (Blood Oxygen Level Dependent), and ladder attenuator, 179 size, 2 543, 557 Calcaneus, 6, 7 sorting, 236 Boltzmann factor, 58, 62–65, 67, 89, 239, 395, Calcium, 295 Cell culture, 480 537 -induced calcium release, 111 Cell survival Boltzmann’s constant, 61 buffer, 110 fractionation curve, 482 Bone scan, 521 diffusion, 110 Cellular automata, 291, 298 Boone, J. M., 438 waves, 111 Center of gravity, 4 Born charging energy, 151, 247 Callaghan, P., 554 Center of mass, 385 Born, C. G., 488 Calland, C. H., 126 Central slice theorem, 351 Boron neutron capture therapy (BNCT), 489 Calorie, 58, 80 Centrifuge, 27, 28, 79 Bouma, B. E., 422, 423 dietary, see Kilocalorie Centripetal acceleration, 27, 215 Boundary layer, 23 Calorimetry, 498 Cerebral cortex, 223 Boundary-element method, 200 Cameron, J. R., 369 Cerutti, S., 328, 329 Bourland, J. D., 202, 555 Cancer Cesium iodide, 470 Boyd, I. A., 597 and power-line-frequency fields, 257 Chamberlain, J. M., 422, 423 prostate, 557 Chance, B., 389, 392, 422, 423 Bracewell, R. N., 315, 321 Candela (unit), 410 Chandler, W. K., 263 Brachytherapy, 484, 489, 523 Capacitance, 149 Chang, W., 422 high-dose-rate, 523 concentric cylinders, 176 Channel Bradshaw, P., 24, 30 cylindrical membrane, 182 selectivity, 252 Bradycardia, 203 membrane, 169 Channelopathies, 250 Bragg peak, 489, 499 resistance and diffusion, 172 Channels Bragg rule, 446 Capillary, 2, 21, 110, 121 calcium, 250 Bragg-Gray relationship, 490 Capillary blockade, 519 chloride, 251 Bramson, M. A., 422 Carbohydrate, 3 delayed rectifier, 250 Braun, T. J., 289 11C, 523 ion, 143 Breathing 14C dating, 532 potassium, 250 energy loss due to, 65 Carbon dioxide sodium, 250 Bremsstrahlung, 437, 441, 461, 462 production, 270 two-state model, 252 energy fluence, 462 regulation, 271, 273, 275 Chaos, 335 Bren,S.P.A.,258 Carbon monoxide, 79 deterministic, 279, 285 Brenner, D. J., 494 Carbonyl group, 252 in heart cells, 291 Brezina,ˇ V., 314 Carcinoma Chaotic behavior, 284 Brezinski, M. E., 392, 422 basal cell, 402, 486 Characteristic x rays, 461 Brightness contrast, 474 squamous cell, 402 Charcoal, activated, 48 Brill, A. B., 532 Cardiac arrest, 76 Charge Brink, S., 39, 51 Cardiac cell, 1 free and bound, 150 Broad, W. J., 478 Cardiac output, 22 Charge distribution Broad-beam geometry, 432 Carlsson, G. A., 431 cylindrically symmetric, 145 Bronzino, J. D., 328, 329 Caro, C. G., 24, 30 line, 145 Brooks, A. L., 494, 500 Carr–Purcell (CP) Sequence, 547 on cell membrane, 147 Brooks, R. A., 478, 480 Carr–Purcell–Meiboom–Gill (CPMG) plane sheet, 145 Brown, J. H., 45, 51 Sequence, 547 point, 145 Brown, J. H. U., 423 Carslaw, H. S., 96, 99, 105, 112, 115 spherically symmetric, 145 Brown, R., 89, 108 Carson, P. L., 371, 379 Charge inversion, 245 Brown, R. W., 548 Carstensen, E. L., 258 Charge screening, 426 Cartilage Brownian motion, 65, 89, 108, 332 Charged-particle equilibrium, 453, 464 articular, 136 Charman, W. N., 413, 422 Buchanan, J. W., 533 Castelli, W. P., 51 Chase, M., 527 Buchsbaum, D., 524 Catalyst, 79 Chavez, A. E., 438 Bucky, G., 474 Cataracts, 118, 480 Cheeseman, J., 403, 422 Budd, T., 449 Catfish, 256 Chemical dosimeter, 469 Budinger, T. F., 511, 512, 514, 517, 533 Catheter, 399 Chemical potential, 66, 92, 120, 121 buffer, 110 Cathode, 202 ideal gas, 67 Bui, T-A., 51 virtual, 203 solute, 67, 73 Buildup factor, 454, 515 “dog bone”, 205 water, 73 Buka, R.L., 403, 422 Cathode-break excitation, 210 Chemical shift, 555, 562 Bulk modulus, 15 Cathode-ray tube, 213 images, 555 Bundle branch block, 198 Cavitation, 375 Chemostat, 40 Bundle of His, 194 Cavity radiation, see Blackbody radiation Chemotaxis, 98, 101 Buonocore, M. H., 350, 362 Cebeci, T., 24, 30 Chemotherapy, 46 Buoyancy, 15 Cell Chen, J., 422 Burch, W. M., 109 eukaryotic, 2 Chen, W., 557 Burnes, J. E., 200 membrane, 2 Cherry, S. R., 517, 520, 522, 523, 531 Bystander effect, 483, 507 producing or absorbing a substance, 101 Cheyne–Stokes respiration, 289 614 Index

Chick, W. L., 295 Computed Radiography (CR), 470, 474 Cox, J. D., 486 Chittka, L., 422 Computed tomography Crank, J., 96, 105, 115 Cho, Z.-H., 351, 357, 362, 548 spiral, 478 Crank-Nicolson method, 111 Cholesterol Concentration Creatinine clearance test, 47 Raman spectrum, 393 and potential difference, 240 Cristy, M., 532 Christian, P., 523, 532 Concentration work, 67 Critical damping, 582 Chromatic aberration, 412 Conductance, 151 Cross correlation Chromosome, 481, 482 Conduction system, 194, 203 and signal averaging, 328 Chronaxie, 179, 202 Conduction velocity Cross product, 6 Chronic granulocytic leukemia, 290 myelinated, 167, 169 Cross section, 90, 388 CIE, 401, 409 unmyelinated, 167, 168 differential, 388 Cilia, 370 Conductivity energy transfer, 431 Circadian rhythm, 341 anisotropic, 200, 201 scattering, 388 Circulation, 20, 294 interior and exterior, 189 total, 388 Circulatory system, 20 non-uniform, 200 Cross-correlation function, 318 Clark, J., 190–192, 200, 208, 220 tensor, 200 Crouzy, S. C., 255 Clark, V. A., 39, 51 Conductor, 148 Crowder, S. W., 142 Clarke, J., 230 Cones (retinal), 409, 413 Crowe, E., 511, 515, 533 Classical electron radius, 430, 443 Conformal radiation therapy Crowell, J. W., 289 Clausius-Clapeyron equation, 83 three-dimensional, 487 CSDA, 447 Clearance, 40, 293, 404 Congestive heart failure, 38 CT, 478 Clement, G. T., 375, 379 Conjunctivitis, 403 Cuevas, J. M., 521 Clostridium, 153 Constant-field model, 248 Cumulated activity, 511, 512 Cloud chamber, 449 Contact lens, 113, 403 Cumulated mean activity per unit mass, 514 Cochlea, 369 Continuity equation, 86 Curie (unit), 512 Cochlear duct, 370 differential form, 88 Curie temperature, 228 Cochlear implant, 202, 370 integral form, 87 Curl, 224 Cochran, W. W., 229 with creation or destruction, 89 Curl in different coordinate systems, 601 Coffey, J. L., 516 Continuous slowing down approximation, 447 Curran, P. F., 126 Cohen, A., 329 Contrast Current Cohen, B. L., 494, 495, 526 brightness, 474 bound, 227 Cohen, D., 228 exposure, 474 electric, 151 Cohen, L. G., 225 film, 465 free, 227 Coherence, 393 noise brightness, 475 total, 17 Coherent scattering, 427, 431 noise exposure, 475 volume, 17, 85 Cole C., 422 Contrast agent, 472 Current density, 17 Collagen, 136 Control system, 278 electric, 151 Collective dose, 493 Convection coefficient, 80 Current dipole, 187 Collett B., 115 Conversion factors, 610 electric and magnetic measurements Collimator, 471 Convolution, 346 compared, 222 gamma camera, 520, 530 theorem, 346, 348 in spherical conductor, 233 multi-leaf, 487 Cook, G., 82, 83 Current dipole moment, 188 Collision kerma, 464 Cook, G. J. R., 557 of the heart, 195 Collision time, 90 Cooley, J. W., 315 Current source, 186 Color blindness, 76, 415 Coordinate system potential due to, 186 Color flow imaging, 375 rotating, 539 Cyan, 415 Color vision, 415 64Cu, 48 Cyclic GMP, 415 Colyvan, M., 297 Cormack, A. M., 478 Cyclotron, 215 Commission Internationale de l’Eclairage, see Cornea, 113, 403, 411 Cyclotron frequency, 215 CIE Cornsweet, T. N., 413, 422 Cylindrical coordinates, 601 Common bundle, 194 Correlation function, 317 Cystic fibrosis, 589 Compass Cortisol, 340 Cytokinesis, 481 in birds, 229 Cosgrove,D.O.,379 Competitive binding assay, 503 Coster-Kronig transition, 435, 507 Complex exponential, 311 Coulomb, 143 da Luz, L. C. Q. P., 511, 533 Complex notation, 322 Coulomb’s Law, 143 da Silva, F. C., 515, 533 Complex numbers, 311 Coulter counter, 177 Damage, see Radiation damage Compound interest, 33 Coulter, W. H., 177 Dasari, R. R., 422 Compound microscope, 420 Countercurrent exchange Data Compounds and mixtures, 434 and heat loss, 138 surrogate, 336, 337 Compressibility, 15, 364 Countercurrent transport, 127 Davis, L., 160 Compressive strength, 13 Coupling medium, 377 de Araujo, F. F., 229 Compton scattering, 427, 428, 466 Couriel D. R., 422 de Boer J. F., 422 cross section, 430 Coursey, B. M., 524, 532 de Boer, J. F., 393 differential cross section, 430 Covino, B. G., 169 de Broglie, L., 383 Compton wavelength, 429 Cowen, A. R., 470 de Jong, N., 379 Index 615

Dead-time correction, 530 DeVita, V. T., 485 storage phosphor, 470 Death rate, 38, 41, 46, 47 Dewaraja, Y. K., 515, 532 thin-film transistor, 470 DeBlois,R.W.,177 DeWerd,L.A.,471 Digital subtraction angiography, 476 Debye (unit), 175 Dextrose, 136 Dimensional analysis, 30 Debye length, 242 Diabetes, 295 Dimethylchlortetracycline, 402 Debye–Hückel model, 244 Diabetes insipidus, 118 Diopter (unit), 412 Decay Dialysis, 123 Dipole constant, 35 renal, 126 energy in a magnetic field, 536 exponential, 35 Diamagnetism, 227 magnetic, 535 Multiple paths, 41 Diamantopoulos, L., 423 Dipole moment rate, 35 Diastole, 20 electric, 175, 245 variable, 38 Diastolic interval, 298 Dirac, P. A. M., 323 with constant input, 41 Diatomic molecule, 385 Direct Radiography (DR), 470, 474 Decibel, 331, 368 DiChiro, G., 478, 480 Discrete Fourier transform, 311 Deckers, R., 379 Dichromate vision, 415 Disease, incidence and prevalence, 47 Deexcitation Dickerson, R. H., 82, 83 Dispersion, 586 atomic, 434 Dielectric, 149 Displacement, 3, 569 DeFelice, L. J., 254, 327 saturation, 245 Displacement current, 217 Deffeyes, K. S., 49, 51, 229 Dielectric constant, 150 Ditto, W. L., 291 Defibrillation, 76 lipid, 151 Diuresis, 123 Defibrillator, 202, 204 lipid bilayer, 169 Divergence, 88 Deformation, 12 water, 151 Divergence in different coordinate systems, Degeneracy, 63 Diem, M., 393, 422 601 Degrees of freedom, 57, 59, 280 Differential equation, 35 Divergence theorem, 108 Delaney, C., 352, 362 characteristic equation, 581 Diving, 15 Delaney, T. F., 489 homogeneous and inhomogeneous, 581 DNA, 78, 481, 483, 507 Delannoy, J., 558 linear, 581 Dobson unit, 401 Delay-differential equation, 288, 290 nonlinear, 279 Dog bone, 205 Delayed rectifier channels, 250 second order, 582 Dogfish, 256 Delmar, M., 193 Diffey, B. L., 401, 403, 422 Doi, K., 470, 474 Delta function, 323 Diffusion, 89, 125, 563 Domains, 227 Delta rays, 449 and capacitance, 172 Donnan equilibrium, 239 Deltoid muscle, 25 and chemical reaction, 111 Doppler effect, 374, 393 Demand pacemaker, 209 and drift, 102 Dore, C. J., 379 Demir, S. S., 165, 172 and electrotonus, 179 dos Santos, D. S., 533 den Boer, J. A., 548, 553 anisotropic, 110 Dosdall, D. J., 202 Dendrites, 141 as random walk, 106 Dose in cerebral cortex, 223 between concentric spheres, 173 absorbed, 452 Denier van der Gon, J. J., 203 between two spheres, 174 effective, 491 Denk, W., 256 circular disk, 174 entrance, 465 Denny, M. W., 15, 30, 65, 80, 82, 83, 113–115, constant, 92 equivalent, 491 366, 367, 371, 379, 418, 421, 422 in one dimension, 98 exit, 465 Density in three dimensions, 98, 107 Dose equivalent, 490 current, 17 in two dimensions, 98, 106 Dose factor, 511 optical, 465 of photons, 390 Dose fractions, 484 Density effect, 444, 446 of spins in MRI, 558 Dose rate Density gradient separation, 27 self, 94 in photochemistry, 419 Density of states factor, 63 sphere to infinity, 173 Doss, M., 494 Density-weighted image, 554 tensor, 110, 558 Dot product, 12 Deoxyhemoglobin, 543 time-dependent, 391 Doubling time, 36 Deoxyribonucleic acid (DNA), 3, 25 to a cell producing or absorbing a Douglas, J. G., 489 Department of Energy, 494 substance, 101 Doyle, D. A., 252 Dephasing, 539, 544 to or from a disk, 99, 112 Drexler, W., 422 Depolarization, 142 to or from a spherical cell, 98 Drift, 89 Depression, 226 trace (of tensor), 558, 564 and diffusion, 102 Depth of field, 412 with a buffer, 110 Driving force, 92, 94 Derivative control, 279 Diffusion constant Driving pressure, 121, 124 Derivative, partial, 61 and viscosity, 94 Droplet keratopathy, 403 Dermis, 402 determination of, 105 Drosophilia melanogaster, 252 Detecting weak magnetic fields, 229 effective, 111 Dubois, A., 423 Detective quantum efficiency (DQE), 476 photon, 390 Ducros M. G., 422 Detector vs. molecular weight, 94 Duderstadt, J. J., 390, 422 thin-film transistor, 474 Diffusion equation, 95 Dummer R., 421 Deterministic effects, 480 general solution, 104 Duncan, W., 489 Detriment, 490, 491 DiFrancesco, D., 172 Durbin, R. P., 133 Deuteron, 506 Digital detector, 470 Duty cycle, 561 616 Index

Dye Electroencephalogram, 185, 205, 210, 211, Ensemble and time averages, 60 voltage-sensitive, 205 223, 329 Ensemble average, 56 Dyer, A. G., 422 average reference recording, 205 Enthalpy, 73 Dynamic pressure, 19 Electromagnetic spectrum, 382 Entrainment, 282, 296 Dynamical system, 280 Electromagnetic wave, 382 Entropy, 62, 68 Dysplasia, 401 Electromotive force, 225 ideal gas, 81 Electromyogram, 185, 202, 329 of mixing, 72 Electron Entry region, 24 e, definition of, 34 Auger, 435 Enzyme, 79 Eames, C. and R., 31 classical radius, 430 Eosinophilic granuloma, 521 Eames, M., 379 for radiation therapy, 488 Epicardium, 200 Ear, 369 rest energy, 432 Epidemiological studies, 257 canal, 369 rest mass, 432 Epidermis, 401 drum, 369 Electron capture, 509 Epiphyses, 521 external, middle, inner, 369 Electron microscope, 383 Epiphysis, 8–10, 521 inner, 256 Electron volt (unit), 62, 151, 382 Epstein, A. E., 291, 292 mechanoreceptors, 256 Electroporation, 257 Equal anisotropy ratios, 201 ossicles, 369 Electrosurgery, 177 Equilibrium sensitivity, 368 Electrotonus, 159, 167 approach to, 273 Early-responding tissue, 483, 484 Elementary charge, 63 charged-particle, 453 Echo planar imaging (EPI), 553 Elias, W. J., 379 transient, 453 Eckerman, K. F., 511, 532 Elliott, D. M., 28, 30 diffusive, 66 Eddy currents, 225 Ellis, S., 375, 379 in volume exchange, 68 Edema, 122 Elliston, C. D., 494 particle, 66 and osmotic pressure, 122 Elson, H. R., 488 radiation, 452 cerebral, 123, 136 Embedding, 335 rotational, 4 pulmonary, 125 dimension, 335 thermal, 60, 66 Eder, 237 time lag, 335 translational, 4 Emission Eder,S.H.K.,229 Equilibrium absorbed dose constant, 511, 514 spontaneous and stimulated, 542 Equilibrium and steady state Edge spread function, 349 Emissivity, 395, 398 distinction, 59 EEG, see Electroencephalogram of skin, 398 Equilibrium constant, 71 Effective dose, 491 Emmer, M., 379 Equilibrium state, 57 Efimov, I. R., 205 Emmetropia, 412 Equipartition of energy, 65 Ehman, R. L., 556 End-plate potential, 597 Equivalent dose, 491 Einstein (unit), 417 Endo, A., 532 Erector spinae muscle, 26 Einthoven’s triangle, 197 Endocardial and epicardial membrane Ergodic hypothesis, 60 Eisberg, R., 227, 230, 383, 386, 387, 422, 505, currents, 193 Erhardt, J. C., 521 508, 509 Endoplasmic reticulum, 111 Error Eisenman, G., 115 Energy mean square, 304 Ejection fraction, 521 equipartition theorem, 65 Error function, 105, 160, 179, 202, 209 El-Samad, H., 279 fluence, 404, 406, 407 Erythema, 401, 419 Elastic recoil pressure, 20 fluence rate, 406, 408 Erythrocyte, 2, 123 Elastography, 375, 399, 556 imparted, 450, 452 Escape peak, 466 Electric dipole moment, 245 ionization, 384, 426 Escherichia coli, 527 Electric displacement, 217 kinetic, 11 Escherichia coli, 2, 85, 101 Electric field lost and imparted, 447 Esenaliev, R. O., 393, 422 definition, 144 radiant, 407 Essenpreis, M., 422 Electrical potential, 147 transferred, 450 Ethanol, 48 Electrical stimulation, 202 net, 452 Eukaryotes, 2 Electrocardiogram, 185 Energy exchange, 60 Eustachian tube, 369 and solid angle theorem, 207 Energy levels, 57, 60, 383 Evans, R. D., 508, 532 leads, 196 atomic electron, 425 Even function, 308 left ventricular hypertrophy, 198 band, 395 Evoked response, 205, 224, 328 normal, 198 hydrogen atom, 383 Excess absolute risk, 493 R-R interval, 208 rotational, 385 Excess relative risk, 493 right bundle branch block, 199 splitting, 395 Exchange term, 446 right ventricular hypertrophy, 198 vibrational, 386 Excitable media, 298 Electrode Energy spectrum waves in, 290 anodal, 203, 209 of a pulse, 324 Exit dose, 465 cathodic, 203, 209 of exponential pulse, 325 Exitance, 406, 408 pacing, 203, 209 Energy-momentum relationship Expectation value, 390 spherical, 177 relativistic, 429 Exponent, 573 Electrode heating Engdahl, J. C., 521, 533 Exponential during magnetic stimulation, 234 Enquist, B. J., 51 complex, 311 in a changing magnetic field, 234 Ensemble, 54 decay, 35 Index 617

function, 34 Feynman, R., 337 Flux, 85 growth, 33 Fibrillation total, 17 Exponentials, fitting, 42 ventricular, 204 volume, 85 Exposure, 464 Fibula,6,7 Flux density, 85 Exposure contrast, 474 Fick tracer method, 108 volume, 17 Extensive variables, 69 Fick’s first law, 92 Flux transporter (magnetic), 230 Exterior potential Fick’s second law, 95, 104, 242, 390, 404 Focal length, 411 and electrocardiogram, 188 Field of view, 349 Follicle-stimulating hormone, 341, 342 and solid angle, 189 Film Food consumption, 44 arbitrary pulse, 190 as x-ray detector, 465 Force, 4 cardiac depolarization, 187 gamma, 465 generalized, 69 from action potential, 189 speed, 465 nuclear, 505 from depolarization, 188 Film-screen combination, 474 surface, 12 general case far from axon, 190 Filtration Ford, M. R., 516, 533 nerve impulse, 187 of x-rays, 471 Foster, K. R., 258, 261 ratio to interior, 188 Filtration coefficient, 123 Fourier integral, 320, 322 Extinction coefficient, 417 Finegold, L., 228 complex, 322 Extracellular fluid, 142 Finlay, J. C., 399, 423 Fourier series, 308 Extracorporeal photopheresis, 404 First law of thermodynamics, 57 complex, 311 Eye Fish discrete, equally-spaced data, 310 emmetropic (normal), 412 breathing, 80 for piecewise continuous function, 315 human, 410 gills, 128 for square wave, 316 hypermetropic (farsighted), 412 Fisher, H. L., 516, 533 sample duration not a period, 313 insect, 410 Fishman, H. M., 256, 266 square wave, 311 myopic (nearsighted), 412 Fishman, R. A., 123 Fourier theorem, 316 resolution, 413 Fission Fourier transform, 320, 322, 543, 551 under water, 421 nuclear, 506, 519 complex, 322 Fit discrete, 311 linear, 304 discrete (2-d), 351 Factorial, 587 polynomial, 305 fast, 315 Faez, T., 371, 379 Fitzgerald, A. J., 394, 422, 423 of exponential pulse, 322 Fainting, 203 FitzHugh-Nagumo model, 296 Fourtanier, A. M., 402, 422 Falen, S. W., 517, 520, 533 Fitzmaurice, M., 422 Fovea, 413 False negative, 587, 605 Fixed point Fowler, P. H., 448, 449 False positive, 587, 605 stable, 276 Fox, J. J., 292 Far field, 372 unstable, 276 Fox, R. A., 524 Farad (unit), 149 Flannery, B. P., 84, 115, 362 Fractional distribution function, 528 Faraday constant, 64 Fletcher D. A., 115 Fractional rate of growth or decay, 38 Faraday induction law, 224, 544 Fletcher, D. A., 85 Fractionation curve, 482 Faraday, M., 224 Flotte, T., 422 Fraden, J., 399, 422 Farmer, J., 392, 422 Flounder Framingham study, 39 Farrell, T. J., 422 ocean, 256 Francis, J. E., 467 Fast Fourier Transform (FFT), 315, 341 Flow Frankel, R. B., 228, 229, 231 Fatt, I., 113, 115 laminar, 15 Fraunhoffer zone, 372 Fédération CECOS, 76 Poiseuille, 17 Free body diagram, 8 Feedback total, 17 Free currents, 227 carbon dioxide regulation, 270 Flow (mathematical), one dimensional, 276 Free expansion, 78 loop, 269 Flow effects (in MRI), 555 Free induction decay (FID), 544, 546 negative, 269 Flow rate, 85 Free radical, 469, 481 one time constant, 273 Fluence Freeston, I. L., 226 positive, 269 energy, 404, 436 Freitas, C., 226 steady-state conditions, 270 particle, 85, 390, 436 French L. E., 422 summary, 288 photon, 390 Frequency, 308 time constant and fixed delay, 287 volume, 85 angular, 308 two time constants, 276 Fluence rate, 85, 390 half-power, 331 Feinstein, A. R., 588 volume, 17, 123 spatial, 346, 349 Feld, M. S., 422 Fluid Frequency response, 295, 330 Femur, 8 Newtonian, 16, 91 Fresnel zone, 372 Feng, T.-C., 422 Fluorescein, 402 Frey, E. C., 532 Fenster, A., 371, 379 Fluorescence, 205, 386, 427, 435 Fricke dosimeter, 469 Fercher, A. F., 392, 422 Fluorescence yield, 435, 437 Friedman, R. N., 223 Ferrara J. L. M., 422 18F, 215, 510, 523 Fritzberg, A. R., 524, 532 Ferrimagnetism, 227 19F, 557 Fruit fly, 252 Ferromagnetism, 227 Fluorodeoxyglucose, 523 shaker mutant, 252 Ferrous sulfate, 469 Fluoroscopy Fuchs, A. F., 31 Feynman’s ratchet, 337 for fitting shoes, 497 Fujimoto, J. G., 422 618 Index

Function Gilhuijs, K. G., 379 Guyton, A. C., 21, 121, 143, 276, 289, 340 piecewise continuous, 315 Gillooly, J. F., 51 Gyromagnetic ratio, 536, 537 Functional MRI, 557 Gimm, H. A., 428 Fundamental constants, 609 Gingl, Z., 337, 338 Fundamental equation of thermodynamics, 69 Ginzburg, L., 297 h gate, 164, 181 Fung, Y. C., 12, 13, 30 Glaser, K. J., 556 Haas, R. A., 589 Furocoumarins, 402 Glass, L., 280–282, 284, 286, 288–291, 300, Hafemeister, D., 257 Fusion 335–337 Hagen, S. J., 40, 43, 51 nuclear, 506 Glazier, D. S., 45, 51 Hair cells, 256, 370 Fuster, V., 422 Glide-Hurst, C. K., 371, 379 Halberg, F., 313, 341 Globulin, 136 Haldane, J. B. S., 48, 51 Glomerulus, 17, 28, 123 Half-life, 35, 507 Gadolinium, 543, 557 Glucose, 3, 72, 123 effective, 513 Gadolinium oxysulfide, 466 Glucose regulation, 295 Half-power frequency, 331 Gain, 272 Glutamate, 143 Half-value layer, 498 open-loop, 272 Gluteus medius, 8 Hall, E. J., 480, 481, 483, 485, 494 67Ga (gallium), 520, 522 Gluteus minimus, 8 Hall, J. E., 121, 123, 128, 143 Gamma (film), 465 Glycine, 143 Hall, M. A., 286 Gamma camera, 520 Glycoaminoglycans, 263 Hallett, M., 225 Gamma decay, 506, 507 GMP, cyclic, 415 Halliday, D., 398 Gamma function, 600 Goff, J. P., 279 Hämäläinen, M., 229, 230 Gammaitoni, L., 337 Goitein, M., 486–489 Hämäläinen, R., 334 Gangrene, 153 Goiter, 292 Hamblen, D. P., 467 Gans, D. S., 423 Goldberg, M. J., 51 Hamill, O. P., 251, 255 Gao, L., 350, 362 Goldman–Hodgkin–Katz equations, 249 Hamilton, L. J., 390, 422 Gap junctions, 193 Gompertz mortality function, 47 Hanlon, E. B., 394, 422 Gap phase, 481 Goodsell, D. S., 3, 30 Hao O-Y., 422 Garfinkel, A., 291, 298 Gorby, Y., 228 Hao, D., 557 Gas Gou, S-y., 101, 115 Hao, O-Y., 403 ideal, 64, 65 Gould, J. L., 229 Hari, R., 224 Gas constant, 64 Gouy-Chapman model, 241 Harisinghani, M. G., 557 Gas multiplication, 468 Gradient, 92, 148 Harmonic images, 374 Gasiorowicz, S., 396, 422 diffusion, 558 Harmonic oscillator, 277, 386, 583 Gaskill, J. D., 349, 362 magnetic, 549 Harmonics, 308 Gastrocnemius, 6, 7 phase-encoding, 550 Harri, R., 334 Gating current, 253 readout, 550 Harris, C. C., 467 Gauss, 213 slice-selection, 548 Harris, J. W., 189 Gauss (unit), 213 Gradient in different coordinate systems, 601 Hart, H., 51 Gauss’s law, 144 Gradiometer, 231 Hartmann, W. M., 370, 379 for magnetic field, 215 Graininess, 475 Hartree–Fock approximation, 444 free and bound charge, 150 Granger, H. J., 121 Haskell, R. C., 422 Gaussian distribution, 96, 104, 592 Grant, R. M., 369 Hasselquist, B., 521, 522 Gawande, A., 589 Gras, J. L., 109 Hastings, H. M., 298 Geddes, L. A., 142, 202 Grass, Inc., 205 Hawkes, D. J., 431 Geiger counter, 468 Gray (unit), 452, 491 Hayes, N. D., 288 Geiger’s rule, 499 Gray body, 396 Haynie, D. T., 54, 83 Geise, R., 475, 477 Gray,D.E.,379 HCl, 417 Gel dosimeter, 469 Gray, R. A., 290, 291 He, B., 235 General thermodynamic relationship, 69 Green (color), 415 Heald, M. A., 410, 422 Generalized force, 69 Green’s function, 104 Health Physics Society, 494 Generalized functions, 323 Gregory, K., 422 Heart block, 203 Gennari, F. J., 123 Greigite, 228 second degree, 209 George, A. L., 193 Greinix H., 422 sinus exit, 209 Germanium, 469 Greivenkamp, J. E., 413, 422 Heart failure, 122, 203 Gersten, K., 23, 31 Grenier, R. P., 531 Heart rate, 50 Gevenois, P. A., 480 Grid Heat capacity, 65, 293 Gevins, A., 224 antiscatter, 474 air, 65, 80 Giaccia, A. J., 480, 481, 483, 485 Griffiths, D. J., 213 at constant volume, 65 Giasson, 422 Grosberg, A. Y., 245 ideal gas, 65 Giasson, C. J., 403 Grossweiner, L., 386, 389, 391, 404, 405, 422 specific, 65 Gibbs factor, 81 Growth rate water, 80 Gibbs free energy, 70 variable, 38 Heat conduction, 109, 112, 404 Gibbs paradox, 72 Grundfest, H., 190, 191, 220 Heat engine, 75 Gibbs phenomenon, 316 Guevara, M. R., 291 Heat flow, 58, 65 Gielen, F. L. H., 223, 236 Gulrajani, R. M., 185, 200 Heat loss Giercksky, K. E., 422 Guttman, R., 46 and countercurrent exchange, 138 Index 619

Heat stroke, 289 Hydrogen spectrum, 383 Insulin, 295 Heaton, C. L., 401 Hydrostatics, 13 Integral control, 279 Hecht, S., 413, 414, 422 Hydroxyl free radical, 481 Integrals Hee, M. R., 422 Hynynen, K., 375, 379 trigonometric, 579 Helical CT, 478 Hyperbolic functions, 174 Intensifying screen, 465 Hellman, L., 51 Hyperopia, 412 Intensity Helmholtz, 142 Hyperplasia, 401 of sound, 367 Hematocrit, 24 Hyperpolarization, 181, 559 radiant, 407 Hemmingsen, A. M., 45, 51 Hyperthermia, 404 reference, 368 Hemochromatosis, 228 Hyperthyroidism, 524 Intensity attenuation coefficient, 370 Hemoglobin, 2, 25, 392, 543, 557 Hypertrophy, 401 Intensity-modulated proton therapy (IMPT), Hemolysis, 400 Hypoproteinemia, 122 489 Hemosiderosis, 228 Hypothalamus, 270, 294 Intensity-modulated radiation therapy, 487 Hempelmann, L. H., 497 Hysteresis, 13, 20 Interferometer, 393 Hendee, W. R., 363, 373, 376, 379, 471, 474, magnetic, 227 Internal conversion, 507 477 Internal radiotherapy, 524 Henriquez, C. S., 192, 200, 201 International Commission on Radiation Units Hereditary spherocytosis, 123 Ice ages, 337 and Measurements, see ICRU Hermann, M., 422 ICRP, 490, 491 International Commission on Radiological Herrick, J. F., 17, 31 ICRU, 370, 374, 379, 390, 438, 439, 441, Protection, see ICRP Hershey, A., 527 444–448, 450, 462–464, 471, 472, 474, International Committee on Non-ionizing Hertz (unit of frequency), 308 492, 501, 511 Radiation Protection (ICNIRP), 261 Hielscher, A., 391, 422 Ideal gas, 78 Interpolation, 352 High-temperature superconductors, 230 Ideal gas law, 64, 67 Interstitial fluid, 121 Higson, D. J., 494 Ideal solution, 121 Intestinal parasites, 589 Hilborn, R. C., 277, 280, 286 Ideker, R. E., 292 Intestinal villi, 128 Hildebrand, J. H., 72, 83, 91, 115, 121 Iliakis, G., 51 Intracellular fluid, 142 Hill, W., 152 Illuminance, 406 Intravascular ultrasound, 399 Hille, B., 31, 162, 247, 250, 256 Ilmoniemi, R. J., 226, 334 Inulin, 417 Hiller J. E., 423 Image Inverse problem, 207 Hine, H. G., 531 density-weighted, 554 Inversion recovery (IR), 546 Hip, 7, 8 T1-weighted, 554 Iodine, 472 Hitzenberger, C. K., 422 T2-weighted, 554 123I, 520 Hobbie, R. K., 194, 197, 198, 450 Image contrast and MR pulse parameters, 554 131I, 522, 524 Hodgkin’s disease, 485 Image quality, 474 Iodine deficiency, 292 Hodgkin, A., 142, 155, 160, 161, 179 Imaginary number, 311 Ion Hodgkin–Huxley model, 3, 161, 163, 165, Impact parameter, 442 Born charging energy, 151 166, 283, 290 Impedance Ionization chamber, 468, 490 Hofbauer G. F., 421 acoustic, 366 Ionization energy, 384, 426 Hoffmann, P. M., 85, 115 characteristic, 367 average, 444 Hogstrom, K. R., 488 reactive component, 367 Ionizing radiation, 425 Hole, 461 transformation in middle ear, 369 192Ir, 524 Homeotherms, 45 Impulse approximation, 440, 443 Iris, 411 Honig, B., 243 Impulse function, 323 Iron oxide, 543 Hormesis, 494 Impulse response, 331, 345 Iron stores in the body, 228 Horowitz, P., 152 Incidence, 402 Irradiance, 406, 408 Hosaka, H., 222 Incidence of disease, 47 Isodose contours, 486 Hot tub, 289 Incoherent scattering, 427, 431 Isosbestic point, 392 Hounsfield (unit), 478 Incremental-signal transfer factor, 466 Isotonic saline, 136 Hounsfield, G. N., 478 Incubator Isotope, 504 Howell, R. W., 507, 533 neonatal, 399 Isotropic radiation, 409 Hu, X-p., 350, 351 Incus, 369 Itoh, M., 423 Huang, D., 392, 422 Index of refraction, 381, 387 Itzkan, I., 422 Huang, S., 53, 84 of aqueous, 411 Izatt, J. A., 394 Hubbell, J. H., 428, 429, 431, 432, 435, 438, of the lens, 411 472 of vitreous, 411 Hubbert’s peak, 49 Inelastic scattering, 427 Jackson, D. F., 431 Human Immunodeficiency Virus (HIV), 2 Inflammatory response, 122, 401 Jackson, J. D., 431 Humm, J. L., 507 Influence function, 104 Jacques, S. L., 422 Hunt, J. G., 515 Infrared Jaeger, J. C., 96, 99, 105, 112, 115 Husband, J. E., 557 photography, 399 Jaksch P., 422 Huxley, A. F., 142, 161 radiation, 382, 398 Jalife, J., 172, 205, 223, 283 HVL, 498 spectroscopy, 391 Jalinous, R., 226 Hydraulic permeability, 123 Ingard, K. U., 363, 379 Jang, I-K., 422 pore model, 129 Inman,V.T.,8,9,31 Janks, D. L., 205, 210 Hydrogen peroxide, 481 Inner ear, 369 Jaramillo, F., 337 620 Index

Jaundice, neonatal, 400 artificial, 126 Late-responding tissue, 483, 484 Jayakumar, T., 422 glomerulus, 17 Latent heat of vaporization, 79 Jeanmonod, D., 375, 379 Kilocalorie, 58, 80 Lateral geniculate nucleus, 557 Jeffrey, K., 203, 209 Kim, B. H., 543 Latitude, 465 Jeffreys, B. S., 159 Kimlin, M. G., 422 Laüger, P., 155 Jeffreys, H., 159 Kinesin, 85 Lauterbur, P. C., 548 Jejunum, 178 Kinetic energy, 11 Law of Laplace, 27, 82 Jewett, J. W., 7, 148, 423 King, K. F., 553 Lazovich D., 422 Jiang, D., 209 Kirchhoff’s laws, 153 Lazovich, D., 403 Jiang, Y., 252 Kirschvink, J. L., 229, 260, 261 Le, T. H., 350, 351 Johnsen, S., 381, 422 Kiss, L. B., 338 Lead zirconate titanate, 371 Johnson noise, 255, 332, 333 Kissel, L. H., 431 Leads Johnson, G. F., 51 Kleiber’s law, 45 ECG, 196 Johnson, J. B., 333 Klein-Nishina limb, 196 Joint probability distribution, 605 factor, 456 augmented, 197 Jones, J. P., 362, 548 formula, 430 precordial, 197 Jones, R. H., 531 Kligman, L. H., 402, 422 Least squares, 211, 303 Jordan, A., 229 Knobler R., 422 linear, 304 Joseph, P. M., 546 Knobler, R., 404 nonlinear, 306 Josephson junction, 230 Knutson, J. S., 202 polynomial, 306 Joule (unit), 11 Knuutila, J., 334 Lebec, M., 423 Juziene, A., 422 Kobayashi, A. K., 260 LeBihan, D., 115 Koh, D.-M., 557 LeDuc, P. R., 279 Kohl, M., 379 Left and right bundles, 194 K edge Köhler, M. O., 379 Left ventricular hypertrophy, 198 in contrast agent, 472 Köhler, T., 26, 31 Legendre polynomial, 189, 192 K shell, 461 Koizumi, H., 423 Lei, H., 555 Kagan, A. R., 485 Kondo, S., 494 leibnitz, 53 Kaiser, I. H., 313, 341 Koo, T-W., 422 Leigh, J., 423 Kalender, W. A., 478–480 Kooiman, K., 379 Lens, 411 Kalmijn, Ad. J., 256 Kooy,H.M.,489 Lenz’s law, 225, 227 Kaluza, G. L., 524, 533 Körner, M., 470 LET, 447 Kaminer, M., 375, 379 Korotkoff sounds, 21 Leuchtag, H. R., 249 Kanal, E., 555 Kovacs,G.T.A.,142 Leukemia, 45, 290 Kandel, S., 193 Kowalsky, R. J., 517, 520, 533 Levenberg-Marquardt method, 307 Kane, B. J., 142 Krämer, U., 422 Levitt, D., 91, 103, 115, 130, 133 Kannel, W. B., 51 Kramer’s law, 497 Levitt, M. H., 543 Kaplan, D., 280, 286, 291, 335, 336 Kramer, E. M., 118 Levy, D., 39, 51 Karhu, J., 226 Kramer, J. R., 422 Lewis number, 109 Karp, J. S., 523, 533 Krane, K. S., 398 Lewis, C. A., 250 Kassell, N. F., 379 Kremkau, F. W., 371, 379 Li, T. K., 48 Kassirer, J. P., 123 Kricker, 421 Liang, Z.-P., 548 Kassis, A. I., 480, 483, 507, 524, 533 Kricker, A., 402 Light Katchalsky, A., 126 Kripke, M. L., 402, 403, 422 photon properties, 382 Kathren, R. L., 494 81mKr, 522 polarized, 393 Katz, B., 141, 143, 161 Kupfer cells, 519 speed, 381 Keener, J. P., 270, 291 Kwong, K. K., 557 wave nature, 382, 392 Keizer, J., 111, 115 Lightfoot, E. N., 94 Kelvin (unit), 61 Lighthill, J., 16, 31 Kempe, C. H., 44, 51 LaBarbera, M., 28, 31 Lighthill, M. J., 316, 323 Keratin, 401 Lahiri, B. B., 422 Likavec, M. J., 123 Keratitis, 403 Lakshminarayanan, A. V., 357, 362 Limb leads, 196 Keratopathy, 403 Lamb, H., 260 Lime peel, 402 Keriakes, J. G., 488 Lambert’s law, 409 Limit cycle Kerma, 452 Laminar flow, 15, 16 bifurcation of, 282 collision, 452, 453 Lanchester, F. W., 296 stable, 281 radiative, 452 Langevin equation, 91 unstable, 281 Kernicterus, 400 Langrill, D. M., 210 Lin, C. P., 422 Ketcham, D., 473 Lapicque, L., 202 Lindemans, F. W., 203 Kevan, P. G., 410, 422 Laplacian, 95 Lindhard, J., 442, 444, 445 Kevles, B. H., 345, 362 Large-signal transfer factor, 466 Lindner, A., 126 Keynes, R. D., 252 Larin, K. V., 422 Lindsay, R. B., 371, 379 Khammash, M., 279 Larina, I. V., 422 Line approximation, 192 Khan, F. M., 486–489, 533 Larmor precession frequency, 538, 542 Line integral, 148 Khurana, V. G., 261 Laser surgery, 404 Line spread function, 349 Kidney, 40, 123, 293 Lasser, T., 422 Linear energy transfer, 447, 481 Index 621

Linear no-threshold model, 493, 494 ordered, 56 Martin, E., 379 Linear regression, 304 random, 56 Mass attenuation coefficient, 433 Linear system, 345 Maddock, J. R., 101, 115 tissue, 471 Linear-no-threshold model, 526 Madronich, S., 400, 401, 422 Mass energy absorption coefficient, 437 Linear-quadratic model, 482 Magenta, 415 table (on the web), 465 Links, J. M., 521, 533 Magic angle, 560 Mass energy transfer coefficient, 437 Liow, J. S., 533 Magleby, K. L., 251 Mass fraction, 434 Lipid bilayer, 169 Magnetic dipole, 226 Mass number, 503 Lissner, H. R., 7, 8, 31 field far from, 542 Mass stopping power, 439 Liter (unit), 21 Magnetic field, 213, 215 Mass unit Lithium iodide battery, 210 crayfish axon, 219 atomic, 505 Lithotripsy, 375 detection, 229 unified, 505 Little, M. P., 494 earth, 229 Mattiello, J., 115, 558 Littmark, U., 439, 442, 445, 446, 448 from a current dipole in a conducting Maughan, W. Z., 315 Liu, H., 392, 422 sphere, 223, 233 Mauricio, C. L., 515, 533 Liver, 122, 228 from axon, 219 Mauro, A., 241, 242 Ljungberg, M., 532 from depolarizing cell, 219 Mavroidis, C., 126 Local control of tumor, 485 from point source of current, 217 Maxwell’s equations, 234 Loevinger, R., 511, 512, 514, 517, 533 lines, 216 May, R. M., 285 Log-log graph paper, 43 remanent, 227 Mayaux, M. J., 76, 77 Logarithm Magnetic field intensity, 227 Mazumdar, J. N., 16, 21, 23, 31 definition, 573 Magnetic flux quantum, 230 McAdams,M. S., 422 natural, 573 Magnetic moment, 214, 535 McCollough, C. H., 490, 491, 494 Logistic equation, 42, 281 and angular momentum, 536 McDonagh, A. F., 400, 422 Logistic map, 284 nuclear, 537 McGrayne, S. B., 228 Longitudinal wave, 363, 366 torque, 214 McKee, P. A., 39, 51 Lopes da Silva, F. H., 205 Magnetic monopole McMahon, T., 45, 51 Lorente de Nó, R., 160 absence of, 215 McNamara, P. M., 51 Lorentz force, 214, 234 Magnetic permeability, 227 Meadowcroft, P. M., 24, 31 Lorenz, E. N., 296 Magnetic resonance imaging, 535 Mean, 585 Lotka-Volterra equations, 49 intravascular, 399 Mean absorbed dose, 511 Lounasmaa, O. V., 334 Magnetic shielding, 225 Mean absorbed dose per unit cumulated Lowe, H. C., 422 Magnetic stimulation, 235 activity, 511 Lu, J., 256, 266 Magnetic susceptibility, 213, 227, 228 Mean energy emitted per unit cumulated Lubin, J. H., 495 Magnetically-shielded rooms, 230 activity, 511 Lumen, 399 Magnetism Mean energy per ion pair, 464 Lumen (unit), 409 and special relativity, 213 Mean energy per unit cumulated activity, 511 Luminance, 406 Magnetite, 228, 229, 235 Mean free path, 90 Luminous Magnetization, 227, 537 Mean initial activity per unit mass, 514 efficacy, 410 Magnetocardiogram, 219, 222 Mean number per disintegration, 507 energy, 406 Magnetoencephalogram, 219, 223, 334 Mean number per transformation, 511 exitance, 406 Magnetopneumography, 228 Mean square error, 304 exposure, 406 Magnetosome, 228, 229, 235, 260, 261 Meandering waves, 292 flux, 406, 409 Magnetotactic bacteria, 228 Mechanoreceptors, 256 intensity, 406, 410 Magnification, 347 Medel, R., 379 Lung transplant, 522 Magnifying glass, 420 Medical Internal Radiation Dose, see MIRD Luo, C. H., 165, 193 Maidment, A. D. A., 379 MEG, 219 Luther, S., 291 Mainardi, L. T., 328, 329 Meidner, H., 101, 115 Lutz, G., 469 Maki, A., 423 Melanin, 401 LVH, see Left ventricular hypertrophy Makita, S., 423 Melanocyte, 401 Lybanon, M., 306 Mali, W. P., 379 Melanoma, 402, 403 Lynch, D. K., 410, 423 Malleus, 369 Mellinger, M. D., 422 Lysaght, M. J., 126 Malmivuo, J., 185 Membrane, 2, 117 Mammography, 477 force on, 134 Mannitol, 123, 136 permeability, 125 m gate, 164 Manoharan, R., 422 permeable, 117 M mode, 374 Manoyan, J. M., 446 semipermeable, 117 Mackenzie, D., 51 Mansfield, P., 548 Membrane capacitance, 169 Mackey, M. C., 280–282, 284, 288–290, 300 Mansfield, T. A., 101, 115 Merckel, L. G., 375, 379 MacKinnon, R., 252 Maor, E., 35, 51 Mermin, N. D., 593 MacNeill, B. D., 399, 422 Mariappan, Y. K., 556 Messner, W. C., 279 Macovski, A., 474 Maris, M., 423 Metabolic rate, 45, 270 Macrophage, 228 Maritan, A., 51 Metaplasia, 401 Macrostate, 56, 59 Markov process, 587 Metastable state, 507 disordered, 56 Marshall, M., 449, 463 Metastases, 485 nonrandom, 56 Martin, A. R., 597 Metric prefixes, 1 622 Index

Metric system, 1 Moses, H. W., 202 Neper, 370 Mettler, F. A., 477, 492 Moskowitz, B. M., 228 Nephron, 123 Metz, C. E., 474 Mosley, M. L., 523, 533 Nephrotic syndrome, 122 Michels, L., 379 Moss, F., 337, 338 Nernst equation, 64, 155, 239 Micron, 1 Moss, W. T., 486 sodium, 161 Microscope, 420, 421 Mossman, K. L., 494 Nernst potential Microstate, 56, 59 Motamedi, M., 422 calcium, 178 Microstates, 68 Mottle, 475 potassium, 162 energy dependence, 61 Motz, J. T., 422 sodium, 162 Microtubules, 85 Moulder, J. E., 257, 258, 261 Nernst-Planck equation, 247 Microwave radiation, 382 Mouritsen, H., 229 Nerve cell, 79 Mielczarek, E. V., 228, 291 Moy, G., 243 Nerves Miller, J. M., 422 Moyal, D. D., 402, 422 Sympathetic and parasympathetic, 194 Milner T. E., 422 MR Image and pulse parameters, 554 Nesson, M. H., 260 Milnor, W. R., 21, 24, 31, 316 Muehllehner, G., 523, 533 Neuromagnetic current probe, 229 Miners, 228 Mugler, J. P., 559 Neurotransmitter, 143 Mines, G. R., 290 Mullin, J. C., 202 Neutrino, 509 Minimum auditory field (MAF), 368 Multipole expansion, 192 Neutron therapy, 489 Minimum auditory pressure (MAP), 368 Mundy L., 423 Newhauser, W. D., 488, 489 Minimum erythemal dose, 401 Murata, K., 118 Newman D., 115 Miralbell, R., 489 Murphy, E. A., 588 Newman, E. B., 369, 379 MIRD, 503, 511, 515, 516, 524 Murphy, M. R., 400, 422 Newton (unit), 4 Mitochondria, 2 Murray’s law, 28 Newton’s law of cooling, 80 Mitosis, 85, 480, 481 Murray, J. D., 43, 49, 51, 270 Newton’s second law, 11, 53 Mitral valve, 193 Muscle Newton’s third law, 12 Mixing gastrocnemius, 6, 7 Newton, I., 381 entropy of, 72 gluteus medius, 8 Newtonian fluid, 16, 24, 91 heat of, 73 gluteus minimis, 8 Nicholls, A., 243 Miyachi Y., 423 soleus,6,7 Nicholls, J. G., 141 Mizuno M., 423 Mutual inductance, 236 Nichols, C. G., 264 ml (unit), 21 Myelinated fiber Nickell, S., 389, 422 Moan, J., 422 time and space constants, 167 Nielsen, P., 228 Mobile phones, 261 Myers, D. R., 118 Nioka, S., 423 Mode, 586 Myers, K. J., 345, 353, 362 NIST, 433, 438, 448, 465 Model (mathematical), 3 Myocardial cells 13N, 523 Models, 3 and nerve cells compared, 193 Nitrogen narcosis, 15 Modulation transfer function, 348, 413, 474 repolarization, 186 No-slip boundary condition, 15 Modulus Myocardial infarct, 39, 48, 222, 399 Noble, D., 172 shear, 13 Myoglobin, 81 Nodal escape beats, 194 Young’s, 13 Myopia, 412 Node of Ranvier, 142, 167 Moe, G. K., 283 Myosin, 85 Noise, 80, 254, 326, 327 Mole, 388 1/f, 335 Mole fraction, 73 autocorrelation function, 334 Molecular dynamics, 244 n gate, 164 Johnson, 255, 332, 333 Moller, J. H., 198, 199 Nag, S., 523 pink, 335 Moller, W., 228 Naito N., 423 power spectrum, 333 99Mo, 519 Narmoneva, D. A., 30 quantum, 475 Molybdenum target x-ray tube, 477 Narrow-beam geometry, 432 shot, 254, 332, 475 Momentarm,5,10 Nath, R., 524, 532 source, 332 Moment of inertia, 385 National Institute of Standards and white, 333 Monte Carlo calculation, 389, 391, 450, 515 Technology, see NIST Noise brightness contrast, 475 Monteith, S., 375, 379 National Nuclear Data Center, 511 Noise equivalent quanta, 476 Montesinos, G. D., 450, 515, 532 National Research Council, 258 Noise exposure contrast, 475 Moonen, C. T., 379 Natural background, 492 Nolte, J., 143 Moore, J. W., 289 Natural logarithm, 573 Noninvasive electrocardiographic imaging, Moran, J., 126 Navier-Stokes equation, 28 200 Morel, A., 379 NCRP, 477, 492, 494 Nonlinear least squares, 306 Morgan, K., 394 Near field, 372 Nonmelanoma skin cancer (NMSC), 402 Morin, D. J., 213, 218 Near-infrared radiation, 392 Norepinephrine, 194 Morin, R., 555 Near-infrared spectroscopy, 399 Normal distribution, 592 Morrison, P., 3, 31 Nedbal, L., 314 Nuclear force, 505 Morse, P. M., 363, 379 Negative resistance, 265, 266 Nuclear scattering, 445 Mortality function, 47 Neher, E., 251 Nunez, P. L., 205 Mortimer, M., 281 Nelson J. S., 422 Nutation, 541, 549 Moseley, H., 497 Neonatal jaundice, 400 Nyenhuis, J. A., 555 Moseley, M. E., 558 Neovascular membrane, 394 Nyquist frequency, 329 Index 623

Nyquist sampling criterion, 309 Ozone, 400 Phase encoding, 548, 551 Nyquist, H., 333 Phase transfer function, 348 Phased array, 373 P wave, 195, 203 Phelps, M. E., 517, 520, 522, 523, 531, 532 O’Brien, D., 44, 51 31P, 555 Phenobarbital, 48 O’Reardon, J. P., 226 Pacemaker, 202, 203, 282 Phenothiazines, 402 O’Sullivan N., 422 Pacing electrode, 203 Philip, J., 422 O’Sullivan, N., 403 Packard, G. C., 308 Phlebotomy, 125 OATZ, 228 Page, C. H., 225 Phosphate, 293 Oberly, L. W., 521 Paine, P. L., 91, 115 Phosphorescence, 387 Ocean flounder, 256 Pair production, 428, 432 Photochemotherapy, 404 Odd function, 308 free electron, 432 Photodynamic therapy, 399 Oellrich, R. G., 400, 422 Pallotta, B. S., 251 Photoelectric effect, 426, 428 Oersted, H. C., 213 Pancreas, 295 Photographic emulsion, 448 Ogawa, S., 557 Panfilov, A. V., 291 Photometry, 405 Ohm (unit), 151 Pankhurst, Q. A., 229 Photomultiplier tube, 520 Ohm’s law, 151, 190, 199 Paramagnetism, 227 Photon, 382 and current density, 152 Paramecium, 1 attenuation coefficient, 433 anisotropic medium, 200 Parasympathetic nerves, 194 characteristic, 435 Oldendorf, W., 478 Parathyroid hormone, 295 diffusion constant, 390 OLINDA/EXM, 511, 515 Parisi, A. V., 422 exitance, 406 One-dimensional flow (mathematical), 276 Parisi, M., 118 fluence, 406 Open-loop gain, 272 Parkinson, J. S., 101, 115 fluence rate, 406 Operating point, 271 Parseval’s theorem, 324 fluorescence, 435 approach to, 273 Partial derivative, 61, 607 flux, 406 Optical coherence microscopy, 393 order of differentiation, 608 flux exposure, 406 Optical coherence tomography (OCT), 392 Particle exchange flux intensity, 406 Optical density, 465 equilibrium, 66 flux irradiance, 406 and exposure, 497 Particular solution, 584 flux radiance, 406 Optical mapping, 205 Pascal (unit), 13 secondary, 436, 437, 454 Optical transfer function, 348 Patch-clamp recording, 251 Photon absorption, 387 Orbital magnetic moment, 227 Patterson, J. C., 523, 533 Photon scattering, 387 Orbital quantum number, 425 Patterson, M. S., 391, 399, 422, 423 Photons, 257 Orear, J., 306 Patton, H. D., 21, 31, 123, 128, 141, 143, 155, Photopheresis, 404 Organelle, 2 271 Photopic vision, 409 Orthogonality relations Pauli exclusion principle, 384, 446 Photosynthesis, 72, 314 for trigonometric functions, 315 Pauli promotion, 446 in aquatic plants, 418 Orton, C., 379, 485, 494 Payandeh, J., 252 Physiological recruitment, 209 Osmolality, 121 Payne, J. T., 357 π, 541 Osmolarity, 121 Pearle P., 115 π/2, 544 Osmoreceptors, 294 Pearle, P., 108 Pickard, W. F., 256 Osmosis Peckham, P. H., 202 Piezoelectric transducer, 371 in ideal gas, 118 Péclet number, 113 Pigeon, 229, 235 in liquid, 121, 123 Pecora, R., 393, 422 Pillsbury, D. M., 401 reverse, 125 Pedley, T. J., 30 Pinna, 369 Osmotic diuresis, 123 Peng, Q., 417, 422 Pirenne, M. H., 413, 414, 422 Osmotic flow Perfumes, 402 Pisano, E. D., 477 difference from diffusion, 117 Perfusion, 404 Pituitary, 270, 292 Osmotic pressure Period doubling, 285 Planck’s constant, 230, 257, 382 and capillaries, 121 in heart cells, 291 Planck, M., 396 and chemical potential, 120 Periodogram, 329 Plane wave, 363, 409 and driving pressure, 121 Perkins, D. H., 448, 449 Plank, G., 205 and interstitial fluid, 121 Permeability Plaque, 399 in ideal gas, 120 hydraulic, 123 vulnerable, 399 in liquid, 121 magnetic, 227 Plasma, 40 Ossicles, 369 membrane, 125 Plasma membrane, 239 Osteoarthritis, 136, 263 Perrin, J., 89, 108, 115 Plasmon, 442 Othmer, J. G., 111, 115 PET, 557 Plasticity, 205 Oudit, G. Y., 194 Peters, R. H., 45, 51 Platzman, R. L., 464 Oval window, 369 Petridou, N., 557 Plonsey, R., 21, 160, 161, 167, 185, 190–192, Øverbø, I., 428 Phagocytosis, 519 200, 202, 208, 220 Oxic-anoxic transition zone, 228 Phantom, 515 Plug flow, 556 15O, 523 Phase, 308 Pneumothorax, 472 18O, 215 resetting, 280–282 Pogue, B. W., 391, 422 Oxygen consumption, 270 singularity, 299 Poikilotherms, 45 Oxyhemoglobin, 543 space, 277 Poincaré, J. H., 279 624 Index

Point charge, 145 diastolic, 20 Radial isochron clock, 3, 281, 296 Point-spread function, 347, 413 dynamic, 19 Radian, 567 for an ideal imaging system, 347 systolic, 20 Radiance, 406–408 Poise (unit of viscosity), 16 Preston, G. M., 118 Radiant Poiseuille flow, 17, 21, 128 Prevalence of disease, 47 energy, 406, 407 departures from, 22 Prickle layer, 401 exposure, 406 Poiseuille, J. L. M., 17 Primary colors, 415 flux, 406 Poisson distribution, 389, 414, 595 Principal quantum number, 425 intensity, 406, 407 and shot noise, 255 Probability, 54 power, 406, 407 Poisson equation, 242 binomial, 55 Radiation Poisson statistics Product biological effects, 480 in cancer incidence, 493 cross, 6 electromagnetic, 257 in cell survival, 485 dot, 12 natural background, 492 in image, 475 scalar, 12 risk, 490 in photon fluence, 480 vector, 6 weighting factor, 490 in radiation damage, 482 Projection, 522 Radiation chemical yield, 463 Poisson’s ratio, 28 Projection reconstruction, 548, 551 Radiation damage Poisson-Boltzmann equation, 242 Projection theorem, 351 type A, 482, 484 analytic solution, 263 Projections, 351 type B, 482, 484 linearized, 242 Prokaryotes, 2 Radiation equilibrium, 452 Polarization, 394 Proportional control, 279 Radiation risk model electric, 246 Proportional counter, 468 hormesis, 494 of dielectric, 149 Protein, 2 linear no-threshold, 493 Polarized light, 393 Proton therapy, 488 threshold, 494 Polk, C., 257, 259–261 Protoplast, 2 Radiation therapy, 486 Polynomial Protozoans, 1 conformal, 487 regression, 306 PRU (peripheral resistance unit), 21 electron, 488 Population inversion, 417 Pryde, J. A., 91, 115 intensity-modulated, 487 Positron, 428, 432, 438 Pseudovector, 6 Radiation yield, 448, 461, 462 decay, 508 Psoralen, 404 Radiative transition, 435 Positron Emission Tomography (PET), 215, Psoriasis, 403 Radiobiology, 480 523 Pterygium, 403 Radiochromic film, 469 Postulates of statistical mechanics, 59 Puliafito, C. A., 422 Radiograph, 470 Potassium channels Pulmonary embolus, 517 Radiographic signal, 475 shaker, 254 Pulse echo techniques, 373 Radioimmunoassay, 503 slow, 193 Pulse oximeter, 392 Radioimmunotherapy, 524 Potassium conductance, 163 Pupil, 411 Radiometry, 405 Potassium gate, 164 Pupil size, 289 Radiopharmaceuticals, 517 Potassium Nernst potential, 162 Purcell, E. M., 23, 31, 101, 114, 115, 213, Radiotherapy Potential 218, 539 internal, 524 226 electric, 147 Purkinje fibers, 194, 282 Ra, 524 outside axon, 185 PUVA, 404 Radius transmembrane, 205 Pyramidal cells, 233 atomic, 504 Potential Alpha Energy Concentration (PAEC) Pyroelectric crystal, 399 nuclear, 504 (unit), 526 Radon, 492, 495 Potential difference, 148 222Rn, 524, 525 and ion concentration, 240 Q10, 79, 164, 165 Radon transformation, 353 Potential energy, 147 QRS wave, 195, 203 Railroad tracks, 46 Powell, C. F., 448, 449 Quadrupole Raizner, A. E., 524, 533 Power, 11, 153 magnetic, 535 Ramachandran, G. N., 357, 362 Power law, 43 Quality factor, 490 Raman scattering, 393 Power spectral density (PSD), 328 Quantum mottle, 475 Raman spectroscopy, 399 Blackman-Tukey method, 329 Quantum number, 57 Raman, C. V., 393 magnetic noise, 334 Quantum numbers Random walk, 106 periodogram, 329 atomic electrons, 383 Range, 439, 447 Power spectrum, 317 rotational, 385 Ranvier Power, average, 324 vibrational, 386 node of, 142 Prausnitz, J. M., 72, 83, 91, 115 Quasiperiodicity, 286 Rao, D. V., 533 Precession, 538 Quatrefoil pattern, 222 Ratchet and pawl, 337 Precordial leads, 197 Québec, 495 Ratliff, S. T., 485 Predator-prey problem, 49 Quinine water, 402 Rattay, F., 209 Prephasing lobe, 562 Rayleigh, 296 Press, W. H., 76, 84, 111, 115, 191, 211, 219, Rayleigh scattering, 431 220, 303, 307, 314, 315, 328, 329, 360, R project, 307 RBE, 490 362 RADAR (Radiation Group Assessment Reactance, 367 Pressure, 14 Resource), 511, 515 Reaction-diffusion process, 111 Index 625

Receptor Respiration of glucose, 72 SA node, 172, 194, 203 mechanical, 142 Response Sacks, O., 370, 379 stretch, 141 impulse, 345 Saedi, N., 375, 379 temperature, 141 Rest energy, 504 Safety in MRI, 555 Receptor field, 413 Rest mass, 429 Saint-Jalmes, H., 393, 423 Reconstruction from projections Restenosis, 524 Sakitt, B., 415, 423 filtered back projections, 352 Resting potential, 142 Sakmann, B., 251 Fourier transform, 351 atrial and ventricular cells, 193 Salmelin, R., 224 Recruitment Restitution, 298 Saltatory conduction, 167 physiological, 209 Restricted linear collision stopping power, 447 Samuels S., 115 Rectifier channels, 248 Retina, 403, 411 Santini, L., 555 Red (color), 415 Reverberation echo, 378 Sap flow, 29 Red blood cell, 2, 25, 123 Reversal potential, 249, 250, 264 SAR, see Specific absorption rate Reddy, A. K. N., 247 Reverse osmosis, 125 Sarcoplasmic reticulum, 111 Reduced mass, 385 Sarvas, J., 233 Reversible process, 69 Reduced scattering coefficient, 389 Sastry,K.S.R.,507 Reynolds number, 22, 30, 49 Reentrant circuit, 194, 204, 207 Sato, S., 236 Rheobase, 179, 202 Reference action spectrum, 401 Saturation, 227 Rieke, F., 415, 422 Reflection, 411 Savage,V.M.,51 Rieke, V., 379 total internal, 421 Savannah sparrow, 229 Reflection at a boundary, 367 Rigel D., 422 Scalar, 570 Reflection coefficient, 124, 125, 367 Riggs, D. S., 42, 51, 270, 293, 295 Scalar product, 12 pore model, 133 Right ventricular hypertrophy, 198 Scaling, 43 Refraction, 411 Rinaldo, A., 51 Scattering Refractory period, 165, 181, 194 Ripplinger, C. M., 205 coherent, 427, 431 Regression Risk Compton, 427, 428 linear, 304 excess, 493 incoherent, 427, 431 polynomial, 306 models, 493 inelastic, 427 Rehm, K., 523, 524, 533 relative, 257 nuclear, 445 Reich, P., 50, 51 Ritenour, E. R., 363, 373, 376, 379, 471, 474, Raman, 427 Reif, F., 54, 57, 59, 60, 68, 72, 84, 90, 115, 477, 479 Rayleigh, 431 385, 587 Roberson, P., 532 Thomson, 430 Reinisch W., 422 Robinson, D. K., 307 Scattering coefficient Relative biological effectiveness, 490 Robitaille, P.-M., 545 reduced, 389 Relative risk, 257 Rodieck, R. W., 410, 413, 423 Scattering cross section, see Cross section Relativity Rodriguez, J., 352, 362 Schaefer, D. J., 555 special, see Special relativity Rods (retinal), 409, 413 Schaefer-Prokop, C., 470 Relaxation oscillator, 194 Roentgen (unit), 464, 497 Schaper, K. A., 533 Relaxation time Roentgen, W., 465 Scher, A. M., 31, 294 experimental, 544 Rohs, R., 243 Scherr, P., 91, 115 longitudinal, 538, 543 Roll-off, 331 Schey, H. M., 88, 93, 102, 115, 144, 601 non-recoverable, 544 Rollins, A. M., 394 Schiff, G., 379 spin-lattice, see Relaxation time, Romberg integration, 191, 220 Schlichting, H., 23, 31 longitudinal Romer, R. H., 46, 225 Schlomka, J. P., 472 spin-spin, see Relaxation time, transverse Rook A. H., 422 Schmidt, C. W., 403, 423 thermal, 405 Rose, A., 415, 423 Schmidt-Nielsen, K., 50, 51, 128, 138 transverse, 538, 544 Rosenbaum, D. S., 205 Schmitt, F., 553 Rem (unit), 490 Schmitt, J. M., 392, 423 Rossi, S., 225 Remanent magnetic field, 227 Schmitt, O. H., 258 Rossi-Fanelli, A., 81 Renal arteries, 479 Schrödinger equation, 53 Rotating coordinate system, 539 Renal dialysis, 123 Schroeder, D. V., 54, 84, 385, 396, 423 Rotation matrices, 540, 560 Renal tubules, 128 Schroter, S. T., 30 Repeller, 276 Roth, B. J., 201, 205, 210, 219, 223, 226, 229, Schueler, B. A., 490, 491 Repolarization, 142, 194 232–234, 236, 418, 423 Schultz, J. S., 134 Reservoir, 62, 70 Rottenberg, D. A., 533 Schulz, R. J., 485 Residence time, 512 Round window, 370 Schuman, J. S., 422 Residuals, 304 Rowlands, J. A., 470 Schwan, H. P., 258 Resistance, 151 Rudy, Y., 165, 193, 200 Schwartz, D., 76, 77 parallel and series, 154 Ruohonen, J., 226 Schwarz T., 422 vascular, 21 Rushton, W. A. H., 160, 167, 179 Schwiegerling, J., 422 Resistive pulse technique, 177 Ruska, E., 383 Scintillation camera, 520 Resnick, R., 227, 230, 383, 386, 387, 398, Ruth, T. J., 503, 533 Scintillation detector, 466, 520 505, 508, 509, 532 Rutherford, E., 527 Scotopic vision, 409, 413 Resolution RVH, see Right ventricular hypertrophy Scott, A. C., 156, 161 and spatial frequency, 349 Rydberg constant, 417 Scott, G. C., 233 Resonance, 584 Ryman, J. C., 532 Scott, R. L., 72, 83, 91, 115, 121 626 Index

Screen, intensifying, 465 Single-channel recording, 251 Spector, R., 51 Screening Single-photon emission computed Spectral efficiency function, 409 atomic electron (charge), 426 tomography, 521 Spectroscopy SCUBA, 15 Sink, 276 infrared, 391 Second law of thermodynamics, 75 Sinoatrial node, see see SA node Spectrum Sedimentation velocity, 27 Sinogram, 353 beta decay, 509 Seed,W.A.,30 Sinus exit block, 209 Wiener, 475 Seibert, J. A., 470 Skin depth, 234 Spherical aberration, 412 Selection rules, 384–386, 461 Skofronick, J. G., 369 Spherical coordinates, 601 Self-diffusion, 94 Slice selection, 548 Spherocytosis, 123 Self-similarity, 286 Slichter, C. P., 538, 543 Spheroid degeneration of the eye, 403 Seltzer, S. M., 438, 472 Slide projector, 420 Sphincter, pre-capillary, 21 Semicircular canals, 369 Smith, M. A., 422 Sphygmomanometer, 20 Semiconductor detector, 469 Smye, S. W., 394, 423 Spider’s thread, 26 Semiempirical mass formula, 508 Smythe, W. R., 174 Spin Semilog paper, 36 Snell’s law, 411 electron, 227, 384 Semipermeable membrane, 117 Snell, J., 379 nuclear, 536 Separation, 23 Sneyd, J., 270, 291 Spin echo (SE), 546, 548 Sequestration, 519 Snow blindness, 403 fast or turbo, 553 Serruys, P. W., 423 Snyder, W. S., 511, 515, 516, 533 Spin quantum number, 425 Serway, R. A., 7, 148, 423 Sobol, W. T., 471 Spin warp encoding, 548 Setton, L. A., 30 Söderberg, P. G., 403, 421 Spiral CT, 478 Sevick, E. M., 391, 423 Sodium Spiral wave, 195, 291, 299 Seymour, R. S., 45, 51 -potassium pump, 155 Spoiler gradient, 562 Sgouros, G., 532 Nernst potential, 162 Spontaneous births, 314 Shadowitz, A., 218 24Na, 509, 510 Spontaneous emission, 542 Shafer, K. E., 422 Sodium gate, 164 Spring constant, 583 Shaker channels, 252 Sodium spectral line, 394 Squamous cell carcinoma (SCC), 402, 403 Shani, G., 469, 470 Soffer, B. H., 410, 423 SQUID, 230, 236 Shapiro, E. M., 263 Solar constant, 400, 419 Squid, 2 Shapiro, L., 101, 115 Soleus, 6, 7 giant axon, 161 Shark Solid angle, 189, 389, 568 SRIM, 439, 446, 448 and earth’s magnetic field, 231 defined, 567 Srinivas S. M., 422 Shaw, C. D., 277 theorem, 207 Srinivasan, R., 205 Shaw, R., 349, 362 Solute, 72, 73 Stabin, M. G., 511, 515, 524, 533 Shear modulus, 13 permeability, 125 Stahlhofen, W., 228 Shear rate, 16 Solute permeability Standard deviation, 586 Shear strain, 13 pore model, 134 Standard deviation of the mean, 586 Shear stress, 13 Solute transport, 125, 128 Standing wave, 366 Shear wave, 365 pore model, 133 Stanfield J., 422 Sheehan, J., 379 Solution Stanley, P. C., 200 Shells ideal, 121 Stapes, 369 atomic, 426 Solvent, 72 Stark, L., 270, 290, 413, 423 Sheppard, A. R., 261 Solvent drag, 89, 125 State space, 277 Sherwood number, 114 in an electric field, 247 Stationarity, 347 Shlaer, S., 413, 414, 422 Sonoda, E., 51 Stationary random process, 327 Shot noise, 254, 332 Sorenson, J. A., 517, 520, 522, 523, 531, 532 Stationary system, 346 Shrier, A., 291 Sorgen, P. L., 193 Statistical mechanics, 54 SI system, 1, 226 Sound level measurement postulates, 59 Sick sinus syndrome, 203, 209 weighting, 369 Staton, D. J., 222, 223 Sidtis, J. J., 533 Source, 276 Steady state and equilibrium Siegel, J. A., 524, 533 Space clamp, 161 distinction, 59 Siemens (unit), 151 Space invariance, 347 Steel,G.G.,482 Sievert (unit), 490, 491 Spano, M. L., 291 Stefan-Boltzmann law, 396 Signal, 80, 326 Sparks, R. B., 511, 515, 533 Stegun, I. A., 209, 263, 359, 361, 362 and noise, 327 Spatial frequency, 346, 349 Stehling, M. K., 553 radiographic, 475 and field of view, 349 Steiner, R. F., 31 Signal averaging, 328 and resolution, 349 Steinmuller, D. R., 136 Signal-to-noise ratio, 476 Special relativity, 213, 231, 234, 429, 504 Steketee, J., 398, 423 Sigworth, F. J., 252–255 Specific absorbed fraction, 511 Stenosis, 399 Silicon, 469 Specific absorption rate (SAR), 555, 561 Stent, 524 Silver halide, 465 Specific heat drug-eluting, 524 Silver, H. K., 44, 51 water, 80 Steradian, 568 Silverstein, M. E., 125 Specific heat capacity, 65, 404 Stereocilia, 256 Sinc function, 549 Specific metabolic rate, 50 Steric factor, 125, 133 Singh, M., 362, 548 SPECT, 521 Stern, R. S., 404, 423 Index 627

Steroid-eluting electrode, 204 Sutoh, Y., 423 “10-20” system for EEG leads, 205 Stewart, W. E., 94 Svassand, L. O., 422 Tendon Stimulated emission, 417, 542 Svedberg (unit), 28 Achilles, 6, 7 Stimulus Swanson, E. A., 422 Tenforde, T. S., 257 brain, 205 Sweating, 293 Tensile strength, 13 current density, 172 Swihart, J. C., 249 Terahertz radiation, 382, 394 strength-duration curve, 179 Swinney, K. R., 219, 220 Teramura T., 423 strength-interval curve, 181 Sympathetic nerves, 194 Teramura, T., 403 Stinson, W. G., 422 Synapse, 110, 141, 143 Tesla (unit), 213 Stirling’s approximation, 591 Syncope, 203 Tetley, I. J., 109 Stochastic resonance, 337 Syncytium, 193 Tetrodotoxin, 251 Stocker, H., 189 Synolakis, C. E., 19, 31 Teukolsky, S. A., 84, 115, 362 Stokes’ law, 23, 91, 94, 260 Synthesis, 481 TFT, 470 Stopping cross section, 439 System, 56 Thalamotomy, 379 Stopping interaction strength, 444 linear, 345 201Tl, 518, 520, 522 Stopping number stationary, 346 Theodoridis, G. C., 413, 423 per atomic electron, 444 System dynamics, 280 Theriot J. A., 115 Stopping power, 439 Systole, 19, 20 Theriot, J. A., 85 electronic, 440 Szabo, A., 101, 115 Thermal conductivity, 109, 404 for compounds, 446 Szabo, Z., 533 Thermal equilibrium, 60, 61 for electrons and positrons, 446 Thermal noise mass, 439 retinal, 415 nuclear, 440 T rays, 394 Thermal penetration depth, 405 radiative, 440 T wave, 195 Thermal relaxation time, 405 Storment, C. W., 142 T-cell lymphoma, 404 Thermodynamic identity, 69 Straggling, 447 T1-weighted image, 554 Thermodynamic relationship, general, 69 Straight-line fit, 303, 304 T2-weighted image, 554 Thermodynamics Strain, 364 Tachycardia first law, 57 normal, 12 ventricular, 195 second law, 75 shear, 13 Tack, D., 480 Thermography, 399 Strasburger, J. F., 222 Tai, C., 209 Thermoluminescent dosimeter, 469 Stratum corneum, 401 Tait C., 422 Thiazide diuretics, 402 Stratum granulosum, 401 Tait, C., 403 Thin-film transistor, 470, 474 Streamline, 15, 18 Takano, M., 422 Thin-lens equation, 411 Strength-duration curve, 179, 202 Takata, M., 51 Thomas, D. L., 558 Strength-interval curve, 181 Takeda, S., 51 Thomas, L., 126 Stress, 12, 364 Talus, 6, 7 Thomas, S. R., 532 normal, 12 Tan, G. A., 222 Thompson, D. J., 281 shear, 13 Tanelian, D. L., 142 Thompson, J. H., 589 Strogatz, S. H., 276, 277, 280, 286 Tang, Y-h., 111, 115 Thomson scattering, 430 Stroink, G., 222, 225, 228 Target entity, 388 Thomson, R. B., 589 90Sr, 524 Taylor’s series, 575 232Th, 532 Strother, S. C., 523, 533 of exponential function, 576 Three-dimensional conformal radiation Structure mottle, 475 Taylor, A. E., 121 therapy, 487 Subchoroidal neovascular membrane, 394 Tearney, G. J., 422, 423 Threshold detection, 337 Subexcitation, 464 99mTc Threshold stimulus strength, 181 Subshells diphosphonate, 521 Thyroid, 270 atomic, 426 pyrophosphate, 518 stimulating hormone (TSH), 270, 292 Suess, C., 499 sulfur colloid, 518 Thyroid hormone, 293 Sulfides, 228 tetrofosmin, 522 Thyroxine (T4), 270, 292 Sulfonamides, 402 99mTc, 35, 507, 519, 522 Tri-iodothyronine (T3), 270 Sulfonureas, 402 albumin, 519 Tibia, 6, 7 Sun protection factor (SPF), 403 diphosphonate, 521 Time gain compensation, 378 Superconducting quantum interference device, generator, 519 Tissue weighting factor, 491 see SQUID pertechnetate, 519 Tittel, F. K., 422 Superconductor, 230 red blood cells, 521 TMS, 225 Superparamagnetic particles, 236 sestamibi, 520 Tobacco, photosynthesis, 314 Superposition, 345 sulfur colloid, 512, 529 Tomography Surface tension, 69, 82 tetrofosmin, 520 computed, 478 Surface-to-volume ratio, 200 Tectorial membrane, 370 optical coherence, 392 Surfactant, 82 Telegrapher’s equation, 159 Tooby, P. F., 27, 31 Surrogate data, 336, 337 Temperature Torque, 5, 535 Surroundings, 56 absolute, 61, 62 on rotating sphere, 260 Survival curve, 480 centigrade or celsius, 62 Torr (unit), 14, 20 Susceptibility negative, 78 Total angular momentum quantum number, electric, 149 Temperature regulation, 293 425 628 Index

Total internal reflection, 421 Units of “power” and “energy”, 329 Vitreous, 411 Total linear attenuation coefficient, see Upton, A. C., 494 Vlaardingerbroek, M. T., 548, 553 Attenuation coefficient 235U, 519, 532 Vogel R. I., 422 Toy model, 3 238U, 524, 525 Vogel, S. V., 16, 20, 23, 27, 30, 31 Trace of a matrix, 110 Urea, 123 Vollrath, F., 26, 31 Transcranial magnetic stimulation, 225 UVA, 400, 402, 403 Volt (unit), 148 Transducer, 141 UVB, 400, 403, 404 Voltage clamp, 161 piezoelectric, 371 UVC, 400, 403 Voltage difference, 147 Transfer factor, 466 Voltage divider, 154 Transfer function, 330 Voltage source modulation, 348 van den Bongard, H. J., 379 ideal, 332 optical, 348 van den Bosch, M. A., 379 Voltage-sensitive dye, 205 phase, 348 van der Pol oscillator, 336 Volume transport, 128 Transformation, 506, 507 van der Steen, A. F. W., 379 pore model, 133 Transient charged-particle equilibrium, 453 van Eijk, C. W. E., 468 through a membrane, 123 Transition, 506, 507 van Ginneken, B., 470 Voorhees, C. R., 202, 203 nonradiation, 435 van Langenhove, G., 423 Vreugdenburg T. D., 423 radiationless, 435 van’t Hoff’s law, 121, 124 Vreugdenburg, T. D., 399 radiative, 435 Vapor pressure, 79, 83 Transmission at a boundary, 367 Variable rate of growth or decay, 38 Transmission coefficient, 367, 377 Variables Wagner, H. N., 467, 533 Transmittance, 465 extensive, 69 Wagner, J., 111, 115 Transport electric charge, 69 Wagner, R. F., 474, 476 countercurrent, 127 Length, 69 Walcott, C. J., 229 solute, 125, 128 Surface area, 69 Walker, G. C., 422 volume, 128 volume, 69 Walker, M. M., 229 Transvenous pacing, 203 Variance, 586 Wang, D., 101, 115 Transverse relaxation time Vascular resistance, 21 Wang, H., 49, 51 recoverable and non-recoverable, 544 Vasodilation, 122 Wang, L., 422 Transverse wave, 366 Vasopressin, 294 Wang, M. D., 114 Traveling wave, 366 Vecchia, P., 261 Warburg equation, 113 Trayanova, N., 192, 200, 205 Vector, 4, 569 Warloe, T., 422 Tree, 29 addition, 569 Warner, G. G., 516, 533 Trichromate vision, 415 components, 570 Warshaw E. M., 422 Tricuspid valve, 193 unit, 570 Warwick, W., 589 Trigonometric integrals, 579 Vector operators in different coordinate Washout (in MRI), 555 Triplet production, 432 systems, 601 Wassermann, E., 226 Trochanter, greater, 8 Vector product, 6 Watanabe, A., 190, 191, 220 Tromberg, B. J., 422 Velocity Water, 3 Trontelj, Z., 223 average, 570 acoustic impedance, 367 Trowbridge, E. A., 24, 31 instantaneous, 570 compressibility, 366 Tsay,T.-T.,422 root-mean-square, 89 density, 366 TTX, 251 Velocity gradient, 16 dielectric constant, 151 Tubiana, M., 494 Vena cava, 21 speed of sound, 366 Tucker, R. D., 258 Ventilation rate, 270 transmission vs. wavelength, 410 Tukey, J. W., 314, 315 Ventricle, 193 Water molecule Tumor eradication, 485 Ventricular fibrillation, 204, 291, 292, 298 dipole moment, 245 Tuna, 229, 235 Ventricular tachycardia, 195, 204, 291 Waterstram-Rich, K. M., 523, 532 Tung, C. J., 448 Venules, 21 Watson, E. E., 511, 512, 514, 516, 517, 524, Turbidity, 389 Vergence, 411 532, 533 Turbulence, 22 Verheye, S., 399, 423 Watson, S. B., 533 Turner, R., 553 Versluis, M., 379 Watt (unit), 11, 153 Tympanic chamber, 370 Vestibular chamber, 370 Wave Vetterling, W. T., 84, 115, 362 longitudinal, 363 Villars, F. M. H., 11, 15, 30, 95, 105, 115 plane, 363 Uehara, M., 239 Virtual cathode, 203 standing, 366 Uffmann, M., 470 “dog-bone”, 205 traveling, 366 Ugurbil, K., 555–557 Virtual image, 420 Wave equation, 181, 363, 364 Ultracentrifuge, 79 Virus, 2 Wave number, 346, 366 Ultrafiltration, 124 Viscoelasticity, 13 Weaver, J. C., 256, 257, 259, 260, 265, 337 Ultrasound Viscosity, 13, 16, 91 Weaver, W., 54, 84 diagnostic, 371 of water, 94 Weaver,W.D.,76 intravascular, 399 Viscous torque on a rotating sphere, 260 Webb, W. W., 256 Ultraviolet spectrum, 400 Visscher, P. B., 315 Weber (unit), 224 Ungar, I. J., 209 Visual cortex, 555, 557 Webster, J. G., 177, 392, 422, 423 Unified mass unit, 505 Visual evoked response, 328 Weighting factor Index 629

radiation, 490 Williamson, S. J., 224 Yaffe, M. J., 477 Weinstock M. A., 422 Willis C. D., 423 Yamashita, Y., 392, 423 Weiss, G., 202 Wilson, B. C., 399, 422, 423 Yasuno, Y., 393, 423 Weiss, J. N., 291 Winfree, A. T., 282–284, 291, 292, 298 Yatagai, T., 423 Weiss-Fogh, T., 113, 115 Wintermark, M., 379 Yazdanfar, S., 393, 394 Welder’s flash, 403 Witkowski, F. X., 292 Yehia, A. R., 298 Wells, P. N. T., 371, 379 Woods, M. C., 233 Yi, M., 244 Werner, B., 379 Woods, R. P., 533 Ying, W., 201 Wessels, B. W., 524, 532 Woosley, J. K., 219, 236 Yodh, A., 389, 392, 422, 423 West, G. B., 45, 51 Work, 11, 58 Young’s modulus, 13, 364 Westfall, R. J., 532 concentration, 67 Young, A. C., 294 Westphalen, A. C., 379 pressure-volume, 19, 21 Young, A. R., 401, 422 White noise, 333 Working level, 525 Young, T., 381 White, C. R., 45, 51 month, 526 Yu, C. X., 487 White, P. D., 39, 51 Worthington, C. R., 171 White, R. J., 123 White-blood-cell count, 290 Zadicario, E., 379 Wick, G. L., 27, 31 X radiation, 382 Zalewski, E. F., 405, 410, 423 Wieben, O., 392, 423 X-ray absorption edge, 428 Zanzonico, P. B., 520, 523, 532, 533 Wiener spectrum, 475 X-ray spectrum Zaret, B. L., 222 Wiener theorem for random signals, 328 bremsstrahlung, 461 Zeng, Z-C., 51 Wiesenfeld, K., 337 characteristic, 461 Zeuthen, T., 118 Wijesinghe, R., 229 thick target, 463 Zhang, X-C., 394, 423 Wikelski, M., 229 thin target, 462 Zhang, Y., 422 Wikswo, J. P., 53, 84, 205, 219, 220, 222–224, X-ray tube, 470 Zhong, J., 557 228–230, 232, 236, 279 Xenon, 499 Zhou, X.-H., 553 Wilders, R., 165, 172 129Xe, 558, 559 Zhu, T. C., 399, 423 Wiley, J. D., 177 133Xe, 522 Ziegler, J. F., 439, 442, 445, 446, 448 Williams, C. R., 497 Xia, Y., 560 Zinovev, N. N., 422 Williams, C. S., 348, 362 Xu,X.G.,489 Zipes, D. P., 223 Williams, L. E., 503 Xu, Y., 235 Zumoff, B., 38, 51 Williams,M.,7,8,31 Xylem, 29 Zwanzig, R., 101, 115