# Bayesian spatial mixture of experts for a flexible dose-response assessment after brain radiotherapy

Théo Silvestre, Florence Forbes, Sophie Ancelet

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## To cite this version:

Théo Silvestre, Florence Forbes, Sophie Ancelet. Bayesian spatial mixture of experts for a flexible dose-response assessment after brain radiotherapy. ISBA 2024 - International Society for Bayesian Analysis, Jul 2024, Venise, Italy. pp.1-1, 2024. [hal-04871355](https://hal.science/hal-04871355v1)

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https://hal.science/hal-04871355v1

Submitted on 7 Jan 2025

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# Bayesian spatial mixture of experts for a flexible dose-response assessment after brain radiotherapy

IRSN

INSTITUT DE RADIOPROTECTION
ET DE SÛRETÉ NUCLÉAIRE

Théo Silvestre

Laboratory: PSE-SANTE/SESANE/LEPID (IRSN)

PhD supervisors: Sophie Ancelet (LEPID, IRSN), Florence Forbes (Statify, INRIA)

## Context and motivation

- **High-grade glioma:** The most frequent high-grade brain tumor in adults.

- **Radiotherapy (RT):** One of the most important treatments, generally combined with surgery and/or chemotherapy:

- increased patient survival,

- but also an increase in side effects like cognitive impairments [Bompaire et al., 2018].

- **Clinical question:** Potential neurotoxicity of brain RT on the central nervous system?

- **Aim:** To estimate the potential association between the occurrence of white-matter hyperintensities (WMH) and the absorbed dose after brain RT.

- **Background:** Mass univariate analysis (voxel-based t-tests, voxel-based logistic regression).

- **Our idea:** Model-based approach at the voxel level ⇒ Extensions of GLiM models [Deleforge et al., 2015] to the context of binary outcomes and zero-inflated mixture components.

## Modelling

For an individual i and a voxel v and a time t, let $Y_{ivt}$ be the outcome variable describing the presence/absence of a WMH, $D_{ivt}$ the absorbed dose, $W_i^{1:P}$ some individual-specific covariates and $C_v$ the unknown class label of the voxel. Pour $k \in \{1, \dots, K\}$:

$$Y_{ivt} | D_{ivt}, W_i^{1:P}, C_v = k \sim \text{Bern}(p_{ivt})$$

$$\text{logit } p_{ivt} = \eta_{ivt} = \beta_0^k + \beta_1^k d_{ivt} + \beta_{2:P+1} w_i^{1:P}$$

$$D_{ivt}|C_v = k \sim \begin{cases} \delta_0 & \text{si } k = 0 \\ \text{Log-}\mathcal{N}(\mu_k, \tau_k^{-1/2}) & \text{si } k \neq 0 \end{cases} \\ C_v \sim \text{Multinomial}(1, \{\pi_1, \dots, \pi_K\})$$

## Data modelling

For all $k \in \{1, \dots, K\}$:

$$\beta^k \sim \mathcal{N}_2(m_0 = 0, \sigma_0 = 10^3)$$

$$\mu_k \sim \mathcal{N}(\mu_0 = 0, \lambda_0 = 10^3)$$

$$\tau_k \sim \mathcal{G}(a_0 = 10^{-3}, b_0 = 10^{-3})$$

$$\pi_k \sim \text{sym-Dirichlet}(\alpha/K)$$

## Prior distributions

## Bayesian inference

- Variational Bayes

- Gibbs Sampler

[Durante and Rigon, 2019] [Polson et al., 2013]

## Difficulties

- Large amount of data: $512 \times 512 \times 121 \sim 10^7$ voxels per MRI, 250 individuals with multiple times.

- Registration of MRI longitudinally and inter-individuals when brains deal with deformations.

## Further modelling

- Add a temporal interaction with the dose in the regressor :

$$\eta_{ivt} = \beta_0^k + (\beta_{10}^k + \beta_{11}^k t) d_{ivt} + \beta_{2:P+1} w_i^{1:P}$$

- Add spatiality

- through a residual $\xi_v$:

$$\eta_{ivt} = \beta_0^k + \beta_1^k d_{ivt} + \dots + \xi_v$$

$$\text{with } \xi_v | \xi_{-v} \sim \mathcal{N}\left(\frac{\sum_{u \in \partial v} \xi_u}{n(\partial v)}, \frac{\sigma^2}{n(\partial v)}\right)$$

- through a centred autologistic model using the pseudo-likelihood:

$$\eta_{ivt} = \beta_0^k + \beta_1^k d_{ivt} + \dots + \gamma \sum_{u \in \partial v} (Y_{iut} - \mu_{iut})$$

$$\text{with } \mu_{iut} = \mathbb{E}[Y_{iut} | \gamma = 0].$$

## Data

There are three types:

- Segmentation map (binary data) of WMHs observed from one 3D brain MRI, observed at multiple time.

- Dosimetric CT-scan, giving the absorbed dose to each voxel of the brain during RT.

- Clinical variables (sex, age, etc).

**Aim:** use our model on the EpiBrain-Rad cohort [Durand et al., 2015].

## Data illustration

<figure><img src="image_4.png" /><figcaption>Figure 1: Example of a patient FLAIR, WMH in blue and dose from black=0 through a red to yellow grandient finishing with white~ 60 Gy. On the left 6 months after RT and on the right 47 months after RT.</figcaption></figure>

## Preliminary results on one patient from simulated data ...

Our model is fitted on 100 simulated datasets. Coverage rate (CR) of the true regression parameters are displayed on the right, estimated mixture parameters are not shown:

<table><thead><tr><th rowspan="2">Method</th><th colspan="3">β<sub>1</sub><sup>1</sup> = 0.02</th><th colspan="3">β<sub>0</sub><sup>1</sup> = -3.90</th></tr><tr><th>CR 95%</th><th>Relative bias</th><th>Var</th><th>CR 95%</th><th>Relative bias</th><th>Var</th></tr></thead><tbody><tr><td>VBEM</td><td>50</td><td>2.20e01</td><td>1.39e00</td><td>54</td><td>2.80e-03</td><td>2.32e-02</td></tr><tr><td>Gibbs</td><td>93</td><td>2.81e01</td><td>8.92e00</td><td>93</td><td>3.80e-03</td><td>1.49e-01</td></tr></tbody></table>

<table><thead><tr><th rowspan="2">Method</th><th colspan="3">β<sub>1</sub><sup>2</sup> = 0.10</th><th colspan="3">β<sub>0</sub><sup>2</sup> = -4.30</th></tr><tr><th>CR 95%</th><th>Relative bias</th><th>Var</th><th>CR 95%</th><th>Relative bias</th><th>Var</th></tr></thead><tbody><tr><td>VBEM</td><td>82</td><td>1.36e-01</td><td>1.90e-04</td><td>75</td><td>5.12e-02</td><td>4.03e-02</td></tr><tr><td>Gibbs</td><td>99</td><td>1.39e-01</td><td>4.80e-04</td><td>96</td><td>5.25e-02</td><td>1.32e-01</td></tr></tbody></table>

<table><thead><tr><th rowspan="2">Method</th><th colspan="3">β<sub>1</sub><sup>3</sup> = 0.15</th><th colspan="3">β<sub>0</sub><sup>3</sup> = -5.45</th></tr><tr><th>CR 95%</th><th>Relative bias</th><th>Var</th><th>CR 95%</th><th>Relative bias</th><th>Var</th></tr></thead><tbody><tr><td>VBEM</td><td>88</td><td>4.78e-02</td><td>8.00e-05</td><td>88</td><td>4.34e-02</td><td>1.08e-01</td></tr><tr><td>Gibbs</td><td>92</td><td>4.86e-02</td><td>1.40e-04</td><td>94</td><td>4.42e-02</td><td>1.77e-01</td></tr></tbody></table>

<table><caption>Table 2: Statistics computed on 100 simulated datasets</caption><thead><tr><th rowspan="2">Method</th><th colspan="3">β<sub>1</sub><sup>4</sup> = 0.20</th><th colspan="3">β<sub>0</sub><sup>4</sup> = -7.70</th></tr><tr><th>CR 95%</th><th>Relative bias</th><th>Var</th><th>CR 95%</th><th>Relative bias</th><th>Var</th></tr></thead><tbody><tr><td>VBEM</td><td>82</td><td>4.66e-02</td><td>9.00e-04</td><td>83</td><td>7.51e-02</td><td>3.24e-00</td></tr><tr><td>Gibbs</td><td>98</td><td>4.45e-02</td><td>3.90e-03</td><td>98</td><td>7.26e-02</td><td>1.29e-01</td></tr></tbody></table>

- Markov chains obtained using MH sampler are strongly auto-correlated, effective sample size is almost 0 and assessing the convergence is not possible.

- Gibbs sampler with Pólya-Gamma latent random variables yields much better Markov chains,

- VBEM: much more faster but a far worse coverage rate compared to the Gibbs sampler.

<table><caption>Table 1: True mixture parameters</caption><thead><tr><th></th><th>μ</th><th>τ</th><th>π</th><th>Mode (Gray)</th></tr></thead><tbody><tr><td>Class 1</td><td>-1.21</td><td>1.00</td><td>0.57</td><td>0.11</td></tr><tr><td>Class 2</td><td>2.63</td><td>0.48</td><td>0.18</td><td>11</td></tr><tr><td>Class 3</td><td>3.72</td><td>0.17</td><td>0.11</td><td>40</td></tr><tr><td>Class 4</td><td>4.17</td><td>0.15</td><td>0.14</td><td>63</td></tr></tbody></table>

## ... and from real data

<figure><img src="image_5.png" /><figcaption>Figure 2: Six classes obtained with the model. Grey voxels indicate a WMH.</figcaption></figure>

- The model assigns a class to each voxel (figure 2).

- To each class is assigned a dose response-relationship, which leads to a piece-wise logistic association (graph 3).

- Thus, it becomes possible to have a relation at the scale of an organ, the white-matter or the whole brain.

<figure><img src="image_6.png" /><figcaption>Figure 3: Dose-response relationship estimated (with the associated uncertainty), depending on classes.</figcaption></figure>

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ISBA | 01/07-07/07 2024

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