Nonlinear effects of localized absorption perturbations on the light distribution in a turbid medium

Harry L. Graber; Raphael Aronson; Randall L. Barbour

doi:10.1364/JOSAA.15.000834

Journal of the Optical Society of America A
Vol. 15,
Issue 4,
pp. 834-848
(1998)
•https://doi.org/10.1364/JOSAA.15.000834

Nonlinear effects of localized absorption perturbations on the light distribution in a turbid medium

Harry L. Graber, Raphael Aronson, and Randall L. Barbour

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Harry L. Graber, Raphael Aronson, and Randall L. Barbour, "Nonlinear effects of localized absorption perturbations on the light distribution in a turbid medium," J. Opt. Soc. Am. A 15, 834-848 (1998)

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Abstract

A theoretical model of photon propagation in a scattering medium is presented, from which algebraic formulas for the detector-reading perturbations $(Δ R)$ produced by one or two localized perturbations in the macroscopic absorption cross section $(Δ μ_{a})$ are derived. Examination of these shows that when $Δ μ_{a}$ is titrated from very small to large magnitudes in one voxel, the curve traced by the corresponding $Δ R$ values is a rectangular hyperbola. Furthermore, while $Δ R^{\infty} \equiv \lim_{Δ μ_{a} \to \infty} Δ R$ is dependent on the location of the detector with respect to the source and the voxel, the ratio $Δ R / Δ R^{\infty}$ is independent of the detector location. We also find that when $Δ μ_{a}$ is varied in two voxels simultaneously, the quantity $Δ R (Δ μ_{a, 1} \land Δ μ_{a, 2})$ is a bilinear rational function of the $Δ μ_{a} s .$ These results apply not only in the case of steady-state illumination and detection but to time-harmonic measurements as well. The validity of the theoretical formulas is demonstrated by applying them to the results of selected numerical diffusion computations. Potential applications of the derived expressions to image-reconstruction problems are discussed.

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Coefficient Multiplying	Numerator	Denominator, Absolute Mutual Coupling	Denominator, Relative Mutual Coupling
1	0	1	0
$z$	0	$2 (T_{0})_{qq}$	$r_{0}^{T} t_{q} ϕ_{q}^{0}$
ν	0	$2 (T_{0})_{pp}$	$r_{0}^{T} t_{p} ϕ_{p}^{0}$
$z^{2}$	0	$(T_{0})_{qq}^{2}$	$r_{0}^{T} t_{q} (T_{0})_{qq} ϕ_{q}^{0}$
$ν^{2}$	0	$(T_{0})_{pp}^{2}$	$r_{0}^{T} t_{p} (T_{0})_{pp} ϕ_{p}^{0}$
$z ν$	$r_{0}^{T} [t_{q} (T_{0})_{qp} ϕ_{p}^{0} + t_{p} (T_{0})_{pq} ϕ_{q}^{0}]$	$4 (T_{0})_{qq} (T_{0})_{pp} - (T_{0})_{qp} (T_{0})_{pq}$	$2 r_{0}^{T} [t_{q} (T_{0})_{pp} ϕ_{q}^{0} + t_{p} (T_{0})_{qq} ϕ_{p}^{0}]$
$z^{2} ν$	$r_{0}^{T} [t_{q} (T_{0})_{qq} (T_{0})_{qp} ϕ_{p}^{0} + t_{p} (T_{0})_{pq} (T_{0})_{qq} ϕ_{q}^{0} - t_{q} (T_{0})_{qp} (T_{0})_{pq} ϕ_{q}^{0}]$	$(T_{0})_{qq} [2 (T_{0})_{qq} (T_{0})_{pp} - (T_{0})_{qp} (T_{0})_{pq}]$	$r_{0}^{T} [2 t_{q} (T_{0})_{qq} (T_{0})_{pp} ϕ_{q}^{0} + t_{p} (T_{0})_{qq}^{2} ϕ_{p}^{0} - t_{q} (T_{0})_{qp} (T_{0})_{pq} ϕ_{q}^{0}]$
$z ν^{2}$	$r_{0}^{T} [t_{p} (T_{0})_{pp} (T_{0})_{pq} ϕ_{q}^{0} + t_{q} (T_{0})_{qp} (T_{0})_{pp} ϕ_{p}^{0} - t_{p} (T_{0})_{pq} (T_{0})_{qp} ϕ_{p}^{0}]$	$(T_{0})_{pp} [2 (T_{0})_{qq} (T_{0})_{pp} - (T_{0})_{qp} (T_{0})_{pq}]$	$r_{0}^{T} [2 t_{p} (T_{0})_{qq} (T_{0})_{pp} ϕ_{p}^{0} + t_{q} (T_{0})_{pp}^{2} ϕ_{p}^{0} - t_{p} (T_{0})_{pq} (T_{0})_{qp} ϕ_{q}^{0}]$
$z^{2} ν^{2}$	$r_{0}^{T} {t_{q} (T_{0})_{qp} (T_{0})_{pp} [(T_{0})_{qq} ϕ_{p}^{0} - (T_{0})_{pq} ϕ_{q}^{0}] + t_{p} (T_{0})_{pq} (T_{0})_{qq} [(T_{0})_{pp} ϕ_{q}^{0} - (T_{0})_{qp} ϕ_{p}^{0}]}$	$(T_{0})_{qq} (T_{0})_{pp} [(T_{0})_{qq} (T_{0})_{pp} - (T_{0})_{qp} (T_{0})_{pq}]$	$r_{0}^{T} [t_{q} (T_{0})_{pp} ϕ_{q}^{0} + t_{p} (T_{0})_{qq} ϕ_{p}^{0}] \times [(T_{0})_{qq} (T_{0})_{pp} - (T_{0})_{qp} (T_{0})_{pq}]$

Detector Located over Cell	$K$	$Δ R^{\infty}$ from Curve Fit	$Δ R^{\infty}$ from Relaxation Computation
(26, 31, 1)	0.68552	$5.3684 \times 10^{- 5}$	$5.3685 \times 10^{- 5}$
(26, 31, 1)	$\pm 2 \times 10^{- 5}$	$\pm 9 \times 10^{- 10}$
(6, 31, 1)	0.68550	$1.2498 \times 10^{- 7}$	$1.2498 \times 10^{- 7}$
(6, 31, 1)	$\pm 1 \times 10^{- 5}$	$\pm 1 \times 10^{- 12}$	$1.2498 \times 10^{- 7}$
(31, 31, 11)	0.685540	$7.9602 \times 10^{- 5}$	$7.9602 \times 10^{- 5}$
(31, 31, 11)	$\pm 2 \times 10^{- 6}$	$\pm 1 \times 10^{- 10}$	$7.9602 \times 10^{- 5}$
(11, 31, 11)	0.685545	$5.2639 \times 10^{- 7}$	$5.2639 \times 10^{- 7}$
(11, 31, 11)	$\pm 1 \times 10^{- 6}$	$\pm 6 \times 10^{- 13}$

Perturbed Cell	λ	$K$
(31, 31, 1)	$0.99979 \pm 6 \times 10^{- 5}$	$0.79427 \pm 6 \times 10^{- 5}$
(31, 31, 5)	$0.99998 \pm 2 \times 10^{- 5}$	$0.68553 \pm 2 \times 10^{- 5}$
(31, 31, 7)	$1.00001 \pm 5 \times 10^{- 5}$	$0.68557 \pm 6 \times 10^{- 5}$
(31, 31, 6)	$1.00001 \pm 2 \times 10^{- 5}$	$0.68444 \pm 2 \times 10^{- 5}$
(31, 30, 6)	$0.99999 \pm 2 \times 10^{- 5}$	$0.68434 \pm 3 \times 10^{- 5}$

Coefficient	Detector $r$ , Coupled	Detector $r$ , Independent	Detector $t$ , Coupled	Detector $t$ , Independent
$a_{n}$	$1.484979 \times 10^{- 4}$	$1.484971 \times 10^{- 4}$	$1.161159 \times 10^{- 4}$	$1.161166 \times 10^{- 4}$
$a_{n}$	$\pm 7 \times 10^{- 10}$	$\pm 9 \times 10^{- 10}$	$\pm 5 \times 10^{- 10}$	$\pm 4 \times 10^{- 10}$
$b_{n}$	$3.70941 \times 10^{- 5}$	$3.70914 \times 10^{- 5}$	$= a_{n}$	$= a_{n}$
$b_{n}$	$\pm 6 \times 10^{- 10}$	$\pm 6 \times 10^{- 10}$	$= a_{n}$	$= a_{n}$
$c_{n}$	$2.40447 \times 10^{- 4}$	$2.70710 \times 10^{- 4}$	$2.75103 \times 10^{- 4}$	$3.38746 \times 10^{- 4}$
$c_{n}$	$\pm 7 \times 10^{- 9}$	$\pm 8 \times 10^{- 9}$	$\pm 4 \times 10^{- 9}$	$\pm 3 \times 10^{- 9}$
$a_{d}$	1.45868	1.45868	1.45871	1.45873
$a_{d}$	$\pm 1 \times 10^{- 5}$	$\pm 2 \times 10^{- 5}$	$\pm 1 \times 10^{- 5}$	$\pm 1 \times 10^{- 5}$
$b_{d}$	1.45870	1.45855	$= a_{d}$	$= a_{d}$
$b_{d}$	$\pm 4 \times 10^{- 5}$	$\pm 4 \times 10^{- 5}$	$= a_{d}$	$= a_{d}$
$c_{d}$	2.08686 $\pm 7 \times 10^{- 5}$	2.12772 $\pm 7 \times 10^{- 5}$	2.08681 $\pm 5 \times 10^{- 5}$	2.12767 $\pm 4 \times 10^{- 5}$

Coefficient Multiplying	Numerator	Denominator, Absolute Mutual Coupling	Denominator, Relative Mutual Coupling
1	0	1	0
$z$	0	$2 (T_{0})_{qq}$	$r_{0}^{T} t_{q} ϕ_{q}^{0}$
ν	0	$2 (T_{0})_{pp}$	$r_{0}^{T} t_{p} ϕ_{p}^{0}$
$z^{2}$	0	$(T_{0})_{qq}^{2}$	$r_{0}^{T} t_{q} (T_{0})_{qq} ϕ_{q}^{0}$
$ν^{2}$	0	$(T_{0})_{pp}^{2}$	$r_{0}^{T} t_{p} (T_{0})_{pp} ϕ_{p}^{0}$
$z ν$	$r_{0}^{T} [t_{q} (T_{0})_{qp} ϕ_{p}^{0} + t_{p} (T_{0})_{pq} ϕ_{q}^{0}]$	$4 (T_{0})_{qq} (T_{0})_{pp} - (T_{0})_{qp} (T_{0})_{pq}$	$2 r_{0}^{T} [t_{q} (T_{0})_{pp} ϕ_{q}^{0} + t_{p} (T_{0})_{qq} ϕ_{p}^{0}]$
$z^{2} ν$	$r_{0}^{T} [t_{q} (T_{0})_{qq} (T_{0})_{qp} ϕ_{p}^{0} + t_{p} (T_{0})_{pq} (T_{0})_{qq} ϕ_{q}^{0} - t_{q} (T_{0})_{qp} (T_{0})_{pq} ϕ_{q}^{0}]$	$(T_{0})_{qq} [2 (T_{0})_{qq} (T_{0})_{pp} - (T_{0})_{qp} (T_{0})_{pq}]$	$r_{0}^{T} [2 t_{q} (T_{0})_{qq} (T_{0})_{pp} ϕ_{q}^{0} + t_{p} (T_{0})_{qq}^{2} ϕ_{p}^{0} - t_{q} (T_{0})_{qp} (T_{0})_{pq} ϕ_{q}^{0}]$
$z ν^{2}$	$r_{0}^{T} [t_{p} (T_{0})_{pp} (T_{0})_{pq} ϕ_{q}^{0} + t_{q} (T_{0})_{qp} (T_{0})_{pp} ϕ_{p}^{0} - t_{p} (T_{0})_{pq} (T_{0})_{qp} ϕ_{p}^{0}]$	$(T_{0})_{pp} [2 (T_{0})_{qq} (T_{0})_{pp} - (T_{0})_{qp} (T_{0})_{pq}]$	$r_{0}^{T} [2 t_{p} (T_{0})_{qq} (T_{0})_{pp} ϕ_{p}^{0} + t_{q} (T_{0})_{pp}^{2} ϕ_{p}^{0} - t_{p} (T_{0})_{pq} (T_{0})_{qp} ϕ_{q}^{0}]$
$z^{2} ν^{2}$	$r_{0}^{T} {t_{q} (T_{0})_{qp} (T_{0})_{pp} [(T_{0})_{qq} ϕ_{p}^{0} - (T_{0})_{pq} ϕ_{q}^{0}] + t_{p} (T_{0})_{pq} (T_{0})_{qq} [(T_{0})_{pp} ϕ_{q}^{0} - (T_{0})_{qp} ϕ_{p}^{0}]}$	$(T_{0})_{qq} (T_{0})_{pp} [(T_{0})_{qq} (T_{0})_{pp} - (T_{0})_{qp} (T_{0})_{pq}]$	$r_{0}^{T} [t_{q} (T_{0})_{pp} ϕ_{q}^{0} + t_{p} (T_{0})_{qq} ϕ_{p}^{0}] \times [(T_{0})_{qq} (T_{0})_{pp} - (T_{0})_{qp} (T_{0})_{pq}]$

Abstract

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Tables (4)

Equations (64)

Journal of the Optical Society of America A