Pol J Med Phys Eng 2007;13(1):1-12. PL ISSN 1425-4689 doi: 10.2478/v10013-007-0001-x website: http://www.pjmpe.waw.pl Grzegorz Domański, Bogumił Konarzewski, Zdzisław Pawłowski, Krzysztof Zaremba, Janusz Marzec, Artur Trybuła, Robert Kurjata A Simple Method of Determining the Effective Attenuation Coefficient Institute of Radioelectronics, Warsaw University of Technology, Nowowiejska 15/19, 00-665 Warsaw, Poland e-mail: [email protected] This paper presents a simple method of determining the effective attenuation coefficient from steady-state diffuse reflectance measurements. Key words: optical parameters, effective attenuation coefficient, reflectance measurements. Introduction Visible and near-infrared (NIR) light interacts with biological tissues by absorption and scattering [1, 2, 3, 4, 5]. It is important to know the optical properties of tissues in order to properly choose the measurement geometry in optical tomography [2, 3]. Also, optical properties themselves can potentially provide information about the tissue’s metabolic status or diagnose a disease [2, 3]. Optical properties of phantom tissues and biological tissues have been described by authors using various methods and techniques [6, 7]. However, it is still a problem of how to determine these optical properties quickly and relatively cheaply [6, 7]. The aim of this work was to develop a simple and cheap method for effective in vivo estimation of the attenuation coefficient. 2 Grzegorz Domański et al. Theory and Modelling In this section a simple model of radiation transport is described, which was used to derive an effective absorption coefficient from spatially resolved measurements of diffuse reflectance. A diffusion model was used to generate analytical expressions for the reflectance. For an infinitive medium the solution of the diffusion equation is given by [3]: I exp(-m r ) F()r =×0 eff (1) 4pD r F 2 where: is the photon fluence rate [W/cm ], I0 is the source intensity [W], D is the 1 diffusion coefficient [6] given by D = , m is an absorption coefficient [cm­1] mm+ ¢ a 3(as ) m ­1 mm¢ =- and s is the scattering coefficient [cm ], ss()1 g being the reduced scattering m coefficient, g is the anisotropy coefficient, and m = a is the effective attenuation eff D coefficient [7]. From equation (1) it can clearly be seen that in the diffusion theory for an m infinitive medium light attenuation in tissue depends only on eff rather than on the m absorption coefficient or the scattering coefficient alone. The knowledge of eff enables us to quickly evaluate light attenuation between a source and a detector. A quotient of two photon fluence rates for two different source-to-detector distances r and r0 is given by F r =--0 exp()m (rr ) F eff 0 0 r Its natural logarithm is æ F ö æ r ö lnç ÷ = lnç 0 ÷ --m (rr ) (2) ç F ÷ eff 0 è 0 ø è r ø For large source-to-detector distances, the above function behaves asymptotically m as a linear function with a slope ­ eff, which makes its determination fast and easy. The Effective Attenuation Coefficient… 3 æ F ö Let the logarithm of the measured quotient of light intensity be denoted by y = lnç ÷ ç F ÷ è 0 ø and the distances from the source to the detector at the measured points by ri, where i = 1, …, n. The values y, measured at points ri, are denoted by yi. Then æ F ö æ r ö fr(,)lnmm= ç ÷ = lnç 0 ÷ -- (rr ) (3) effç F ÷ eff 0 è 0 ø è r ø m In order to determine the value of eff, the sum of squares of the deviations of the m theoretical signal values y=f( eff,r)from the measured signals yi is minimized. Let the sum be written as n =-m 2 Syfrå()i (,)eff i (4) i =1 m Minimization of S in view of eff leads to dS = 0 (5) m d eff By computing the derivative we obtain n ¶Sf¶ 2 =-=å []()yf(,r)m ¶m ¶m i eff i effi =1 eff n é ¶f ù =-å 2()yf(,r)m - (,r)m = i eff i ê ¶m eff i ú i =1 ë eff û n =-(,r)m - 2()å()yfi eff ii rr0 i =1 After comparing it to zero, we obtain n æ æ r ö ö 2åç y - lnç 0 ÷ + m=(r - r )÷ (r - r ) 0 (6) ç i ç r ÷ eff ii00÷ i =1 è è i ø ø From the above we finally have n æ æ r öö n åç y - lnç 0 ÷÷(r -r )+m å (r -r )2 = 0 (7) ç i ç r ÷÷ i 00eff i i =1 è è i øø i=1 4 Grzegorz Domański et al. m The value of eff, which minimizes the cost function S, can be computed from the following formula n æ æ r öö ç - ç 0 ÷÷(r - r ) åç yi lnç ÷÷ i 0 = è r ø m =- i 1 è i ø eff n (8) (r - r ) 2 å i 0 i=1 m With the aim of finding the uncertainty of the estimated parameter eff, let us assume that the yi measurement is a random variable with the variance var(yi), and the m random variables yi are independent for different values of i. The variance var( eff) is then given by 2 n æ ¶m ö var(m ) = åç eff ÷ var(y ) (9) eff ç ¶ ÷ i i =1 è yi ø Its derivative is given by ¶m rr- eff =- i 0 (10) n ¶ 2 yi - ) å(rrk 0 k=1 m The variance var( eff)is n - ) 2 å(var()rri 0 yi m = i =1 var(eff ) 2 (11) æ n ö - ) 2 ç å(rri 0 ÷ è i =1 ø s m The standard deviation ( eff)is n - ) 2 å(var()rri 0 yi sm()var()== m i =1 (12) eff eff n - ) 2 å(rri 0 i =1 The Effective Attenuation Coefficient… 5 The variance var(yi) is expressed by the measured light intensity 2 é æ F öù é ¶ æ æ F ööù var(y )= var lnç i ÷ = ê ç lnç i ÷÷ú var(F ) = i ê ç FF÷ú ¶ ç ç F ÷÷ i ë è 00øû ë i è è øøû 2 æ F 1 ö var(F ) = ç 0 ÷ var(F ) = i ç FF÷ i F 2 è i 0 ø i If we assume that the main source of the uncertainty of the measured photon fluence rate F is random optical contact of the detector with the tissue, then the standard deviation of the measured value of F is proportional to F, with the coefficient of proportionality of a sa()FF= (13) The variance var(yi) is expressed by the variance of the measured photon fluence rate var(F ) var(y ) ==i a 2 (14) i 2 F i m The variance of the estimated effective attenuation coefficient var( eff)is n å(rr- ) 22a i 0 a 2 var(m ) = i =1 = (15) eff 2 n n 2 æ 2 ö - ) - ) å(ri r0 ç å(rri 0 ÷ i =1 è i =1 ø m The standard deviation of eff is then given by a sm( = eff ) (16) n - ) 2 å(rri 0 i =1 For a semi-infinitive medium the solution of the diffusion equation is given by é --+m 22 -m ++22 + ù I exp() (zz ) r exp( ()zz0 2 z r) F(,rz )= 0 ×-ê eff 0 eff b ú (17) 4pD ê -+22 ++22 + ú ë ()zz0 r ()zz0 2 zb r û 6 Grzegorz Domański et al. 1 1+ R where z = , z is the distance to the extrapolated boundary, zD=2 eff , 0 m¢ b b - s 1 Reff n =---21 + + + = tissue Rnneff1440.... rel 0 710 rel 0668 00636 n rel , nrel is the relative refraction nair index of the tissue compared to that of air. m For this model the estimated value of parameter x = eff is found by minimizing the cost function E(x) n =-2 Ex()å() f (,) rii x y (18) i =1 where æ F(rx,)ö fr(,)ln x= ç i ÷ (19) i ç F( ÷ è rx0 ,)ø dE The condition for the minimum is =0. dx Then, the value of the derivative is given by n ¶ dE =-f å 2()fr(,)ii x y (20) dx i =1 ¶x ¶f F(rx,) 1 æ ¶F(rx,) ¶F(r ,)x ö =-0 ç i F(rx,) 0 F(rx,)÷ ¶ F( F(2 ¶ 0 ¶ i x rxi ,) rx0 ,)è x x ø ¶f 1 æ ¶F(rx,) ¶F(rx,) ö =-ç i F(rx,) 0 F(rx,)÷ ¶ F( F( ¶ 0 ¶ i xrxrxi ,)0 ,)è x x ø ¶f ¶F(rx,) 11¶F(rx,) =-i 0 (21) ¶ ¶ F( ¶ F( x xrxi ,) xrx0 ,) The value of the second derivative is given by 2 dE2 n éæ ¶f ö ¶ 2 f ù = å 2êç ÷ +-()fr(,) x y ú (22) 2 ¶ ii¶ 2 dx i =1 ëêè x ø x ûú The Effective Attenuation Coefficient… 7 m The approximate value of x = eff obtained from (8) can be improved by iteration æ dE ö ç ÷ è dx ø xx¢ =- (23) æ 2 ö ç dE÷ ç 2 ÷ è dx ø Let replace function F from (17) by solving the diffusion equation for a semi-infinitive medium, written as - - I éexp()xR exp()xR2 ù F(,rx )=×1 ê 1 - ú (24) p 4 D ë R1 R2 û =-22 + =++22 + where Rzzr10(), Rzzzr20()2 b . If R10 = R1(r0), R1i = R(ri), R20 = R2(r0), R2i = R2(ri), function f can be written as é exp(-xR ) exp(-xR )ù ê 1i - 2i ú R R = ê 1i 2i ú fr(,)lni z (25) êexp(-xR ) exp(-xR )ú ê 10 - 20 ú ë R10 R20 û If exp(-xR ) exp(-xR ) j = 1i - 2i i R1i R2i exp(-xR ) exp(-xR ) j = 10 - 20 0 R10 R20 then æ j ö f = lnç i ÷ ç j ÷ è 0 ø 8 Grzegorz Domański et al.