Article Contents

2024, Volume 18, Issue 5: 1052-1077. Doi: 10.3934/ipi.2024006

This issue Previous Article Next Article Quantitative analysis of the electric field in a two-layer mixed structure and its application

Estimating absorption and scattering in quantitative photoacoustic tomography with an adaptive Monte Carlo method for light transport

1.
Department of Technical Physics, University of Eastern Finland, Finland
2.
Department of Computer Science, University College London, United Kingdom

^*Corresponding author: Niko Hänninen
^*Corresponding author: Niko Hänninen

Received: September 2023

Revised: February 2024

Early access: February 2024

Published: October 2024

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Abstract

In this work, the optical inverse problem of quantitative photoacoustic tomography is studied. Maximum a posteriori estimates for absorption and scattering are computed from absorbed energy density, and the reliability of the estimates is evaluated utilizing the Laplace's approximation. The forward operator and evaluation of the search direction in the minimization algorithm are based on the Monte Carlo method for light transport. Monte Carlo is a stochastic method where the solution of a light transport model is approximated by simulating paths of photon packets as they undergo absorption and scattering events in a scattering medium. This makes evaluation of the search direction also stochastic. In this work, we study an adaptive approach where the number of simulated photon packets on each iteration is determined by utilizing a so-called norm test. In the norm test, the variance of the gradient of the objective function is assessed and used to control the number of photon packets. The approach is studied with numerical simulations. The results show that the adaptive approach can be used to control the number of photon packets during an iterative solution of the inverse problem of quantitative photoacoustic tomography. Further, the adaptive approach can reduce the computational cost compared to a conventional approach where the number of photon packets is fixed.

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Mathematics Subject Classification: Primary: 65Z05; Secondary: 35R30.

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Figure 1. Simulated absorption (left) and scattering (right) distributions in the highly scattering (top row) and the low scattering 2D target (bottom row). The dashed lines indicate the cross sections where the results and credibility intervals are visualized in Section 6.1

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Figure 2. MAP estimates of absorption coefficient $ \mu_{ \rm{a}} $ (first column), standard deviation of the absorption estimate $ \sigma_{ \rm{a}} $ (second column), MAP estimates of scattering coefficient $ \mu_{ \rm{s}} $ (third column) and standard deviation of the scattering estimate $ \sigma_{ \rm{s}} $ (fourth column). Results computed with the adaptive A-SGN approach (first row), with fixed number of photon packets with S-SGN$ _1 $ (second row), S-SGN$ _2 $ (third row), S-SGN$ _3 $ (fourth row) and with the adaptive A-SGN$ _{ \rm{GN}} $ (fifth row) approaches

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Figure 3. Absorption coefficient (first column) and scattering coefficient (second column) along the cross section shown in Figure 1. True values are presented with solid blue line, MAP estimates with dashed black line and credibility intervals (CI) with gray area. Results computed with the adaptive A-SGN approach (first row), with fixed number of photon packets with S-SGN$ _1 $ (second row), S-SGN$ _2 $ (third row), S-SGN$ _3 $ (fourth row), and with adaptive A-SGN$ _{ \rm{GN}} $ (fifth row) approaches

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Figure 4. MAP estimates of absorption coefficient $ \mu_{ \rm{a}} $ (first column), standard deviation of the absorption estimate $ \sigma_{ \rm{a}} $ (second column), MAP estimates of scattering coefficient $ \mu_{ \rm{s}} $ (third column) and standard deviation of the scattering estimate $ \sigma_{ \rm{s}} $ (fourth column). Results computed with the adaptive A-SGN approach (first row), with fixed number of photon packets with S-SGN$ _1 $ (second row), S-SGN$ _2 $ (third row), S-SGN$ _3 $ (fourth row) and with the adaptive A-SGN$ _{ \rm{GN}} $ (fifth row) approaches

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Figure 5. Absorption coefficient (first column) and scattering coefficient (second column) along the cross section shown in Figure 1. True values are presented with solid blue line, MAP estimates with dashed black line and credibility intervals (CI) with gray area. Results computed with the adaptive A-SGN approach (first row), with fixed number of photon packets with S-SGN$ _1 $ (second row), S-SGN$ _2 $ (third row), S-SGN$ _3 $ (fourth row), and with adaptive A-SGN$ _{ \rm{GN}} $ (fifth row) approaches

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Figure 6. Value of the objective function $ u(x) $ (first column), relative error of the absorption estimate $ E_{ \rm{a}} $ (second column) and scattering estimate $ E_{ \rm{s}} $ (third column) in the case of the highly scattering target (top row) and low scattering target (bottom row). Results computed with the A-SGN (red, solid), S-SGN$ _1 $ approach (blue, dotted), S-SGN$ _2 $ (blue, dashed) and S-SGN$ _3 $ (blue, solid) approaches

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Figure 7. Cumulative number of simulated photon packets $ \sum_i P_i $ on each iteration with the highly scattering target (left) and low scattering target (right). Results computed with the A-SGN (red, solid), S-SGN$ _1 $ (blue, dotted), S-SGN$ _2 $ (blue, dashed), S-SGN$ _3 $ (blue, solid) and A-SGN$ _{ \rm{GN}} $ (red, dashed) approaches

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Figure 8. Values of the objective function $ u(x) $ on the final iteration (first column), relative errors of the absorption estimates $ E_{\rm a} $ (second column) and the scattering estimates $ E_{\rm s} $ (third column), and total number of simulated photon packets $ P_b $ (fourth column) for the highly scattering target (top row) and low scattering target (bottom row) with A-SGN, S-SGN$ _1 $, S-SGN$ _2 $, S-SGN$ _3 $ and A-SGN$ _{ \rm{GN}} $ approaches. The black vertical lines (whiskers) denotes all samples excluding outliers, box denotes the 25th and 75th percentile, horizontal line denotes the median and $ + $ symbol denotes outliers

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Figure 9. Values of the objective function $ u(x) $ on the final iteration (first column), relative errors of the absorption estimates $ E_{\rm a} $ (second column) and the scattering estimates $ E_{\rm s} $ (third column), and total number of simulated photon packets $ P_b $ (fourth column) with the adaptive A-SGN approach using different values of the threshold parameter $ \gamma_{\nabla} $ (horizontal axis). The black vertical lines (whiskers) denotes all samples excluding outliers, box denotes the 25th and 75th percentile, horizontal line denotes the median and $ + $ symbol denotes outliers

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Table 1. Parameters of the minimization algorithms utilized in this study. $ P_i $ is the number of photon packets per illumination on each iteration, $ L $ is the number of samples utilized in the norm test, $ P_1 $ the initial number of photon packets and $ \gamma $ is the threshold parameter ($ \gamma_{\nabla} $ or $ \gamma_{ \rm{GN}} $). Norm test direction indicates which minimization direction is used to evaluate the norm test

Approach	$ P_i $	Norm test direction	$ L $	$ P_1 $	$ \gamma $
A-SGN	Adaptive	Stochastic gradient	10	$ 10^4 $	$ 0.4-0.8 $
S-SGN$ _1 $	$ 10^4 $	-	-	-	-
S-SGN$ _2 $	$ 10^7 $	-	-	-	-
S-SGN$ _3 $	$ 10^9 $	-	-	-	-
A-SGN$ _{ \rm{GN}} $	Adaptive	Stochastic GN	10	$ 10^4 $	0.8

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Table 2. Mean and standard deviation of the objective function value on the final iteration $ u(x) $, the relative error of absorption estimates $ E_{\rm a}\, (\%) $ and scattering estimates $ E_{\rm s}\, (\%) $, the total number of simulated photon packets $ P_b $ and the percentage of the true values that are within the credibility intervals for the absorption $ \rm{CI}_{ \mu_{\mathrm{a}}} $ and scattering $ \rm{CI}_{ \mu_{\mathrm{s}}} $ for the A-SGN, S-SGN$ _1 $, S-SGN$ _2 $, S-SGN$ _3 $ and A-SGN$ _{ \rm{GN}} $ approaches

	Highly scattering target
	A-SGN	S-SGN$ _1 $	S-SGN$ _2 $	S-SGN$ _3 $	A-SGN$ _{\rm GN} $
$ u(x) $	$ 1523 \pm 30 $	$ 8821 \pm 710 $	$ 1441 \pm 2 $	$ 1432.9 \pm 0.1 $	$ 1467 \pm 14 $
$ E_{\rm a}\, (\%) $	$ 7.547 \pm 0.009 $	$ 8.6 \pm 0.2 $	$ 7.537 \pm 0.001 $	$ 7.5353 \pm 0.0002 $	$ 7.538 \pm 0.002 $
$ E_{\rm s}\, (\%) $	$ 8.0 \pm 0.2 $	$ 16.4 \pm 1.6 $	$ 7.8 \pm 0.1 $	$ 7.8 \pm 0.1 $	$ 7.9 \pm 0.2 $
$ P_b $	$ (3.3 \pm 0.9) \cdot 10^6 $	$ (9 \pm 3) \cdot 10^5 $	$ (7\pm 0) \cdot 10^7 $	$ (5 \pm 0) \cdot 10^9 $	$ (1.1 \pm 0.4) \cdot 10^7 $
CI$ _{ \mu_{\mathrm{a}}}\, (\%) $	$ 99.9 $	$ 85.9 $	$ 99.9 $	$ 100 $	$ 100 $
CI$ _{ \mu_{\mathrm{s}}}\, (\%) $	$ 99.6 $	$ 75.3 $	$ 99.8 $	$ 99.8 $	$ 99.8 $
	Low scattering target
	A-SGN	S-SGN$ _1 $	S-SGN$ _2 $	S-SGN$ _3 $	A-SGN$ _{\rm GN} $
$ u(x) $	$ 1343.5 \pm 0.6 $	$ 10 186 \pm 444 $	$ 1367 \pm 4 $	$ 1352.8 \pm 0.4 $	$ 1354 \pm 1 $
$ E_{\rm a}\, (\% $)	$ 7.4642 \pm 0.0008 $	$ 8.4 \pm 0.1 $	$ 7.470 \pm 0.009 $	$ 7.4639 \pm 0.0003 $	$ 7.465 \pm 0.001 $
$ E_{\rm s}\, (\% $)	$ 10.13 \pm 0.03 $	$ 22.3 \pm 1.8 $	$ 10.2 \pm 0.1 $	$ 10.15 \pm 0.02 $	$ 10.15 \pm 0.03 $
$ P_b $	$ (7.5 \pm 2.5) \cdot 10^8 $	$ (1.02 \pm 0) \cdot 10^6 $	$ (7.7\pm 2.6) \cdot 10^8 $	$ (7.2 \pm 0.5) \cdot 10^9 $	$ (1.9 \pm 2.4) \cdot 10^9 $
CI$ _{ \mu_{\mathrm{a}}}\, (\%) $	$ 99.4 $	$ 82.3 $	$ 99.4 $	$ 99.4 $	$ 99.4 $
CI$ _{ \mu_{\mathrm{s}}}\, (\%) $	$ 99.8 $	$ 67.6 $	$ 99.8 $	$ 99.8 $	$ 99.8 $

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