Appendix a Plane and Solid Angles

Appendix A Plane and Solid Angles

A.1 Plane Angles 1.0

The angle θ between two intersecting lines is shown in y = tan θ y = sin θ Fig. A.1. It is measured by drawing a circle centered on 0.8 the vertex or point of intersection. The arc length s on y = θ (radians) that part of the circle contained between the lines measures the angle. In daily work, the arc length is marked 0.6 off in degrees. In some cases, there are advantages to measuring the y angle in radians. This is particularly true when trigono- 0.4 metric functions have to be differentiated or integrated. The angle in radians is defined by s 0.2 θ = . (A.1) r

Since the circumference of a circle is 2πr, the angle 0.0 corresponding to a complete rotation of 360 ◦ is 2πr/r = 0 20 40 60 80 2π. Other equivalences are θ (degrees)

Degrees Radians FIGURE A.2. Comparison of y =tanθ, y = θ (radians), and y =sinθ. 360 2π (A.2) 57.2958 1 1 0.01745 specify that something is measured in radians to avoid Since the angle in radians is the ratio of two distances, it confusion. is dimensionless. Nevertheless, it is sometimes useful to One of the advantages of radian measure can be seen in Fig. A.2. The functions sin θ, tan θ,andθ in radians are plotted 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% error at 15 ◦)andy =sinθ (1.2% error at 15 ◦).

A.2 Solid Angles

FIGURE A.1. A plane angle θ is measured by the arc length A plane angle measures the diverging of two lines in two s on a circle of radius r centered at the vertex of the lines dimensions. Solid angles measure the diverging of a cone deﬁning the angle. of lines in three dimensions. Figure A.3 shows a series of 544 Appendix A. Plane and Solid Angles

FIGURE A.5. For small angles, the arc length is very nearly equal to the length of the tangent to the circle.

sphere subtends a solid angle of 4π steradians, since the surface area of a sphere is 4πr2. When the included angle in the cone is small, the dif- FIGURE A.3. A cone of rays in three dimensions. ference between the surface area of a plane tangent to the sphere and the sphere itself is small. (This is diﬃ- cult 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. A 3 × 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 It is not necessary to calculate the surface area on a sphere of 72 in. radius.

FIGURE A.4. The solid angle of this cone is Ω = S/r2. S is the surface area on a sphere of radius r centered at the vertex. Problems rays diverging from a point and forming a cone. The solid Problem 1 Convert 0.1 radians to degrees. Convert angle Ω is measured by constructing a sphere of radius r 7.5 ◦ to radians. centered at the vertex and taking the ratio of the surface area S on the sphere enclosed by the cone to r2: 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 S Ω= . (A.3) table and see how accurate the approximation is. r2 This is shown in Fig. A.4 for a cone consisting of the Problem 3 What is the solid angle subtended by the planes deﬁned by adjacent pairs of the four rays shown. pupil of the eye (radius =3mm) at a source of light The unit of solid angle is the steradian (sr). A complete 30 m away? Appendix B Vectors; Displacement, Velocity, and Acceleration

B.1 Vectors and Vector Addition Displacements can be added in any order. In Fig. B.2, either of the vectors A represents the same displacement. A displacement describes how to get from one point to Displacement B can first be made from point 1 to point another. A displacement has a magnitude (how far point 4, followed by displacement A from4to3.Thesumis 2 is from point 1 in Fig. B.1) and a direction (the direc- still C: tion one has to go from point 1 to get to point 2). The displacement of point 2 from point 1 is labeled A.Dis- C = A + B = B + A. (B.2) placements can be added: displacement B from point 2 puts an object at point 3. The displacement from point The sum of several vectors can be obtained by first 1 to point 3 is C and is the sum of displacements A and adding two of them, then adding the third to that sum, B: and so forth. This is equivalent to placing the tail of each C = A + B. (B.1) 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 A displacement is a special example of a more general to the head of the last. quantity called a vector. One often finds a vector defined The negative of vector A is that vector which, added as a quantity having a magnitude and a direction. How- to A, yields zero: ever, the complete definition of a vector also includes the requirement that vectors add like displacements. The rule A +(−A)=0. (B.3) for adding two 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. It has the same magnitude as A and points in the oppo- A displacement is a change of position so far in such a site direction. direction. It is independent of the starting point. To know Multiplying a vector A by a scalar (a number with no where an object is, it is necessary to specify the starting associated direction) multiplies the magnitude of vector point as well as its displacement from that point. A by that number and leaves its direction unchanged.

FIGURE B.1. Displacement C is equivalent to displacement A followed by displacement B: C = A + B. FIGURE B.2. Vectors A and B can be added in either order. 546 Appendix B. Vectors; Displacement, Velocity, and Acceleration

5 Sum 3 4

FIGURE B.5. Addition of components in three dimensions. FIGURE B.3. Addition of several vectors. become diﬃcult to keep the distinction straight. There- fore, it is customary to write xˆ, yˆ,andˆz to mean vectors of unit length pointing in the x, y,andz directions. (In some books, the unit vectors are denoted by ˆı, ˆ,andkˆ instead of xˆ, yˆ,andˆz.) With this notation, instead of Ax, one would always write Axxˆ. The addition of vectors is often made easier by using components. The sum A + B = C can be written as

Axxˆ + Ayyˆ + Azˆz + Bxxˆ + Byyˆ + Bzˆz

= Cxxˆ + Cyyˆ + Czˆz.

Like components can be grouped to give FIGURE B.4. Vector A has components Ax and Ay.

(Ax + Bx)xˆ +(Ay + By)yˆ +(Az + Bz)ˆz

B.2 Components of Vectors = Cxxˆ + Cyyˆ + Czˆz.

Consider a vector in a plane. If we set up two perpen- Therefore, the magnitudes of the components can be dicular axes, we can regard vector A as being the sum added separately: of vectors parallel to each of these axes. These vectors, Cx = Ax + Bx, Ax and Ay in Fig. B.4, are called the components of A along each axis1. If vector A makes an angle θ with the x Cy = Ay + By, (B.7) axis and its magnitude is A, then the magnitudes of the Cz = Az + Bz. components are A = A cos θ, x (B.4) Ay = A sin θ. B.3 Position, Velocity, and 2 2 The sum of the squares of the components is Ax + Ay = Acceleration A2 cos2 θ + A2 sin2 θ = A2(sin2 θ +cos2 θ). Since, by Pythagoras’ theorem, this must be A2, we obtain the The position of an object at time t is deﬁned by specifying trigonometric identity its displacement from an agreed-upon origin:

cos2 θ +sin2 θ =1. (B.5) R(t)=x(t)xˆ + y(t)yˆ + z(t)ˆz.

In three dimensions, A = Ax + Ay + Az. The magni- The average velocity vav(t1,t2) between times t1 and t2 tudes can again be related using Pythagoras’ theorem, as is defined to be shown in Fig. B.5. From triangle OPQ, A2 = A2 + A2. xy x y − From triangle OQR, R(t2) R(t1) vav(t1,t2)= . t2 − t1 2 2 2 2 2 2 A = Axy + Az = Ax + Ay + Az. (B.6) This can be written in terms of the components as In our notation, Ax means a vector pointing in the x − − direction, while Ax is the magnitude of that vector. It can x(t2) x(t1) y(t2) y(t1) vav = − xˆ + − yˆ t2 t1 t2 t1 1 z(t ) − z(t ) Some texts define the component to be a scalar, the magnitude + 2 1 ˆz. of the component defined here. t2 − t1 Problems 547

dv dv dv dv The instantaneous velocity is a(t)= = x xˆ + y yˆ + z ˆz. dt dt dt dt dR dx dy dz v(t)= = xˆ + yˆ + ˆz dt dt dt dt = vx(t)xˆ + vy(t)yˆ + vz(t)ˆz. (B.8) Problems

The x component of the velocity tells how rapidly the x Problem 1 At t =0, the position of an object is given component of the position is changing. by R =10xˆ +5yˆ,whereR is in meters. At t =3s, the The acceleration is the rate of change of the velocity position is R =16xˆ−10yˆ. What was the average velocity with time. The instantaneous acceleration is between t =0and3s? Appendix C Properties of Exponents and Logarithms

In the expression am, a is called the base and m is The most useful property of logarithms can be derived called the exponent. Since a2 = a × a, a3 = a × a × a,and by letting

m a =(a × a × a ×···×a), y = am, m times z = an, it is easy to show that w = am+n, m n a a =(a × a × a ×···×a)(a × a × a ×···×a), so that m times n times m = loga y, m n m+n a a = a . (C.1) n = loga z,

If m>n, the same technique can be used to show that m + n = loga w. m Then, since am+n = aman, a m−n n = a . (C.2) a w = yz, If m = n, this gives loga(yz) = loga w = loga y + loga z. (C.6) am 1= = am−m = a0, This result can be used to show that am log(ym) = log(y × y × y ×···×y) 0 a =1. (C.3) = log(y) + log(y) + log(y)+···+ log(y), (C.7) The rules also work for m

(am)n =(am × am × am ×···×am), n times Problems (am)n = amn. (C.5) Problem 1 What is log2(8)? If y = ax, then by deﬁnition, x is the logarithm of Problem 2 If log10(2) = 0.3, what is log10(200)? y to the base a: x = loga(y). If the base is 10, since × 5 2 log10(2 10 )? 100 = 10 , 2 = log10(100). Similarly, 3 = log10(1000), √ 4 = log10(10000), and so forth. Problem 3 What is log10( 10)? Appendix D Taylor’s Series

Consider the function y(x) shown in Fig. D.1. The is value of the function at x1, y1 = y(x1), is known. We 2 dy 1 d y 2 wish to estimate y(x +∆x). y(x1 +∆x) ≈ y(x1)+ ∆x + (∆x) . 1 dx 2 dx2 The simplest estimate, labeled approximation 0 in Fig. x1 x1 D.1, is to assume that y does not change: y(x1 +∆x) ≈ That this is the best approximation can be derived in y(x1). A better estimate can be obtained if we assume the following way. Suppose the desired approximation is n n that y changes everywhere at the same rate it does at x1. more general and uses terms up to (∆x) =(x − x1) : Approximation 1 is y = A +A (x−x )+A (x−x )2 +···+A (x−x )n approx 0 1 1 2 1 n 1 dy (D.1) y(x1 +∆x) ≈ y(x1)+ ∆x. dx x1 The constants A0,A1,...,An are determined by making The derivative is evaluated at point x1. the value of yapprox and its first n derivatives agree with An even better estimate is shown in Fig. D.2. Instead of the value of y and its first n derivatives at x = x1. When fitting the curve by the straight line that has the proper x = x1, all terms with x − x1 in yapprox vanish, so that first derivative at x1, we fit it by a parabola that matches both the first and second derivatives. The approximation yapprox(x1)=A0.

y(x) y(x) y y

Actual

Approximation 2

(dy/dx)| ∆x x1 + (1/2)(d2y/dx2)| (∆x)2 Approximation 1 x1 (dy/dx)| ∆x x1 y1 Approximation 0

x + ∆x ∆ x1 1 x x1 x1 + x x

FIGURE D.2. The second-order approximation ﬁts y(x)with FIGURE D.1. The zeroth-order and ﬁrst-order approxima- a parabola. tions to y(x). 552 Appendix D. Taylor’s Series

TABLE D.1. y = e2x and its derivatives. 20 2x Function or derivative Value at x1 =0 y = e 2x y = e 1 2 3 y = 1 + 2x + 2x + 4x /3 dy 15 =2e2x 2 y = 1 + 2x + 2x2 dx d2y y = 1 + 2x =4e2x 4 10 dx2 y d3y =8e2x 8 dx3 5 y = 1 0

TABLE D.2. Values of y and successive approximations. 1+2x+ -5 2x 2 2 4 3 xy= e 1+2x 1+2x +2x 2x + 3 x -2 -1 0 1 2 3 4 5 −20.0183 −3.05.0 −5.67 x −1.50.0498 −2.02.5 −2.0 −10.1353 −1.01.0 −0.33 FIGURE D.3. The function y = e2x with Taylor’s series ex- −0.40.4493 0.2000 0.5200 0.4347 pansions about x = 0 of degree 0, 1, 2, and 3. −0.20.6703 0.6000 0.6800 0.6693 −0.10.8187 0.8000 0.8200 0.8187 01.0000 1.0000 1.0000 1.0000 2x 0.11.2214 1.2000 1.2200 1.2213 3.0 y = e 0.21.4918 1.4000 1.4800 1.4907 y = 1 + 2x + 2x2 + 4x3/3 0.42.2255 1.8000 2.1200 2.2053 2.5 1.07.389 3.0000 5.0000 6.33 2.054.60 5.013.023.67 y = 1 + 2x + 2x2 2.0 y

The ﬁrst derivative of yapprox is 1.5 y = 1 + 2x

d(yapprox) − y = 1 = A +2A (x−x )+3A (x−x )2+···+nA (x−x )n 1. 1.0 dx 1 2 1 3 1 n 1 The second derivative is 0.5 n−2 2A2 +3× 2A3(x − x1)+···+ n(n − 1)An(x − x1) , 0.0 and the nth derivative is -0.5 0.0 0.5 1.0 x

n(n − 1)(n − 2) ···2An = n!An. FIGURE D.4. An enlargement of Fig. D.3 near x =0. Evaluating these at x = x1 gives d(yapprox) = A1, dx Combining these expressions for An with Eq. D.1, we get x1 2 N d (yapprox) n × × 1 d y n =2 1 A2, y(x +∆x) ≈ y(x )+ (∆x) . (D.2) dx2 1 1 n x1 n! dx n=1 x1 3 d (yapprox) × × × 3 =3 2 1 A3, Tables D.1 and D.2 and Figs. D.3 and D.4 show how dx x 1 the Taylor’s series approximation gets better over a larger n d (yapprox) and larger region about x1 as more terms are added. The = n!An. dxn function being approximated is y = e2x. The derivatives x1 Problems 553 are given in Table D.1. The expansion is made about Problems x1 =0. Finally, the Taylor’s series expansion for y = ex about Problem 1 Make a Taylor’s series expansion of y = a+ x = 0 is often useful. Since all derivatives of ex are ex, bx + cx2 about x =0. Show that the expansion exactly the value of y and each derivative at x = 0 is 1. The reproduces the function. series is Problem 2 Repeat the previous problem, making the ex- ∞ pansion about x =1. 1 1 xm ex =1+x + x2 + x3 + ···= . (D.3) 2! 3! m! Problem 3 (a) Make a Taylor’s series expansion of m=0 the cosine function about x =0. Remember that d(sin x)/dx =cosx and d(cos x)/dx = − sin x. (Note that 0! = 1 by deﬁnition.) (b) Make a Taylor’s series expansion of the sine function. Appendix E Some Integrals of Sines and Cosines

The average of a function of x with period T is defined 1 to be 1 x +T (a) (b) f = f(x) dx. (E.1) T x The sine function is plotted in Fig. E.1(a). The integral 0 over a period is zero, and its average value is zero. The 0 2π area above the axis is equal to the area below the axis. Figure E.1(b) shows a plot of sin2 x. Since sin x varies between −1 and +1, sin2 x varies between 0 (when sin x = -1 0) and +1 (when sin x = ±1). Its average value, from FIGURE E.2. Plot of one period of (a) y =sinx sin 2x;(b) 1 inspection of Fig. E.1(b) is 2 . If you do not want to trust y =sinx cos x. 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 shifted along the axis, These could be used to show that the average value of their squares must also look similar. Therefore, sin2 θ and sin x or cos x is zero. Then Eq. E.2 could be used to show 2 1 2 cos θ must have the same average. But if their sum is that the average of sin x is 2 . 2 always 1, the sum of their averages must be 1. If the two The integral of sin x over a period is its average value 1 times the length of the period: averages are the same, then each must be 2 . These same results could have been obtained analyti- T T T cally by using the trigonometric identity sin2 xdx= cos2 xdx= . (E.4) 1 1 0 0 2 sin2 x = − cos 2x. (E.2) 2 2 We will also encounter integrals like The integrals of sin x and cos x are T 1 sin ax dx = − cos ax, sin mx sin nx dx, m = n, a 0 (E.3) 1 T cos ax dx = sin ax. a cos mx cos nx dx, m = n, (E.5) 0 1 1 T Av = 1/2 cos mx sin nx dx, m = n, m = n. 0 Av = 0 0 0 All these integrals are zero. This can be shown using integral tables. Or, you can see why the integrals vanish by considering the specific examples plotted in Fig. E.2. Each integrand has equal positive and negative contribu- -1 -1 tions to the total integral. FIGUREE.1.(a)Plotofy =sinx. (b) Plot of y =sin2 x. Appendix F Linear Differential Equations with Constant Coefficients

The equation is called the characteristic equation of this differential dy equation. It can be written in a much more compact form + by = a (F.1) dt using summation notation: is called a linear differential equation because each term involves 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 − ··· N + bN 1 N−1 + + b1 + b0y = f(t). (F.2) dt dt dt with bN =1. For Eq. F.1, the characteristic equation is s + b =0or The highest derivative is the Nth derivative, so the equa- s = −b, and a solution to the homogeneous equation is tion is of order N. It has been written in standard form y = Ae−bt. by dividing through by any b that was there originally, N If the characteristic equation is a polynomial, it can so that the coefficient of the highest term is one. If all have up to N roots. For each distinct root s , y = A esnt the b’s are constants, this is a linear differential equation n n is a solution to the differential equation. (The question with constant coefficients. The right-hand side may be of solutions when there are not N distinct roots will a function of the independent variable t, but not of y.If be taken up below.) This is still not the solution to the f(t) = 0, it is a homogeneous equation; if f(t) is not zero, equation we need to solve. However, one can prove1 that it is an inhomogeneous equation. the most general solution to the inhomogeneous equation Consider first the homogeneous equation consists of the homogeneous solution, dN y dN−1y dy + b − + ···+ b + b y =0. (F.3) dtN N 1 dtN−1 1 dt 0 N snt y = Ane , st The exponential e (where s is a constant) has the n=1 property that d(est)/dt = sest, d2(est)/dt2 = s2est, dn(est)/dtn = snest. The function y = Aest satisfies Eq. plus any solution to the inhomogeneous equation. The F.3 for any value of A and certain values of s. The equa- values of the arbitrary constants A are picked to satisfy tion becomes n some other conditions that are imposed on the problem. N st N−1 st st st If we can guess the solution to the inhomogeneous equa- A s e + b − s e + ···+ b se + b e =0, N 1 1 0 tion, that is fine. However we get it, we need only one such solution to the inhomogeneous equation. We will N N−1 ··· st A s + bN−1s + + b1s + b0 e =0. not prove this assertion, but we will apply it to the first- This equation is satisfied if the polynomial in parentheses and second-order equations and see how it works. is equal to zero. The equation 1See, for example, G. B. Thomas. Calculus and Analytic Geom- N N−1 s + bN−1s + ···+ b1s + b0 = 0 (F.4) etry, Reading, MA, Addison-Wesley (any edition). 558 Appendix F. Linear Differential Equations with Constant Coefficients

F.1 First-order Equation The only way that the equation can be satisfied for all times is if the coefficient of the sin(ωt + φ) term and the The homogeneous equation corresponding to Eq. F.1 has coefficient of the cos(ωt+ φ) term separately are equal to solution y = Ae−bt. There is one solution to the inhomo- zero. This means that we have two equations that must 2 geneous equation that is particularly easy to write down: be satisfied (call b0 = ω0): when y is constant, with the value y = a/b, the time derivative vanishes and the inhomogeneous equation is 2αω = b1ω, satisfied. The most general solution is therefore of the 2 − 2 − 2 form α ω b1α + ω0 =0. a y = Ae−bt + . b From the first equation 2α = b1, while from this and the second, α2 − ω2 − 2α2 + ω2 =0,orω2 = ω2 − α2. Thus, If the initial condition is y(0) = 0, then A can be deter- 0 0 the solution to the equation mined from 0 = Ae−b0 + a/b. Since e0 = 1, this gives A = −a/b. Therefore 2 d y dy 2 +2α + ω0y = 0 (F.9) a − dt2 dt y = 1 − e bt . (F.6) b is A physical example of this is given in Sec. 2.7. y = Ae−αt sin(ωt + φ) (F.10a) where 2 2 − 2 F.2 Second-order Equation ω = ω0 α ,α<ω0. (F.10b) Solution F.10 is a decaying exponential multiplied by The second-order equation a sinusoidally varying term. The initial amplitude A and the phase angle φ are arbitrary and are determined by d2y dy + b + b y = 0 (F.7) other conditions in the problem. The constant α is called dt2 1 dt 0 the damping. Parameter ω0 is the undamped frequency, 2 has a characteristic equation s + b1s + b0 = 0 with roots the frequency of oscillation when α =0.ω is the damped # frequency. 2 −b1 ± b − 4b0 When the damping becomes so large that α = ω , then s = 1 . (F.8) 0 2 the solution given above does not work. In that case, the This equation may have no, one, or two solutions. solution is given by If it has two solutions s and s , then the general solu- − 1 2 y =(A + Bt)e αt,α= ω . (F.11) s1t s2t 0 tion of the homogeneous equation is y = A1e + A2e . 2 − If b1 4b0 is negative, there is no solution to the equa- This case is called critical damping and represents the tion for a real value of s. However, a solution of the form case in which y returns to zero most rapidly and with- −αt y = Ae sin(ωt + φ) will satisfy the equation. This can out multiple oscillations. The solution can be verified by be seen by direct substitution. Differentiating this twice substitution. shows that If α>ω0, then the solution is the sum of the two dy exponentials that satisfy Eq. F.8: = −αAe−αt sin(ωt + φ)+ωAe−αt cos(ωt + φ), dt −at −bt 2 y = Ae + Be , (F.12a) d y 2 −αt − −αt 2 = α Ae sin(ωt + φ) 2αωAe cos(ωt + φ) dt where , − 2 −αt ω Ae sin(ωt + φ). 2 − 2 a = α + α ω0, (F.12b) If these derivatives are substituted in Eq. F.7, one gets , − 2 − 2 the following results. The terms are written in two b = α α ω0. (F.12c) columns. One column contains the coefficients of terms with sin(ωt + φ), and the other column contains the co- When α = 0, the equation is efficients of terms with cos(ωt + φ). The rows are labeled d2y on the left by which term of the differential equation they + ω2y =0. (F.13) dt2 0 came from. The solution may be written either as Term Coefficients sin(ωt + φ)cos(ωt + φ) y = C sin(ω0t + φ) (F.14a) d2y/dt2 α2 − ω2 −2αω b1(dy/dt) −b1αb1ω or as b0yb0 0 y = A cos(ω0t)+B sin(ω0t). (F.14b) Problems 559

Underdamped Critically damped Overdamped

t t t

FIGURE F.1. Different starting points on the sine wave give different combinations of the initial position and the initial FIGURE F.2. Plot of y(t) (solid line) and dy/dt (dashed line) velocity. for different values of α.

The simplest physical example of this equation is a mass on a spring. There will be an equilibrium position TABLE F.1. Solutions of the harmonic oscillator equation. of the mass (y = 0) at which there is no net force on the d2y dy +2α + ω2y =0 mass. If the mass is displaced toward either positive or dt2 dt 0 negative y, a force back toward the origin results. The Case Criterion Solution force is proportional to the displacement and is given −αt by F = −ky. The proportionality constant k is called Underdamped α<ω0 y = Ae sin(ωt + φ) 2 2 − 2 the spring constant. Newton’s second law, F = ma,is ω = ω0 α −αt m(d2y/dt2)=−ky or, defining ω2 = k/m, Critically damped α = ω0 y =(A + Bt)e 0 −at −bt Overdamped α>ω0 y = Ae + Be 2 2 − 2 1/2 d y 2 a = α +(α ω0) + ω0y =0. − 2 − 2 1/2 dt2 b = α (α ω0) This (as well as the equation with α = 0) is a second-order differential equation. Integrating it twice introduces two constants of integration: C and φ,orA and B. They are The second-order equation we have just studied is usually found from two initial conditions. For the mass on called the harmonic oscillator equation. Its solution is the spring, they are often the initial velocity and initial needed in Chap. 8 and is summarized in Table F.1. position of the mass. The equivalence of the two solutions can be demonstrated by using Eqs. F.14a and a trigonometric identity Problems to write C sin(ω0t + φ)=C[sin ω0t cos φ +cosω0t sin φ]. Comparison with Eq. F.14b shows that B = C cos φ, A = Problem 1 From Eq. F.14 with ω0 =10, find A, B, C, C sin φ. Squaring and adding these gives C2 = A2 + B2, and φ for the following cases: while dividing one by the other shows that tan φ = A/B. (a) y(0) = 5, (dy/dt)(0) = 0. Changing the initial phase angle changes the relative (b) y(0) = 5, (dy/dt)(0) = 5. values of the initial position and velocity. This can be (c) y(0) = 0, (dy/dt)(0) = 50. seen from the three plots of Fig. F.1, which show phase (d) What values of A, B,andC would be needed to angles 0, π/4, and π/2. When φ = 0, the initial position have the same φ as in case (b) and the same amplitude is zero, while the initial velocity has its maximum value. as in case (a)? When φ = π/4, the initial position has a positive value, and so does the initial velocity. When φ = π/2 the initial Problem 2 Verify Eq. F.11 in the critically damped position has its maximum value and the initial velocity case. is zero. The values of A and B are determined from the Problem 3 Find the general solution of the equation initial position and velocity. At t = 0, Eq. F.14b and its $ derivative give y(0) = A, dy/dt(0) = ω0B. d2y dy 0,t≤ 0 +2α + ω2y = The term in the differential equation equal to 2 0 2 ≥ dt dt ω0y0,t 0 2α(dy/dt) corresponds to a drag force acting on the mass and damping the motion. Increasing the damping coef- subject to the initial conditions y(0) = 0, (dy/dt)(0) = 0 ficient α increases the rate at which the oscillatory be- (a) for critical damping, α = ω0, havior decays. Figure F.2 shows plots of y and dy/dt for (b) for no damping, and different values of α. (c) for overdamping, α =2ω0. Appendix G The Mean and Standard Deviation

TABLE G.1. Quiz scores. In many cases the frequency distribution gives more information than we need. It is convenient to invent some Student No. Score Student No. Score quantities that will answer the questions: Around what 180 1671values do the results cluster? How wide is the distribution 268 1783of results? Many diﬀerent quantities have been invented 390 1888for answering these questions. Some are easier to calculate 472 1975or have more useful properties than others. 565 2069 The mean or average shows where the distribution is 681 2150centered. It is familiar to everyone: add up all the scores 785 2281and divide by the number of students. For the data given 893 2394above the mean is x =77.8. 976 2473 It is often convenient to group the data by the value 10 86 25 79 obtained, along with the frequency of that value. The 11 80 26 82 data of Table G.1, are grouped this way in Table G.2. 12 88 27 78 The mean is calculated as 13 81 28 84 14 72 29 74 1 f x x = f x = i i i , 15 67 30 70 N i i f i i i

where the sum is over the diﬀerent values of the test scores that occur. For the example in Table G.2, the sums are In many measurements in physics or biology there may i fi = 30, i fixi = 2335, so x = 2335/30 = 77.8. If a be several possible outcomes to the measurement. Dif- large number of trials are made, fi/N can be called the ferent values are obtained when the measurement is repeated. For example, the measurement might be the number of red cells in a certain small volume of blood, whether 6 a person is right-handed or left-handed, the number of _ 5 x radioactive disintegrations of a certain sample during a σσ 4 5-min interval, or the scores on a test. Table G.1 gives the scores on an examination admin- 3 istered to 30 people. These results are also plotted as a 2 histogram in Fig. G.1. 1

The table and the histogram give all the information Frequency of score, f 0 that there is to know about the experiment unless the 0 20 40 60 80 100 result depends on some variable that was not recorded, Score, x such as the age of the student or where the student was sitting during the test. FIGURE G.1. Histogram of the quiz scores in Table G.1 562 Appendix G. The Mean and Standard Deviation

TABLE G.2. Quiz scores grouped by score The width of the distribution is often characterized by the dispersion or variance: Score Score x Frequency of f x i i i number i score, fi 2 2 2 (∆x) = (x − x) = pi(xi − x) . (G.3) 1501 50 i 2651 65 3671 67This is also sometimes called the mean square variation: 4681 68the mean of the square of the variation of x from the 5691 69mean. A measure of the width is the square root of this, 6701 70which is called the standard deviation σ. The need for 7711 71taking the square root is easy to see since x may have 8 72 2 144 units associated with it. If x is in meters, then the vari- 9731 73ance has the units of square meters. The width of the 10 74 1 74 distribution in x must be in meters. 11 75 1 75 A very useful result is 12 76 1 76 (x − x)2 = x2 − x2. 13 78 1 78 14 79 1 79 − 2 2 − 2 To prove this, note that (xi x) = xi 2xix + x .The 15 80 2 160 variance is then 16 81 3 243 17 82 1 82 2 2 − 2 (∆x) = pixi 2 xi xpi + pix . 18 83 1 83 i i i 19 84 1 84 20 85 1 85 The first sum is the definition of x2. The second sum has a 21 86 1 86 number x in every term. It can be factored in front of the − 22 88 2 176 sum, to make the second term 2x xipi,whichisjust 2 2 2 23 90 1 90 −2(x) .Thelasttermis(x) pi =(x) . Combining all 24 93 1 93 three sums gives Eq. G.4. In summary, 25 94 1 94 , 2 σ = (∆x) , (G.4) σ2 = (∆x)2 = (x − x)2 = x2 − x2. probability pi of getting result xi. Then This equation is true as long as the pi’s are accurately x = x p . (G.1) known. If the pi’s have only been estimated from N exper- i i 2 − i imental observations, the best estimate of σ is N/(N 1) times the value calculated from Eq. G.4. Note that pi =1. For the data of Fig. G.1, σ =9.4. This width is shown The average of some function of x is along with the mean at the top of the figure. g(x)= g(xi)pi. (G.2) i Problems For example, 2 2 Problem 1 Calculate the variance and standard devia- x = (xi) pi. i tion for the data in Table G.2. Appendix H The Binomial Probability Distribution

Consider an experiment with two mutually exclusive gives directly outcomes that is repeated N times, with each repetition N! 3! 3 × 2 × 1 6 being independent of every other. One of the outcomes − = = × = =3. is labeled “success”; the other is called “failure.” The n!(N n)! 1!2! (1)(2 1) 2 experiment could be throwing a die with success being a The remaining factor, pn(1 − p)N−n, is the probability three, flipping a coin with success being a head, or placing of taking n tries in a row and having success and taking a particle in a box with success being that the particle is N − n tries in a row and having failure. located in a subvolume v. The binomial distribution applies if each “try” is in- In a single try, call the probability of success p and the dependent of every other try. Such processes are called probability of failure q. Since one outcome must occur Bernoulli processes (and the binomial distribution is of- and both cannot occur at the same time, ten called the Bernoulli distribution). In contrast, if the probability of an outcome depends on the results of the p + q =1. (H.1) previous try, the random process is called a Markov process. Although such processes are important, they are Suppose that the experiment is repeated N times. The more difficult to deal with and are not discussed here. probability of n successes out of N tries is given by the bi- Some examples of the use of the binomial distribu- nomial probability distribution, which is stated here with- tion are given in Chap. 3. As another example, consider out proof.1 We can call the probability P (n; N), since the problem of performing several laboratory tests on a it is a function of n and depends on the parameter N. patient. In the 1970s it became common to use auto- Strictly speaking, it depends on two parameters, N and mated machines for blood-chemistry evaluations of pa- p: P (n; N,p). It is2 tients; such machines automatically performed (and reported) 6, 12, 20, or more tests on one small sample of a patient’s blood serum, for less cost than doing just one N! − P (n; N)=P (n; N,p)= pn(1 − p)N n. or two of the tests. But this meant that the physician n!(N − n)! (H.2) got a large number of results—many more than would The factor N!/[n!(N − n)!] counts the number of dif- have been asked for if the tests were done one at a time. ferent ways that one can get n successful outcomes; the When such test batteries were first done, physicians were probability of each of these ways is pn(1 − p)N−n.Inthe surprised to find that patients had many more abnormal example of three particles in Sec. 3.1, there are three ways tests than they expected. This was in part because some to have one particle in the left-hand side. The particle can tests were not known to be abnormal in certain diseases, be either particle a or particle b or particle c. The factor because no one had ever looked at them in that disease. But there still was a problem that some tests were abnormal in patients who appeared to be perfectly healthy. 1 A detailed proof can be found in many places. See, for example, We can understand why by considering the following F. Reif (1964). Statistical Physics. Berkeley Physics Course, Vol. 5, New York, McGraw-Hill, p. 67. idealized situation. Suppose that we do N independent 2N!isN factorial and is N(N − 1)(N − 2) ···1. By definition, tests, and suppose that in healthy people, the probabil- 0! = 1. ity that each test is abnormal is p. (In our vocabulary, 564 Appendix H. The Binomial Probability Distribution

1.0 tests. The data have the general features predicted by this simple model. 34 Clinically normal patients We can derive simple expressions to give the mean and 0.8 Binomial distribution standard deviation if the probability distribution is bino- P(n,12) if p = 0.05 mial. The mean value of n is defined to be N N N!n 0.6 n = nP (n; N)= pn(1 − p)N−n. n!(N − n)! n=0 n=0 The first term of each sum is for n = 0. Since each term 0.4 is multiplied by n, the first term vanishes, and the limits

Fraction of patients of the sum can be rewritten as N 0.2 N!n pn(1 − p)N−n. n!(N − n)! n=1 − 0.0 To evaluate this sum, we use a trick. Let m = n 1and 0 1 2 3 M = N − 1. Then we can rewrite various parts of this Number of abnormal tests expression as follows: n 1 1 FIGURE H.1. Measurement of the probablity that a clini- = = , n! (n − 1)! m! cally normal patient having a battery of 12 tests done has n abnormal tests (solid line) and a calculation based on the pn = ppm, binomial distribution (dashed line). The calculation assumes N!=(N)(N − 1)!, that p =0.05 and that all 12 tests are independent. Several of − − − − − the tests in this battery are not independent, but the general (N n)! = [N 1 (n 1)]! = (M m)!. features are reproduced. The limits of summation are n =1orm =0,andn = N or m = M. With these substitutions M M! having an abnormal test is “success”!). The probability n = Np pm(1 − p)M−m. − m!(M − m)! of not having the test abnormal is q =1 p.Inaper- m=0 fect test, p would be 0 for healthy people and would be 1 This sum is exactly the sum of a binomial distribution in sick people; however, very few tests are that discrim- over all possible values of m and is equal to one. We have inating. The deﬁnition of normal vs abnormal involves the result that, for a binomial distribution, a compromise between false positives (abnormal test results in healthy people) and false negatives (normal test n = Np. (H.3) results in sick people). Good reviews of this problem have This says that the average number of successes is the total 3 4 been written by Murphy and Abbey and by Feinstein. number of tries times the probability of a success on each In many cases, p is about 0.05. Now suppose that p is the try. If 100 particles are placed in a box and we look at half same for all the tests and that the tests are independent. 1 the box so that p = 2 , the average number of particles Neither of these assumptions is very good, but they will × 1 in that half is 100 2 = 50. If we put 500 particles in show what the basic problem is. Then, the probability 1 the box and look at 10 of the box, the average number for all of the N tests to be normal in a healthy patient is of particles in the volume is also 50. If we have 100 000 given by the binomial probability distribution: particles and v/V = p =1/2000, the average number is still 50. N! P (0; N,p)= p0qN = qN . For the binomial distribution, the variance σ2 can be 0!N! expressed in terms of N and p using Eq. G.4. The average 2 If p =0.05, then q =0.95, and P (0; N,p)=0.95N .Typi- of n is cal values are P (0; 12) = 0.54, and P (0; 20) = 0.36. If the N N! n2 = P (n; N)n2 = n2pn(1 − p)N−n. assumptions about p and independence are right, then n!(N − n)! only 36% of healthy patients will have all their tests nor- n n=0 mal if 20 tests are done. The trick to evaluate this is to write n2 = n(n − 1) + n. Figure H.1 shows a plot of the number of patients in a With this substitution we get two sums: series who were clinically normal but who had abnormal N N! n2 = n(n − 1)pnqN−n n!(N − n)! 3E. A. Murphy and H. Abbey (1967). The normal range—a com- n=0 20 N mon misuse. J. Chronic Dis. : 79. N!n 4A. R. Feinstein (1975). Clinical biostatistics XXVII. The de- + pnqN−n. rangements of the normal range. Clin. Pharmacol. Therap. 15: 528. n!(N − n)! n=0 Problems 565

0.20 0.6 (a) N = 100 (c)

p = 0.05 0.15 p = 0.95 _ _ n = Np = 5 n = Np = 95 0.4 1/2 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

[p(1-p)] 0.2 0.05 σ = 2.18 σ = 2.18

0.00 0.0 0 20 40 60 80 100 0.0 0.2 0.4 0.6 0.8 1.0 n -3 100x10 (b) p 80 N = 100 1/2 p = 0.5 FIGURE H.2. Plot of [p(1 − p)] . 60 _ P(n) n = 50 40 σ = 5 20 The second sum is n = Np. The ﬁrst sum is rewritten by 0 0 20 40 60 80 100 noticing that the terms for n =0andn = 1 both vanish. n Let m = n − 2andM = N − 2: 100x10-3

80 (d) M M! − 2 − 2 m M m 60 N = 1,000,000 n = Np+ N(N 1) p p q P(n) p = 0.00005 m!(M − m)! _ m=0 40 n = 50 σ = 7.07 = Np+ N(N − 1)p2 = Np+ N 2p2 − Np2. 20 0 0 20 40 60 80 100 Therefore, n FIGURE H.3. Examples of the variation of σ with N and p. (∆n)2 = n2 − n2 = Np− Np2 = Np(1 − p)=Npq. (a), (b), and (c) show variations of with p when N is held For the binomial distribution, then, 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) σ = Npq = nq. and (d) shows the variation of σ as p and N change together in such a way that n remains equal to 50. The standard deviation for the binomial distribution for# fixed p goes as N 1/2. For fixed N, it is proportional Problem 3 The Mayo Clinic reported that a single stool to p(1 − p), which is plotted in Fig. H.2. The maximum specimen in a patient known to have an intestinal parasite 1 yields positive results only about 90% of the time [R. B. value of σ occurs when p = q = 2 .Ifp is very small, the event happens rarely; if p is close to 1, the event nearly Thomson, R. A. Haas, and J. H. Thompson, Jr. (1984). always happens. In either case, the variation is reduced. Intestinal parasites: The necessity of examining multiple On the other hand, if N becomes large while p becomes stool specimens. Mayo Clin. Proc. 59: 641–642]. What small in such a way as to√ keep n fixed, then σ increases is the probability of a false negative if two specimens are to a maximum value of n. This variation of σ with N examined? Three? and p is demonstrated in Fig. H.3. Figures H.3(a)–H.3(c) show how σ changes as N is held fixed and p is varied. Problem 4 The Minneapolis Tribune on October 31, For N = 100, p is 0.05, 0.5, and 0.95. Both the mean 1974 listed the following incidence rates for cancer in the and σ change. Comparing Fig. H.3(b) with H.3(d) shows Twin Cities greater metropolitan area, which at that time two different cases where n = 50. When p is very small had a total population of 1.4 million. These rates are com- because N is very large in Fig. H.3(d), σ is larger than pared to those in nine other areas of the country whose in Fig. H.3(b). total population is 15 million. Assume that each study was for one year. Are the differences statistically significant? Show calculations to support your answer. How would Problems your answer differ if the study were for several years? Type of cancer Incidence per 100 000 Problem 1 Calculate the probability of throwing population per year 0, 1,...,9 heads out of a total of nine throws of a coin. Twin Cities Other Problem 2 Assume that males and females are born Colon 35.6 30.9 with equal probability. What is the probability that a cou- Lung (women) 34.2 40.0 ple will have four children, all of whom are girls? The Lung (men) 63.6 72.0 couple has had three girls. What is the probability that Breast (women) 81.3 73.8 they will have a fourth girl? Why are these probabilities Prostate (men) 69.9 60.8 so different? Overall 313.8 300.0 566 Appendix H. The Binomial Probability Distribution

Problem 5 The probability that a patient with cystic ﬁ- probability is ten times less.5 Show that the probability of brosis gets a bad lung illness is 0.5% per day. With treat- not having an illness in a year is 16% without treatment ment, which is time consuming and not pleasant, the daily and 83% with treatment.

5These numbers are from W. Warwick, MD, private commu- nication. See also A. Gawande, The bell curve. The New Yorker, December 6, 2004, pp. 82–91. Appendix I The Gaussian Probability Distribution

Appendix H considered a process that had two mutu- 3 ally exclusive outcomes and was repeated N times, with the probability of “success” on one try being p.Ifeach y=ln(m) try is independent, then the probability of n occurrences 2 of success in N tries is y 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 the two para- m meters N and p. We have seen two other parameters, the mean, which roughly locates the center of the distribution, and the standard deviation, which measures its FIGURE I.1. Plot of y =lnm used to derive Stirling’s ap- width. These parameters, n and σ, are related to N and proximation. p by the equations n = Np, To take the logarithm of Eq. I.1, we need a way to han- σ2 = Np(1 − p). dle the factorials. There is a very useful approximation to the factorial, called Stirling’s approximation: It is possible to write the binomial distribution formula in terms of the new parameters instead of N and p.At ln(n!) ≈ n ln n − n. (I.2) best, however, it is cumbersome, because of the need to evaluate so many factorial functions. We will now develop To derive it, write ln(n!) as an approximation that is valid when N is large and which n allows the probability to be calculated more easily. ln(n!) = ln 1 + ln 2 + ···+lnn = ln m. The procedure is to take the log of the probability, y =ln(P ) and expand it in a Taylor’s series (Appendix m=1 D) about some point. Since there is a value of n for which The sum is the same as the total area of the rectangles P has a maximum and since the logarithmic function is in Fig. I.1, where the height of each rectangle is ln m and monotonic, y has a maximum for the same value of n.We the width of the base is one. The area of all the rectangles will expand about that point; call it n0. Then the form is approximately the area under the smooth curve, which of y is is a plot of ln m. The area is approximately 2 n dy 1 d y 2 n y = y(n0)+ (n − n0)+ (n − n0) + ··· . ln mdm=[m ln m − m] = n ln n − n +1. dn 2 dn2 1 n0 n0 1

Since y is a maximum at n0, the ﬁrst derivative vanishes This completes the proof of Eq. (I.2). Table I.1 shows val- and it is necessary to keep the quadratic term in the ex- ues of n! and Stirling’s approximation for various values pansion. of n. The approximation is not too bad for n>100. 568 Appendix I. The Gaussian Probability Distribution

TABLE I.1. Accuracy of Stirling’s approximation. nn!ln(n!) n ln n − n Error % Error 5 120 4.7875 3.047 1.74 36 10 3.6 × 106 15.104 13.026 2.08 14 20 2.4 × 1018 42.336 39.915 2.42 6 100 9.3 × 10157 363.74 360.51 3.23 0.8

FIGURE I.2. Evaluating the normalization constant. We can now return to the task of deriving the binomial distribution. Taking logarithms of Eq. I.1, we get y =lnP =ln(N!) − ln(n!) − ln(N − n)! + n ln p +(N − n)ln(1− p). 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 FIGURE I.3. The allowed values of x are closely spaced in this case. dy = − ln n +ln(N − n)+lnp − ln(1 − p). dn To evaluate C , note that the sum of P (n) for all n The second derivative is 0 is the area of all the rectangles in Fig. I.2. This area is 2 d y − 1 − 1 approximately the area under the smooth curve, so that 2 = . dn n N − n ∞ −(n−n)2/2σ2 The point of expansion n0 is found by making the first 1=C0 e dn. −∞ derivative vanish: (N − n)p It is shown in Appendix K that half of this integral is 0=ln . " n(1 − p) ∞ −bx2 1 π − − dx e = . Since ln 1 = 0, this is equivalent to (N n0)p = n0(1 p) 0 2 b or n0 = Np. The maximum of y occurs when n is equal − to the mean. At n = n0 the value of the second derivative Therefore the normalization integral is (letting x = n n) is 2 ∞ √ d y 1 1 1 − 2 2 = − − = − . e x /2σ dx = 2πσ2. dn2 Np N(1 − p) Np(1 − p) −∞ It is still necessary to evaluate y = y(n ). If we try to √ 0 0 2 The normalization constant is C0 =1/ 2πσ , so that do this by substitution of n = n0 in Eq. I.3, we get zero. The reason is that the Stirling approximation we used is the Gaussian or normal probability distribution is too crude for this purpose. (There are additional terms 1 − − 2 2 in Stirling’s approximation that make it more accurate.) P (n)=√ e (n n) /2σ . (I.4) 2πσ2 The easiest way to find y(n0) is to call it y0 for now and determine it from the requirement that the probability It is possible, as in the case of the random-walk prob- be normalized. Therefore, we have lem, that the measured quantity x is proportional to n 1 with a very small proportionality constant, x = kn,so y = y − (n − Np)2 0 2Np(1 − p) that the values of x appear to form a continuum. As shown in Fig. I.3, the number of different values of n [each so that, in this approximation, with about the same value of P (n)] in the interval dx is

2 proportional to dx. The easiest way to write down the y y0 −(n−Np) /[2Np(1−p)] P (n)=e = e e . Gaussian distribution in the continuous case is to recog- With Np = n, ey0 = C ,andNp(1 − p)=σ2, this is nize that the mean is x = kn, and the standard deviation 0 2 − 2 2 − 2 2 2 − 2 2 2 2 is σx = (x x) = x x = k n k n = k σ . −(n−n)2/2σ2 P (n)=C0e . The term P (x)dx is given by P (n) times the number of diﬀerent values of n in dx. This number is dx/k. Appendix I. The Gaussian Probability Distribution 569

Therefore, To recapitulate: the binomial distribution in the case of large N can be approximated by Eq. I.4, the Gaussian or dx 1 2 2 P (x)dx = P (n) = dx √ e−(x/k−x/k) /2σ normal distribution, or Eq. I.5 for continuous variables. k k 2πσ2 The original parameters N and p are replaced in these 1 −(x−x)2/2σ2 approximations by n (or x)andσ. = dx√ e x . (I.5) 2πσx Appendix J The Poisson Distribution

Appendix H discussed the binomial probability distrib- If these factors are multiplied out, the first term is N x, ution. If an experiment is repeated N times, and has two followed by terms containing N x−1, N x−2,...,downto possible outcomes, with “success” occurring with proba- N 1. But there is also a factor N x in the denominator of bility p in each try, the probability of getting that out- the expression for P , which, combined with this gives come x times in N tries is 1 + (something)N −1 + (something)N −2 + ··· . N! x − N−x P (x; N,p)= − p (1 p) . x!(N x)! As long as N is very large, all terms but the first can The distribution of possible values of x is characterized be neglected. With these substitutions, Eq. J.1 takes the by a mean value x = Np and a variance σ2 = Np(1 − p). form − 2 x It is possible to specify x and σ instead of N and p to 1 − x P (x)= xxe x 1 − . (J.2) define the distribution. x! N Appendix I showed that it is easier to work with the Gaussian or normal distribution when N is large. It is The values of x for which P (x) is not zero are near specified in terms of the parameters x and σ2 instead of x, which is much less than N. Therefore, the last term, − x N and p: which is really [1/(1 p)] , can be approximated by one, while such a term raised to the Nth power had to be ap- − 1 2 2 x P (x; x, σ2)= e−(x−x) /2σ . proximated by e . If this is difficult to understand, con- (2πσ2)1/2 sider the following numerical example. Let N = 10 000 and p =0.001, so x = 10. The two terms we are con- The Poisson distribution is an approximation to the sidering are (1 − 10/10 000)10 000 =4.517 × 10−5,which binomial distribution that is valid for large N and for is approximated by e−10 =4.54 × 10−5, and terms like small p (when N gets large and p gets small in such a way (1 − 10/10 000)−10 =1.001, which is approximated by 1. that their product remains finite). To derive it, rewrite With these approximations, the probability is P (x)= the binomial probability in terms of p = x/N: [(x)x/x!]e−x or, calling x = m, N! x N−x P (x)= (x/N) (1 − x/N) x − m − x!(N x)! P (x)= e m. (J.3) − x! N! 1 x N x x = xx 1 − 1 − . x!(N − x)! N x N N This is the Poisson distribution and is an approximation (J.1) to the binomial distribution for large N and small p, such that the mean x = m = Np is defined (that is, it does not It is necessary next to consider the behavior of some of go to infinity or zero as N gets large and p gets small). − these factors as N becomes very large. The factor (1 This probability, when summed over all values of x, N −x →∞ x/N) approaches e as N , by definition (see p. should be unity. This is easily verified. Write 32). The factor N!/(N − x)! can be written out as ∞ ∞ x − − ··· − m N(N 1)(N 2) 1 − − ··· − P (x)=e m . = N(N 1)(N 2) (N x+1). x! (N − x)(N − x − 1) ···1 x=0 x=0 572 Appendix J. The Poisson Distribution

TABLE J.1. Comparison of the binomial, Gaussian, and Pois- 1.0 son distributions. N! x − N−x 0.8 Binomial P (x; N,p)= − p (1 p) x!(N x)! P(0) x = m = Np 0.6 2 σ = Np(1 − p)=m(1 − p) P 0.4 P(1) 1 − − 2 2 Gaussian P (x; m, σ)= e (x m) /2σ P(2) (2πσ2)1/2 P(3) 0.2 mx Poisson P (x; m)= e−m x! 0.0 m = Np 0 1 2 3 4 5 6 σ2 = m m = Nλt

FIGURE J.1. Plot of P (0) through P (3) vs Nλt.

But the sum on the right is the series for em,and − e mem =1. The same trick can be used to verify that The last example is worth considering in greater detail. the mean is m: The probability p that each nucleus decays in time dt is ∞ ∞ ∞ x x proportional to the length of the time interval: p = λdt. m − m − xP (x)= x e m = x e m. The average number of decays if many time intervals are x! x! x=0 x=0 x=1 examined is The index of summation can be changed from x to y = m = Np = Nλdt. x − 1: The probability of x decays in time dt is ∞ ∞ ∞ y (y +1) y −m m −m xP (x)= m me = m e = m. (Nλdt)x (y + 1)! y! P (x)= e−Nλdt. x=0 y=0 y=0 x! One can show that the variance for the Poisson distribu- As dt → 0, the exponential approaches one, and tion is σ2 = (x − m)2 = m. Table J.1 compares the binomial, Gaussian, and Pois- (Nλdt)x son distributions. The principal difference between the P (x) → . x! binomial and Gaussian distributions is that the latter is valid for large N and is expressed in terms of the mean The overwhelming probability for dt → 0 is for there to and standard deviation instead of N and p. Since the be no decays: P (0) ≈ (Nλdt)0/0! = 1. The probability Poisson distribution is valid for very small p, there is only of a single decay is P (1) = Ndt; the probability of two one parameter left, and σ2 = m rather than m(1 − p). decays during dt is (Nλdt)2/2, and so forth. The Poisson distribution can be used to answer ques- If time interval t is finite, it is still possible for the tions like the following: Poisson criterion to be satisfied, as long as p = λt is small. Then the probability of no decays is 1. How many red cells are there in a small square in a hemocytometer? The number of cells N is large; P (0) = e−m = e−Nλt. the probability p of each cell falling in a particular square is small. The variable x is the number of cells The probability of one decay is per square. −Nλt 2. How many gas molecules are found in a small volume P (1) = (Nλt)e . of gas in a large container? The number of tries is This probability increases linearly with t at first and then each molecule in the larger box. The probability that decreases as the exponential term begins to decay. The an individual molecule is in the smaller volume is reason for the lowered probability of one decay is that p = V/V , where V is the small volume and V is the 0 0 it is now more probable for two or more decays to take volume of the entire box. place in this longer time interval. As t increases, it is more 3. How many radioactive nuclei (or excited atoms) de- probable that there are two decays than one or none; for cay (or emit light) during a time dt? The probability still longer times, even more decays become more proba- of decay during time dt is proportional to how long ble. The probability that n decays occur in time t is P (n). dt is: p = λdt. The number of tries is the N nuclei Figure J.1 shows plots of P (0), P (1), P (2), and P (3), vs that might decay during that time. m = Nλt. Problems 573

Problems found that the electrical response could be interpreted as resulting from the release of packets of acetylcholine by Problem 1 In the United States one year, 400 000 peo- the nerve. In terms of this model, they obtained the fol- ple were killed or injured in automobile accidents. The lowing data: total population was 200 000 000. If the probability of being killed or injured is independent of time, what is the Number of packets reaching Number of times probability that you will escape unharmed from 70 years the end plate observed of driving? 018 144 Problem 2 Large proteins consist of a number of 255 smaller subunits that are stuck together. Suppose that an 336 error is made in inserting an amino acid once in every 425 105 tries; p =10−5. If a chain has length 1000, what is 512 the probability of making a chain with no mistakes? If the 65 chain length is 105? 72 Problem 3 The muscle end plate has an electrical re- 81 sponse whenever the nerve connected to it is stimulated. 90 I. A. Boyd and A. R. Martin [The end plate potential in mammalian muscle. J. Physiol. 132: 74–91 (1956)] Analyze these data in terms of a Poisson distribution. Appendix K Integrals Involving e−ax2

The desired integral is, therefore, " ∞ 2 π I = e−ax dx = . (K.1) −∞ a

This integral is one of a general sequence of integrals of the general form

∞ n −ax2 In = x e dx. 0 FIGURE K.1. An element of area in polar coordinates. From Eq. K.1, we see that

2 " Integrals involving e−ax appear in the Gaussian dis- I 1 π I = = . (K.2) tribution. The integral 0 2 2 a ∞ 2 I = e−ax dx The next integral in the sequence can be integrated −∞ directly with the substitution u = ax2: can also be written with y as the dummy variable: ∞ ∞ −ax2 1 −u 1 ∞ I1 = xe dx = e du = . (K.3) 2 0 2a 0 2a I = e−ay dy. −∞ A value for I2 can be obtained by integrating by parts: These can be multiplied together to get ∞ 2 −ax2 ∞ ∞ ∞ ∞ I2 = x e dx. 2 2 2 2 I2= dxdy e−ax e−ay = dxdy e−a(x +y ). 0 −∞ −∞ −∞ −∞ − 2 − 2 Let u = x and dv = xe ax dx = −(1/2a)d(e ax ). A point in the xy plane can also be speciﬁed by the polar Since udv = uv − vdu, coordinates r and θ (Fig. K.1). The element of area dxdy is replaced by the element rdrdθ: ∞ −ax2 3 −ax2 xe 1 −ax2 x e dx = − + e dx. 2π ∞ ∞ 2a 2a 2 2 0 I2 = dθ rdre−ar =2π rdre−ar . 0 0 0 This expression is evaluated at the limits 0 and ∞.The 2 term xe−ax vanishes at both limits. The second term is To continue, make the substitution u = ar2, so that du = I /2a. Therefore, 2ardr. Then 0 ∞ " 1 π ∞ π 1 π I2 =2π e−u du = −e−u = . I = . 0 2 × 0 2a a a 2 2a a 2 576 Appendix K. Integrals Involving e−ax

∞ ∞ 2 2 This process can be repeated to get other integrals in J = y2ne−ay 2ydy=2 y2n+1e−ay dy. the sequence. The even members build on I0; the odd 0 0 members build on I . General expressions can be written. 1 Therefore Note that 2n and 2n + 1 are used below to assure even and odd exponents: ∞ n! Γ(n +1) xne−axdx = = . (K.6) " n+1 n+1 ∞ × × × − 0 a a 2n −ax2 1 3 5 (2n 1) π x e dx = n+1 n , (K.4) 0 2 a a The gamma function Γ(n)=(n−1)! if n is an integer. Un- ∞ like n!, it is also deﬁned for noninteger values. Although 2n+1 −ax2 n! we have not shown it, Eq. K.6 is correct for noninteger x e dx = n+1 , (a>0). (K.5) 0 2a values of n as well, as long as a>0andn>−1. The integrals in Appendix I are of the form ∞ 2 2 e−x /2σ dx. Problems −∞ Problem 1 Use integration by parts to evaluate This integral is 2I with a =1/(2σ2). Therefore, the in- √ 0 2 ∞ tegral is 2πσ . 3 −ax2 Integrals of the form I3 = x e dx. 0 ∞ J = xne−axdx, Compare this result to Eq. K.5. 0 ∞ −ax2 Problem 2 Show that −∞ xe dx =0. Note the can be transformed to the forms above with the substi- lower limit is −∞, not 0. There is a hard way and an tution y = x1/2, x = y2, dx =2ydy. Then easy way to show this. Try to ﬁnd the easy way. Appendix L Spherical and Cylindrical Coordinates

FIGURE L.1. Spherical coordinates. FIGURE L.2. The volume element and element of surface area in spherical coordinates. It is possible to use coordinate systems other than the rectangular (or Cartesian) (x, y, z): In spherical coordinates (Fig. L.1) the coordinates are radius r and angles calculation1 shows that the divergence is θ and φ: 1 ∂ 1 ∂ x = r sin θ cos φ, div J = ∇·J = (r2J )+ (sin θJ ) r2 ∂r r r sin θ ∂θ θ y = r sin θ sin φ, (L.1) 1 ∂ + (J ). (L.3) r sin θ ∂φ φ z = r cos θ. The gradient, which appears in the three-dimensional In Cartesian coordinates a volume element is defined diffusion equation (Fick’s first law), can also be written by surfaces on which x is constant (at x and x + dx), y is in spherical coordinates. The components are constant, and z is constant. The volume element is a cube with edges dx, dy,anddz. In spherical coordinates, the ∂C (∇C) = , cube has faces defined by surfaces of constant r, constant r ∂r θ, and constant φ (Fig. L.2). A volume element is then 1 ∂C (∇C) = , (L.4) θ r ∂θ dV =(dr)(rdθ)(r sin θdφ)=r2 sin θdθdφdr. (L.2) 1 ∂C (∇C)φ = . To calculate the divergence of vector J, resolve it into r sin θ ∂φ components Jr, Jθ,andJφ, as shown in Fig. L.2. These components are parallel to the vectors defined by small 1H. M. Schey (2005). Div, Grad, Curl, and All That.4th.ed. displacements in the r, θ,andφ directions. A detailed New York, Norton. 578 Appendix L. Spherical and Cylindrical Coordinates

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

Laplacian ∂2C 1 ∂ ∂C 1 ∂ ∂C ∇2C = ∇2C = r ∇2C = r2 ∂x2 r ∂r ∂r r2 ∂r ∂r ∂2C 1 ∂2C ∂2C 1 ∂ ∂C + + + + sin θ ∂y2 r2 ∂φ2 ∂z2 r2 sin θ ∂θ ∂θ ∂2C 1 ∂2C + + ∂z2 r2 sin2 θ ∂φ2 Divergence ∂j 1 ∂(rj ) 1 ∂(r2j ) FIGURE L.3. A cylindrical coordinate system. ∇·j = x ∇·j = r ∇·j = r ∂x r ∂r r2 ∂r ∂j ∂j 1 ∂j ∂j 1 ∂(sin θj ) + y + z + φ + z + θ Figure L.2 also shows that the element of area on the ∂y ∂z r ∂φ ∂z r sin θ ∂θ 2 surface of the sphere is (rdθ)(r sin θdφ)=r sin θdθdφ. 1 ∂j + φ The element of solid angle is therefore r sin θ ∂φ Curl dΩ=sinθdθdφ. ∂jz 1 ∂jz 1 (∇×j)x = (∇×j)r = (∇×j)r = ∂y r ∂φ r sin θ This is easily integrated to show that the surface area of ∂j ∂j ∂(sin θj ) ∂(j ) a sphere is 4πr2 or that the solid angle is 4π sr. − y − φ × φ − θ ∂z ∂z ∂θ ∂φ π 2π π ∂jx ∂jr 1 2 2 (∇×j)y = (∇×j)φ = (∇×j)θ = S = r sin θdθ dφ =2πr sin θdθ ∂z ∂z r sin θ 0 0 0 ∂j ∂j ∂j sin θ∂(rj ) 2 − π 2 − z − z × r − φ =2πr [ cos θ]0 =4πr . ∂x ∂r ∂φ ∂r

∂jy 1 ∂(rjφ) Similar results can be written down in cylindrical co- (∇×j)z = (∇×j)z = (∇×j) ∂x r ∂r φ ordinates (r, φ, z), shown in Fig. L.3. ∂j 1 ∂j 1 ∂(rj ) ∂j Table L.1 shows the divergence, gradient and curl in − x − r = θ − r rectangular, cylindrical, and spherical coordinates, along ∂y r ∂φ r ∂r ∂θ with the Laplacian operator ∇2. 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 probability 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 assume only two values. The ﬁrst might be the patients health, with values healthy and sick; the other might by the results of some laboratory test, with results normal and abnormal. Table M.1 shows the values of the two variables for a sample of 100 patients. The joint probability that a patient is healthy 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. FIGURE M.1. The results of measuring two continuous vari- The probability of a false positive test is P (0, 1) = 0.20; ables simultaneously. Each experimental result is shown as a the probability of a false negative is P (1, 0) = 0.05. point. The probability that a patient is healthy regardless of the test result is obtained by a summing over all possible test outcomes: P (x =0)=P (0, 0)+P (0, 1) = 0.6+0.2= independent of yP (x), and so forth. Then 0.8. x In a more general case, we can call the joint probability P (x, y), the probability that x has a certain value Px(x)= y P (x, y) (M.1) Py(y)= x P (x, y).

TABLE M.1. The results of measurements on 100 patients Since any measurement must give some value for x and showing whether they are healthy or sick and whether a lab- y, we can write oratory test was normal or abnormal. Healthy (x =0) Sick(x =1) 1= x Px(x)= x y P (x, y), Normal test (y =0) 60 5 (M.2) Abnormal test (y =1) 20 15 1= y Py(y)= y x P (x, y). 580 Appendix M. Joint Probability Distributions

interval. We will call it px(x)dx. The extension to joint probability in two dimensions is p(x, y)dxdy. This is the probability that x is in the interval (x, dx)andy 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 ver- tical strip in Fig. M.1. Normalization requires that

FIGURE M.2. Perspective drawing of p(x, y). 1= px(x)dx = dx dy p(x, y). (M.4)

The ﬁrst strip could be taken horizontally: M.2 Continuous Variables

When a variable can take on a continuous range of values, 1= py(y)dy = dy dx p(x, y). it is quite unlikely that the variable will have precisely the value x. Instead, there is a probability that it is in Figure M.2 shows a perspective drawing of p(x, y). The the interval (x, dx), meaning that it is between x and volume of the shaded column is p(x, y)dxdy. The volume x + dx. For small values of dx, the probability that the of the slice is px(x)dx. The entire volume under the sur- value is in the interval is proportional to the width of the face is equal to one. Appendix N Partial Derivatives

When a function depends on several variables, we may Suppose now that we allow small changes in both r and want to know how the value of the function changes when h. The difference in volume is one or 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 the cylinder is kept fixed? ∆V = V (r +∆r, h +∆h) − V (r, h +∆h) 2 2 2 V (r +∆r)=π(r +∆r) h = π(r +2r∆r +∆r )h. + V (r, h +∆h) − V (r, h) V (r +∆r, h +∆h) − V (r, h +∆h) Subtracting the original volume, we have = ∆r ∆r ∆V = π(2r∆r +∆r2)h. V (r, h +∆h) − V (r, h) + ∆h. ∆h In the limit of small ∆r, this is → dV =2πhrdr. In the limit as ∆r and ∆h 0, the first term is This is the same answer we would have gotten if h had ∂V ∆r been regarded as a constant. The partial derivative of V ∂r with respect to r is defined to be h ∂V V (r +∆r, h) − V (r, h) evaluated at h +∆h. If the derivatives are continuous at = lim =2πrh. ∆r→0 (r, h), the derivative evaluated at (r, h +∆h) is negligibly ∂r h ∆r different from the derivative evaluated at (r, h). There- The subscript h in the partial derivative symbol means fore, we can write that h is held fixed during the differentiation. Sometimes it is omitted; when it is not there, it is understood that ∂V ∂V all variables except the one following the ∂ are held fixed. dV = dr + dh. If the cylinder radius is held fixed while the height is ∂r h ∂h r varied, we can write This result is true for several variables. For a function − 2 ∆V = V (r, h +∆h) V (r, h)=πr ∆h. w(x, y, z) The partial derivative is ∂w ∂w ∂w − dw = dx + dy + dz. (N.1) ∂V V (r, h +∆h) V (r, h) 2 = lim = πr . ∂x y,z ∂y x,z ∂z x,y → ∂h r ∆h 0 ∆h 582 Appendix N. Partial Derivatives

The derivatives are evaluated as though the variables The mixed partials are being held ﬁxed were ordinary constants. If w =3x2yz4, ∂2w ∂f f(x, y +∆y) − f(x, y) = = lim = lim ∂y∂x ∂y ∆y→0 ∆y ∆x→0 ∂w ∆y→0 =6xyz4, − − ∂x y,z × w(x +∆x, y +∆y) w(x, y +∆y) w(x +∆x, y)+w(x, y) ∆x ∆y ∂w =3x2z4, ∂y ∂2w ∂g x,z = = lim ∂x∂y ∂x ∆y→0 ∆x→0 ∂w 2 3 =12x yz . w(x +∆x, y +∆y) − w(x +∆x, y) − w(x, y +∆y)+w(x, y) ∂z × . x,y ∆x ∆y

It is also possible to take higher derivatives, such as The right side of each of these equations is the same, ∂2w/∂x2 or ∂2w/∂x∂y. One important result is that the except for the order of the terms. Thus order of differentiation is unimportant, if the function, its ∂ ∂w ∂ ∂w first derivatives, and the derivatives in question are con- = ∂x ∂y ∂y ∂x tinuous at the point where they are evaluated. Without filling in all the details of a rigorous proof, we will simply note that Problems ∂w w(x +∆x, y) − w(x, y) f = = lim Problem 1 If w =12x3y+z, find the three partial deriv- → ∂x ∆x 0 ∆x atives ∂w/∂x,∂w/∂y, and ∂w/∂z. ∂w w(x, y +∆y) − w(x, y) Problem 2 If V = xyz and x =5, y =6, z =2, find g = = lim . dV when dx =0.01, dy =0.02,anddz =0.03.Makea ∂y ∆y→0 ∆y geometrical interpretation of each term. Appendix O Some Fundamental Constants and Conversion Factors

The values of the fundamental constants are Some of the more useful conversion factors for con- from the 2002 least-squares adjustment, available at verting from older units to SI units are listed. They are physics.nist.gov/PhysRefData/contents-constants.html. taken from Standard for Metric Practice, ASTM E 380- 76, Copyright 1976 by the American Society for Testing Symbol Constant Value SI units and Materials, Philadelphia. c Velocity of light in 2.997 925 × 108 ms−1 vacuum To convert from To Multiply by e Elementary charge 1.602 177 × 10−19 C × −10 F Faraday constant 9.648 53 × 104 Cmol−1 angstrom meter 1.000 000 10 × 5 g Standard acceleration 9.806 65 m s−2 atmosphere (standard) pascal 1.013 250 10 × 5 of free fall bar pascal 1.000 000 10 2 −28 h Planck’s constant 6.626 069 × 10−34 Js barn meter 1.000 000 × 10 Planck’s constant 1.054 572 × 10−34 Js calorie (thermochemical) joule 4.184 000 (reduced) centimeter of pascal 1.333 22 × 103 ◦ 6.582 119 × 10−16 eV s mercury (0 C) −23 −1 1 kB Boltzmann’s constant 1.380 651 × 10 JK centimeter of pascal 9.806 38 × 10 8.617 343 × 10−5 eV K−1 water (4 ◦C) −31 −3 me Electron rest mass 9.109 382 × 10 kg centipoise pascal second 1.000 000 × 10 2 −14 mec Electron rest energy 8.187 105 × 10 J curie becquerel 3.700 000 × 1010 5 5.109 99 × 10 eV dyne newton 1.000 000 × 10−5 × −27 − mp Proton rest mass 1.672 622 10 kg electron volt joule 1.602 18 × 10 19 × 23 −1 − NA Avogadro’s number 6.022 142 10 mol erg joule 1.000 000 × 10 7 × −15 re Classical electron radius 2.817 940 10 m fermi (femtometer) meter 1.000 000 × 10−15 −1 R Gas constant 8.314 47 J mol × −4 −1 gauss tesla 1.000 000 10 K − liter meter3 1.000 000 × 10 3 u Mass unit (12C standard) 1.660 538 × 10−27 kg 2 × 8 mho siemens 1.000 000 uc Mass unit (energy units) 9.314 94 10 eV × 2 −12 2 −1 millimeter of mercury pascal 1.333 22 10 0 Electrical permittivity 8.854 19 × 10 C N − − poise pascal second 1.000 000 × 10 1 of space m 2 −4 9 2 −2 roentgen coulomb 2.58 × 10 1/4π0 8.987 55 × 10 Nm C −8 −2 per kilogram σSB Stefan Boltzmann 5.670 40 × 10 Wm − × 2 constant K 4 torr pascal 1.333 22 10 −12 λC Compton wavelength 2.426 31 × 10 m of electron −24 −1 µB Bohr magneton 9.274 009 × 10 JT −7 −1 µ0 Magnetic permeability 4π × 10 TmA of space −27 −1 µN Nuclear magneton 5.050 783 × 10 JT Index

A scan, 352 ADP, 67, 534 AAPM, 383, 384, 397, 445, 500 Afterloading, 504 Abbey, H., 564 Ahlen, S. P., 418, 420, 423, 434 Abductor muscle, 7 Aine, C., 225 Aberration Air chromatic, 389 acoustic impedance, 346 spherical, 389 attenuation, 350 Ablation, 381 density and specific heat, 61 Able, K. P., 217, 224 speed of sound, 345 Able, M. A., 217, 224 Alberts, B., 459 Abramowitz, M., 199, 201, 250, 253, 338, 340 Albumin, 129, 130 Absolute temperature, 57 Aldroubi, A., 542 Absorbed dose, 427 Algae, 217 Absorption coefficient, 365 Aliasing, 294, 309 Absorption edge, 404 in an image, 331 Acceleration, 547 Allen, A. P., 47 Accommodation, 389 Allen, R. D., 103 Acetabulum, 7 Allison, J., 542 Acetylcholine, 104, 137, 185, 573 Allos, S. H., 225 ACHD, 376 α particle, 424, 484, 505 Achilles tendon, 6 Alternans, 282 Acoustic impedance, 346 Alveoli, 1, 19, 77, 104, 256 Actinometry, 383 American Association of Physicists in Medicine, see Action potential, 135 AAPM foot, 174 American Heart Association, 195 Gaussian approximation, 182 Ampere, 145 propagating, 159 Ampere’s law, 206, 213, 216 space-clamped, 158 and Biot-Savart law, 220 Activating function, 200 Amplitude attenuation coefficient, 350 Active transport, 497 Ampullae of Lorenzini, 244, 252 Activity, 76, 484, 489 Anaplasia, 378 cumulated, 488, 489 Anderka, M., 293, 323 Activity vector, 180 Anderson, H. L., 357 Adair, R. K., 244, 247, 253, 317, 323 Anderson, J. R., 513 Adenosine triphosphate (ATP), 3 Anesthetic, 163 Adiabatic approximation, 416 Angelakos, J. D., 175 Adiabatic process, 54 Angiography, 376, 453 586 Index

Angioplasty, 504 cascade, 411, 438 Angioscopy, 376 Augmented limb leads, 188 Angstrom, 3 Austin-Seymour, M., 478 Angular momentum, 516 Autocorrelation function, 299, 522 Anisotropy, 191, 192 of exponential pulse, 305 Annihilation radiation, 414, 488 and energy spectrum, 305 Anode, 193 and power spectrum, 301 Anode break excitation, 173 of noise, 314 Anomalous rectification, 251 of sine wave, 300 Antineutrino, 486 of square wave, 300 Antiscatter grid, 450 AV node, 185, 194 Antonini, E., 76 Average, 561 Antzelevitch, C., 185, 201 ensemble, 52 Anumonwo, J. B., 165, 175 time, 52 Aorta, 20, 456 Average reference recording, 196, 201 Apoptosis, 458 Avogadro’s number, 59, 85, 409 Aquaspirillum magnetotacticum, 217 definition in SI units, 366 Aqueous, 388 Axel, L., 526, 535, 542 Arai, S., 224 Axelrod, D., 96, 108, 109 Armstrong, B. K., 380, 397 Axon, 2, 135 Arqueros, F., 425, 434, 494, 512 cable model, 149 Arteriole, 20 electric field, 148 Artificial insemination, 72 membrane capacitance, 149, 163 Artificial kidney, 120 membrane capacitance and conductance, 150 Ascites, 116 membrane equivalent circuit, 155 Ashley, J. C., 435 membrane time constant, 150 Astigmatism, 389 myelinated, 136, 160 Astumian, R. D., 244, 247, 252–254, 317–319, 323 potassium gate, 157 Ataxia-tangliectasia, 461 potassium Nernst potential, 155 Athens, J. W., 323 potential outside, 177 Atherosclerosis, 376 sodium conductance, 157 Atkins, P. W., 70, 79 sodium gate, 158 Atmosphere, 43 sodium Nernst potential, 155 pressure variation, 60, 75 space-clamped, 154, 158 Atmosphere (pressure unit), 14 surface charge density, 149 Atomic deexcitation, 410 unmyelinated, 136, 160 Atomic energy levels, 402 voltage-clamped, 154 Atomic number, 482 Axoplasm, 136 Atoms per unit volume, 366 Ayotte, P., 473, 477 ATP, 67, 534 Atrioventricular node, see see AV node B scan, 352, 356 Attenuation Background of sound wave, 349 natural, 470 water, 350 Backscatter Attenuation coefficient, 365 factor, 430 amplitude, 350 Backx, P. H., 202 effective, 368 Bacteria, 2 intensity, 350 magnetotactic, 217, 222 linear, 409 orientation in a magnetic field, 223 mass, 409 Bacteriophage, 2 Attenunator Badeer, H. S., 18, 29 ladder, 171 Bainton, C. R., 261 Attix, F. H., 407, 414, 420, 422, 426, 429, 430, 434, Balduzzi, A., 398 440, 445, 467, 477 Balloon angioplasty, 504 Attractor, 262, 269 Bambynek, W., 411, 434 Auditory evoked response, 308 Banavar, J. R., 41, 47 Auger electron, 410, 425, 485, 500, 504 Bank, R. A., 133 Index 587

Bar, 12 Berne, B. J., 372, 397 Barach, J. P., 212, 218, 221, 224 Bernoulli equation, 18 Barentsz, J., 542 Bernoulli process, 563 Barium, 449 Bernstein, M. A., 532, 533, 542 Barium fluorobromide, 446 Berry, E., 398, 399 Barker, A. T., 214, 224 Berson, A. S., 225 Barkovich, A. J., 542 Bessel function, 250, 338, 340, 352 Barlow, H. B., 392, 397 modified, 199 Barn (unit), 404 Beta decay, 484, 485 Barnes, F. S., 244, 253 spectrum, 487 Barneston, R. StC., 399 Bethe, H., 418 Barold, S. S., 194, 201 Bethe–Bloch formula, 420 Barr, G., 87, 108 Bevington, P. R., 289, 323 Barr, R. C., 154, 160, 175 Beyer, R. T., 350, 357 Barrett, H. H., 325, 333, 341 Bianchi, A. M., 308, 310, 323 Barrett, J. N., 240 Bicudo, C. E. M., 224 Barth, R. F., 467, 477 Bidomain model, 191, 192, 195, 199, 221 Bartlett, A. A., 39, 47, 274, 283 Bieber, M. T., 107, 108 Basal cell, 378, 381 Biersack, J. P., 415, 418–423, 435 Basal cell carcinoma, 464 Bifocal lenses, 389 Basal cell carcinoma (BCC), 380, 381 Bifurcation, 266 Basal metabolic rate, 61 diagram, 270 Base, 549 Bilirubin, 377 Basilar membrane, 349 Binding energy Basin of attraction, 266 electronic, 484 Bass, M., 397, 399 nuclear, 483 Basser, P. J., 104, 108, 130, 133, 537, 542 per nucleon, 483 Bastian, J., 243, 253 Binomial probability distribution, 51, 101, 563 Battocletti, J. H., 535, 542 Bioheat equation, 382 Baum, M. J., 398 Biot-Savart law, 206 Baumgertner, C., 225 and Ampere’s law, 220 Baylor, D. A., 392, 399 and magnetic field around an axon, 208 Bazylinski, D. A., 217, 224 current in, 207 Beam hardening, 448 Bipolar electrode, 194 Bean, C. P., 87, 108, 123, 127, 128, 133 Birch, R., 439, 477 Beaurepaire, E., 399 Bird, R. B., 90 Beck, R. E., 128, 133 Birds, 217 Becker, D. V., 500 Birefringence, 371 Becklund, O. A., 328, 341 Births, 321 Becquerel (unit), 472, 489 spontaneous, 293, 294 Bee, 217, 222 Bishop, C. R., 323 Beer’s law, 365 Blackbody, 372 Begon, M., 266, 283 Blackbody radiation BEIR, 471–473, 477, 505, 506, 512 and heat loss, 375 Bell, C., 542 vs. frequency, 375 Bell, G. I., 313 vs. wavelength, 373 Bellon, E. M., 224 Blackford, B., 225 Belousov–Zhabotinsky reaction, 275 Blackman, R. B., 295, 323 Bender, M. A., 511 Blackman-Tukey method, 309 Benedek, G. B., 10, 29, 90, 91, 100, 108 Blair, D. F., 96, 109 Bennison, L. J., 44 Blakemore, N., 217 Berg, H. C., 96, 108 Blakemore, R. P., 217, 224 Berg, M. J., 44, 47 Blank, M., 253, 254 Berger, M. J., 494, 512 Bloch equations, 519 Berggren, K-F., 434 Bloch, F., 418, 519 Bergmanson, J. P. G., 381, 397 Blood flow Berlinger, W. G., 47 pulmonary artery, 296 588 Index

Blood pressure, 255 Brown, B., 225 Blood-brain barrier, 116 Brown, J. H., 42, 47, 79 Blount, W. P., 10 Brown, J. H. U., 399 Blume, S., 376, 397 Brown, R., 85 Boas, D. A., 398 Brown, R. T., 434 Boccara, A. C., 399 Brownian motion, 61, 85, 312 Bockris, J. O’M., 234, 253 Bruglio, A., 398 Bodenschatz, E., 283 Bub, G., 275, 283 Bohr, N., 418 Buchanan, J. W., 500, 513 Boice, J. D., 468, 477 Buchsbaum, D., 505, 512 BOLD (Blood Oxygen Level Dependent), 523, 536 Bucky, G., 450 Boltzmann factor, 54, 58–62, 85, 227, 372, 517 Budd, T., 425, 434 Boltzmann’s constant, 57 Budinger, T. F., 488, 490, 491, 498, 513 Bone scan, 501 Buffer Boone, J. M., 414, 434 calcium, 104 Borgstede, J. P., 542 Buildup factor, 429, 494 Born charging energy, 144, 234 Buka, R.L., 380, 397 Born, C. G., 466 Bundle branch block, 189 Boron neutron capture therapy (BNCT), 467 Bundle of His, 185 Borthakur, A., 254 Buoyancy, 14 Bostrom, R. G., 480 Burch, W. M., 103 Bouma, B. E., 370, 397, 399 Burnes, J. E., 191, 202 Boundary-element method, 191 Bush, D. A., 479 Bourland, J. D., 193, 201, 202, 534, 542 Bystander effect, 460, 500 Boyd, I. A., 573 Bracewell, R. N., 295, 323 Cable equation, 152 Brachytherapy, 461, 467, 503 and ladder attenuator, 171 high-dose-rate, 504 Cable model, 149 Bradley, W. G., 542 Cabral, J. M., 253 Bradshaw, L. A., 225 Cadene, M., 254 Bradshaw, P., 23, 29 Calamante, F., 542 Bradycardia, 194 Calcaneus, 6 Bragg peak, 466, 477 Calcium, 279 Bragg rule, 422 -induced calcium release, 105 Bragg-Gray relationship, 468 diffusion, 105 Bramson, M. A., 373, 397 waves, 105 Brandts, B., 225 Calland, C. H., 120, 133 Brauer, F., 225 Calore, E., 398 Braun, T. J., 274, 283 Calorie, 54, 75 Breathing dietary, see Kilocalorie energy loss due to, 61 Calorimetry, 475 Bremsstrahlung, 413, 417, 437 Cameron, J. R., 348, 349, 378, 397 energy fluence, 438 Campbell, D. L., 185, 201 Bren, S. P. A., 245, 253 Cancer Brennan, J. N., 224 prostate, 536 Brenner, D. J., 472, 477 Cancer and power-line-frequency fields, 244 Bˇrezina, V., 295 Candle (unit), 387 Bˇrezina, V., 294, 323 Capacitance, 143 Briggs, E. A., 434 concentric cylinders, 169 Brightness contrast, 451 cylindrical membrane, 174 Brill, A. B., 500 membrane, 163 Brittenham, G. N., 217, 224 resistance and diffusion, 165 Broad, W. J., 455, 477 Capillary, 2, 20, 104, 115 Broad-beam geometry, 408 Capillary blockade, 497 Bronzino, J. D., 308, 310, 323 Carbohydrate, 3 Brooks, A. L., 471, 478 11C, 502 Brooks, R. A., 455, 457, 478 14C dating, 512 Index 589

Carbon dioxide Chang, S-B. R., 225 production, 256 Chang, W., 398 regulation, 257, 259, 261 Channel Carbon monoxide, 74 selectivity, 240 Carbonyl group, 240 Channels Carcinoma calcium, 238 basal cell, 380, 464 chloride, 238 squamous cell, 380 delayed rectiﬁer, 238 Cardiac arrest, 72 potassium, 238 Cardiac cell, 2 sodium, 238 Cardiac output, 21 two-state model, 240 Carlsson, C. A., 434 Chaos, 316 Carlsson, G. A., 407, 434 deterministic, 264, 270 Caro, C. G., 23, 29 in heart cells, 276 Carr–Purcell (CP) Sequence, 526 Chaotic behavior, 269 Carr–Purcell–Meiboom–Gill (CPMG) Sequence, 527 Characteristic x rays, 438, 485 Carslaw, H. S., 91, 94, 100, 106, 108 Charcoal, activated, 44 Carstensen, E. L., 245, 253 Charge Cartilage free and bound, 144 articular, 130 Charge distribution Casper, B. M., 68, 79 cylindrically symmetric, 139 Castelli, W. P., 47 line, 139 Catalyst, 74 on cell membrane, 141 Catﬁsh, 243 plane sheet, 139 Catheter, 376 point, 139 Cathode, 194 spherically symmetric, 139 virtual, 194 Charge screening, 403 “dog bone”, 196 Charged-particle equilibrium, 428, 440 Cathode-ray tube, 203 Charman, W. N., 390, 397 Cavitation, 353 Charnov, E. L., 79 Cavity radiation, see Blackbody radiation Charra, B., 133 Cebeci, T., 23, 29 Chavez, A. E., 414, 434 Cell Cheeseman, J., 380, 397 eukaryotic, 2 Chemical dosimeter, 445 membrane, 3 Chemical potential, 62, 88, 114, 115 producing or absorbing a substance, 96 ideal gas, 62 prokaryotic, 2 solute, 62, 69 size, 2 water, 69 Cell culture, 458 Chemical shift, 534, 541 Cell sorting, 223 images, 535 Cell survival Chemotaxis, 94, 96 fractionation curve, 460 Chemotherapy, 42 Cella, L., 479 Chen, C. N., 542 Cellular automata, 275, 282 Chen, J., 254 Center of gravity, 3 Chen, P.-S., 283 Center of mass, 362 Chen, W., 536, 542 Central slice theorem, 331 Cherry, S. R., 501, 502, 511, 513 Centrifuge, 26, 75 Cheyne–Stokes respiration, 274 Cerebral cortex, 212 Chick embryo, 213 Cerutti, S., 308, 310, 323 Chick, W. L., 279 Cesaro, S., 398 Chien, A., 478 Cesium iodide, 446 Chileuitt, L., 542 Chait, B. T., 253, 254 Chittka, L., 398 Chamberlain, J. M., 398, 399 Cho, Z. H., 335 Chance, B., 367–369, 398, 399 Cho, Z.-H., 331, 341, 455, 478, 527, 542 Chand, B., 435 Cholesterol Chandler, W. K., 250, 253 Raman spectrum, 372 590 Index

Christodolou, E., 478 Compounds and mixtures, 410 Chromatic aberration, 389 Compressibility, 15, 344 Chromosome, 458, 459 Compressive strength, 12 Chronaxie, 171, 193 Compton scattering, 403, 405, 442 Chronic granulocytic leukemia, 275 cross section, 406 Chung, S-H., 254 differential cross section, 406 CIE, 379, 387 Compton wavelength, 405 Cilia, 349 Computed radiography, 450 Circulation, 20, 278 Computed tomography Circulatory system, 20 spiral, 455 Clark, J., 182, 183, 191, 199, 201, 209 Concentration Clark, J. W., 175 and potential difference, 228 Clark, V. A., 36, 47 Concentration work, 63 Clarke, G. D., 542 Conductance, 145 Clarke, J., 218, 224 Conduction system, 185, 194 Classical electron radius, 406, 419 Conduction velocity Clausius-Clapeyron equation, 78 myelinated, 160, 162 Clearance, 36, 278, 382 unmyelinated, 160, 162 Clostridium, 146 Conductivity Cloud chamber, 425 anisotropic, 191, 192 Cocco, C., 398 interior and exterior, 181 Cochlea, 349 non-uniform, 191 Cochlear duct, 349 tensor, 191 Cochlear implant, 192 Conductor, 142 Cochran, W. W., 217, 224 Cones (retinal), 387, 390 Coffey, J. L., 495, 513 Conformal radiation therapy Cohen, A., 309, 310, 323 three-dimensional, 465 Cohen, B. L., 472, 473, 478, 506 Congestive heart failure, 35 Cohen, D., 216, 224 Conjunctivitis, 381 Cohen, L., 478 Constant-field model, 235 Cohen, L. G., 214, 224, 225 Contact lens, 107, 381 Cohen, S. L., 253 Continuity equation, 82 Cohen, Y., 542 differential form, 84 Coherence, 370 integral form, 83 Coherent scattering, 403, 407 with creation or destruction, 85 Cokelet, G. R., 133 Continuous slowing down approximation, 423 Collagen, 130 Contrast Collective dose, 471 brightness, 451 Collimator, 447 exposure, 451 gamma camera, 501, 510, 511 film, 441 multi-leaf, 464 noise brightness, 451 Collision kerma, 440 noise exposure, 451 Collision time, 86 Contrast agent, 448 Color blindness, 72 Convection coefficient, 76 Colton, C. K., 133 Conversion factors, 583 Colyvan, M., 281, 283 Convolution, 326, 327 Comer, M. B., 201 theorem, 326, 328 Commission Internationale de l’Eclairage, see CIE Cook, G., 77, 79 Common bundle, 185 Cook, G. J. R., 536, 542 Compass Cooley, J. W., 295, 323 in birds, 217 Coordinate system Competitive binding assay, 481 rotating, 519 Complex exponential, 291 64Cu, 44 Complex notation, 303 Cormack, A. M., 455, 478 Complex numbers, 291 Cornea, 107, 381, 388 Compound interest, 32 Cornsweet, T. N., 390, 397 Compound microscope, 396 Correlation function, 298 Index 591

Corry, B., 254 Current dipole moment, 180 Cortisol, 321 of the heart, 186 Cosset, J. M., 478 Current source, 178 Coster-Kronig transition, 411, 500 potential due to, 178 Cotton, P., 376, 397 Cyclic GMP, 392 Coulomb, 137 Cylindrical coordinates, 578 Coulomb’s Law, 137 Cystic fibrosis, 566 Countercurrent exchange Cytokinesis, 459 and heat loss, 132 Countercurrent transport, 121 da Luz, L. C. Q. P., 489, 513 Coupling medium, 355 da Silva, F. C., 495, 513 Coursey, B. M., 505, 513 Dainty, J. C., 452, 478 Covino, B. G., 163, 175 Dall’Amico, R., 398 Cox, J. D., 464 Damage, see Radiation damage Crank, J., 91, 100, 108 Danish, E. H., 224 Crasemann, B., 434 Dasari, R. R., 371, 398 Creatinine clearance test, 44 Data Cristy, M., 513 surrogate, 316, 317 Critical damping, 558 Davidenko, J. M., 186, 201 Cromer, D. T., 434 Davidson, H., 399 Cross correlation Davis, L., 153, 175, 284 and signal averaging, 309 de Araujo, F. F., 217, 224 Cross product, 5 De Boer, J. F., 371, 397 Cross section, 86, 365 De Kepper, P., 283 differential, 366 de la Rosette, J., 542 energy transfer, 407 De Stefano, P., 398 scattering, 366 Dead-time correction, 510 total, 366 Death rate, 35, 38, 43 Cross-correlation function, 298 Debye, 168 Crouzy, S. C., 243, 253 Debye length, 230 Crowder, S. W., 136, 175 Debye–Hückel model, 231 Crowe, E., 488, 489, 494, 513 Decay Crowell, J. W., 274, 284 constant, 33 Crues, J. V., 542 exponential, 33 CSDA, 423 Multiple paths, 37 CT, 455 rate, 33 Cuevas, J. M., 502 variable, 35 Cuffin, B. N., 224 with constant input, 38 Cumulated activity, 488, 489 Decibel, 311, 347 Cumulated mean activity per unit mass, 492 Declercq, E. R., 323 Curie (unit), 489 Deexcitation Curie temperature, 216 atomic, 410 Curl, 213 DeFelice, L. J., 242, 253, 307, 323 in different coordinate systems, 578 Deffeyes, K. S., 217, 224 Curran, P. F., 120, 133 Defibrillation, 72 Current Defibrillator, 192, 195 bound, 216 Degeneracy, 59 electric, 145 Degrees of freedom, 53, 55, 265 free, 216 Del Fante, C., 398 total, 16 Delaney, C., 332, 341 volume, 16, 81 Delannoy, J., 536, 542 Current density, 16 Delay-differential equation, 272, 275 electric, 145 Delayed rectifier channels, 238 Current dipole, 179 Delmar, M., 184, 201 electric and magnetic measurements compared, Delta function, 304 210 Delta rays, 424 in spherical conductor, 221 Demand pacemaker, 200 592 Index den Boer, J. A., 527, 533, 542 non-linear, 263 Dendrites, 135 second order, 558 in cerebral cortex, 212 Diffey, B. L., 378–380, 397, 398 Denier van der Gon, J. J., 195 Diffusion, 85, 119, 541 Denk, W., 243, 253 and capacitance, 165 Dennis, S. S., 158, 165, 175 and chemical reaction, 105 Denny, E. W., 480 and drift, 98 Denny, M. W., 15, 29, 61, 75–77, 79, 107, 108, 345, and electrotonus, 171 346, 350, 357, 394, 397 anisotropic, 104 Density as random walk, 100 current, 16 between concentric spheres, 166 optical, 441 between two spheres, 166 Density effect, 420, 422 circular disk, 166 Density gradient separation, 26 constant, 88 Density of states factor, 59 in one dimension, 93 Density-weighted image, 533 in three dimensions, 93, 101 Denyer, J. C., 175 in two dimensions, 93, 101 Deoxyribonucleic acid (DNA), 3, 24 of photons, 367 Department of Energy, 471 of spins in MRI, 536 Dephasing, 519, 523 self, 90 Depolarization, 136 sphere to infinity, 166 Depth of field, 390 tensor, 104, 537 Derivative, partial, 57 time-dependent, 368 Dermis, 380 to a cell producing or absorbing a substance, 96 Deserno, W. M., 542 to or from a disk, 94, 106 Detecting weak magnetic fields, 218 to or from a spherical cell, 94 Detective quantum efficiency (DQE), 452 trace (of tensor), 537, 541 Detector with a buffer, 105 thin-film transistor, 450 Diffusion constant Detriment, 468 and viscosity, 89 Deuteron, 484 determination of, 100 DeVita, V. T., 463, 478 effective, 105 DeWerd, L. A., 446 photon, 367 Dextrose, 130 vs. molecular weight, 90 Di Diego, J. M., 201 Diffusion equation, 91 Diabetes, 312 general solution, 99 Dialysis, 116 DiFrancesco, D., 165, 175 renal, 120 Digital detector, 446 Diamagnetism, 215, 216 storage phosphor, 446 Diamantopoulos, L., 399 thin-film transistor, 446 Diastole, 20 Digital subtraction angiography, 453 Diastolic interval, 282 Dillman, L. T., 485, 487–489, 498, 513 Diathermy, 350 Dimethylchlortetracycline, 379 Diatomic molecule, 362 Dini, G., 398 DiChiro, G., 455, 457, 478 Diopter (unit), 389 Dickerson, R. H., 77, 79 Dipole Dielectric, 143 electric, 484 saturation, 233 energy in a magnetic field, 516 Dielectric constant, 144 magnetic, 484, 516 lipid, 144 Dipole moment lipid bilayer, 163 electric, 168, 233 water, 144 Dirac, P. A. M., 304 Diem, M., 372, 397 Direct radiography, 450 Differential equation Discrete Fourier transform, 291 characteristic equation, 557 Disease, incidence and prevalence, 44 homogeneous and inhomogeneous, 557 Dispersion, 562 linear, 557 Displacement, 3, 545 Index 593

Displacement current, 207 mechanoreceptors, 243 Ditto, W. L., 276, 283, 284 ossicles, 348 Diuresis, 117 sensitivity, 347 Divergence, 84 Early-responding tissue, 461, 462 in different coordinate systems, 578 Echo planar imaging (EPI), 532 Dixon, R. L., 523, 534, 542 Eckerman, K. F., 489, 498, 513 Dizon, A. E., 225 Eddy currents, 214 DNA, 74, 459, 461, 500 Edema, 116 Dobson unit, 378 and osmotic pressure, 116 Dog bone, 196 cerebral, 116, 129 Dogfish, 244 pulmonary, 119 Doi, K., 442, 451, 478, 479 Edge-spread function, 329 Donnan equilibrium, 227 Edwards, F. M., 477, 479 Doppler effect, 353, 370 EEG, 196, see Electroencephalogram Dose average reference recording, 196 absorbed, 427 Effective dose, 469 effective, 469 Eidelman, S., 486 entrance, 441 Einstein (unit), 393 equivalent, 468 Einthoven’s triangle, 188 exit, 441 Eisberg, R., 216, 218, 224, 364, 483, 486, 487, 513 Dose equivalent, 468 Eisberg, R. M., 364 Dose factor, 489 Eisenman, G., 108 Dose fractions, 461 Ejection fraction, 501 Dose rate Ekstrand, K. E., 542 in photochemistry, 395 Elastography, 376 Dot product, 11 Electric dipole moment, 233 Doubling time, 34 Electric displacement, 207 Douglas, J. G., 467, 478 Electric field Doyle, D. A., 239, 253 definition, 138 Drift, 85 Electrical potential, 141 and diffusion, 98 Electrical stimulation, 192 Driving force, 88, 90 Electrocardiogram, 177 Driving pressure, 114, 118 and solid angle theorem, 198 Droplet keratopathy, 381 leads, 187 Drosophilia melanogaster, 239 left ventricular hypertrophy, 189 Drost, D. J., 542 normal, 189 Dubois, A., 399 R-R interval, 199 Ducros, M. G., 397 right bundle branch block, 189 Duda, J., 542 right ventricular hypertrophy, 189 Duderstadt, J. J., 368, 398 Electrode Duffy, H. S., 201 anodal, 194, 200 Duncan, W., 467, 478 cathodic, 194, 200 Dunn, R. A., 225 pacing, 194, 199 Durbin, R. P., 127, 128, 133 spherical, 170 Dye Electrode heating voltage-sensitive, 196 during magnetic stimulation, 222 Dyer, A. G., 398 in a changing magnetic field, 222 Dynamical system, 265 Electroencephalogram, 177, 196, 201, 212, 309 Dysplasia, 378 Electromagnetic spectrum, 360 Electromagnetic wave, 360 e, definition of, 33 Electromotive force, 214 Eames, C. and R., 29 Electromyogram, 177, 192, 310 Ear, 348 Electron canal, 348 Auger, 410 drum, 348 classical radius, 406 external, middle, inner, 348 for radiation therapy, 465 inner, 243 rest energy, 407 594 Index

rest mass, 407 Entry region, 22 Electron capture, 488 Enzyme, 74 Electron volt, 57, 144, 360 Eosinophilic granuloma, 503 Electroporation, 244 Epicardium, 191 Electrosurgery, 170 Epidemiological studies, 245 Electrotonus, 153, 161 Epidermis, 378 Eller, M. S., 398 Epiphysis, 7, 9, 501 Elliott, D. M., 27, 29 Epstein, A. E., 275, 276, 283 Elliston, C. D., 472, 477 Epstein, I. R., 283 Elson, H. R., 466 Equal anisotropy ratios, 192 Embedding, 316 Equilibrium dimension, 316 approach to, 259 time lag, 316 charged-particle, 428 Emission transient, 429 spontaneous and stimulated, 522 diffusive, 62 Emissivity, 373, 375 in volume exchange, 63 of skin, 375 particle, 62 Emmetropia, 389 radiation, 428 End-plate potential, 573 rotational, 4 Endocardial and epicardial membrane currents, 184 thermal, 56, 62 Endoplasmic reticulum, 105 translational, 3 Energy Equilibrium absorbed dose constant, 489, 494 equipartition theorem, 60 Equilibrium and steady state fluence, 382, 384, 385 distinction, 55 fluence rate, 384, 385 Equilibrium constant, 67 imparted, 426, 427 Equilibrium state, 52 ionization, 362, 403 Equipartition of energy, 60, 61 kinetic, 10 Equivalent dose, 468 lost and imparted, 423 Ergodic hypothesis, 56 radiant, 383 Erhardt, J. C., 502 transferred, 426 Erne, S. N., 225 net, 428 Error Energy exchange, 56 mean square, 286 Energy level, 53 Error function, 100, 154, 171, 193, 199 Energy levels, 56, 361 Erythema, 379, 395 atomic electron, 402 Erythrocyte, 2, 117 band, 372 Escape peak, 442 hydrogen atom, 361 Escherichia coli,2,96 rotational, 363 Esenaliev, R. O., 371, 398 splitting, 372 Esselle, K. P., 214, 224 vibrational, 364 Essenpreis, M., 398 Energy spectrum Ethanol, 44 of a pulse, 304 Eukaryotes, 2 of exponential pulse, 305 Eustachian tube, 348 Energy-momentum relationship Evans, R. D., 438, 478, 486, 513 relativistic, 405 Evans, S. J., 284 Engdahl, J. C., 501, 513 Even function, 290 Englehardt, R., 224 Evoked response, 196, 212, 308 Enquist, B. J., 47 Excess absolute risk, 470 Ensemble, 50 Excess relative risk, 470 Ensemble and time averages, 56 Excitable media, 282 Ensemble average, 52 waves in, 275 Enthalpy, 68 Exit dose, 441 Entrainment, 266, 281 Exitance, 384, 385 Entropy, 58, 64 Expectation value, 367 ideal gas, 76, 77 Exponent, 549 of mixing, 67 Exponential Index 595

complex, 291 Feld, M. S., 371, 398 decay, 33 Feldman, E. S., 224 function, 32 Felmlee, J. P., 542 growth, 31 Femur, 7 Exponentials, fitting, 38 Fenton, F. H., 284 Exposure, 440 Ferrimagnetism, 216 Exposure contrast, 451 Ferromagnetism, 216 Extensive variables, 64 Ferrous sulfate, 445 Exterior potential Feynman’s ratchet, 318 and electrocardiogram, 180 Feynman, R., 318, 323 and solid angle, 181 Fibrillation arbitrary pulse, 182 ventricular, 195 cardiac depolarization, 179 Fibula, 6 from action potential, 181 Fick tracer method, 103 from depolarization, 180 Fick’s law, 88, 91, 99 general case far from axon, 183 Fick’s second law, 230, 367, 382 nerve impulse, 179 Field ratio to interior, 180 radiation treatment, 464 Extinction coefficient, 394 Field of view, 329 Extracellular fluid, 136 Film Extracorporeal photopheresis, 381 as x-ray detector, 440 Eye gamma, 441 emmetropic (normal), 389 graininess, 452 human, 388 speed, 441 hypermetropic (farsighted), 389 Film-screen combination, 450 insect, 387 Filtration myopic (near-sighted), 389 of x-rays, 446 resolution, 390 Filtration coefficient, 117 under water, 397 Finite-element method, 191 Fink, R. W., 434 Factorial, 563 First law of thermodynamics, 53 Fainting, 194 Fischer, R., 224 False negative, 564, 579 Fish False positive, 564, 579 breathing, 76 Far field, 352 Fish gills, 122 Faraci, M., 398 Fisher, H. L., 495, 496, 513 Farad, 143 Fishman, H. M., 244, 252, 254 Faraday constant, 60 Fishman, R. A., 116, 133 Faraday induction law, 213, 524 Fission Farmer, J., 369, 398 nuclear, 484, 497 Farr, P. M., 379, 398 Fit Farrell, D. E., 224 linear, 287 Farrell, T. J., 398 polynomial, 287 Fast Fourier Transform (FFT), 295, 321 Fitzgerald, A. J., 372, 398, 399 Fatt, I., 107, 108 FitzHugh–Nagumo model, 281 Fédération CECOS, 72 Fitzmaurice, M., 371, 398 Feedback Fixed point carbon dioxide regulation, 256 stable, 262 loop, 255 unstable, 262 negative, 255 Flannery, B. P., 109, 202, 225, 323, 341 one time constant, 259 Flotte, T., 398 positive, 255 Flounder steady-state conditions, 256 ocean, 244 summary, 273 Flow time constant and fixed delay, 272 laminar, 15 two time constants, 262 Poiseuille, 17 Feinstein, A. R., 564 total, 16 596 Index

Flow (mathematical), one dimensional, 262 Fractional rate of growth or decay, 35 Flow effects (in MRI), 535 Fractionation curve, 460 Flow rate, 81 Fraden, J., 376, 398 Fluence Framingham study, 36 energy, 382, 412 Francis, J. E., 443 particle, 81, 367, 412 Frankel, R. B., 217, 220, 224 photon, 367 Fraunhoffer zone, 352 volume, 81 Freake, M., 225 Fluence rate, 81, 367 Free body diagram, 7 volume, 16, 117 Free energy Fluid Gibbs, 66 Newtonian, 15, 87 Free expansion, 73 Fluorescein, 380 Free induction decay (FID), 524, 525 Fluorescence, 196, 364, 403, 410 Free radical, 445, 458 Fluorescence yield, 411, 413, 492 Freeston, I. L., 214, 224 18F, 488, 492, 502 Freier, S., 68, 79 fluorodeoxyglucose, 502 Frequency, 289 Fluoroscopy, 454 angular, 289 for fitting shoes, 475 half-power, 311 Flux, 81 spatial, 326, 329 total, 16 Frequency response, 280, 310 volume, 81 glucose and insulin, 312 Flux density, 81 Fresnel zone, 352 volume, 16 Freund, H. U., 434 Flux transporter (magnetic), 218 Fricke dosimeter, 445 Flynn, E., 225 Friedman, R. N., 212, 224, 225 Focal length, 388 Fritzberg, A. R., 504, 513 Follicle-stimulating hormone, 322 Froelich, J. W., 542 Food consumption, 40 Fruit fly, 239 Force, 3 shaker mutant, 239 generalized, 64 Fuchs, A. F., 29, 133, 175, 284 nuclear, 483 Fujimoto, J. G., 370, 398 surface, 12 Function Ford, C. A., 119, 133 piecewise continuous, 295 Ford, M. R., 495, 496, 513 Functional MRI, 536 Foster, K. R., 245, 246, 253, 254 Fundamental constants, 583 Fourier integral, 301, 303 Fundamental equation of thermodynamics, 65 complex, 303 Fung, Y. C., 12, 13, 29 Fourier series, 290 Furocoumarins, 380 complex, 292 Fusion discrete, equally-spaced data, 291 nuclear, 484 for piecewise continuous function, 295 Fuster, V., 398 for square wave, 296 sample duration not a period, 293 Gabbe, E. E., 224 square wave, 292 Gadolinium, 523, 536 Fourier transform, 301, 303, 523, 530 Gadolinium oxysulfide, 441 complex, 303 Gain, 258 discrete, 291 open-loop, 258 discrete (2-d), 332 Galanski, M., 479 fast, 295 Galletti, P. M., 120, 133 of exponential pulse, 303 67Ga, 498, 502 Fourtanier, A. M., 380, 398 Gamma (film), 441 Fovea, 390 Gamma camera, 500 Fowler, P. H., 424, 425, 435 Gamma decay, 484 Fox, J. J., 277, 283 Gamma function, 576 Fox, R. A., 504, 513 Gammiatoni, L., 317, 323 Fractional distribution function, 508 Gangrene, 146 Index 597

Gans, D. S., 399 Glomerulus, 17, 27, 116 Gap junctions, 184 Glucose, 3, 67, 117 Gap phase, 459 Glucose regulation, 279 Gardner, M. J., 225 Glucose tolerance, 312 Garfinkel, A., 276, 282–284 Glutamate, 137 Garg, M. L., 435 Gluteus medius, 7 Garg, R. R., 435 Gluteus minimus, 7 Garrard, C. L., 225 Glycine, 137 Gas Glycoaminoglycans, 249 ideal, 60, 61 GMP, cyclic, 392 Gas constant, 60 Goiter, 277 Gas multiplication, 444 Goldberg, M. J., 47 Gasiorowicz, S., 374, 398 Goldman–Hodgkin–Katz equations, 236 Gaskill, J. D., 329, 341 Golzarian, J., 225 Gastric juice, 63 Gompertz mortality function, 44 Gastrocnemius, 6 Goodsell,D.S.,3,29 Gating current, 241 Goodsitt, M., 457, 478 Gatland, I. R., 288, 323 Gorby, Y., 217 Gauss, 204 Gou, S-y., 96, 109 Gauss’s law, 138 Gougoutas, A., 254 for magnetic field, 205 Gould, J. L., 217, 224, 225 free and bound charge, 144 Gouy-Chapman model, 229 Gaussian distribution, 91, 99, 568 Gradient, 88, 142 Gauvin, D., 477 diffusion, 537 Gawande, A., 566 in different coordinate systems, 578 Geddes, L. A., 136, 175, 193, 201, 202 magnetic, 528 Geetha, J., 284 phase-encoding, 529 Geiger counter, 444 readout, 529 Geiger’s rule, 477 slice-selection, 528 Geise, R., 451, 453, 454 Gradiometer, 219 Gel dosimeter, 445 Granger, H. J., 115, 133 Geller, A. C., 398 Grant, R. M., 348, 349 Genant, H. K., 442, 478 Gras, J. L., 103 General thermodynamic relationship, 65 Grass, Inc., 196 Generalized force, 64 Gray (unit), 427, 468 Gennari, F. J., 117, 133 Gray body, 373 Germanium, 445 Gray, D. E., 357 Gevins, A., 212, 226 Green’s function, 100 Giasson, C. J., 381, 398 Gregory, K., 398 Gibbs factor, 76 Greigite, 217 Gibbs free energy, 65, 66 Grenier, R. P., 511 Gibbs paradox, 68 Grid Gibbs phenomenon, 296 antiscatter, 450 Gielen, F. L. H., 212, 223, 224 Grievenkamp, J. E., 390, 398 Gilchrist, B. A., 380, 398 Grimes, D. I. F., 212, 224 Giles, W. R., 175 Grossweiner, L., 364, 367, 368, 381–383, 398 Gillooly, J. F., 47, 74, 79 Grove, R. I., 479 Gilmour, R. F., 283 Growth rate Gimm, H. A., 404, 434 variable, 35 Gingl, Z., 318, 323 Grundfest, H., 182, 202, 209 Gingras, S., 477 Guevara, M. R., 276, 283 Ginzburg, L., 281, 283 Gulbis, J. M., 253 Giorgiani, G., 398 Gulrajani, R. M., 177, 191, 201 Gittelson, K., 284 Guttman, R., 43 Glass, L., 264, 266–268, 271, 273–276, 283, 284, 316, Guyton, A. C., 20, 115, 117, 122, 133, 137, 175, 261, 317, 323 274, 284, 321, 323 Globulin, 129 Gyromagnetic ratio, 516, 517 598 Index h gate, 157, 173 Heat engine, 70 Haas, R. A., 565 Heat flow, 54, 61 Habrand, J. L., 467, 478 Heat loss Haddad, J. G., 381, 398 and countercurrent exchange, 132 Hademenos, G. J., 26, 29 Heat stroke, 274 Hafemeister, D., 244, 253 Heaton, C. L., 378 Hahn, P. F., 542 Hecht, S., 390, 391, 398 Haie, C., 478 Hee, M. R., 370, 398 Hair cells, 243, 349 Helfre, S., 478 Halbach, R. E., 542 Helical CT, 455 Halberg, F., 293, 321, 323 Heller, L., 214, 224 Haldane, J. B. S., 45, 47 Hellman, L., 35, 36, 47 Half-life, 34, 484 Helmholtz, 136 effective, 490 Hematocrit, 23 Half-power frequency, 311 Hemmingsen, A. M., 41, 47 Half-value layer, 476 Hemochromatosis, 217 Hall, E. J., 457–462, 472, 478 Hemoglobin, 24, 369, 523, 536 Hall, J. E., 175 Hemosiderosis, 217 Hall, K., 283 Hempelmann, L. H., 475 Hall, M. A., 270, 271 Hendee, W. R., 343, 352, 355, 357, 446, 447, 450, 454, Hallett, M., 214, 224, 225 477–479 Halliday, D., 7, 29, 142, 175, 363, 373, 375, 398 Henderson, L. W., 119, 133 Halliday, G. M., 399 Henriquez, C. S., 184, 191, 192, 202 Hämäläinen, M., 212 Hereditary spherocytosis, 117 Hämäläinen, R., 212, 213, 218, 224, 315 Hermann, M., 398 Hamblen, D. P., 443 Herrick, J. F., 17, 29 Hamill, O. P., 238, 243, 253 Hertz (unit of frequency), 289 Hamilton, L. J., 368, 398 Hexadecapole Hänggi, P., 323 magnetic, 484 Hanlon, E. B., 371, 372, 398 Hidajat, N., 479 Harisinghani, M. G., 536, 542 Hielscher, A., 369, 398 Harmonic images, 353 Higson, D. J., 471, 478 Harmonic oscillator, 263, 363, 559 Hilborn, R. C., 263–265, 271, 284 Harmonics, 289 Hildebrand, J. H., 67, 79, 87, 108, 114, 133 Harri, R., 213, 224, 315 Hill, B. C., 196, 202 Harris, C. C., 443 Hille, B., 29, 133, 155, 175, 235, 238, 243, 253, 284 Harris, J. W., 181, 184, 202, 224 Hinds, M., 202 Hart, H., 35, 36, 47 Hine, H. G., 511 Hasselquist, B., 503 Hip, 7, 8 Hastings, H. M., 277, 282, 284 Hobbie, R. K., 185, 188, 189, 425, 434 Hawkes, D. J., 407, 435 Hodgkin’s disease, 463 Hayes, N. D., 273, 284 Hodgkin, A., 136, 148, 153, 154, 171, 175, 253 HCl, 393 Hodgkin–Huxley model, 268, 275 Heald, M. A., 387, 398 Hoke, M., 225 Heart block, 194 Hole, 438 intermittent, 200 Holmes, T. W., 465, 478 second degree, 200 Honig, B., 231, 254 sinus exit, 200 Horacek, B. M., 224, 225 Heart failure, 116, 194 Hormesis, 472 Heart rate, 46 Horowitz, P., 175 Heat capacity, 61, 278 Hosaka, H., 211, 224 air, 61, 76 Hot tub, 274 at constant volume, 61 Hotomaroglu, O., 284 ideal gas, 61 Hounsfield (unit), 456 specific, 61 Hounsfield, G. N., 455, 478 water, 75 Howell, R. W., 500, 513 Heat conduction, 103, 106, 382 Howerton, R. J., 434 Index 599

Hu, X-p., 330, 331 Impulse response, 312, 326 Huang, D., 370, 398 Incidence, 380 Hubbell, J. H., 404, 407, 408, 411, 414, 434, 435, 448, Incidence of disease, 44 478 Incoherent scattering, 403, 406 Huda, W., 457, 478 Incremental-signal transfer factor, 442 Human immunodeficiency virus (HIV), 2 Incubator Humm, J. L., 500, 513 neonatal, 376 Hunt, D. C., 454, 478 Incus, 348 Hunt, J. G., 495, 513 Index of refraction, 359, 364 Husband, J. E., 536, 542 of aqueous, 388 Huxley, A. F., 136, 154, 175 of crystalline lens, 388 HVL, 476 of vitreous, 388 Hydraulic permeability, 117 Inelastic scattering, 403 pore model, 123 Inflammatory response, 116, 378 Hydrogen Influence function, 100 spectrum, 361 Infrared Hydrogen peroxide, 458 photography, 376 Hydroxyl free radical, 458 radiation, 360, 375 Hyperbolic functions, 166 spectroscopy, 368 Hypermetropia, 389 Ingard, K. U., 343, 357 Hyperplasia, 378 Inman, V. T., 7, 9, 29 Hyperpolarization, 173 Inner ear, 349 Hyperthermia, 381 Insulin, 312 Hyperthyroidism, 504 Integrals Hypertrophy, 378 trigonometric, 555 Hypoproteinemia, 116 Intensifying screen, 441 Hysteresis, 13, 19 Intensity magnetic, 216 of sound, 347 radiant, 383 Ice ages, 317 reference, 347 ICRP, 468, 469, 478 Intensity attenuation coefficient, 350 ICRU, 350, 352, 357, 367, 414, 415, 417, 420–424, 426, Intensity-modulated proton therapy, 467 435, 437, 439, 440, 447, 448, 450, 478 Intensity-modulated radiation therapy, 465 Ideal gas, 73 Interferometer, 370 Ideal gas law, 60, 63 Internal conversion, 484, 492 Ideal solution, 114 coefficient, 485 Ideker, R. E., 196, 202, 276, 283 electron, 492 Illuminance, 384 Internal radiotherapy, 504 Ilmoniemi, R. J., 213, 214, 224, 315 International Commission on Radiation Units and Image Measurements, see ICRU density-weighted, 533 International Commission on Radiological Protection, T1-weighted, 533 see ICRP T2-weighted, 533 Interstitial fluid, 115 Image contrast Intestinal parasites, 565 and MR pulse parameters, 533 Intestinal villi, 122 Image intensifier tube, 454 Intracellular fluid, 136 Image quality, 450 Intravascular ultrasound, 376 Imaginary number, 292 Inulin, 393 Impact parameter, 418 Inverse problem, 198 Impedance Inversion recovery (IR), 525 acoustic, 346 Iodine, 449 characteristic, 346 123I, 498 reactive component, 346 131I, 502, 504 transformation in middle ear, 349 Iodine deficiency, 277 IMPT, 467 Ion Impulse approximation, 416, 419 Born charging energy, 144 Impulse function, 304 Ionization chamber, 444, 468 600 Index

Ionization energy, 362, 403 Kaminski, E. M., 542 average, 420 Kanal, E., 534, 542 192Ir, 504 Kane, B. J., 136, 175 Iris, 388 Kannel, W. B., 47 Iron stores Kaplan, D., 264, 271, 275, 284, 316, 317, 323 in the body, 217 Karagueuzian, H. S., 283 Irradiance, 384, 385 Karhu, J., 214, 224 Isodose contours, 464 Kassirer, J. P., 117, 133 Isosbestic point, 369 Kassis, A. I., 458, 461, 479, 500, 505, 513 Isotonic saline, 130 Katchalsky, A., 120, 133 Isotope, 482 Kathren, R. L., 472, 479 Isotropic radiation, 386 Katila, T., 225 Itoh, M., 399 Kato, T., 542 Itzkan, I., 371, 398 Katz, B., 135, 137, 154, 175 Izatt, J. A., 370, 371, 399 Kaufman, L., 224, 225 Kavanagh, K. M., 284 J. H. Brown, 47 Kay, A. R., 542 Jabinsek, V., 225 Kazerooni, E., 478 Jabola, B. R., 479 Keeler, E. K., 542 Jackson, D. F., 407, 435 Keener, J. P., 275, 284 Jacques, S. L., 398 Keizer, J., 105, 109 Jaeger, J. C., 91, 94, 100, 106, 108 Kelvin, 57 Jalife, J., 165, 175, 194–196, 201, 202, 212, 225, 268, Kempe, C. H., 41, 47 275, 283, 284 Keratin, 378 Jalinous, R., 214, 224 Keratitis, 381 James, R. B., 445, 479 Keratopathy, 381 Jang, I-K., 398 Keriakes, J. G., 466 Janka, G. E., 224 Kerma, 427 Jaramillo, F., 317, 323 collision, 428, 429 Jaundice, neonatal, 377 radiative, 428 Jeffrey, K., 194, 200, 202 Kernicterus, 377 Jeffreys, B. S., 153, 175 Ketcham, D., 449 Jeffreys, H., 153, 175 Kevan, P. G., 387, 398 Jejunum, 170 Keynes, R. D., 239, 254 Jiang, D., 200, 202 Khan, F. M., 464–467, 479 Jiang, Y., 239, 254 Khoo, M. C. K., 256, 284 Johnson noise, 242, 312, 314 Kidney, 37, 116, 278 Johnson, G. F., 47 artificial, 120 Johnson, J. B., 314 glomerulus, 17 Joint probability distribution, 579 Kil, J. R., 283 Jones, J. P., 341, 455, 478, 527, 542 Kilocalorie, 54, 75 Jones, R. H., 511 Kim, Y.-H., 283 Jongsma, H. J., 175 Kimlin, M. G., 380, 398 Joseph, P. M., 526, 527, 542 Kinetic energy, 10 Josephson junction, 218 King, K. F., 532, 533, 542 Joule, 11 Kirby, S. S., 454, 478 Jung, P., 323 Kirchhoff’s laws, 147 Kirschvink, J. L., 217, 224, 225, 247, 248, 254 K edge Kiss, L. B., 318, 323 in contrast agent, 448 Kissel, L. H., 407, 435 K shell, 438 Kleiber, M., 41 Köhler, T., 25, 29 Klein, L. S., 195, 202 Kagan, A. R., 463, 479 Klein-Nishina Kaiser, I. H., 293, 321, 323 factor, 432 Kalender, W. A., 455, 456, 478, 479 formula, 406 Kalmijn, Ad. J., 243, 254 Kligman, L. H., 380, 398 Kaluza, G. L., 504, 513 Knisley, S. B., 196, 202 Index 601

Knobler, E., 381, 398 precordial, 188 Knuutila, J., 213, 224, 315 Least squares, 201, 285 Kobayashi, A. K., 247, 254 linear, 287 Kobayashi-Kirschvink, A., 254 nonlinear, 288 Koh, D.-M., 536, 542 polynomial, 288 Koh, W. J., 478 Lebec, M., 399 Koizumi, H., 399 LeBihan, D., 108, 542 Konarzewski, M., 42, 47 Lee, M.-H., 283 Kondo, S., 472, 479 Lee, T. M., 542 Koo, T-W., 371, 398 Left and right bundles, 185 Kovacs, G. T. A., 136, 175 Left ventricular hypertrophy, 189, 190 Kozlowski, J., 42, 47 Legendre polynomial, 181, 184 Krämer, U., 398 Lei, H., 534, 542 Kramer, J. R., 371, 398 Leigh, J., 399 Krane, K. S., 7, 29, 142, 175, 363, 373, 375, 398 Leighton, R. B., 323 Kricker, A., 380, 397 Leiter, D., 284 Kripke, M. L., 380, 398 Lenir-Cohen-Solal, C., 478 81mKr, 502 Lennard, R. F., 212, 224 Kucharczyk, J., 542 Lens, 388 Kuo, A., 253 Lenz’s law, 214, 215 Kupfer cells, 497 Lester, J. W., 542 Kustin, K., 283 LET, 423 Kuyucak, S., 254 Létourneau, E. G., 477 Leuchtag, H. R., 236, 254 Laüger, P., 148, 175 Leukemia, 42, 275 LaBarbera, M., 27, 29 Levenberg-Marquardt method, 289 Lakshminarayanan, A. V., 335, 341 Lever, L. R., 380, 398 Lamb, H., 248, 254 Lévesque, B., 477 Lambert’s law, 386 Levin, R. L., 542 Laminar flow, 15, 16 Levitt, D., 87, 98, 109, 124, 127, 133 Lamothe, M. J. R., 225 Levy, R. P., 479 Lanchester, F. W., 280 Lewis number, 104 Langevin equation, 87 Lewis, C. A., 238, 254 Lanino, E., 398 Li,T.K.,44 Lapicque, L., 193, 202 Liang, Z.-P., 527, 542 Laplacian, 91 Light Laramore, G. E., 478 photon properties, 360 Large-signal transfer factor, 442 polarized, 371 Larin, K. V., 398 speed, 359 Larina, I. V., 398 wave nature, 360, 370 Larmor precession frequency, 518, 522 Lightfoot, E. N., 90 Larsson, S. A., 502, 513 Lighthill, J., 15, 29 Laser surgery, 381 Lighthill, M. J., 304, 323 Late-responding tissue, 461, 462 Likavek, M. J., 116, 133 Latent heat of vaporization, 75 Limb leads, 187 Lateral geniculate nucleus, 536 Lime peel, 380 Latitude, 441 Limit cycle Lauterbur, P. C., 527, 542 bifurcation of, 266 Law of Laplace, 26, 77 stable, 266 Lawrence, C. M., 380, 398 unstable, 266 Le, A., 254 Lin, C. P., 370, 398 Le, T. H., 330, 331 Lindemans, F. W., 195, 202 Lead zirconate titanate, 351 Lindhard, J., 418, 420 Leads Lindner, A., 120, 133 ECG, 187 Lindsay, R. B., 350, 357 limb, 187 Line approximation, 184 augmented, 188 Line integral, 142 602 Index

Line-spread function, 329 M mode, 353 Linear energy transfer, 423, 458 Maccaferri, G., 165, 175 Linear least squares MacDose, 434 weighted, 288 Maciel, H. S., 254 Linear regression, 287 Mackay, J., 398 Linear system, 325 Mackenzie, D., 47 Linear-nonthreshold model, 471, 472, 506 Mackey, M. C., 264, 266–268, 273–275, 283, 284 Linear-quadratic model, 460 Mackie, T. R., 465, 478 Links, J. M., 501, 513 MacKinnon, R., 239, 240, 253, 254 Liow, J. S., 513 Macklem, P. T., 13, 29 Lipid bilayer, 163 MacNeill, B. D., 376, 398 Lissner, H. R., 7, 29 Macovski, A., 451, 479 Lithotripsy, 350, 353 Macrophage, 216 Littmark, U., 415, 418–423, 435 Macrostate, 52, 55 Liu, D-W., 201 disordered, 52 Liu, H., 369, 398 nonrandom, 52 Liver, 116, 217 ordered, 52 Local control of tumor, 463 random, 52 Locatelli, F., 398 Maddock, J. R., 96, 109 Loevinger, R., 488, 490, 491, 498, 513 Madronich, S., 377, 378, 398 Log-log graph paper, 39 Maer, W., 479 Logarithm Magleby, K. L., 240 definition, 549 Magnetic dipole, 215 natural, 549 field far from, 522 Logistic equation, 39, 265, 266 Magnetic field, 203, 205 Logistic map, 269 crayfish axon, 208 Lomax, A., 479 detection, 218 Longitudinal wave, 344, 346 earth, 218 Lopatin, A. N., 254 from a current dipole in a conducting sphere, 212 LorentedeNó, R., 153, 175 from axon, 208 Lorentz force, 204, 222 from depolarizing cell, 210 Lotka-Volterra equations, 46 from point source of current, 207 Lounasmaa, O. V., 213, 224, 315 lines, 205 Lowe, H. C., 398 of a current dipole in a conducting sphere, 221 Lu, J., 244, 252, 254 remanent, 216 Lubin, J. H., 473, 479 Magnetic field intensity, 216 Lukas, A., 201 Magnetic moment, 204, 516 Lumen, 376 and angular momentum, 516 Lumen (unit), 387 nuclear, 517 Luminance, 384 torque, 205 Luminous Magnetic monopole efficacy, 387 absence of, 205 energy, 384 Magnetic permeability, 216 exitance, 384 Magnetic resonance imaging exposure, 384 intravascular, 377 flux, 384, 387 Magnetic shielding, 214 intensity, 384, 387 Magnetic stimulation, 222 Lung transplant, 503 Magnetic susceptibility, 203, 216 Luo, C. H., 158, 175 Magnetism Lutz, G., 445, 479 and special relativity, 203 LVH, see Left ventricular hypertrophy Magnetite, 216, 217, 222 Lybanon, M., 288, 323 Magnetization, 216, 517 Lynch, D. K., 387, 399 Magnetocardiogram, 209, 211 Lysaght, M. J., 119, 120, 133 Magnetoencephalogram, 209, 212, 314 Lythgoe, M. F., 542 Magnetopneumography, 216 Magnetosome, 217, 222, 247, 248 m gate, 157 Magnetotactic bacteria, 217 Index 603

Magnification, 327 Mean absorbed dose, 489 Magnifying glass, 396 Mean absorbed dose per unit cumulated activity, 489 Magnoni, D. M., 175 Mean energy emitted per unit cumulated activity, 489 Magnoni, M., 165 Mean energy per ion pair, 439 Mainardi, L. T., 308–310, 323 Mean energy per unit cumulated activity, 488 Makhina, E. N., 254 Mean free path, 85 Maki, A., 399 Mean initial activity per unit mass, 492 Makita, S., 399 Mean number per disintegration, 484, 489 Malleus, 348 Mean square error, 286 Malmivuo, J., 177, 202 Meandering waves, 276 Mammography, 453 Mechanoreceptors, 243 Mannitol, 116, 129 Medical Internal Radiation Dose, see MIRD Manoharan, R., 371, 398 MEG, 209 Manoyan, J. M., 422, 435 Mehta, D., 435 Mansfield, P., 527 Meidner, H., 96, 109 Mansfield, T. A., 96, 109 Melanin, 378 Maor, E., 33, 47 Melanocyte, 378 Marchesoni, F., 323 Melanoma, 380, 381 Maris, M., 399 Mellinger, M. D., 398 Maritan, A., 47 Membrane, 111 Mark, H., 434 force on, 128 Markov process, 563 permeable, 111 Maroudas, A., 133 semipermeable, 111 Marshall, M., 425, 434, 439, 477 Membrane capacitance, 163 Martel, R., 477 Membrane permeability, 120 Martin, A. R., 175, 573 Mercader, M. A., 275, 284 Marty, A., 253 Messina, C., 381, 398 Mass attenuation coefficient, 409 Metabolic rate, 41, 256 tissue, 447 Metaplasia, 378 Mass energy absorption coefficient, 413 Metastable state, 484 table (on the web), 441 Metastases, 463 Mass energy transfer coefficient, 413 Metric prefixes, 1 Mass fraction, 410 Metric system, 1 Mass number, 482 Metz, C. E., 451, 479 Mass stopping power, 414 Meves, H., 253 Mass unit Michaels, D. C., 284 atomic, 483 Microscope, 396 unified, 483 Microstate, 52, 55 Massoth, R. J., 542 Microstates, 64 Mattiello, J., 108, 537, 542 energy dependence, 57 Maughan, W. Z., 295, 323 Microwave radiation, 360 Mauricio, C. L., 495, 513 Mielczarek, E. V., 217, 224, 275, 284 Mauro, A., 229, 230, 254 Miles, W. M., 194, 195, 202 Maxwell’s equations, 222 Miller, B., 398 May, R. M., 269, 284 Miller, B. D., 202 Mayaux, M. J., 72 Miller, C. E., 191, 202 Mazal, A., 478 Miller, J. M., 398 Mazumdar, J. N., 15, 20, 22, 29 Milner, T. E., 397 McCollough, C. H., 468, 469, 479 Milnor, W. R., 20, 23, 29, 296, 297, 323 McDonagh, A. F., 377, 398 Milonni, P. W., 364, 398 McGrayne, S. B., 217, 224 Miners, 216 McGregor, R. G., 477 Mines, G. R., 275, 284 McKee, P. A., 36, 47 Minimum auditory field (MAF), 347 McMahon, T., 41, 47 Minimum auditory pressure (MAP), 347 McNamara, P. M., 47 Minimum erythemal dose, 379 Meadowcroft, P. M., 23, 29 Mintorovitch, J., 542 Mean, 561 Miralbell, R., 467, 479 604 Index

MIRD, 481, 489, 494–496, 505 Murray, J. D., 39, 46, 47 Mitochondria, 2 Muscle Mitosis, 458, 459 gastrocnemius, 6 Mitrani, R. D., 195, 202 gluteus medius, 7 Mixing gluteus minimis, 7 entropy of, 67 soleus, 6 heat of, 68 Mutual inductance, 223 Modulation transfer function, 328, 390, 450 Myelinated ﬁber Modulus time and space constants, 161 shear, 13 Myers, K. J., 325, 333, 341 Young’s, 12 Myocardial cells Moe, G. K., 268 and nerve cells compared, 184 Mole, 366 repolarization, 179 Mole fraction, 69 Myocardial infarct, 36, 44, 211, 376 Moller, J. H., 189, 190 Myoglobin, 76 Moller, W., 216, 225 Myopia, 389 99Mo, 497, 498 Molybdenum target x-ray tube, 454 n gate, 157 Moment arm, 5, 10 Nag, S., 504, 513 Moment of inertia, 363 Nagai, Y., 283 Monte Carlo calculation, 367, 369, 425, 494 Nagel, H. D., 457, 458, 479 Montesinos, G. D., 425, 434, 494, 512 Narrow-beam geometry, 408 Moore, J. W., 274, 284 Nath, R., 505, 513 Moran, J., 120, 133 National Institute of Standards and Technology, see Moran, P. R., 542 NIST Morin, R., 534 National Nuclear Data Center, 489 Morrison, P., 3, 29 National Research Council, 245, 254 Morrow, M. N., 202 Nattel, S., 283 Morse, P. M., 343, 357 Natural background, 470 Mortality function, 44 Natural logarithm, 549 Mortimer, J. T., 202 Navier-Stokes equation, 27 Mortimer, M., 266, 283 NCRP, 454, 470, 471, 479 Moseley, H., 380, 398, 474 Near ﬁeld, 352 Moseley, M. E., 536, 542 Near-infrared radiation, 369 Moses, H. W., 194, 202 Near-infrared spectroscopy, 377 Moskowitz, B. M., 217, 224 Nedbal, L., 294, 295, 323 Mosley, M. L., 502, 513 Negative resistance, 252 Moss, F., 317–319, 323 Neher, E., 238, 253, 254 Moss, W. T., 464 Nelson, J. S., 397 Mossman, K. L., 472, 479 Nemoto, I., 224 Motamedi, M., 398 Neonatal jaundice, 377 Mottle, 452 Neper, 350 Motz, J. T., 371, 398 Nephron, 116 Moulder, J. E., 244, 245, 254, 458, 479 Nephrotic syndrome, 116 Moulton, K. P., 202 Nernst equation, 60, 148, 228 Mouritsen, H., 217, 224 sodium, 155 Moy, G., 231, 254 Nernst potential Moyal, D. D., 380, 398 calcium, 170 Moyers, M., 466, 467, 479 potassium, 155 MR Image sodium, 155 and pulse parameters, 533 Nernst-Planck equation, 235 Muir, W. A., 224 Nerve cell, 75 Multipole expansion, 184 Nerves Murphy, C. R., 175 Sympathetic and parasympathetic, 185 Murphy, E. A., 564 Nesson, M. H., 247, 254 Murphy, M. R., 377, 398 Nesterenko, V. V., 201 Murray’s law, 27 Neunlist, M., 196, 202 Index 605

Neurotransmitter, 137 OATZ, 217 Neutrino, 486 Oberly, L. W., 502 rest mass, 486 Ocean flounder, 244 Neutron therapy, 467 Octupole Newhauser, W. D., 466, 467 electric, 484 Newman, E. B., 349, 357 Odd function, 290 Newton’s law of cooling, 76 Oellrich, R. G., 377, 398 Newton’s second law, 10, 49 Oersted, H. C., 203 Newton’s third law, 12 Ogawa, S., 536, 542 Newton, I., 359 Ohm, 145 Newtonian fluid, 15, 23, 87 Ohm’s law, 145, 182, 190 Nicholls, A., 231, 254 and current density, 146 Nichols, C. G., 251, 254 anisotropic medium, 191 Nickel skin allergy, 380 Okada, T. C., 225 Nickell, S., 367, 398 Okada, Y., 225 Nielsen, P., 217, 224 Oldendorf, W., 455 Nijsten, T. E., 381, 399 OLINDA/EXM, 489, 494 Nilson, J., 284 One-dimensional flow (mathematical), 262 Nioka, S., 399 Open-loop gain, 258 NIST, 410, 414, 423, 441 Operating point, 257 13N, 502 approach to, 259 Noble, D., 165, 175 Optical coherence microscopy, 371 Nodal escape beats, 185 Optical coherence tomography, 370, 376 Node of Ranvier, 136, 160 Optical density, 441 Noise, 75, 242, 306, 308 and exposure, 475 1/f, 315 Optical transfer function, 328 autocorrelation function, 314 Orban, M., 283 Johnson, 242, 312, 314 Orbital quantum number, 402 pink, 315 Ordidge, R. J., 542 power spectrum, 314 Orear, J., 288, 323 quantum, 451 Organelle, 2 shot, 242, 312, 451 Orthogonality relations source, 313 for trigonometric functions, 296 white, 314 Orton, C., 458, 462, 479 Noise brightness contrast, 451 Osmolality, 115 Noise equivalent quanta, 452 Osmolarity, 115 Noise exposure contrast, 451 Osmosis Nolte, J., 137, 175 in ideal gas, 112 Noninvasive electrocardiographic imaging, 191 in liquid, 114, 117 Nonlinear least squares, 288 reverse, 119 Nonmelanoma skin cancer (NMSC), 380 Osmotic diuresis, 116 Norepinephrine, 185 Osmotic flow Normal distribution, 568 difference from diffusion, 111 Normoneva, D. A., 29 Osmotic fragility, 117 Northern Hemisphere, 217 Osmotic pressure Nuclear force, 483 and capillaries, 115 Nuclear scattering, 421 and chemical potential, 114 Nunally, R. L., 534 and driving pressure, 114 Nunez, P. L., 196, 202 and interstitial fluid, 115 Nutation, 521, 528 in ideal gas, 114 Nyenhuis, J. A., 534, 542 in liquid, 114 Nyquist frequency, 310 Ossicles, 348 Nyquist sampling criterion, 290 Osteoarthritis, 130, 249 Nyquist, H., 314, 323 Othmer, J. G., 105, 109 Oudit, G. Y., 185, 202 O’Brien, D., 41, 47 Oval window, 348, 349 O’Meara, N. M., 313 Øverbø, I., 404, 434 606 Index

Oxic-anoxic transition zone, 217 magnetic, 216 15O, 502 membrane, 120 Oxygen consumption, 256 Perotti, C., 398 Ozone, 377 Perrin, J. E., 85 Pertsov, A. M., 275, 284 P wave, 186, 194 PET, 502, 536 31P, 534 Peters, R. H., 41, 47 Pacemaker, 192, 194, 267 Petrie, R. J., 213, 225 Pacing electrode, 194 Pfuetzner, R. A., 253 Page, C. H., 214, 224 Phagocytosis, 497 Paine, P. L., 87, 109 Phantom, 494 Pair production, 403, 407 Phase, 289 free electron, 407 resetting, 264–267 Pajevic, S., 542 singularity, 283 Paliwal, B., 478 space, 263 Palotta, B. S., 240 Phase encoding, 528, 530 Panfilov, A. V., 275, 284 Phase transfer function, 328 Paramagnetism, 215, 216 Phased array, 353 Paramecium, 1 Phelps, M. E., 501, 502, 511, 513 Parasympathetic nerves, 185 Phenobarbital, 44 Parathyroid hormone, 279 Phenothiazines, 379 Parisi, A. V., 380, 398 Phlebotomy, 119 Parkinson, J. S., 96, 109 Phosphate, 278 Parseval’s theorem, 305 Phosphorescence, 364 Partial derivative, 57 Photochemotherapy, 381 Partial derivatives Photoelectric effect, 403, 404 order of differentiation, 582 Photographic emulsion, 424 Particle Data Group, 486 Photometry, 383 Particle exchange Photomultiplier tube, 500 equilibrium, 62 Photon, 360 Particles per transition, 489 attenuation coefficient, 409 Pascal, 12 characteristic, 410 Pascual-Leone, A., 225 diffusion constant, 367 Patch-clamp recording, 238 exitance, 384 Patterson, J. C., 502, 513 fluence, 384 Patterson, M. S., 368, 369, 398, 399 fluence rate, 384 Patton, H. D., 20, 29, 116, 122, 133, 135, 137, 148, 175, fluorescence, 410 257, 284 flux, 384 Paul, C. F., 381, 399 flux exposure, 384 Pauli exclusion principle, 362, 421 flux intensity, 384 Pauli promotion, 421 flux irradiance, 384 Payne, J. T., 337 flux radiance, 384 Pearson, W. L., 254 secondary, 413, 429 Pecora, R., 372, 397 Photon absorption, 365 Pedley, T. J., 29 Photon scattering, 365 Pell, G. S., 542 Photons, 244 Penetration depth Photopheresis, 381 thermal, 383 Photopic vision, 387 Penkoske, P. A., 284 Photosynthesis, 67, 294 Perfumes, 380 in aquatic plants, 394 Perfusion, 382 Physiological recruitment, 200 Period doubling, 269 Pi pulse, 521 in heart cells, 276 Pi/2 pulse, 524 Periodogram, 309 Pickard, W. F., 244, 254 Perkins, D. H., 424, 425, 435 Pierpaoli, C., 542 Permeability Piezoelectric transducer, 350 hydraulic, 117 Pigeon, 217, 222 Index 607

Pilkington, T. C., 202 Postulates of statistical mechanics, 55 Pillon, M., 398 Potassium channels Pillsbury, D. M., 378 shaker, 241 Pinna, 348 slow, 185 Pirenne, M. H., 390, 391, 398, 399 Potassium conductance, 156 Pires, M. A., 224 Potassium gate, 157 Pituitary, 256, 277 Potassium Nernst potential, 155 Planck’s constant, 244, 360 Potential Plane wave, 344, 386 electric, 141 Plaque, 376 outside axon, 177 vulnerable, 376 transmembrane, 196 Plasma, 37 Potential Alpha Energy Concentration (PEAC) (unit), Plasmon, 418 505 Plasticity, 196 Potential difference, 142 Platt, J., 478 and ion concentration, 228 Platzman, R. L., 440, 479 Potential energy, 141 Plonsey, R., 20, 153, 154, 160, 175, 177, 182–184, 191, Potish, R. A., 479 193, 199, 201, 202, 209, 284 Potts, T. J., 398 Plug flow, 535 Powell, C. F., 424, 425, 435 Pneumothorax, 448 Power, 10, 11, 146 Pogue, B. W., 369, 399 Power law, 39 Poincaré, J. H., 263 Power spectral density (PSD), 309 Point charge, 139 Blackman-Tukey method, 309 Point-spread function, 327, 390 magnetic noise, 314 for an ideal imaging system, 327 periodogram, 309 Poise (unit of viscosity), 15 Power spectrum, 297 Poiseuille flow, 17, 20, 123 Power, average, 304 departures from, 22 Pratt, R. H., 407, 435 Poiseuille, J. L. M., 17 Prausnitz, J. M., 67, 79, 87, 108 Poisson distribution, 366, 391, 571 Precession, 518 and shot noise, 242 Precordial leads, 188 Poisson equation, 230 Predator-prey problem, 46 Poisson statistics Press, W. H., 105, 109, 182, 201, 202, 208, 209, 225, in cancer incidence, 471 285, 289, 295, 308, 309, 323, 339, 341 in cell survival, 463 Pressure, 13 in image, 451 diastolic, 20 in photon fluence, 457 systolic, 20 in radiation damage, 460 Preston, W., 479 Poisson’s ratio, 27 Prevalence of disease, 44 Poisson-Boltzmann equation, 230 Price, R. E., 434 analytic solution, 250 Price, R. R., 536, 542 linearized, 230 Prickle layer, 378 Polacin, A., 479 Principal quantum number, 402 Polarization, 372 Probability, 50 electric, 234 binomial, 51 of dielectric, 143 Product Polarized light, 371 cross, 5 Polk, C., 244, 247, 248, 254 dot, 11 Polonsky, K. S., 313 scalar, 11 Polynomial vector, 5 regression, 288 Projection, 501 Pontvert, D., 478 Projection reconstruction, 528, 530 Poon, T. S. C., 380, 399 Projection theorem, 331 Positron, 403, 407, 414 Projections, 331 decay, 485 Prokaryotes, 2 Positron emission tomgraphy, 502 Proportional counter, 444 Postow, E., 254 Protein, 2 608 Index

Proton therapy, 466 risk, 468 Protoplast, 2 weighting factor, 468 Protozoans, 1 Radiation chemical yield, 439 PRU (peripheral resistance unit), 20 Radiation damage Pryde, J. A., 87, 109 type A, 460, 462 Pryor, T. A., 323 type B, 460, 462 Psoralen, 381 Radiation equilibrium, 428 Psoriasis, 381 Radiation risk model Pterygium, 381 hormesis, 472 Puliafito, C. A., 370, 398 linear non-threshold, 471 Pulmonary embolus, 500 threshold, 472 Pulse echo techniques, 352 Radiation therapy, 464 Pulse oximeter, 369 conformal, 465 Punske, B. B., 211, 225 electron, 465 Pupil, 388 intensity-modulated, 465 Pupil size, 274 Radiation yield, 424, 437 Purcell, C. J., 211, 225 Radiative transition, 410 Purcell, E. M., 22, 29, 96, 107–109, 203, 207, 215, 225, Radiobiology, 457 519 Radiochromic film, 445 Puri, S., 435 Radiograph, 446 Purkinje fibers, 185, 268 Radiographic signal, 451 PUVA, 381 Radioimmunoassay, 481 Pyramidal cells, 221 Radioimmunotherapy, 504 Radiometry, 383 Q10, 74, 157, 159 Radiopharmaceuticals, 495 QRS wave, 186, 194 Radiotherapy Qu, Z., 283 internal, 504 Quadrupole 226Ra, 505 electric, 484 Radius magnetic, 516 atomic, 482 Quality factor, 468 nuclear, 482 Quantum mottle, 452 Radon, 469, 470, 472 Quantum number, 53 222Rn, 505 Quantum numbers Radon transformation, 333 atomic electrons, 361 Railroad tracks, 42 rotational, 363 Raizner, A. E., 504, 513 vibrational, 364 Ramachandran, G. N., 335, 341 Quasiperiodicity, 271 Raman scattering, 371 Quatrefoil pattern, 211 Raman spectroscopy, 377 Québec, 473 Ramirez, R. J., 202 Quinine water, 379 Random walk, 100 Range, 414, 423 RADAR (Radiation Group Assessment Resource), Ranvier 489, 495 node of, 136 Radial isochron clock, 265, 266, 281 Rao, D. V., 513 Radian, 543 Rao, P. Y., 434 Radiance, 384, 386 Rappoport, S. I., 117 Radiant Rardon, D. P., 194, 195, 202 energy, 383, 384 Rasmusson, R. L., 201 exposure, 384 Ratchet and pawl, 318 flux, 384 Rattay, F., 200, 202 intensity, 383, 384 Rayleigh scattering, 407 power, 383, 384 RBE, 468 Radiation Reactance, 346 biological effects, 457 Reaction-diffusion process, 105 electromagnetic, 244 Receptor natural background, 470 mechanical, 136 Index 609

stretch, 135 parallel and series, 147 temperature, 135 vascular, 20 Receptor field, 390 Resnick, R., 7, 29, 142, 175, 216, 218, 224, 363, 364, Reckwerdt, P., 465, 478 373, 375, 398, 483, 486, 487, 513 Reconstruction from projections Resolution filtered back projections, 332 and spatial frequency, 329 Fourier transform, 331 Respiration of glucose, 67 Recruitment Response physiological, 200 impulse, 326 Rectification Rest energy, 482 anomalous, 251 Rest mass, 405 Rectifier channels, 236 Restenosis, 504 Red blood cell, 3, 24 Resting potential, 136 Red cells, 117 atrial and ventricular cells, 184 Reddy, A. K. N., 234, 253 Restitution, 282 Reddy, R., 254 Restricted linear collision stopping power, 423 Reduced mass, 363 Retina, 381, 388 Reduced scattering coefficient, 367 Reversal potential, 237, 238, 251 Reentrant circuit, 186, 195, 198 Reverse osmosis, 119 Reference action spectrum, 379 Reversible process, 65 Reflection, 388 Reynolds number, 22, 29, 45 total internal, 396 Rheobase, 171, 193 Reflection at a boundary, 347 Ribberfors, R., 434 Reflection coefficient, 118, 120, 347 Richards, W. O., 213, 225 pore model, 127 Rieke, F., 392, 399 Refraction, 388 Riggs, D. S., 39, 47, 256, 278, 279, 284 Refractory period, 158, 173, 186 Right ventricular hypertrophy, 189 Regression Rinaldo, A., 47 linear, 287 Risk polynomial, 288 excess, 470 Rehm, K., 503, 504, 513 models, 471 Reich, P., 46, 47 relative, 245 Reif, F., 50, 53, 55, 64, 79, 86, 109, 563 Ritchie, R. H., 435 Relative biological effectiveness, 468 Ritenour, E. R., 343, 352, 355, 357, 446, 447, 450, 454– Relative risk, 245 456, 478 Relativity Rivabella, L., 398 special, 482 Robinson, D. K., 289, 323 Relaxation oscillator, 185 Rodriguez, J., 332, 341 Relaxation time Rods (retinal), 387, 390 experimental, 524 Roentgen (unit), 440, 475 longitudinal, 518, 523 Roll-off, 312 non-recoverable, 524 Rollins, A. M., 371, 399 spin-lattice, see Relaxation time, longitudinal Romani, G.L., 225 spin-spin, see Relaxation time, transverse Romberg integration, 182, 209 thermal, 382 Romer, R. H., 42, 214, 225 transverse, 518, 523 Rose, A., 392, 399 rem (unit), 468 Rosenbaum, D. S., 196, 202 Remanent magnetic field, 216 Rossi, B., 406, 435 Renal arteries, 456 Rossi, C. J., 479 Renal dialysis, 116 Rossi-Fanelli, A., 76 Renal tubules, 122 Rossman, K., 442, 478 Repeller, 262 Rotating coordinate system, 519 Repolarization, 136, 185 Rotation matrices, 519, 538 Reservoir, 58, 65 Roth, B. J., 192, 195, 202, 208, 212, 214, 218, 221–226 Residence time, 490 Rottenberg, D. A., 513 Residuals, 286 Round window, 349 Resistance, 145 Rowlands, J. A., 446, 450, 454, 478, 479 610 Index

Roy, S.C., 435 Schneidermann, R., 133 Rudy, Y., 158, 175, 191, 202 Schrödinger equation, 49 Ruohonen, J., 214, 224 Schroter, S. T., 29 Rushton, W. A. H., 153, 160, 171, 175 Schueler, B. A., 468, 469, 479 Ruta, V., 254 Schultz, J. S., 128, 133 RVH, see Right ventricular hypertrophy Schulz, R. J., 463, 479 Ryman, J. C., 498, 513 Schuman, J. S., 370, 398 Schwan, H. P., 246, 253 SA node, 164, 185, 194 Schwartz, D., 72 Safety in MRI, 534 Schwartz, L., 478 Saint-Jalmes, H., 371, 399 Schwiegerling, J., 398 Sakane, K. K., 254 Scintillation camera, 500 Sakitt, B., 392, 399 Scintillation detector, 442, 500 Sakmann, B., 238, 253, 254 Scotopic vision, 387, 390, 454 Salles-Cunha, S. X., 542 Scott, A. C., 150, 154, 175 Saltatory conduction, 160 Scott, R. L., 67, 79, 87, 108, 114, 133 Sands, M., 323 Scoumis, E. A., 542 Sap flow, 27 Screen SAR, see Specific absorption rate intensifying, 441 Sarcoplasmic reticulum, 105 Screening Sarvas, J., 221, 225 atomic electron (charge), 403 Sastry, K. S. R., 500, 513 Scribner, B. H., 133 Sato, S., 224, 225 Second law of thermodynamics, 70 Saturation, 216 Sedimentation velocity, 26 Savage, V. M., 41, 47, 79 Seed, W. A., 29 Savannah sparrow, 217 Seibert, J. A., 446, 479 Scalar, 545 Selection rules, 362–364, 438, 484 Scalar product, 11 Self-diffusion, 90 Scaling, 40 Self-similarity, 270 Scattering Seltzer, S. M., 414, 434, 435, 448, 478 coherent, 403, 407 Semicircular canals, 349 Compton, 403, 405 Semiconductor detector, 445 incoherent, 403, 406 Semilog paper, 34 inelastic, 403 Semipermeable membrane, 111 nuclear, 421 Sepulveda, N. G., 195, 202 Raman, 403 Sequestration, 497 Rayleigh, 407 Serruys, P. W., 399 Thomson, 406 Setton, L. A., 29 Scattering coefficient Sevick, E. M., 369, 399 reduced, 367 Seymour, R. S., 41, 47 Scattering cross section, see Cross section Sha, Q., 254 Schaefer, B. E., 378, 399 Shabashon, L., 458, 479 Schaefer, D. J., 534, 542 Shadley, J. D., 458, 479 Schaper, K. A., 513 Shadowitz, A., 207, 225 Scher, A. M., 29, 133, 175, 278, 284 Shafer, K. E., 371, 398 Scherr, P., 87, 109 Shaker channels, 239 Schey, H. M., 84, 89, 97, 109, 138, 175, 577 Shani, G., 445, 479 Schlaer, S., 390, 391, 398 Shapiro, E. M., 250, 254 Schlesinger, T. E., 445, 479 Shapiro, L., 96, 109 Schlienger, P., 478 Shark Schmidt, T., 479 and earth’s magnetic field, 220 Schmidt-Nielsen, K., 46, 47, 122, 131, 133 Shaw, R., 329, 341, 452, 478 Schmitt, F., 533, 542 Shear modulus, 13 Schmitt, J. M., 370, 399 Shear rate, 15 Schmitt, O. H., 245, 254 Shellock, F. G., 534, 542 Schneider, J. A., 202 Shells Schneider, U., 479 atomic, 402 Index 611

Sherrard, D. J., 133 Soleus, 6 Sherwood number, 107 Solid angle, 181, 366, 544 Sherwood, B. A., 364, 399 defined, 543 Shimizu, H., 542 theorem, 198 Shot noise, 242, 312 Solute, 67, 69 Shrier, A., 276, 283 Solute permeability, 120 Sick sinus syndrome, 194, 200 pore model, 128 Sicouri, S., 201 Solute transport, 119, 122 Sidtis, J. J., 513 pore model, 127 Siegel, J. A., 505, 513 Solution Siemens, 145 ideal, 114 Sievert (unit), 468 Solvent, 67 Signal, 75, 306 Solvent drag, 85, 119 and noise, 308 in an electric field, 235 radiographic, 451 Sorenson, J. A., 501, 502, 511, 513 Signal averaging, 308 Sorgen, P. L., 201 Signal-to-noise ratio, 452 Sound level measurement Sigworth, F. J., 239, 241, 243, 253, 254 weighting, 347 Silicon, 445 Source, 262 Silver halide, 441 Space invariance, 327 Silver, H. K., 41, 47 Spano, M. L., 276, 283, 284 Silver, S., 175 Sparks, R. B., 488, 489, 494, 513 Silverstein, M. E., 119, 133 Spatial frequency, 326, 329 Sinc function, 528 and field of view, 329 Singh, M., 341, 455, 478, 527, 542 and resolution, 329 Singh, N., 435 Special relativity, 203, 220, 222, 405, 482 Singh, S., 435 Specific absorbed fraction, 489 Single-channel recording, 238 Specific absorption rate, 534, 539 Single-photon emission computed tomography, 501 Specific heat Sink, 262 water, 75 Sinoatrial node, see see SA node Specific heat capacity, 61, 382 Sinogram, 333 Specific metabolic rate, 46 Sinus exit block, 200 SPECT, 501 Skofronick, J. G., 348, 349 Spector, R., 47 Slater, J. D., 467, 479 Spectral efficiency function, 387 Slater, J. M., 479 Spectroscopy Slice selection, 527, 528 infrared, 368 Slichter, C. P., 518, 523, 542 Spectrum Slide projector, 396 beta decay, 487 Small intestine, 213 Wiener, 452 Smith, M. A., 398 Spherical aberration, 389 Smith, W., 323 Spherical coordinates, 577 Smye, S. W., 372, 399 Spherocytosis, 117 Smythe, W. R., 166, 175 Spheroid degeneration of the eye, 381 Snell’s law, 388 Sphincter, pre-capillary, 20 Snow blindness, 381 Spider’s thread, 25 Snyder, W. S., 489, 494–496, 513 Spin Sobol, W. T., 446, 479 electron, 215, 361 Söderberg, P. G., 381, 397 spin Sodium nuclear, 517 -potassium pump, 148 Spin echo (SE), 526, 528 Nernst potential, 155 fast or turbo, 532 24Na, 487 Spin quantum number, 402 Sodium gate, 157 Spin warp encoding, 528 Sodium spectral line, 372 Spiral CT, 455 Soffer, B. H., 387, 399 Spiral wave, 186, 275, 283 Solar constant, 377, 395 Spontaneous births, 294 612 Index

Spontaneous emission, 522 per atomic electron, 420 Spray, D. C., 201 Stopping power, 414 Squamous cell carcinoma (SCC), 380, 381 electronic, 416 SQUID, 218, 223 for compounds, 422 Squid, 2 for electrons and positrons, 422 giant axon, 154 mass, 414 SRIM, 415, 422, 423 nuclear, 416 Srinivas, S. M., 397 radiative, 416 Srinivasan, R., 196, 202 Storment, C. W., 136, 175 Stabin, M. G., 488, 489, 494, 505, 513 Straggling, 423 Stahlhofen, W., 216, 225 Straight-line ﬁt, 285, 287 Standard deviation, 562 Strain, 344 Standing wave, 345 normal, 12 Stanley, P. C., 191, 202 shear, 13 Stapes, 348 Stratum corneum, 378 Stark, L., 256, 275, 284 Stratum granulosum, 378 Stark, L. C., 390, 399 Strauss, H. C., 201 State space, 263 Streamline, 15, 18 Stationarity, 327 Strength-duration curve, 171, 193 Stationary random process, 307 Strength-interval curve, 173 Stationary system, 326 Stress, 12, 344 Statistical mechanics, 50 normal, 12 postulates, 55 shear, 13 Staton, D. J., 211, 212, 225 Strogatz, S. H., 262–265, 271, 284 Steady state and equilibrium Stroink, G., 211, 214, 216, 225 distinction, 55 Stroke volume, 21 Steel, G. G., 458, 459, 479 90Sr, 504 Stefan-Boltzmann law, 374 Strother, S. C., 503, 513 Stegun, I. A., 199, 201, 250, 253, 338, 340 Strouse, P., 478 Stehling, M. K., 533, 542 Structure mottle, 452 Steiner, R., 29 Strupp, J., 542 Steiner, R. F., 133, 175, 284 Stuchly, M. A., 214, 224 Steinmuller, D. R., 130, 133 Studniberg, H. M., 381, 399 Steketee, J., 375, 399 Sturis, J., 313 Stenosis, 376 Subexcitation, 440 Stent, 504 Subshells drug-eluting, 504 atomic, 402 Steradian, 544 Suess, C., 477, 479 Stereocilia, 243 Suit, H., 467, 479 Steric factor, 120, 127 Sulﬁdes, 217 Stern, R. S., 381, 399 Sulfonamides, 379 Steroid-eluting electrode, 195 Sulfonureas, 379 Stewart, W. E., 90 Sun protection factor (SPF), 380 Stimulated emission, 522 Superconducting quantum interference device, 218 Stimulus Superconductor, 218 brain, 196 Superposition, 325 current density, 165 Surface tension, 65, 77 strength-duration curve, 171 Surface-to-volume ratio, 191 strength-interval curve, 173 Surfactant, 77 Stinson, W. G., 398 Surrogate data, 316, 317 Stirling’s approximation, 567 Surroundings, 52 Stochastic resonance, 317 Survival curve, 458 Stocker, H., 181, 184, 202 Susceptibility Stokes’ law, 87, 90, 248 electric, 144 Stopping cross section, 415 Sutoh, Y., 399 Stopping interaction strength, 420 Svedberg, 26 Stopping number Swanson, E. A., 370, 398 Index 613

Sweating, 278 celsius, 58 Sweeney, J. D., 202 negative, 73 Swift, C. D., 434 Temperature regulation, 278 Swihart, C. J., 236, 254 “10-20” system for EEG leads, 196 Swinney, K. R., 208, 209, 225 Tendon Swithenby, S. J., 212, 224, 225 Achilles, 6 Sympathetic nerves, 185 Tenforde, T. S., 244, 245, 254 Synapse, 104, 135, 136 Tensile strength, 12 Syncope, 194 Terahertz radiation, 361, 372 Syncytium, 184 Tesla, 204 Synolakis, C. E., 18, 29 Tetley, I. J., 103 Synthesis, 459 Tetrodotoxin, 238 System, 52 Teukolsky, S. A., 109, 202, 225, 323, 341 linear, 325 201Tl, 498, 500, 502 stationary, 326 Theodoris, G. C., 390, 399 System dynamics, 265 Thermal conductivity, 103, 382 Systole, 18, 20 Thermal equilibrium, 56, 57 Szabo, A., 96, 109 Thermal noise Szabo, Z., 500, 513 retinal, 392 Thermal penetration depth, 383 T rays, 372 Thermal relaxation time, 382 T wave, 186 Thermodynamic identity, 65 T1-weighted image, 533 Thermodynamic relationship, general, 65 T2-weighted image, 533 Thermodynamics Tabatabaei, S., 542 first law, 53 Taccardi, B., 211, 225 second law, 70 Tachycardia Thermography, 376 ventricular, 186 Thermoluminescent dosimeter, 445 Taffet, S. M., 201 Thiazide diuretics, 379 Tai, C., 200, 202 Thin-film transistor, 450 Takano, M., 398 Thin-lens equation, 388 Talajic, M. T., 283 Thomas, D. L., 537, 542 Talus, 6, 7 Thomas, G. B., 557 Tan, G. A., 211, 225 Thomas, I. M., 213, 225 Tanaka, A., 479 Thomas, L., 120, 133 Tanelian, D. L., 136, 175 Thomas, S. R., 534 Tang, Y-h., 105, 109 Thompson, D. J., 266, 283 Tank, D. W., 542 Thompson, J. H., 565 Target entity, 365 Thompson, W. J., 288, 323 Taylor’s series, 551 Thomson scattering, 406 of exponential function, 553 Thomson, R. B., 565 Taylor, A. E., 115, 133 232Th, 512 Tearney, G. J., 370, 397, 399 Three-dimensional conformal radiation therapy, 465 99mTc, 33, 484, 492, 497, 498, 502 Threshold detection, 317 albumin, 498 Threshold stimulus strength, 173 diphosphonate, 501, 503 Thyroid, 256 generator, 497 stimulating hormone (TSH), 256, 277 pertechnetate, 497 Thyroid hormone, 278 pyrophosphate, 500 Thyroxine (T4), 256, 277 red blood cells, 501 Tri-iodothyronine (T3), 256 sestamibi, 498 Tibia, 6 sulfur colloid, 490, 500, 510 Tissue weighting factor, 469 tetrofosmin, 498, 503 Tittel, F. K., 398 Tectorial membrane, 349 TMS, 214 Telegrapher’s equation, 152 Tobacco, photosynthesis, 294 Temperature Tomanaga, M., 479 absolute, 57 Tomography 614 Index

computed, 455 Turnbull, G., 225 optical coherence, 370 Turner, J. S., 284 Tomotherapy, 465 Turner, R., 533, 542 Tondury, P., 224 Tympanic chamber, 349 Tooby, P. F., 26, 29 Torque, 4, 516 Uderzo, C., 398 on rotating sphere, 248 Uehara, M., 227, 254 Torr, 20 Ugurbil, K., 534–536, 542 Total angular momentum quantum number, 402 Ultracentrifuge, 75 Total internal reflection, 396 Ultrafiltration, 119 Total linear attenuation coefficient, see Attenuation Ultrasound coefficient diagnostic, 350 Trace of a matrix, 104 intravascular, 376 Transcranial magnetic stimulation, 214 Ultraviolet spectrum, 377 Transducer, 135 Ungar, I. J., 200, 202 piezoelectric, 350 Units of “power” and “energy”, 309 Transfer factor, 442 Upton, A. C., 471, 480 Transfer function, 311 235U, 497, 512 modulation, 328 238U, 505 optical, 328 Urea, 116 phase, 328 Urie, M., 467, 479 Transient charged-particle equilibrium, 429 Urruchi, W. I., 254 Transition UVA, 377, 380, 381 Coster-Kronig, 411 UVB, 377, 380, 381 nonradiation, 410 UVC, 377, 381 radiationless, 410 radiative, 410 van de Kaa, C. H., 542 Transmission at a boundary, 347 van der Gon, J. J. D., 202 Transmission coefficient, 347, 355 van der Pol oscillator, 316 Transmittance, 441 van Eijk, C. W. E., 444, 480 Transport van Ginnekin, A. C. G., 175 countercurrent, 121 van Hulsteyn, D. B., 214, 224 solute, 119, 122 Van Langenhove, G., 399 volume, 122 van Leeuwen, P., 225 Transvenous pacing, 194 van’t Hoff’s law, 115, 118 Transverse relaxation time Vapor pressure, 75, 78 recoverable and non-recoverable, 524 Variable rate of growth or decay, 35 Transverse wave, 346 Variables Traveling wave, 345 extensive, 64 Trayanova, N., 184, 191, 202 electric charge, 64 Tree, 27 Length, 64 Trehan, P. N., 435 Surface area, 64 Trigonometric integrals, 555 volume, 64 Triplet production, 407 Variance, 562 Tripp, J. H., 224 Vascular resistance, 20 Trochanter, greater, 7 Vasodilation, 116 Trontelj, Z., 212, 225 Vector, 3, 545 Trowbridge, E. A., 23, 29 addition, 545 TTX, 238 components, 546 Tucker, R. D., 245, 254 unit, 546 Tukey, J. W., 295, 323 Vector operators Tumor eradication, 463 in different coordinate systems, 578 Tuna, 217, 222 Vector product, 5 Tung, C. J., 423, 435 Veigele, W. J., 434 Tung, L., 196, 202 Veldhuyzen van Zanten, S. J. O., 225 Turbidity, 367 Velocity Turbulence, 22 average, 546 Index 615

instantaneous, 547 Warmuth, I., 398 root-mean-square, 85 Warner, G. G., 495, 496, 513 Velocity gradient, 15 Warwick, W., 566 Vena cava, 20 Washout (in MRI), 535 Ventilation rate, 256 Watanabe, A., 182, 202, 209 Ventricular fibrillation, 195, 275, 276, 282 Water Ventricular tachycardia, 186, 195, 275 acoustic impedance, 346 Venules, 20 compressibility, 345 Vergence, 389 density, 345 Verheye, S., 376, 399 dielectric constant, 144 Vestibular chamber, 349 speed of sound, 345 Vetterling, W. T., 109, 202, 225, 323, 341 transmission vs. wavelength, 387 Villars, F. M. H., 10, 29, 90, 91, 100, 108 Water molecule Virtual cathode, 194 dipole moment, 233 “dog-bone”, 196 Watson, E. E., 488, 490, 491, 495, 498, 505, 513 Virtual image, 396 Watson, S. B., 513 Virus, 2 Watt, 11 Viscoelasticity, 13 Wave Viscosity, 13, 15, 87 longitudinal, 344 of water, 90 plane, 344 Viscous torque on a rotating sphere, 248 standing, 345 Vision traveling, 345 scotopic, 454 Wave equation, 173, 343, 344 Visscher, P. B., 295, 323 Wave number, 345 Visual cortex, 534, 536 Weaver, J. C., 244, 247, 252–254, 317, 323 Visual evoked response, 308 Weaver, K. E., 480 Vitamin D, 381 Weaver, W., 50, 79 Vitreous, 388 Weaver, W. D., 72 Vlaardingerbroek, M. T., 527, 533, 542 Web site for this book, xvi Vogel, S. V., 15, 20, 29 Webb, S., 465, 480 Vollrath, F., 25, 29 Webb, W. W., 243, 253 Volt, 142 Weber, D. A., 489, 498, 513 Voltage difference, 141 Webster, J., 369, 398, 399 Voltage divider, 147 Webster, J. G., 170, 176 Voltage source Weighting factor ideal, 313 radiation, 468 Voltage-sensitive dye, 196 Weinberg, H., 225 Volume transport, 122 Weinstein, P. R., 542 pore model, 127 Weinstock, H., 225 through a membrane, 117 Weiss, G., 193 Von der Lage, F. C., 485, 487–489, 513 Weiss, J. N., 276, 283 Voorhees, C. R., 193, 202 Weiss-Fogh, T., 107, 109 Voorhees, W. D., 202 Weissleder, R., 542 Voroshilovsky, O., 283 Welder’s flash, 381 Weller, P., 381, 399 Wachtel, E., 133 Wendland, M. F., 542 Wagner, H. N., 443, 500, 513 Wessels, B. W., 504, 505, 512, 513 Wagner, J., 105, 109 West, G. B., 41, 42, 47, 79 Wagner, R. F., 451, 452, 478, 480 Westfall, R. J., 513 Walcott, C. J., 217, 225 Whinnery, J. R., 175 Walker, G. C., 398 White noise, 314 Walker, M. M., 217, 225 White, C. R., 41, 47 Walker, W. B., 477 White, R. J., 116, 133 Wang, D., 96, 109 White-blood-cell count, 275 Wang, L., 398 Wick, G. L., 26, 29 Wang, M. D., 108 Wieben, O., 369, 399 Warburg equation, 106 Wiener spectrum, 452 616 Index

Wiener theorem for random signals, 309 Xenon, 477 Wiesenfeld, K., 317, 323 133Xe, 502, 508 Wikelski, M., 217, 224 Xia, Y., 539, 542 Wikswo, J. P., 195, 202, 208, 209, 211, 212, 217, 218, Xylem, 27 221, 223–226 Wilders, R., 158, 165, 175 Yaar, M., 398 Wiley, J. D., 170, 176 Yamashita, Y., 369, 399 Williams, C. R., 475 Yasuno, Y., 371, 399 Williams, C. S., 328, 341 Yatagai, T., 399 Williams, M., 7, 29 Yazdanfar, S., 370, 371, 399 Williamson, S. J., 212, 226 Yield Wilson, B. C., 368, 399 fluorescence, 411 Winfree, A. T., 267, 268, 275, 276, 282, 284 Yodh, A., 367, 369, 398, 399 Winsor, R., 446, 480 Yonas, H., 479 Witkowski, F. X., 276, 284 Yonemoto, L. T., 479 Wolfe, R. S., 217, 224 Young’s modulus, 12, 344 Woodford, B. J., 254 Young, A. C., 278 Woods, C., 225 Young, A. R., 378, 398 Woods, M. C., 221 Young, T., 359 Woods, R. P., 513 Wooodruff, W. H., 47 Zacchello, G., 398 Woosley, J. K., 208, 223, 225, 226 Zalewski, E. F., 383, 387, 399 Work, 10, 54 Zanesco, L., 398 concentration, 63 Zanzonico, P. B., 500 pressure-volume, 19, 21 Zaremba, L. A., 542 Working level, 505 Zaret, B. L., 225 month, 505 Zecca, M., 398 Worthington, C. R., 164, 176 Zhang, X-C., 372, 399 Wortman, R. J., 398 Zhang, X.-L., 542 Zhang, Y., 398 X radiation, 360 Zhou, X.-H., 532, 533, 542 X-ray absorption edge, 404 Zhu, X.-H., 542 X-ray spectrum Ziegler, J. F., 415, 418–423, 435 bremsstrahlung, 437 Zinninger, M. D., 542 characteristic, 437, 438 Zinonev, N. N., 398 thick target, 439 Zipes, D. P., 194, 195, 201, 202, 212, 225, 283, 284 thin target, 438 Zorec, R., 225 X-ray tube, 446 Zumoff, B., 35, 36, 47 Zwanzig, R., 96, 109