Method for estimating closed-form solutions of the light diffusion equation for turbid media of any boundary shape

Umar Alqasemi; Hassan S. Salehi; Quing Zhu

doi:10.1364/JOSAA.33.000205

Journal of the Optical Society of America A
Vol. 33,
Issue 2,
pp. 205-213
(2016)
•https://doi.org/10.1364/JOSAA.33.000205

Method for estimating closed-form solutions of the light diffusion equation for turbid media of any boundary shape

Umar Alqasemi, Hassan S. Salehi, and Quing Zhu

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Umar Alqasemi, Hassan S. Salehi, and Quing Zhu, "Method for estimating closed-form solutions of the light diffusion equation for turbid media of any boundary shape," J. Opt. Soc. Am. A 33, 205-213 (2016)

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Table of Contents Category
- Medical Optics and Biotechnology
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About this Article
History
- Original Manuscript: October 21, 2015
- Revised Manuscript: November 16, 2015
- Manuscript Accepted: November 17, 2015
- Published: January 18, 2016
Virtual Issues
Virtual Journal for Biomedical Optics Vol. 11, Iss. 4

Abstract

This paper reports a method of estimating an approximate closed-form solution to the light diffusion equation for any type of geometry involving Dirichlet’s boundary condition with known source location. It is based on estimating the optimum locations of multiple imaginary point sources to cancel the fluence at the extrapolated boundary by constrained optimization using a genetic algorithm. The mathematical derivation of the problem to approach the optimum solution for the direct-current type of diffuse optical systems is described in detail. Our method is first applied to slab geometry and compared with a truncated series solution. After that, it is applied to hemispherical geometry and compared with Monte Carlo simulation results. The method provides a fast and sufficiently accurate fluence distribution for optical reconstruction.

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GAOPTIMSET Parameter Name	Value
Generations	$1 \times 10^{4}$
Termination tolerance on fitness function value	$1 \times 10^{- 30}$
Termination tolerance on constraints	$1 \times 10^{- 20}$
Function used in selecting parents for next generation	@selectionroulette
Function used in mutating genes	@mutationadaptfeasible
Function used to do crossover	@crossoverheuristic

Slab Thickness (cm)		Truncated-Series Solution		GA-Optimized Solution
Slab Thickness (cm)		Coordinates	$J^{5}$	Coordinates	$J^{5}$
0.5	IS1	(0.00,0.00,0.39)	$6.1911 \times 10^{+ 03}$	(0.00,0.00,0.41)	$1.6400 \times 10^{- 06}$
0.5	IS2	(0.00,0.00,−0.89)	$6.1911 \times 10^{+ 03}$	(0.00,0.00,−1.20)	$1.6400 \times 10^{- 06}$
1.0	IS1	(0.00,0.00,0.39)	$8.4027 \times 10^{- 04}$	(0.00,0.00,0.39)	$1.7453 \times 10^{- 12}$
1.0	IS2	(0.00,0.00,−1.89)	$8.4027 \times 10^{- 04}$	(0.00,0.00,−2.39)	$1.7453 \times 10^{- 12}$
1.2	IS1	(0.00,0.00,0.39)	$1.3481 \times 10^{- 05}$	(0.00,0.00,0.39)	$1.7984 \times 10^{- 14}$
1.2	IS2	(0.00,0.00,−2.29)	$1.3481 \times 10^{- 05}$	(0.00,0.00,−2.84)	$1.7984 \times 10^{- 14}$
1.5	IS1	(0.00,0.00,0.39)	$6.3510 \times 10^{- 08}$	(0.00,0.00,0.39)	$4.4459 \times 10^{- 17}$
1.5	IS2	(0.00,0.00,−2.89)	$6.3510 \times 10^{- 08}$	(0.00,0.00,−3.51)	$4.4459 \times 10^{- 17}$
2.0	IS1	(0.00,0.00,0.39)	$2.9520 \times 10^{- 11}$	(0.00,0.00,0.39)	$6.5276 \times 10^{- 21}$
2.0	IS2	(0.00,0.00,−3.89)	$2.9520 \times 10^{- 11}$	(−0.01,−0.02,−4.6)	$6.5276 \times 10^{- 21}$

Number of IS	$J^{5}$	Time Latency (Approx.)
Number of IS	$J^{5}$	Minutes	Seconds
1	$4.1500 \times 10^{- 13}$	0	5
2	$2.4600 \times 10^{- 22}$	0	30
3	$7.0792 \times 10^{- 21}$	1	36
4	$6.6556 \times 10^{- 16}$	1	49
5	$4.4276 \times 10^{- 16}$	2	22
6	$3.3447 \times 10^{- 15}$	2	22
7	$6.3438 \times 10^{- 15}$	2	41
8	$2.3305 \times 10^{- 14}$	3	52
9	$9.0311 \times 10^{- 14}$	4	16
10	$6.9967 \times 10^{- 15}$	4	41

GAOPTIMSET Parameter Name	Value
Generations	$1 \times 10^{4}$
Termination tolerance on fitness function value	$1 \times 10^{- 30}$
Termination tolerance on constraints	$1 \times 10^{- 20}$
Function used in selecting parents for next generation	@selectionroulette
Function used in mutating genes	@mutationadaptfeasible
Function used to do crossover	@crossoverheuristic

Slab Thickness (cm)		Truncated-Series Solution		GA-Optimized Solution
Slab Thickness (cm)		Coordinates	$J^{5}$	Coordinates	$J^{5}$
0.5	IS1	(0.00,0.00,0.39)	$6.1911 \times 10^{+ 03}$	(0.00,0.00,0.41)	$1.6400 \times 10^{- 06}$
0.5	IS2	(0.00,0.00,−0.89)	$6.1911 \times 10^{+ 03}$	(0.00,0.00,−1.20)	$1.6400 \times 10^{- 06}$
1.0	IS1	(0.00,0.00,0.39)	$8.4027 \times 10^{- 04}$	(0.00,0.00,0.39)	$1.7453 \times 10^{- 12}$
1.0	IS2	(0.00,0.00,−1.89)	$8.4027 \times 10^{- 04}$	(0.00,0.00,−2.39)	$1.7453 \times 10^{- 12}$
1.2	IS1	(0.00,0.00,0.39)	$1.3481 \times 10^{- 05}$	(0.00,0.00,0.39)	$1.7984 \times 10^{- 14}$
1.2	IS2	(0.00,0.00,−2.29)	$1.3481 \times 10^{- 05}$	(0.00,0.00,−2.84)	$1.7984 \times 10^{- 14}$
1.5	IS1	(0.00,0.00,0.39)	$6.3510 \times 10^{- 08}$	(0.00,0.00,0.39)	$4.4459 \times 10^{- 17}$
1.5	IS2	(0.00,0.00,−2.89)	$6.3510 \times 10^{- 08}$	(0.00,0.00,−3.51)	$4.4459 \times 10^{- 17}$
2.0	IS1	(0.00,0.00,0.39)	$2.9520 \times 10^{- 11}$	(0.00,0.00,0.39)	$6.5276 \times 10^{- 21}$
2.0	IS2	(0.00,0.00,−3.89)	$2.9520 \times 10^{- 11}$	(−0.01,−0.02,−4.6)	$6.5276 \times 10^{- 21}$

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Journal of the Optical Society of America A