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]:
F( r ) =
I 0 exp( -m eff r )
×
r
4pD
(1)
where: F is the photon fluence rate [W/cm2], I0 is the source intensity [W], D is the
1
diffusion coefficient [6] given by D =
, ma is an absorption coefficient [cm­1]
3(m a + m ¢s )
and ms is the scattering coefficient [cm­1], m ¢s = (1 - g ) m s being the reduced scattering
coefficient, g is the anisotropy coefficient, and m eff =
ma
is the effective attenuation
D
coefficient [7]. From equation (1) it can clearly be seen that in the diffusion theory for an
infinitive medium light attenuation in tissue depends only on meff rather than on the
absorption coefficient or the scattering coefficient alone. The knowledge of meff 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
r
F
= 0 exp( -m eff ( r - r0 ))
F0 r
Its natural logarithm is
æ F
lnçç
èF0
ö
ær
÷÷ = lnç 0
è r
ø
ö - m (r - r )
÷
0
eff
ø
(2)
For large source-to-detector distances, the above function behaves asymptotically
as a linear function with a slope ­meff, which makes its determination fast and easy.
The Effective Attenuation Coefficient…
3
æ F
Let the logarithm of the measured quotient of light intensity be denoted by y = lnçç
è F0
ö
÷
÷
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
f (m eff , r ) = lnçç
èF0
ö
ær ö
÷÷ = lnç 0 ÷ - m eff ( r - r0 )
è r ø
ø
(3)
In order to determine the value of meff, the sum of squares of the deviations of the
theoretical signal values y = f(meff, r) from the measured signals yi is minimized. Let the
sum be written as
n
S = å ( y i - f (m eff , ri ))
2
(4)
i =1
Minimization of S in view of meff leads to
dS
=0
dm eff
(5)
By computing the derivative we obtain
n
¶S
¶f
=å
¶m eff i =1 ¶m eff
[( y
i
- f (m eff ,ri ))
2
]=
n
é ¶f
ù
= å 2( y i - f (m eff ,ri ))ê (m eff ,ri )ú =
i =1
ë ¶m eff
û
n
= 2å ( y i - f (m eff ,ri ))( ri - r0 )
i =1
After comparing it to zero, we obtain
n æ
ær
2å ç y i - lnçç 0
ç
i =1 è
è ri
ö
ö
÷÷ + m eff (ri - r0 ) ÷÷(ri - r0 ) = 0
ø
ø
(6)
From the above we finally have
n
æ
å çç y
i =1
è
i
ær
- lnçç 0
è ri
öö
÷÷ ÷÷ (ri - r0 ) + m eff
øø
n
å (r
i =1
i
- r0 ) 2 = 0
(7)
4
Grzegorz Domański et al.
The value of meff, which minimizes the cost function S, can be computed from the
following formula
n
m eff = -
æ
å çç y
i =1
è
i
öö
÷÷ ÷÷ (ri - r0 )
øø
ær
- lnçç 0
è ri
n
å (ri - r0 ) 2
(8)
i =1
With the aim of finding the uncertainty of the estimated parameter meff, let us
assume that the yi measurement is a random variable with the variance var(yi), and the
random variables yi are independent for different values of i. The variance var(meff) is then
given by
var(m eff
æ ¶m
) = å çç eff
i = 1 è ¶y i
n
2
ö
÷÷ var(y i )
ø
(9)
Its derivative is given by
¶m eff
=¶y i
ri - r0
n
å( r
k
(10)
- r0 )
2
k =1
The variance var(meff) is
n
å( r
i
var(m eff ) =
- r0 ) 2 var( y i )
i =1
æ n
2 ö
ç å( ri - r0 ) ÷
è i =1
ø
(11)
2
The standard deviation s(meff) is
n
å( r
i
s(m eff ) = var(m eff ) =
- r0 ) 2 var( y i )
i =1
n
å( r
i
i =1
(12)
- r0 )
2
The Effective Attenuation Coefficient…
5
The variance var(yi) is expressed by the measured light intensity
é æF
var( y i ) = var ê lnçç i
ë èF0
2
öù é ¶ æ æ F i ö öù
ç lnçç
÷÷ ÷÷ ú var(F i ) =
÷÷ ú = ê
ç
ø û ë ¶F i è è F 0 ø ø û
2
var(F i )
æF 1 ö
= çç 0
÷÷ var(F i ) =
F 2i
èFi F0 ø
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
s(F ) = aF
(13)
The variance var(yi) is expressed by the variance of the measured photon fluence rate
var( y i ) =
var(F i )
=a 2
2
Fi
(14)
The variance of the estimated effective attenuation coefficient var(meff) is
n
å( r
i
var (m eff ) =
- r0 ) 2 a 2
i =1
æ n
2 ö
ç å( ri - r0 ) ÷
è i =1
ø
2
=
a2
n
å( r
i
- r0 )
(15)
2
i =1
The standard deviation of meff is then given by
s(m eff ) =
a
n
å( r
i
(16)
- r0 )
2
i =1
For a semi-infinitive medium the solution of the diffusion equation is given by
2
2
2
2
é exp( -m
( z - z 0 ) + r ) exp( -m eff ( z + z 0 + 2 z b ) + r ) ù
eff
F( r , z ) =
×ê
ú (17)
2
2
2
2
4pD ê
(
z
z
)
+
r
(
z
+
z
+
2
z
)
+
r
úû
0
0
b
ë
I0
6
Grzegorz Domański et al.
where z 0 =
1 + R eff
1
, zb is the distance to the extrapolated boundary, z b = 2 D
,
m ¢s
1 - R eff
-2
-1
.
+ 0.710 nrel
+ 0.668 + 0.0636 nrel , nrel =
R eff = -1440
nrel
ntissue
is the relative refraction
nair
index of the tissue compared to that of air.
For this model the estimated value of parameter x = meff is found by minimizing the
cost function E(x)
n
E( x ) = å ( f ( ri , x ) - y i )
2
(18)
i =1
where
æ F( ri , x ) ö
f ( ri , x ) = lnçç
÷÷
è F( r0 , x ) ø
(19)
dE
= 0.
dx
Then, the value of the derivative is given by
The condition for the minimum is
¶f
dE n
= å 2 ( f ( ri , x ) - y i )
¶x
dx i =1
(20)
¶F( r0 , x )
¶f F( r0 , x )
1
æ ¶F( ri , x )
ö
=
F( r0 , x ) F( ri , x )÷
ç
2
¶x
¶x F( ri , x ) F ( r0 , x ) è
¶x
ø
¶F( r0 , x )
1
¶f
ö
æ ¶F( ri , x )
=
F( r0 , x ) F( ri , x )÷
ç
¶x F( ri , x )F( r0 , x ) è
¶x
¶x
ø
¶F( r0 , x )
1
1
¶f ¶F( ri , x )
=
F( ri , x )
F( r0 , x )
¶x
¶x
¶x
(21)
The value of the second derivative is given by
2
d 2 E n éæ ¶f ö
¶ 2f
(
,
)
=
2
+
f
r
x
y
å êç ÷ ( i
i)
dx 2 i =1 êëè ¶x ø
¶x 2
ù
ú
úû
(22)
The Effective Attenuation Coefficient…
7
The approximate value of x = meff obtained from (8) can be improved by iteration
æ dE ö
ç
÷
dx ø
è
x¢ = x æ d2Eö
çç 2 ÷÷
è dx ø
(23)
Let replace function F from (17) by solving the diffusion equation for a
semi-infinitive medium, written as
F( r , x ) =
é exp( - xR1 ) exp( - xR2 ) ù
×ê
ú
R1
R2
4pD ë
û
I1
(24)
where R1 = ( z - z 0 )2 + r 2 , R2 = ( z + z 0 + 2z b )2 + r 2 .
If R10 = R1(r0), R1i = R(ri), R20 = R2(r0), R2i = R2(ri),
function f can be written as
é exp( - xR1 i ) exp( - xR2 i ) ù
ê
ú
R1 i
R2 i
ú
f ( ri , z ) = ln ê
ê exp( - xR10 ) exp( - xR20 ) ú
ê
ú
R20
R10
ë
û
If
ji =
exp( - xR1 i ) exp( - xR2 i )
R1 i
R2 i
j0 =
exp( - xR10 ) exp( - xR20 )
R10
R20
then
æj
f = lnçç i
èj0
ö
÷÷
ø
(25)
8
Grzegorz Domański et al.
¶j i
¶j 0
j0 ji
¶j i 1 ¶j 0 1
¶f j 0 ¶x
¶x
=
=
2
¶x j i
¶x j 0
¶x j i
j0
(26)
¶j i
= - exp( - xR1 i ) + exp( - xR2 i )
¶x
(27)
¶j 0
= - exp( - xR10 ) + exp( - xR20 )
¶x
(28)
¶ 2 f é ¶ 2 j i 1 ¶j i
=ê
+
¶x
¶x 2 ë ¶x 2 j i
æ 1 ¶j i ö ù é ¶ 2 j 0 1 ¶j 0
+
÷÷ ú - ê
çç - 2
2
¶x
è j i ¶x ø û ë ¶ x j 0
2
¶ 2f
1 é¶ 2j i
1 æ ¶j i ö ù 1
=
ç
÷ úê
j i è ¶x ø úû j 0
¶x 2 j i êë ¶x 2
æ 1 ¶j 0 ö ù
÷÷ ú
çç - 2
è j 0 ¶x ø û
2
é¶ 2j 0
1 æ ¶j 0 ö ù
ç
÷ ú
ê
2
j 0 è ¶x ø úû
êë ¶x
(29)
¶ 2j i
= R1 i exp( - xR1 i ) - R2 i exp( - xR2 i )
¶x 2
¶ 2j i
= R10 exp( - xR10 ) - R20 exp( - xR20 )
¶x 2
The variance of x can be computed from
dE
n
¶ dx
var( x ) = å
2
i = 1 ¶y i d E
dx 2
dE
¶ dx
1
=
2
¶y i d E æ d 2 E ö 2
dx 2 ç dx 2 ÷
è
ø
2
s 2 ( yi )
é ¶ æ dE ö d 2 E
¶ æ d 2 E ö dE ù
ê ¶y ç dx ÷ dx 2 ¶y ç dx 2 ÷ dx ú
ø
ø
i è
ë i è
û
(30)
The Effective Attenuation Coefficient…
9
¶ dE
df
= -2
¶y i dx
dx
¶ d2E
d 2f
=
2
¶y i dx 2
dx 2
4a 2
¶f d 2 E dE d 2 f
var( x ) =
4 å
2
dx dx 2
æ d 2 E ö i =1 ¶x dx
ç 2 ÷
è dx ø
n
2
(31)
The above equations serve as a background theory for estimating the measuring error.
Materials and Methods
The surface light intensity distributions were measured in vivo for human forearms by
means of a one-channel microprocessor system. The system consists of a light detector
(silicon photodiode), a switched integrator and a microprocessor with a built-in 10-bit
analogue-to-digital converter. The tissue under examination was coupled to a light
detector by fiber optics. The device was controlled from a personal computer via a serial
interface RS232. The light source was a LED-emitting light of the wavelength of about
660 nm. The block schematic of the microprocessor system for the measurements of
optical parameters is shown in Figure 1.
Figure 1. A block schematic of the microprocessor system for measurements of optical
parameters.
10
Grzegorz Domański et al.
Figure 2. Fixing the source on the tissue.
Table 1. The results of meff coefficient in vivo determination for human forearms.
meff [cm
­1
Person
Forearm
A
Left
2.03 ± 0.19
A
Right
2.20 ± 0.19
B
Left
2.52 ± 0.19
B
Right
2.40 ± 0.19
C
Left
2.25 ± 0.19
C
Right
2.16 ± 0.19
D
Left
2.59 ± 0.18
D
Right
2.37 ± 0.19
E
Right
2.05 ± 0.19
]
The Effective Attenuation Coefficient…
11
Figure 2 shows how the source was placed on the tissue examined. Both the source
and the detector were placed on top of the forearm. The plastic cover served as a
positioning support.
Results
The optical parameter meff was measured in vivo on human forearms by means of our
method. The persons studied were 25-56 year-old males. The surface light intensity
distributions were measured by means of our system, and then the meff coefficient
estimations were computed using the theory described in (8). These values were used as
starting values for an iteration procedure given in (23). The final meff coefficient results
for human forearms in vivo are shown in Table 1.
The range of the estimated values of the effective attenuation coefficient was 2-3
cm­1. These results agree with those reported in literature [6, 7]. However, our values
are probably slightly overestimated due to the inaccuracies in the applied model of light
transport in tissue, given by equation (17). Using a more accurate model of light
propagation, for example the Monte Carlo simulation, better values of the effective
attenuation coefficient could be obtained.
Discussion and Conclusions
Our method is simple, fast and inexpensive. It allows finding the effective attenuation
coefficient from light intensity measurements for at least two places on the tissue
examined. A measurement theory was developed and tested with real data. The
estimated values of the optical parameters can be used as starting values for an iteration
procedure in the Monte Carlo simulation procedure.
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Grzegorz Domański et al.
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Bellingham, 2002.
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