# Initial Condition Assessment for Reaction-Diffusion Glioma Growth Models: A Translational MRI-Histology (In)Validation Study

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Clinical Case

#### 2.2. MR Image Acquisitions

#### 2.3. Slicer Design and Tissue Sampling

#### 2.4. Sample Processing and Analysis

#### 2.5. Cell Density Maps

- cell nuclei are approximately spherical;
- cell nuclei distribution is locally isotropic;
- tile dimensions are sufficiently large to contain multiple cells;
- tile dimensions are sufficiently small for the cell density to be considered homogeneous and isotropic;

#### 2.6. Cell Density Maps to Ex Vivo T1 Registration

#### 2.7. Edema Delineation

#### 2.8. Distance Map

#### 2.9. Cell Density Model

## 3. Results

## 4. Discussion

^{18}F]fluoroethyl-L-tyrosine and used in [42] along with conventional MRI to personalize a reaction-diffusion tumor growth model. Finally, combined [

^{11}C]methionine and [

^{18}F]fluorodeoxyglucose PET imaging was used in [43] to improve the detection of glioma cell infiltration. As can be noticed, many imaging techniques have been proposed to assess glioma cell density distributions in vivo but no consensual clinical procedure has seemed to emerge so far, leaving the question open.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

ADC | Average diffusion coefficient |

AFM | Anisotropic fast marching |

ASSD | Average symmetric surface distance |

BBB | Blood–brain barrier |

DTI | Diffusion tensor imaging |

DW | Diffusion-weighted |

FA | Flip angle |

GBM | Glioblastoma |

GVP | Glomeruloid vascular proliferation |

HE | Hematoxylin and eosin |

IHC | Immunohistochemistry |

MR(I) | Magnetic resonance (imaging) |

PD | Proton density |

PET | Positron emission tomography |

PPN | Pseudo-palisading necrosis |

ReLU | Rectified linear unit |

Susp. | Suspected |

T1Gd | T1-weighted sequence with injection of gadolinium-based contrast agent |

T2-FLAIR | T2-weighted sequence with fluid-attenuated inversion recovery |

TE | Echo time |

TI | Inversion time |

TMZ | Temozolomide |

TR | Repetition time |

VEGF | Vascular endothelial growth factor |

WHO | World Health Organization |

## Appendix A. Slicer Design

**Figure A1.**Brain slicer design superimposed to the in vivo T2-FLAIR image used as template in axial (

**a**), sagittal (

**b**), coronal (

**c**) and 3D (

**d**) views.

## Appendix B. Deep Learning-Based Nuclei Counting

**Figure A2.**U-Net architecture with its feature map dimensions used for nuclei detection. The network takes $100\text{}\mathrm{px}\times 100\text{}\mathrm{px}$ RGB tiles with pixel size $1\text{}\mathsf{\mu}\mathrm{m}\times 1\text{}\mathsf{\mu}\mathrm{m}$ as input and predicts a nuclei presence probability map with the same spatial dimensions.

^{−4}, ${\beta}_{1}$: 0.9, ${\beta}_{2}$: 0.999, $\u03f5$: 10

^{−6}). An evaluation was performed on the validation set at the end of each epoch and the training was stopped early after no improvement of the validation metric in 100 epochs. The network parameter values that gave the best validation metric value (0.041) were kept, which was reached at epoch 96. A similar weakly supervised deep learning approach was used in [55] for cell nuclei detection in histological slides, but Euclidian distance to user-pointed nuclei locations and the mean squared error loss were used instead of Gaussian nuclei presence probability maps and the cross-entropy loss.

## Appendix C. In Vivo to Ex Vivo Registration

## Appendix D. Pathological Results

**Table A1.**Results of the pathological examination and numerical tile processing. PPN: pseudo-palisading necrosis, GVP: glomeruloid vascular proliferation, susp.: suspected.

Cell Density [${10}^{3}$ $\mathbf{cell}$ $\mathbf{m}$${\mathbf{m}}^{-2}$] | Distance [$\mathbf{m}\mathbf{m}$] | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

Block | PPN | Tumor Cells | GVP | Edema | Min | Max | Mean | Min | Max | Mean |

1 | No | No | No | No | 1.08 | 20.46 | 11.77 | 33.40 | 40.40 | 34.95 |

2 | No | No | No | No | 2.25 | 23.88 | 13.80 | 35.05 | 46.76 | 40.12 |

3 | No | Infiltrative (susp.) | No | Yes | 1.52 | 25.25 | 14.96 | 7.24 | 21.18 | 14.00 |

4 | No | No | No | No | 1.36 | 20.64 | 12.50 | 29.36 | 41.66 | 36.84 |

5 | No | No | No | Yes | 1.19 | 24.85 | 15.40 | 11.74 | 30.27 | 21.57 |

6 | No | Infiltrative (susp.) | No | No | 3.72 | 28.44 | 18.70 | 32.92 | 40.35 | 36.52 |

7 | No | Infiltrative (susp.) | No | Yes | 0.89 | 38.95 | 21.44 | 38.59 | 61.70 | 49.04 |

8 | Yes | Block | Yes | No | 3.37 | 37.29 | 23.65 | 0.50 | 5.17 | 2.71 |

9 | Yes | Infiltrative | Yes | Yes | 1.02 | 44.37 | 31.71 | 0.50 | 4.72 | 1.91 |

10 | No | Infiltrative (susp.) | No | Yes | 5.52 | 37.15 | 25.53 | 0.50 | 3.71 | 1.06 |

11 | No | No | No | No | 1.73 | 25.68 | 15.21 | 19.85 | 38.74 | 30.74 |

12 | No | Infiltrative | Yes | Yes | 0.89 | 36.08 | 19.70 | 0.50 | 31.46 | 14.24 |

13 | Yes | Block | Yes | No | 1.40 | 52.46 | 21.69 | 0.50 | 22.85 | 9.72 |

14 | No | No | No | No | 1.10 | 19.25 | 11.78 | 17.73 | 32.20 | 25.51 |

15 | No | No | No | Yes | 2.08 | 22.90 | 11.41 | 28.63 | 54.74 | 40.14 |

16 | No | No | No | No | 0.99 | 26.47 | 13.52 | 20.37 | 48.79 | 35.91 |

17 | No | No | No | No | 1.42 | 23.69 | 13.10 | 28.30 | 56.12 | 42.88 |

18 | No | No | Yes | Yes | 2.46 | 39.29 | 20.71 | 7.94 | 27.01 | 16.83 |

19 | Yes | Block | Yes | Yes | 0.99 | 44.12 | 19.82 | 0.50 | 18.50 | 6.92 |

20 | Yes | Block | Yes | No | 6.25 | 61.05 | 28.40 | 0.50 | 7.37 | 2.06 |

21 | No | Infiltrative | Yes | Yes | 4.75 | 34.20 | 23.05 | 0.50 | 14.62 | 7.70 |

22 | No | No | No | Yes | 0.86 | 30.37 | 10.79 | 2.99 | 36.87 | 21.39 |

23 | No | No | No | No | 0.94 | 25.61 | 13.23 | 21.14 | 49.61 | 34.48 |

24 | No | No | No | No | 1.37 | 26.72 | 13.72 | 57.27 | 90.93 | 71.08 |

25 | No | Infiltrative (susp.) | No | Yes | 0.87 | 39.07 | 21.03 | 1.71 | 21.41 | 10.87 |

26 | Yes | Block | Yes | Yes | 6.21 | 54.24 | 22.79 | 0.50 | 8.81 | 2.96 |

27 | No | No | No | Yes | 0.93 | 37.96 | 21.33 | 12.74 | 30.35 | 18.40 |

28 | No | No | No | No | 0.95 | 34.79 | 20.76 | 14.35 | 35.85 | 22.57 |

## Appendix E. Isotropic vs. Anisotropic Metric Tensor

**Figure A3.**Segmented brain map (

**first row**) and arrival time maps from a sphere with radius 2.5 mm (red dot) obtained using an adapted implementation of the anisotropic fast marching algorithm in [23] for an isotropic (

**second row**) and a DTI-derived anisotropic (

**third row**) metric tensor in a healthy volunteer.

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**Figure 1.**Brain slicing and sample collection procedure. (

**a**) The brain is placed inside the 3D-printed slicer. (

**b**) Sagittal slices are cut carefully. (

**c**) Each brain slice is placed inside its cutting guide. (

**d**) Sample blocks are cut with a scalpel along the grooves and placed into standard cassettes.

**Figure 2.**Cell density map computation procedure. (

**a**) $3\times 3$ adjacent tiles (dotted squares) with dimensions $100\text{}\mathrm{px}\times 100\text{}\mathrm{px}$ and pixel size $1\text{}\mathsf{\mu}\mathrm{m}\times 1\text{}\mathsf{\mu}\mathrm{m}$ extracted from the resampled slide in panel (

**c**). Cell nuclei detected by the deep convolutional neural network are indicated with cyan dots. (

**b**) Corresponding $3\times 3$ pixels (dotted squares) of the cell density map with pixel size $0.1\text{}\mathrm{m}\mathrm{m}\times 0.1\text{}\mathrm{m}\mathrm{m}$ whose value is given by the corresponding tile cell count divided by the true tissue area. (

**c**) Whole hematoxylin and eosin stained slide (slide 13, see Table A1). (

**d**) Corresponding whole computed cell density map.

**Figure 3.**Cell density profile analysis. (

**a**) Brain slice inside its 3D-printed cutting guide. (

**b**) Corresponding slice of the registered in vivo T2-FLAIR image with segmented edema outlines. The blue and red segments of the outline respectively correspond to free and non-free to diffuse parts of the edema boundary (see Section 4). (

**c**) Corresponding slice of the ex vivo T1 image (grayscale) and superimposed registered cell density maps (colored) with their slide number (see Table A1). (

**d**) Corresponding slice of the geodesic distance map to the tumor core across white matter.

**Figure 4.**Example of registered cell density maps with their slide number (see Table A1) (

**1st and 3rd columns**) and corresponding slices of the geodesic distance map to the tumor core (

**2nd and 4th columns**) superimposed to the ex vivo T1 image.

**Figure 5.**Scatter plot of the surface cell density (

**a**) and the extrapolated volume cell density (

**b**) versus distance for each available value pairs among white matter voxels (blue dots) with superimposed fitted model curves (red curves).

**Figure 6.**Inverse cumulative distribution of the geodesic distance values along the edema outlines. The expected distribution under the hypothesis of iso-distance edema outlines is plotted in red.

**Figure 7.**Edema region (red) with superimposed thresholded region of the distance map whose contour minimizes the Hausdorff distance (blue,

**1st row**) and average symmetric surface distance (blue,

**2nd row**) to the edema contour in axial (

**1st column**), sagittal (

**2nd column**), and coronal (

**3rd column**) planes.

${\mathit{c}}_{\mathbf{core}}$ [${10}^{3}$ $\mathbf{cell}$ $\mathbf{m}$${\mathbf{m}}^{-2}$] | ${\mathit{\lambda}}_{\mathbf{white}}$ [$\mathbf{m}\mathbf{m}$] | ${\mathit{c}}_{\mathbf{white}}$ [${10}^{3}$ $\mathbf{cell}$ $\mathbf{m}$${\mathbf{m}}^{-2}$] |
---|---|---|

1.47 | 10.55 | 1.43 |

${\mathit{c}}_{\mathbf{core}}$ [${10}^{5}$ $\mathbf{cell}$ $\mathbf{m}$${\mathbf{m}}^{-3}$] | ${\mathit{\lambda}}_{\mathbf{white}}$ [$\mathbf{m}\mathbf{m}$] | ${\mathit{c}}_{\mathbf{white}}$ [${10}^{5}$ $\mathbf{cell}$ $\mathbf{m}$${\mathbf{m}}^{-3}$] |
---|---|---|

1.05 | 8.46 | 0.59 |

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**MDPI and ACS Style**

Martens, C.; Lebrun, L.; Decaestecker, C.; Vandamme, T.; Van Eycke, Y.-R.; Rovai, A.; Metens, T.; Debeir, O.; Goldman, S.; Salmon, I.;
et al. Initial Condition Assessment for Reaction-Diffusion Glioma Growth Models: A Translational MRI-Histology (In)Validation Study. *Tomography* **2021**, *7*, 650-674.
https://doi.org/10.3390/tomography7040055

**AMA Style**

Martens C, Lebrun L, Decaestecker C, Vandamme T, Van Eycke Y-R, Rovai A, Metens T, Debeir O, Goldman S, Salmon I,
et al. Initial Condition Assessment for Reaction-Diffusion Glioma Growth Models: A Translational MRI-Histology (In)Validation Study. *Tomography*. 2021; 7(4):650-674.
https://doi.org/10.3390/tomography7040055

**Chicago/Turabian Style**

Martens, Corentin, Laetitia Lebrun, Christine Decaestecker, Thomas Vandamme, Yves-Rémi Van Eycke, Antonin Rovai, Thierry Metens, Olivier Debeir, Serge Goldman, Isabelle Salmon,
and et al. 2021. "Initial Condition Assessment for Reaction-Diffusion Glioma Growth Models: A Translational MRI-Histology (In)Validation Study" *Tomography* 7, no. 4: 650-674.
https://doi.org/10.3390/tomography7040055