# Optimized reconstruction framework in fluorescence diffuse optical tomography using radiative transfer equation

Pascal Nguyen, Martin Rodriguez-Vega, Julien Wojak, Paul Dorval, Rémi André, Maxime Henry, Véronique Josserand, Xavier Le Guével, Anabela Da Silva

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## ► To cite this version:

Pascal Nguyen, Martin Rodriguez-Vega, Julien Wojak, Paul Dorval, Rémi André, et al., Optimized reconstruction framework in fluorescence diffuse optical tomography using radiative transfer equation. European Conference on Biomedical Optics, OPTICA/ECBO, Jun 2025, Munich, Germany. hal-05194785

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Submitted on 31 Jul 2025

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# Optimized reconstruction framework in fluorescence diffuse optical tomography using radiative transfer equation

Pascal Nguyen,¹,²,* Martin Rodriguez-Vega,¹ Julien Wojak,¹ Paul Dorval,² Rémi André,¹ Maxime Henry,³ Véronique Josserand,³ Xavier Le Guével,³ and Anabela Da Silva¹

¹ Aix Marseille Univ, CNRS, Centrale Med, Institut Fresnel, Marseille, France

² Kaer Labs, Nantes, France

³ Institute for Advanced Biosciences, University Grenoble Alpes, INSERM U1209, CNRS UMR 5309, 38000 Grenoble, France

*pascal.nguyen@fresnel.fr

**Abstract:** We present a fluorescence diffuse optical tomography reconstruction framework based on the radiative transfer equation. Validated against the diffusion approximation framework, it proves to be a viable alternative for second near-infrared window tomography reconstruction. © 2025 The Author(s)

## 1. Introduction

Fluorescence Diffuse Optical Tomography (FDOT) is a non-invasive imaging technique that uses near-infrared wavelengths to visualize deep tissues marked with fluorescent probes [1] and is particularly useful in pre-clinical murine models [2]. The second near-infrared window NIR II (900 nm-1700 nm), also known as shortwave infrared (SWIR), offers higher spatial resolution with reduced scattering and higher absorption [3], but the commonly used Diffusion Approximation (DA) [4] is invalid in this range. Instead, the more accurate but more complex Radiative Transfer Equation (RTE) is considered. This work presents a reconstruction framework using a Monte Carlo-based RTE solver (Monte Carlo eXtreme, MCX) [5] to compute the forward model. An inversion process based on Non-negative Matrix Factorization (NMF) [6] is considered, which removes the need for hyperparameter tuning and enforces the non-negativity constraint intrinsic to FDOT. The forward and inversion processes are GPU-accelerated.

## 2. Methodology

### 2.1. Generation of synthetic data

We consider a continuous-wave FDOT acquisition that scans the sample in transillumination mode at discrete points [7]. At each point, the light distribution in transmission at both excitation and fluorescence wavelengths is measured on the surface of the sample.

To simulate the FDOT data, these two following steps are carried out using MCX with $10^6$ photons for each step using a GPU:

* Excitation: Compute $\phi_{ex}$ (the excitation fluence) using $\delta_{source}$ as the source term. $\delta_{source}$ is a Dirac delta function representing the source point.

* Fluorescence: Compute $\phi_{fluo}$ (the fluorescence fluence) using $\phi_{ex}$ $1_F X$ as the source term. $1_F$ is the fluo-rophore indicator function and $X$ is the fluorescence yield.

### 2.2. Reconstruction framework

The inverse problem is linearized using the normalized Born approximation as [7]:

$$y = \frac{U_{fluo}}{U_{ex}} = \frac{\sum_m G_{sm}^{ex} G_{md}^{fluo} X}{G_{sm}^{ex}} = Ax \quad (1)$$

where $U_{ex}$ and $U_{fluo}$ are the data associated with excitation and fluorescence, $G_{sm}^{ex}$ and $G_{md}^{fluo}$ are Green's functions for excitation and fluorescence wavelengths, $s$ is the index of the source, $d$ is the index of the detector and $m$ is the index of the voxel of the reconstruction mesh. Due to the ill-conditioned nature of the inverse problem, the
fluorescence images are filtered using Otsu thresholding [8] to retain only high SNR pixels. The inversion is solved with NMF, which ensures non-negativity for x since y is measured as positive and A is positive by construction. The update rule for NMF is [6]:

$$x^{n+1} = x^n \odot \frac{(A^\top y)}{(A^\top A)x^n} \quad (2)$$

With $^\top$ as the transpose operator and $\odot$ as the Hadamard product operator.

This algorithm of NMF is chosen for its efficient GPU-parallelized term-wise multiplication. The reconstruction quality is assessed using quantitative metrics (Normalized Mean Square Error - NMSE and Mean Absolute Deviation - MAD) and localization metrics (Intersection over Union - IoU computed via Otsu thresholding and Structural SIMilarity - SSIM). The computation time (matrix A construction and inversion process) is recorded.

### 3. Results

All simulations were performed on a desktop PC (Intel i7-13700KF, 64Gb RAM, Nvidia RTX4080 16Gb VRAM). The performance of our reconstruction scheme is illustrated here on a synthetic sample described hereafter.

#### 3.1. Simulation sample

The synthetic sample mimics a 1 mm fluorescent capillary tube embedded in an intralipid block(40 mm x 40 mm x 15 mm) placed on a plexiglass plate, with a highly absorbing sphere above it, as shown in Figure 1. Optical properties were chosen (see Table 1) to be similar to those of a biological tissue at $\lambda_x \sim 800$ nm (excitation) and $\lambda_m \sim 1000$ nm (fluorescence) [3]. The fluorescence yield was set to $X = 0.01$ for the simulation.

<figure><img src="image_3.png" /><figcaption>Fig. 1: Simulation framework</figcaption></figure>

<table><caption>Table 1: Optical properties for the simulation</caption><thead><tr><th rowspan="2">Components</th><th colspan="4">Excitation ($\lambda_x \sim 800$ nm)</th><th colspan="4">Fluorescence ($\lambda_m \sim 1000$ nm)</th></tr><tr><th>$\mu_a$ (mm<sup>-1</sup>)</th><th>$\mu_s$ (mm<sup>-1</sup>)</th><th>g</th><th>n</th><th>$\mu_a$ (mm<sup>-1</sup>)</th><th>$\mu_s$ (mm<sup>-1</sup>)</th><th>g</th><th>n</th></tr></thead><tbody><tr><td>Intralipid</td><td>0.01</td><td>10</td><td>0.9</td><td>1.33</td><td>0.01</td><td>10</td><td>0.9</td><td>1.33</td></tr><tr><td>Capillary</td><td>0.01</td><td>10</td><td>0.9</td><td>1.33</td><td>0.01</td><td>10</td><td>0.9</td><td>1.33</td></tr><tr><td>Absorbing sphere</td><td>0.1</td><td>10</td><td>0.9</td><td>1.33</td><td>0.1</td><td>10</td><td>0.9</td><td>1.33</td></tr><tr><td>Plexiglass plate</td><td>0</td><td>0</td><td>0</td><td>1.5</td><td>0</td><td>0</td><td>0</td><td>1.5</td></tr></tbody></table>

#### 3.2. Reconstruction results

For the reconstruction, we assumed that the physical model is known. The sensitivity matrix is constructed using the optical properties described in Table 1 at both wavelengths. A is constructed using either RTE using Monte
Carlo simulations with $10^6$ photons (MCX [5]) or DA with a GPU-based finite difference solver (built in-house, but alternative DA solvers exist like NIRFAST [9] and others). The inversion (NMF) is carried out with a maximum of 100 000 iterations and an early stopping tolerance of $10^{-4}$.

<figure><img src="image_4.png" /><figcaption>(a) Radiative Transfer Equation based reconstruction</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>(b) Diffusion Approximation based reconstruction</figcaption></figure>

<caption>Fig. 2: 3D visualisation of RTE-based and DA-based reconstructions</caption>

While DA offers faster computing (44 s for DA vs 95 s for RTE), the speed gain is not overwhelming considering the stochastic nature of the RTE solver, making RTE-based reconstruction a viable alternative to DA-based reconstruction. Both RTE and DA-based reconstructions yield similar results in NMSE, MAD and SSIM. DA-based reconstruction achieves better IoU, likely because of the stochastic noise induced by Monte Carlo simulation in RTE-based reconstruction which affects Otsu thresholding (see Figure 2 and Table 2).

<table><caption>Table 2: Reconstruction results</caption><thead><tr><td>Metrics</td><td>RTE-based reconstruction</td><td>DA-based reconstruction</td></tr></thead><tbody><tr><td>NMSE</td><td>4.5e-5</td><td>4.2e-5</td></tr><tr><td>MAD</td><td>1.1e-4</td><td>9.8e-05</td></tr><tr><td>IoU</td><td>0.38</td><td>0.48</td></tr><tr><td>SSIM</td><td>0.90</td><td>0.90</td></tr></tbody></table>

## 4. Conclusion & Future work

We have developed a reconstruction framework using a RTE solver to construct the sensitivity matrix. The RTE-based reconstruction is relevant compared to the DA-based reconstruction in terms of speed and accuracy, making it a viable alternative for reconstruction farther in the NIR II range where DA is not valid. Future work will include complex geometries closer to murine model with fluorescence wavelengths farther in the NIR II range.

## 5. Acknowledgements

This work received support from the French government under the France 2030 investment plan, as part of the Initiative d'Excellence d'Aix-Marseille Université – A*MIDEX (AMX-20-TRA-053) and was partially funded by France Life Imaging (grant ANR-11-INBS-0006).

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