# Advancements in Numerical Methods for Forward and Inverse Problems in Functional near Infra-Red Spectroscopy: A Review

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Modeling of Light Transport in Biological Tissue as Forward Problem

## 3. Methods for Forward Model Simulation

#### 3.1. Analytical Methods

#### 3.2. Numerical Methods

#### 3.2.1. Finite Difference Method

#### 3.2.2. Finite Volume Method

#### 3.2.3. Boundary Element Method

#### 3.2.4. Finite Element Method

#### 3.3. Stochastic Methods

## 4. Types of Toolboxes for Forward Model Simulation

#### 4.1. MCML

#### 4.2. NIRFAST

#### 4.3. TOAST++

#### 4.4. MCX/MMC

#### 4.5. ValoMC

## 5. Inverse Problem

## 6. Methods for Inverse Problem Solution

#### 6.1. Back Projection

#### 6.2. Singular Value Decomposition (SVD) and Truncated Singular Value Decomposition (tSVD)

#### 6.3. Least Square by QR Decomposition (LSQR) and Regularized LSQR (rLSQR)

#### 6.4. Minimum Norm Estimate (MNE) and Weighted Minimum Norm Estimate (WMNE)

#### 6.5. Low-Resolution Electromagnetic Tomographt (LORETA)

#### 6.6. L1-Norm

#### 6.7. Hierarchical Bayesian as MAP Estimate

- i.
- Considering the measurement noise $\gamma $ as a Gaussian distribution $N\left(0,\nu \right)$ and the forward problem as a probabilistic model as$$P\left(y/x\right)~N\left(Ax,\nu \right)$$
- ii.
- Assuming the data prior distribution and likelihood function as $logP\left(x/y\right)$ and $logP\left(x/C\right)$ respectively.
- iii.
- Computation of the posterior distribution of the unknown as$${x}_{MAP}=argmax\left\{logP\left(x/y\right)+logP\left(x/C\right)\right\}$$$$P\left(x,y,\theta ,\vartheta \right)=P\left(y/x\right)P\left(x/\theta ,\vartheta \right)P\left(\theta \right)P\left(\vartheta \right)$$
- iv.
- By applying the variational Bayesian (VB) method, the posterior could be written as variational free energy$$F\left(Q\left(x,\theta ,\vartheta \right)\right)=\int Q\left(x,\theta ,\vartheta \right)log\left(\frac{P\left(x,y,\theta ,\vartheta \right)}{Q\left(x,\theta ,\vartheta \right)}\right)d\beta dadx$$$$Q\left(x,\theta ,\vartheta \right)=Q\left(x\right)Q\left(\theta \right)Q\left(\vartheta \right)$$

#### 6.8. Expectation-Maximization (EM)

- ⮚
- E-step given the observed data $y$ and the current estimate ${\mu}^{k}$, the conditional anticipation of the whole log-likelihood could be computed as$${x}^{k}={\mu}^{k}+\frac{{\beta}^{2}}{{\delta}^{2}}{A}^{T}\left(y-A{\mu}^{k}\right)$$
- ⮚
- M-step: Update the estimated value of ${x}^{k}$$${x}^{k+1}=argmax\left\{-\Vert \mu -{x}^{k}{\Vert}^{2}-2{\delta}^{2}\alpha \Vert x{\Vert}^{1}\right\}$$Equation can be explained separately for each element ${x}_{l}^{k+1}$ as$${x}_{l}^{k+1}=argmax\left\{-{\mu}_{l}^{2}+2{\mu}_{l}{x}_{l}-2{\delta}^{2}\alpha \Vert x{\Vert}^{1}\right\}$$${x}_{l}$ is the element. It can be resolved using the soft threshold technique.

#### 6.9. Maximum Entropy on the Mean (MEM)

#### 6.10. Bayesian Model Averaging

- i.
- Consider the basic assumption of the Bayesian formulization of the given problem as a normal probability density function as$$p\left(y/x,\phi \right)=N\left(Ax,\phi \right)$$
- ii.
- The estimation of the parameter as the first level of inference using the Bayes theorem is described as the posterior probability density function a$$\left(x/y,\phi ,{H}_{k}\right)=\frac{p\left(y/x,\phi ,{H}_{k}\right)p\left(x/\phi ,{H}_{k}\right)}{\int p\left(y/x,\phi ,{H}_{k}\right)p\left(x/\phi ,{H}_{k}\right)d\phi}$$
- iii.
- The estimation of the hyperparameters as 2nd level of inference is describing as the posterior probability density function as$$p\left(\phi /y,{H}_{k}\right)=\frac{p\left(y/\phi ,{H}_{k}\right)p\left(\phi /{H}_{k}\right)}{\int p\left(y/\phi ,{H}_{k}\right)p\left(\phi /{H}_{k}\right)d\phi}$$
- iv.
- The estimation of the model as the third level of inference as the posterior probability density function$$p\left({H}_{k}/y\right)=\frac{p\left(y/{H}_{k}\right)p\left({H}_{k}\right)}{\int p\left(y/{H}_{k}\right)p\left({H}_{k}\right)d\phi}$$
- v.
- Lastly, marginalizing the first, second, and third level of inference as posterior pdf as$$p\left(x/y\right)={{\displaystyle \int}}_{forall{H}_{k}}p\left(x/y,{H}_{k}\right)p\left({H}_{k}\right)={{\displaystyle \sum}}_{k}p\left(x/y,{H}_{k}\right)p\left({H}_{k}\right)$$

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Visual representation of the case (

**c**) continuous wave, case (

**b**) frequency domain, and case (

**a**) (adapted from Ref. [6]).

**Figure 2.**A graphical representation of the forward and inverse problems (adapted from Ref. [7]).

**Figure 3.**Fundamental steps of MCML technique (adapted from Ref. [53]).

**Figure 4.**Fundamental steps of the NIRFAST technique for the forward problem (adapted from Ref. [54]).

**Figure 5.**Libraries for Toast++ technique for forward model simulation (adapted from Ref [55]).

**Figure 6.**Basic steps of MCX technique (adapted from Ref. [56]).

**Table 1.**Details about the various methods and types of toolboxes/software used for the simulation forward problem in fNIRs measurements.

References | Forward Simulation Method | Simulation Software/Toolbox | Data Type |
---|---|---|---|

B. W. Pogue et al., 1995 [61] | FDM | N/A | N/A |

M. A. Ansari et al., 2014 [62] | BEM | N/A | N/A |

Dehghani, Hamid, et al., 2009 [54] | FEM | NIRFAST | Breast model data |

Yalavarthy, Phaneendra K. et al., 2007–2008 [7,63,64] | FEM | N/A | Phantom |

Brigadoi, Sabrina, et al., [65] | FEM | Toast++ | Real resting-state data |

Chiarelli, Antonio M., et al., 2016 [66] | FEM | NIRFAST | Phantom |

Lu, Wenqi, Daniel Lighter, and Iain B. Styles. 2018 [67] | FEM | NIRFAST | Realistic simulation data |

Machado, A., et al., 2018 [68] | MC | MCX | Realistic simulation data |

Yu, Leiming, et al., 2018 [58] | MC | MCX | Phantom |

Jiang, Jingjing, et al., 2020 [69] | MC and FEM | MCX and Toast++ | Silicon phantom experiment |

Fu, Xiaoxue, and John E. Richards. 2021 [70] | MC | MCX | Realistic simulation data |

Cai, Zhengchen, et al., 2021 [71] | MC | MCX | Realistic |

Mazumder, Dibbyan, et al., 2021 [72] | MC | MCX | Realistic simulation data |

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**MDPI and ACS Style**

Hussain, A.; Faye, I.; Muthuvalu, M.S.; Tang, T.B.; Zafar, M.
Advancements in Numerical Methods for Forward and Inverse Problems in Functional near Infra-Red Spectroscopy: A Review. *Axioms* **2023**, *12*, 326.
https://doi.org/10.3390/axioms12040326

**AMA Style**

Hussain A, Faye I, Muthuvalu MS, Tang TB, Zafar M.
Advancements in Numerical Methods for Forward and Inverse Problems in Functional near Infra-Red Spectroscopy: A Review. *Axioms*. 2023; 12(4):326.
https://doi.org/10.3390/axioms12040326

**Chicago/Turabian Style**

Hussain, Abida, Ibrahima Faye, Mohana Sundaram Muthuvalu, Tong Boon Tang, and Mudasar Zafar.
2023. "Advancements in Numerical Methods for Forward and Inverse Problems in Functional near Infra-Red Spectroscopy: A Review" *Axioms* 12, no. 4: 326.
https://doi.org/10.3390/axioms12040326