# Mesh Optimization for Monte Carlo-Based Optical Tomography

^{*}

## Abstract

**:**

## 1. Introduction

^{6}–10

^{9}) per simulated optode, depending on the data type, in preclinical settings [14]. As the accuracy of the MC-based Jacobian is dependent on the local statistics of the forward problem, coarsening or refining the mesh requires recomputing the Jacobian, possibly with a greater photon packet number to satisfy smaller discretizations. The iterative nature of adaptive mesh techniques may then lead to hours of computations, even in a massively parallel environment. Herein, we investigate the application of fast and efficient forward mesh optimization approaches for time resolved MC-based FMT. We propose a mesh optimization methodology in which the initial mesh and MC forward models are analytically rescaled at each iteration, allowing for fast computation without loss of accuracy.

## 2. Methods

#### 2.1. Optical Inverse Problem

#### 2.2. Mesh-Based Monte Carlo

#### 2.3. Mesh Adaptation

#### 2.4. Jacobian Rescaling

**Figure 3.**The possible cases in the output node positioning. The upper row corresponds to the input discretization and lower row to the output mesh.

#### 2.5. In Silico Model

**Figure 4.**(

**a**) The digimouse model with slice highlighted next to (

**b**) the mesh section used with positions of sources (solid black) and detectors (solid grey). The slice is 4 mm thick.

## 3. Results and Discussion

#### 3.1. Size Factors

Size Factor | Iterations | Nodes | Elements | Max. Elem. Vol. (mm^{3}) |
---|---|---|---|---|

Initial | -- | 2396 | 7574 | 0.08 |

1.15 | 17 | 1332 | 3904 | 0.51 |

1.25 | 21 | 777 | 2200 | 1.49 |

1.35 | 20 | 612 | 1615 | 2.97 |

**Figure 5.**The mesh under different maximum size factors after (

**a**) one iteration and (

**b**) seventeen iterations, the greatest iteration all meshes achieve before convergence. Maximum size factors are provided as subtitles (color bar corresponds to volume of elements in mm

^{3}).

#### 3.2. Forward Model-Based Solution Fields

Field | Iterations | Nodes | Elements | Max. Elem. Vol. (mm^{3}) |
---|---|---|---|---|

∑Jacobian | 21 | 777 | 2200 | 1.49 |

Log(∑Jacobian) | 12 | 1487 | 4335 | 0.25 |

Normalized ∑J | 21 | 1053 | 3074 | 0.48 |

Curvature | 15 | 447 | 1105 | 1.55 |

Log(Curv.) | 20 | 668 | 1651 | 1.37 |

^{3}, the smallest maximum volumes achieved at convergence. This allows for comparison of all solution fields as summarized in Table 3.

Field | Iterations | Nodes | Elements | Max. Elem. Vol. (mm^{3}) |
---|---|---|---|---|

∑Jacobian | 2 | 1614 | 4872 | 0.26 |

Log(∑Jacobian) | 5 | 1611 | 4792 | 0.25 |

Normalized ∑J | 5 | 1457 | 4334 | 0.25 |

Curvature | 2 | 1136 | 3304 | 0.31 |

Log(Curv.) | 2 | 1608 | 4846 | 0.25 |

**Figure 6.**Upper row: mesh at convergence (

**a**) ${{\displaystyle \sum}}^{\text{}}Jacobian$, (

**b**) $log\left({{\displaystyle \sum}}^{\text{}}Jacobian\right)$, (

**c**) $normalized{{\displaystyle \sum}}^{\text{}}J$; (

**d**) $u$ and (

**e**) $Log\left(u\right)$; lower row: mesh at 0.25 mm

^{3}stopping criterion for the respective corresponding input fields.

#### 3.3. Geometry-Based Solution Fields

Field | Iterations | Nodes | Elements | Max. Elem. Vol. (mm^{3}) |
---|---|---|---|---|

∑Jacobian | 21 | 777 | 2200 | 1.49 |

Dist. | 11 | 1675 | 5166 | 0.28 |

Att. | 17 | 699 | 1999 | 1.80 |

#### 3.4. Jacobian Accuracy and Computational Efficiency

^{9}photons) prior to adaptation and from the analytically rescaled Jacobian after adaptation (10

^{9}photons). Moreover, new mMC simulations were computed to obtain a mMC Jacobian on the new adapted mesh. An example of TPSFs produced for one specific source detector pair is provided in Figure 8a) (central pair). The TPSFs simulated under the original and mesh adaptation conditions are matching remarkably. Furthermore, the error in the TPSF from the rescaled Jacobian is similar to the error in the TPSF from the recomputed Jacobian. Throughout all time gates that are typically employed to cast the inverse problem, the maximum error is less than 1.5% for late gates and below 0.5% for early gates (Figure 8b). This error level is similar for all 49 simulated optode combinations. These results indicate that the mesh adaptation and analytical rescaling methodology described above lead to optimized non-homogenous discretization that does not affect light propagation accuracy, even in the most challenging cases, i.e., early gates.

**Figure 8.**Temporal Point Spread Function (TPSF) of Jacobians rescaled to a new mesh and the associated error at each gate.

**Figure 9.**(

**a**) Elements of volume employed to compute the error $e\left(r\right)$ (red label); (

**b**) Computation time for one forward simulation versus number of photons.

^{10}photons per optode. The error was computed for the Jacobian at the time corresponding to the 25% rising gate of the TPSF. A summary of the $e\left(r\right)$ calculated for this configuration and for different photon packets is provided in Table 5.

**Table 5.**Errors in the forward model central nodes before and after ${{\displaystyle \sum}}^{\text{}}Jacobian$ mesh optimization.

Photons | Initial TG | Final TG ∑Jacobian | Final TG $Att.$ |
---|---|---|---|

10^{9} | 10.84% | 11.71% | 13.99% |

10^{8} | 36.83% | 31.39% | 35.55% |

10^{7} | 61.01% | 60.13% | 58.57% |

10^{6} | 86.71% | 90.92% | 151.43% |

^{6}photons, $e\left(r\right)$ was similar between initial and rescaled meshes/Jacobians. Note that repeating 10

^{10}photons simulations on the initial mesh led to $e\left(r\right)~2\%$. Overall, 10

^{9}produced the least $e\left(r\right)$ with both field metric performing well, though they had 32% (∑Jacobian) and 34% (Attenuation Field) less nodes. The reduction in nodes and elements is mainly achieved in the central part of the mesh where the elements of volume are less visited by photons. The coarsening of the mesh in this region leads to improved statistics as seen in $e\left(r\right)$ for early gates (note that $e\left(r\right)$ is the average error as computed by the mean of the error at each node). In the case of 10

^{8}photons, which is the typical number of photons used successfully for preclinical studies [14], the coarsening leads to a 14% reduction in error $e\left(r\right)$. This suggests that mesh optimization as described herein can lead to reduction of the size of photon packets simulated and reduction in nodes/elements of volume for a more tractable inverse problem.

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflict of Interest

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**MDPI and ACS Style**

Edmans, A.; Intes, X.
Mesh Optimization for Monte Carlo-Based Optical Tomography. *Photonics* **2015**, *2*, 375-391.
https://doi.org/10.3390/photonics2020375

**AMA Style**

Edmans A, Intes X.
Mesh Optimization for Monte Carlo-Based Optical Tomography. *Photonics*. 2015; 2(2):375-391.
https://doi.org/10.3390/photonics2020375

**Chicago/Turabian Style**

Edmans, Andrew, and Xavier Intes.
2015. "Mesh Optimization for Monte Carlo-Based Optical Tomography" *Photonics* 2, no. 2: 375-391.
https://doi.org/10.3390/photonics2020375