# Machine Learning Diffuse Optical Tomography Using Extreme Gradient Boosting and Genetic Programming

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Simulating Photon Migration in Digital Breast Phantoms to Generate Dataset

#### 2.2. Detecting Tumors in the Compressed Breast Using Extreme Gradient Boosting

^{2}metrics perform relatively poorly when the prediction window is narrow, as is the case with the radius of the spheres. Furthermore, the RMSE and the CS metrics are better suited to solving inverse problems where the prediction interval is narrow and continuous [8,43]. A computer with an i9 series 9900k processor and two NVIDIA GeForce RTX 2080Ti graphics processors were used to build and train the XGBoost algorithm. Each GPU contains 11 GB of VRAM, and the two GPUs are connected through an NVlink. The network is by Adam Optimizer with an initial learning rate of 0.0001. For the XGB predictions, 50% of the total dataset is used, out of which 60% is used for training and validation, and 40% of the data is used for testing. The XGBoost algorithm learns the features from the input and the labels and accurately predicts the relationship between the input (measurement data) and the labels (tumor location) by regression. Due to variations in the dimensions of the compressed breast and the nature of the coordinate system used to model the compressed breast, the model prediction is made separately for the three different coordinates, the absorption coefficient, and the radius. The best results obtained by the XGBoost algorithm are RMSE values of 0.1862 ± 0.0018 for the x coordinate, 0.1678 ± 0.0042 for the y coordinate, 0.1505 ± 0.0009 for the z coordinate, 0.1131 ± 0.0091 for the absorption coefficient, and 0.2157 ± 0.0103 for the radius. Some of the predictions of the XGBoost algorithm are shown below in Figure 2.

#### 2.3. Enhancing Tumor Detection Capabilities Using Genetic Programming

- Randomly generate an initial population of solutions called individuals. Each individual is generated as a random tree of limited depth, consisting of nodes taken from the terminal set and the function set. The terminal set contains constants and variables, and the function set consists of various operators, for example, mathematical operations, logical operators, etc.
- While the termination criterion is not fulfilled, the following sub-steps are repeated:
- a.
- Evaluate the individuals in the current population according to the fitness function, which outputs a numerical value representing the quality of the individual as a solution.
- b.
- Select individuals from the population using a selection method, where the probability for selection is related to fitness values, for producing the next set of individuals.
- c.
- Apply the following genetic operators to produce new individuals with predetermined probabilities:
- I.
- Reproduction: clone an individual selected by the sub-step ”b” to the population.
- II.
- Crossover: randomly recombine two selected individuals to produce two new offspring.
- III.
- Mutation: randomly alter one selected individual to produce one new offspring.

- Output the best individual found during the run as the output.

## 3. Discussion

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**The reconstructed tumor locations using predictions from the XGBoost machine learning algorithm. (

**a**) showing the reconstructed tumor locations at (z = 5.325 mm), and (

**b**) showing the reconstructed tumor locations at (z = 11.289 mm), are two different examples of the XGBoost predictions. The color bar shows the absorption coefficient in (${\mathrm{cm}}^{-1}$).

**Figure 3.**Schematic block diagram showing the workflow of the proposed methodology. The black and blue boxes show the working of the XGBoost and GP algorithms, respectively.

**Figure 4.**The reconstructed tumor locations using predictions from the GP post-processing algorithm. (

**a**) showing the reconstructed tumor locations at (z = 14.375 mm), (

**b**) showing the reconstructed tumor locations at (z = 5.325 mm), and (

**c**) showing the reconstructed tumor locations at (z = 11.289), are different examples of the XGBoost and GP predictions. The color bar shows the absorption coefficient in (${\mathrm{cm}}^{-1}$).

Parameter | Range of Values |
---|---|

Population size | Between 10,000 and 15,000 |

Generation count | Between 100 and 250 |

Reproduction probability | 0.35 |

Crossover probability | 0.5 |

Mutation probability | 0.15 (including ERC) |

Tree depth | Between 2 and 6 |

Tournament size | 4 |

Values (Units) | RMSE (After XGBoost) | RMSE (After GP) |
---|---|---|

X coordinate (mm) | 0.1862 ± 0.0018 | 0.1808 ± 0.0014 |

Y coordinate (mm) | 0.1678 ± 0.0042 | 0.1539 ± 0.0057 |

Z coordinate (mm) | 0.1505 ± 0.0009 | 0.1340 ± 0.0032 |

Radius (mm) | 0.2157 ± 0.0103 | 0.2017 ± 0.0126 |

$\langle {\mathsf{\mu}}_{\mathrm{a}}\rangle $(mm^{−1}) | 0.1131 ± 0.0091 | 0.0975 ± 0.0065 |

S. No. | Article | Research Type | Background Type | RMSE |
---|---|---|---|---|

P. | Proposed algorithm (XGBoost + GP) | Simulation | Inhomogeneous background mesh (DigiBreast [40]) | 0.12 |

1. | Jaejun Yoo et al. [15] (Neural network for inverting Lippman–Schwinger equation) | Simulation and Experiment | Homogeneous background mesh (breast mesh and full body rat mesh) | 0.66 |

2. | Yun Zou et al. [8](ML-PC model) | Simulation and Experiment | Homogeneous background mesh | 0.30 |

3. | GM. Balasubramaniam et al. [17] (Cascaded feed-forward neural network) | Simulation | Homogeneous background mesh | 0.17 |

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**MDPI and ACS Style**

Hauptman, A.; Balasubramaniam, G.M.; Arnon, S.
Machine Learning Diffuse Optical Tomography Using Extreme Gradient Boosting and Genetic Programming. *Bioengineering* **2023**, *10*, 382.
https://doi.org/10.3390/bioengineering10030382

**AMA Style**

Hauptman A, Balasubramaniam GM, Arnon S.
Machine Learning Diffuse Optical Tomography Using Extreme Gradient Boosting and Genetic Programming. *Bioengineering*. 2023; 10(3):382.
https://doi.org/10.3390/bioengineering10030382

**Chicago/Turabian Style**

Hauptman, Ami, Ganesh M. Balasubramaniam, and Shlomi Arnon.
2023. "Machine Learning Diffuse Optical Tomography Using Extreme Gradient Boosting and Genetic Programming" *Bioengineering* 10, no. 3: 382.
https://doi.org/10.3390/bioengineering10030382