Mathematical and numerical challenges in diffuse optical tomography inverse problems

Andrea Aspri; Alessandro Benfenati; Paola Causin; Cecilia Cavaterra; Giovanni Naldi

doi:10.3934/dcdss.2023210

Article Contents

2024, Volume 17, Issue 1: 421-461. Doi: 10.3934/dcdss.2023210

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Mathematical and numerical challenges in diffuse optical tomography inverse problems

1.
Department of Mathematics, University of Milan, via Saldini 50, 20133 Milan, Italy
2.
Department of Environmental Science and Policy, University of Milan, via Celoria 2, 20133 Milan, Italy
3.
IMATI, CNR, Via Ferrata 1, 27100 Pavia, Italy

^*Corresponding author: Cecilia Cavaterra
^*Corresponding author: Cecilia Cavaterra

Dedicated to Pierluigi Colli on the occasion of his 65th birthday

Received: July 2023

Revised: November 2023

Early access: November 2023

Published: January 2024

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Abstract

Computed Tomography (CT) is an essential imaging tool for medical inspection, diagnosis and prevention. While X-rays CT is a consolidated technology, there is nowadays a strong drive for innovation in this field. Between the emerging topics, Diffuse Optical Tomography (DOT) is an instance of Diffuse Optical Imaging which uses non-ionizing light in the near-infrared (NIR) band as investigating signal. Non-trivial challenges accompany DOT reconstruction, which is a severely ill-conditioned inverse problem due to the highly scattering nature of the propagation of light in biological tissues. Correspondingly, the solution of this problem is far from being trivial. In this review paper, we first recall the theoretical basis of NIR light propagation, the relevant mathematical models with their derivation in the perspective of a hierarchy of modeling approaches and the analytical results on the uniqueness issue and stability estimates. Then we describe the state-of-the-art in analytic theory and in computational and algorithmic methods. We present a survey of the few contributions regarding DOT reconstruction aided by machine learning approaches and we conclude providing perspectives in the mathematical treatment of this highly challenging problem.

Keywords:

Mathematics Subject Classification: Primary: 35R30, 68T05, 68T07, 92C55, 65M32.

Citation:

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Figure 1. a) Main physical phenomena occurring in light propagation in biological tissues, turbid media characterized by a highly complex structure; b) absorption by a single particle; c) scattering by a single particle

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Figure 2. Notation for a single scattering event: light incides at position $ {\bf{r}} $ from direction $ \widehat{ {\bf{s}}}^\prime $ and is scattered in direction $ \widehat{ {\bf{s}}} $ around a cone of angle $ d\widehat{ {\bf{s}}}^\prime $

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Figure 3. Geometry and position of sources and detectors used in the numerical tests in the: (a) 2D setting and (b) 3D setting

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Figure 4. Reconstruction with contrast regions with the same (a) or different (b) strength. In the second case the algorithm struggles to obtain a good quality both in terms of localization and intensity

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Figure 5. Results for test cases ⅰ), ⅱ), ⅲ) in column–wise order. Top to bottom: Tikhonov regularization, Bregman procedure coupled with 2–norm, pure LASSO approach, Bregman coupled with 1–norm and elastic–net procedure with $ \alpha = 0.5 $

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Figure 6. Results in the 2D test setting with: (a) no enforcement of Robin boundary conditions on the background solution; (b): enforcement of Robin boundary conditions on the background solution via the MFS strategy. The red circle represents the exact location of the contrast region

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Figure 7. Results in the 3D test setting with: (a) no enforcement of Robin boundary conditions on the background solution; (b): enforcement of Robin boundary conditions on the background solution via the MFS strategy. The red circle represents the exact location of the contrast region

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Figure 8. Improvement (right) of the spatial resolution in the 3D geometry using as soft-prior a first rough estimation of the position of the contrast region from a coarse voxelization (left)

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Figure 9. Classes of existing DL-based approach for DOT reconstruction: a) methods of this family are fully data-driven. Sometimes the loss function used in the training of the net can be enriched with physically motivated constraints; b) methods of this family partially rely on physical models that can produce a first guess for the solution and then are supported by nets which improve it

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Table 1. Symbols (organized into macro-areas) and relative significance as used in the text

Nomenclature	Description
Geometrical quantities
$ d $	spatial dimension (=2, 3)
$ {\bf{r}} $	position vector
$ \widehat{ {\bf{s}}}\, ^\prime, \widehat{ {\bf{s}}} $	in, out coming light directions
$ d\widehat{ {\bf{s}}} $	solid angle around direction $ \widehat{ {\bf{s}}} $
$ S^{d-1} $	unit sphere in $ \mathbb{R}^d $
Physical model parameters
$ \nu $	refraction coefficient
$ \mu_s, \mu_s^\prime $	scattering coefficient, reduced scattering coefficient
$ D $, $ \mu_a $	diffusion and absorption coefficient
$ L $, $ u $	light radiance (RTE model), fluence (DA model)
$ p_s(\widehat{ {\bf{s}}}, \widehat{ {\bf{s}}}\, ^\prime) $, $ g $	normalized scattering phase function, avg cosine of scatter
$ \delta $	noise level
Functional spaces and operators
$ Q, U, Y $	optical properties, optical field and data Banach spaces
$ \Theta $	generic parameter space
$ \mathcal{M}: D(\mathcal{M}) \subset (Q \times U) \rightarrow Y $	measurement map
$ \mathcal{F}: \Theta \rightarrow Y $	forward map
$ \mathcal{D}: Y \rightarrow [0, \infty] $	discrepancy measure
$ \mathcal{R}: \Theta \rightarrow [0, \infty] $	regularization functional (with parameter $ \lambda $)
Other mathematical symbols
$ \Omega $, $ \partial \Omega $	computational domain, boundary of the domain
$ {\bf{n}} $	unit normal vector on $ \partial \Omega $
Discretization parameters
$ h $	mesh size

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Table 2. ACR (binned according to the intensity of the contrast region) and TPR metrics for different noise levels for the Learned-SVD [31] and for comparison the Elastic Net and Bregman variational approaches. Values are averaged over 150 test samples. Noise level 5% is not considered for the variational approaches due to the extreme difficulty to obtain a sensible reconstruction

Noise Level	$ {3\mu_{a, 0}} $ (GT: 3e-2)	$ {4\mu_{a, 0}} $ (GT: 4e-2)	$ {5\mu_{a, 0}} $ (GT: 5e-02)	TPR
	Learned-SVD
0%	3.05e-02 $ \pm $ 2.00e-03	4.08e-02 $ \pm $ 3.72e-03	4.74e-02 $ \pm $ 2.19e-03	0.94
1%	3.07e-02 $ \pm $ 2.77e-03	3.76e-02 $ \pm $ 5.54e-03	4.38e-02 $ \pm $ 4.01e-03	0.76
3%	3.06e-02 $ \pm $ 5.56e-03	3.61e-02 $ \pm $ 6.10e-03	4.11e-02 $ \pm $ 4.65e-03	0.52
5%	3.00e-02 $ \pm $ 3.82e-03	3.15e-02 $ \pm $ 4.44e-03	3.39e-02 $ \pm $ 6.04e-03	0.46
	Elastic Net
0%	2.73e-02 $ \pm $ 4.74e-03	3.45e-02 $ \pm $ 4.96e-03	3.90e-02 $ \pm $ 8.69e-03	0.45
1%	2.91e-02 $ \pm $ 9.36e-03	4.30e-02 $ \pm $ 8.70e-03	5.01e-02 $ \pm $ 9.80e-03	0.17
3%	9.55e-02 $ \pm $ 1.92e-02	1.25e-01 $ \pm $ 2.83e-02	1.23e-01 $ \pm $ 3.36e-02	0.05
	Bregman
0%	4.02e-02 $ \pm $ 8.95e-03	5.93e-02 $ \pm $ 1.61e-02	8.54e-02 $ \pm $ 2.98e-02	0.26
1%	4.77e-02 $ \pm $ 1.81e-02	5.85e-02 $ \pm $ 1.84e-02	8.34e-02 $ \pm $ 2.85e-02	0.17
3%	1.28e-01 $ \pm $ 5.50e-02	1.45e-01 $ \pm $ 9.06e-02	1.36e-01 $ \pm $ 6.80e-02	0.03

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