# Mechanistic Learning to predict the results of clinical trials

Sébastien Benzekry

► To cite this version:

Sébastien Benzekry. Mechanistic Learning to predict the results of clinical trials. AMLD 2024 - Applied Machine Learning Days, Mar 2024, Lausanne, Switzerland. hal-04990478

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https://inria.hal.science/hal-04990478v1

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Distributed under a Creative Commons Attribution 4.0 International License
# Mechanistic Learning to predict the results of clinical trials

S. Benzekry

Inria - Inserm team COMPO

COMPutational pharmacology and clinical Oncology

Centre Inria Sophia Antipolis - Méditerranée

Center for Research on Cancer of Marseille
# Mechanistic learning

<figure><img src="image_2.png" /></figure>
# Outline: mechanistic learning

## Prediction of OS and response after ICI in NSCLC

1. Data

2. Modeling the kinetics of longitudinal data

1. Tumor size kinetics

2. Blood markers kinetics

3. Machine learning

1. Model development

2. Individual predictions

3. Trial-level predictions

4. ML for deep phenotyping

5. Cell-free DNA
# Clinical challenge: predicting survival in NSCLC immuno-oncology

* Lung cancer is the leading cause of cancer-related death (25.3 % of all cancer deaths, 13.6 % of new cancer cases)

* 57 % are metastatic at diagnosis (stage IV), 5-year overall survival rate is poor (18.6 %)

* Introduction of new immunotherapies (immune checkpoint inhibitors) in the 2010's has revolutionized treatment of advanced lung cancer

* 20–30 % of patients have long term response (> 10 years)

* Prediction of overall survival from baseline or early data could help

* guide treatment decision during clinical drug development (e.g., combo trials, predict OS from tumor phase I/II data)

* inform personalized health care

* Current state of the art

* Only predictive biomarker used in the clinic: PDL1 expression from immunohistochemistry<fnref n="1" /> (AUC = 0.601)

* Predictive model from baseline clinical and biological data: ROPRO score<fnref n="2" />

* 27 variables

* Developed on 120k+ patients over multiple cancer types

* C-index = 0.69 on OAK as independent test set

* Predictive value of transcriptomic and mutation data is unclear

* Tumor kinetics model parameter growth rate (KG) has important predictive power of hazard ratio (HR)<fnref n="3" />

* Predictive value of kinetics of pharmacodynamic biomarkers is unclear

<figure><img src="image_3.png" /></figure>

Brahmer et al., NEJM 2015

<footnote n="1">¹ Rizvi, H. et al., J Clin Oncol Molecular Determinants of Response to Anti-Programmed Cell Death (PD)-1 and Anti-Programmed Death-Ligand 1 (PD-L1) Blockade in Patients with Non-Small-Cell Lung Cancer Profiled with Targeted Next-Generation Sequencing. J. Clin. Oncol. 2018, 36, 633–641.</footnote>

<footnote n="2">² Becker, T. et al. An enhanced prognostic score for overall survival of patients with cancer derived from a large real-world cohort. Ann Oncol 31, 1561–1568 (2020).</footnote>

<footnote n="3">³ Claret, L. et al. A Model of Overall Survival Predicts Treatment Outcomes with Atezolizumab versus Chemotherapy in Non-Small Cell Lung Cancer Based on Early Tumor Kinetics. Clin Cancer Res 24, 3292–3298 (2018).</footnote>
# kinetics – Machine learning (kML) for prediction of OS

Roche

## Raw longitudinal data

Tumor Kinetics (TK)
k = 8,635

Blood markers Kinetics (BK)
k = 108,551
Albumin, CRP, Neutrophils, LDH

<figure><img src="image_4.png" /></figure>

NLME

<figure><img src="image_5.png" /></figure>

## Model-based
individual parameters

TK

BK

<figure><img src="image_6.png" /></figure>

## Outputs

### Individual prognosis

* C-index

* Survival

## Raw baseline data

Clinical characteristics
p = 73

Genomic and Transcriptomic
p = 395 and 58,311

Feature
selection

## Minimal signature

Clinical
characteristics

Genomic and
Transcriptomic

## Machine
Learning

### Clinical trial

* Survival curve

* Hazard ratio
Training and test sets

<figure><img src="image_7.png" /></figure>
# Outline

1. Data

2. Modeling the kinetics of longitudinal data

1. Tumor size kinetics

2. Blood markers kinetics

3. Machine learning

1. Model development

2. Individual predictions

3. Trial-level predictions
# Tumor kinetics

100 random patients

<figure><img src="image_8.png" /></figure>

## Individual kinetics

Double (sum of two) exponential model

<figure><img src="image_9.png" /></figure>

$$S^i(t) = S0^i (e^{-KS^i t} + e^{KG^i t} - 1)$$
# Mixed-effects modeling for population longitudinal data

* **Observation model**

$$y_j^i = M(t_j^i; \theta^i) + \varepsilon_j^i, \varepsilon_j^i \sim \mathcal{N}(0, \sigma_j^i)$$

* **Inter-individual variability model**

$$\ln(\theta^i) = \ln(\theta_{\text{pop}}) + \eta^i, \eta^i \sim \mathcal{N}(0, \omega^2)$$

$M$ = structural model

$y_j^i$ = observation at time $t_j^i$ in

patient $i$

$\varepsilon_j^i$ = error model (constant (TK))

or proportional (BK))

<figure><img src="image_10.png" /></figure>

* Estimation of the population parameters using the SAEM algorithm for likelihood maximization

* Estimation of the individual empirical Bayes estimates (EBEs) from the maximum a posteriori estimator

* Fits performed using Monolix 2020R1 and scripted using the R Monolix API
# The double exponential model was able to describe individual TK

<figure><img src="image_11.png" /><figcaption>## 2 best fits in each response group</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>## 20 randomly selected individual fits</figcaption></figure>

<figure><img src="image_13.png" /></figure>
<figure><img src="image_14.png" /></figure>

# Kinetics of longitudinal markers

Roche

* Tumor kinetics (TK) : double-exponential model<fnref n="1" />,²

* BK time courses : empirical models³

* Is there any significant kinetics?

* Does it carry predictive value?

* constant? linear? hyperbolic? double-exponential?

<figure><img src="image_15.png" /></figure>

<figure><img src="image_16.png" /></figure>

<figure><img src="image_17.png" /></figure>

* Individual TK and PD model parameters = inputs of the Machine Learning (ML) algorithm

* Model-derived baseline parameters were excluded (baseline markers already in clinical category)

<footnote n="1">¹Stein et al., 2008 (PSA), ²Claret et al. 2018 (SLD, atezolizumab), ³Gavrilov et al. 2020 (NLR, durvalumab)</footnote>
# Is there any kinetic pattern? Yes !

Corrected Bayesian Information Criterion (BICc x 10³) for four empirical kinetic models of BK. b : standard deviation of the proportional error model

<table><thead><tr><th rowspan="2">Model</th><th colspan="2">Albumin</th><th colspan="2">CRP</th><th colspan="2">LDH</th><th colspan="2">Neutrophils</th></tr><tr><th>BICc</th><th>b</th><th>BICc</th><th>b</th><th>BICc</th><th>b</th><th>BICc</th><th>b</th></tr></thead><tbody><tr><td>Double-exponential</td><td>48.4</td><td>0.058</td><td>28.8</td><td>0.21</td><td>39.9</td><td>0.56</td><td>102</td><td>0.14</td></tr><tr><td>Hyperbolic</td><td>48.0</td><td>0.056</td><td>29.7</td><td>0.22</td><td>40.9</td><td>0.62</td><td>103</td><td>0.14</td></tr><tr><td>Linear</td><td>49.4</td><td>0.063</td><td>30.0</td><td>0.23</td><td>42.4</td><td>0.70</td><td>105</td><td>0.17</td></tr><tr><td>Constant</td><td>49.7</td><td>0.065</td><td>31</td><td>0.25</td><td>43.0</td><td>0.74</td><td>106</td><td>0.18</td></tr></tbody></table>

<figure><img src="image_18.png" /></figure>
Does it carry predictive value? Yes !

<figure><img src="image_19.png" /></figure>
# Outline

1. Data

2. Modeling the kinetics of longitudinal data

1. Tumor size kinetics

2. Blood markers kinetics

3. Machine learning

1. Model development

2. Individual predictions

3. Trial-level predictions
# Methodology - Machine learning model development

* **Preprocess**

Drop zero-variance or >25% NA columns, dummification, NA imputation, scaling

* **Dimensionality reduction for RNAseq data (bootstrap LASSO)**

* **Features selection:**

* 5 methods: **LASSO**, random survival forest (RSF) importance, Cox-based and stepwise forward/backward

* 3 strategies : i) all variables, ii) per feature set and iii) pooled selected sets

* **4 survival algorithms tested:**

Cox, Cox and accelerated failure time with gradient boosting and **random survival forest (RSF)**

* **Evaluation of machine learning models**

* **Model development:** 10-fold cross-validation

C-index, calibration curves and 12-months survival classification metrics

* **Study-level predictions:** survival curves, **hazard ratios**

**Working principle:** need for a **minimal signature** model with limited number of easily measurable variables
# Minimal baseline (mBSL) signature kinetics-ML (kML) model

## Minimal baseline signature

<figure><img src="image_20.png" /></figure>

<figure><img src="image_21.png" /></figure>

* All features sorted using LASSO

* Incremental models with increasing number of features

* Minimal set of features that reaches the plateau

⇒ 11 mBSL variables

⇒ easily clinically accessible

⇒ RNAseq does not carry predictive value

⇒ discarded
Predictive performances on the test set

<figure><img src="image_22.png" /></figure>

<figure><img src="image_23.png" /></figure>

<figure><img src="image_24.png" /></figure>

6 months

<figure><img src="image_25.png" /></figure>

12 months

<figure><img src="image_26.png" /></figure>

24 months

<figure><img src="image_27.png" /></figure>

c-index = 0.79

AUC-12 months = 0.86

Current state of the art:
ROPRO (BSL) c-index = 0.7

<figure><img src="image_28.png" /></figure>

<figure><img src="image_29.png" /></figure>
# Outline

1. Data

2. Modeling the kinetics of longitudinal data

1. Tumor size kinetics

2. Blood markers kinetics

3. Machine learning

1. Model development

2. Individual predictions

3. Trial-level predictions
# kML for prediction of the outcome of the OAK phase III trial

* The model **predicted well** survival curves of ATZ and **control arm** (docetaxel) from OAK

* The model was able to **predict ATZ survival benefit** over chemotherapy in OAK

<table><thead><tr><th>Observed HR (95%CI)</th><th>kML Prediction HR (95%PI)</th></tr></thead><tbody><tr><td>0.765<br>(0.64 - 0.913)</td><td>0.765<br>(0.692-0.829)</td></tr></tbody></table>

* Note: the **model PI** is **narrower than the data CI** (!!)

* The relationship between TK and PD metrics and survival is **not drug specific**

<figure><img src="image_30.png" /></figure>
<figure><img src="image_31.png" /></figure>

# kML predicts phase 3 trial results from early data

<figure><img src="image_32.png" /></figure>

arm

<figure><img src="image_33.png" /></figure>

Model prediction mean
Data

<figure><img src="image_34.png" /></figure>

Model

<figure><img src="image_35.png" /></figure>

Data

kML Predicted

10 weeks
(2.5 months)

<figure><img src="image_36.png" /></figure>

25 weeks
(6.25 months)

<figure><img src="image_37.png" /></figure>

60 weeks
(15 months)

<figure><img src="image_38.png" /></figure>

Full kinetics
(end of study)

<figure><img src="image_39.png" /></figure>

<figure><img src="image_40.png" /></figure>

<figure><img src="image_41.png" /></figure>

<figure><img src="image_42.png" /></figure>

<figure><img src="image_43.png" /></figure>

<table><tr><td>Data HR (95% CI)</td><td>NA (NA - NA)</td><td>0.486 (0.0863 - 2.73)</td><td>0.697 (0.496 - 0.979)</td><td>0.778 (0.65 - 0.931)</td></tr><tr><td>Predicted HR (95% PI)</td><td>0.836 (0.547 - 1.4)</td><td>0.814 (0.64 - 0.994)</td><td>0.849 (0.746 - 0.934)</td><td>0.784 (0.7 - 0.842)</td></tr><tr><td>Number of patients (DTX - ATZ)</td><td>23 - 30</td><td>125 - 127</td><td>342 - 368</td><td>354 - 377</td></tr><tr><td>Data points (TK/Alb/CRP/LDH/ Neutrophils)</td><td>1/3/3/2/2.5 - 1/2/2/2/2</td><td>2/4/4/4/4 - 2/4/4/4/4</td><td>3/6/5/6/7 - 3/7/6/7/7</td><td>4/8/6/7/8 - 4/9.5/9/9/10</td></tr></table>
# Summary

* Kinetic models based on simple, routinely available blood markers have prognostic value

* Could be used as non-invasive surrogate markers of response

* Modeling >> data only

* The combination of NLME and ML allowed to take the best of the two approaches in order to predict survival

* NLME to account for longitudinal data

* ML and data science methods to build multivariable models from a large number of features

* Minimal signature: 11 baseline clinical features + TK (1 variable – 2 parameters) + BK (4 variables – 8 parameters)

* Application to clinical oncology could improve personalized health care

* Application to early prediction of trial outcome could:

* Save a lot of effort (..... and money!)

* Be used to better stratify patients

* Identify signal in early-phase trials
<figure><img src="image_44.png" /></figure>

Beyond classical biomarkers:
PIONeeR – QUANTIC and SChISM
Inria

THE
PIONEER
PROJECT

# The PIONeeR and QUANTIC project

Precision Immuno-Oncology for advanced Non-small cell lung cancer patients with PD-1 ICI Resistance

<figure><img src="image_45.png" /></figure>

<figure><img src="image_46.png" /></figure>

INSTITUT
NATIONAL
DU CANCER

<figure><img src="image_47.png" /></figure>

Inserm

Institut national
de la santé et de la recherche médicale

Aviesan

Alliance nationale
pour les sciences de la vie et de la santé
ITMO Cancer

450 pts

<figure><img src="image_48.png" /></figure>

24,287 variables

1,581 markers

<figure><img src="image_49.png" /></figure>

<figure><img src="image_50.png" /></figure>
# Multimodal data integration for prediction of early progression from baseline variables through supervised ML: 2nd line monotherapy

## Predictive metrics (logistic regression)

<figure><img src="image_51.png" /></figure>

**Current state of the art:**
AUC = 0.64 (PDL1), 0.65 (TMB)

## Early progressors signature = 37 biomarkers selected by LASSO

<figure><img src="image_52.png" /></figure>

**Color code**

- Circulating blood cells

- Tumor biomarker

- Vascular origin

- Routine blood tests

- Soluble immune marker
# Challenges in biomarker predictive modeling

<table><thead><tr><td>Difficulties</td><td>Remedies</td></tr></thead><tbody><tr><td>A priori complex interactions</td><td>Nonlinear models</td></tr><tr><td>Many, correlated data</td><td>Feature selection</td></tr><tr><td>Missing data</td><td>Imputation</td></tr><tr><td>Small n, Large p setting</td><td>Generative modelling</td></tr></tbody></table>

Tackle these problems one-by-one. Use theory if available, do simulations otherwise.
# FEATURE SELECTION
## SIGNATURE - METHODO

### Signature :

* **Problem = stability – reproductibility**

* **Solution = double bootstrap LASSO = BOBO**

* **Problem = correlation and hierarchical structure**

* **Solution 1 = clustering-based and biologically informed dimension reduction**

<figure><img src="image_53.png" /></figure>

inconclusive

* **Solution 2 = hierarchical neural network**

* **Sol.**

<figure><img src="image_54.png" /></figure>

<figure><img src="image_55.png" /></figure>

<figure><img src="image_56.png" /></figure>

## BOBO

<figure><img src="image_57.png" /></figure>

## LASSO

<figure><img src="image_58.png" /></figure>
# Missing data: simulation studies
## Deep learning survival models

Simulation data with 2 groups, interaction impacting survival, control of the rate of missing data

<figure><img src="image_59.png" /></figure>

<figure><img src="image_60.png" /></figure>

C = Harrel's
C-statistic

⇒ simple median imputation can be no worse than more complicated methods
# Data augmentation

## Synthetic data in ML loop

<figure><img src="image_61.png" /></figure>

Data augmentation

<figure><img src="image_62.png" /></figure>

Generative model*

<figure><img src="image_63.png" /></figure>

Sample

Synthetic train set (n=?)

*Including survival time and censorship
# Data augmentation
## Preliminary results

<figure><img src="image_64.png" /></figure>

P. Dufosse

**Learn** better discriminative models
(ML efficacy > 1)

<figure><img src="image_65.png" /><figcaption>Predict OS from PIONeeR clinical biomarkers, all treatment lines</figcaption></figure>

ML efficacy < 1

<figure><img src="image_66.png" /><figcaption>Predict OS from German Breaster Cancer Study group dataset. (Open data)</figcaption></figure>

ML efficacy ≈ 1

<figure><img src="image_67.png" /><figcaption>Predict OS from PIONeeR basic clinical variables, all treatment lines</figcaption></figure>
# Summary: machine learning lessons for biomarkers analysis

* Detecting effects (and generalize) is hard:

* **few**, possibly **missing** observations,

* But learned models **still predict well**

* many **correlated** biomarkers,

* possibly **complex interactions**

* Many tests, methods, models to try

* Need for **reproducibility** and **benchmarking**

* **Generative modeling** can help

* Specific constraints of biomarker discovery

* Need for **interpretability**

* Need for **minimal signature**
# The QUANTIC project: methodology

<figure><img src="image_68.png" /></figure>

<figure><img src="image_69.png" /></figure>

<figure><img src="image_70.png" /></figure>

$$p = 407 + 20,000$$

$$p = 284 \times 6$$

## Methodological challenges

* To date, no validated model of tumor-immune-drug interaction on clinical data

* Deal with data heterogeneity

* Integrate high-dimensional data:

* at baseline

* longitudinally

## Leads

* Mechanistic models on reduced number of key immune players

* Include individual lesion kinetics and appearance

* For high-dimensional static data: use ML algorithms as covariate models in mixed-effects models

* For high-dimensional dynamic data: combine dimension reduction, mechanistic modeling and mixed-effects

* Use Bayesian and joint modeling
# SChISM
## Size CfDNA Immunotherapies Signature Monitoring

**APHM: S. Salas, Inria / Inserm: S. Benzekry, L. Nguyen, Adelis**

<figure><img src="image_71.png" /></figure>

<figure><img src="image_72.png" /></figure>

<figure><img src="image_73.png" /></figure>

<figure><img src="image_74.png" /></figure>

Institut
Laènnec,
sciences numériques
et IA pour la santé
Aix-Marseille Université
<figure><img src="image_75.png" /></figure>

# Intra-patient variability over time

<figure><img src="image_76.png" /><figcaption>## Patient 700-13
Time from first immunotherapy (months): 0</figcaption></figure>
# Outline

## Kinetics-machine learning for prediction of OS

1. Data

2. Modeling the kinetics of longitudinal data

1. Tumor size kinetics

2. Blood markers kinetics

3. Machine learning

1. Model development

2. Individual predictions

3. Trial-level predictions

## Modeling metastasis

1. Clinical problems and data

2. Mathematical modeling

3. Results

1. Experimental model validation

2. Brain metastasis in NSCLC

3. Prediction of breast metastatic relapse
# Modeling metastasis

<figure><img src="image_77.png" /></figure>

## Mathematical models

$$ egin{cases} 
\partial_t \rho(t, v) + \partial_v (g(v)\rho(t, v)) = 0 \\ 
g(V_0)\rho(t, V_0) = d(V_p(t)) \left( + \int_{V_0}^{+\infty} d(v)\rho(t, v)dv \right) \\ 
\rho(0, v) = \rho^0 
\end{cases} $$

## Clinical problem

<figure><img src="image_78.png" /></figure>

<figure><img src="image_79.png" /></figure>

<figure><img src="image_80.png" /></figure>

## Data

<figure><img src="image_81.png" /></figure>
<figure><img src="image_82.png" /></figure>

# Clinical problem

<figure><img src="image_83.png" /></figure>

## Early-stage breast cancer

* 94% of cases are local or regional at diagnosis but 30% will relapse

* Personalize the adjuvant (chemo)-therapy

<figure><img src="image_84.png" /></figure>

## Brain metastases in non-small cell lung cancer (NSCLC)

* Decide whether to use whole brain radiation therapy or just (stereotactic) surgery

<figure><img src="image_85.png" /></figure>
Clinical problem

<figure><img src="image_86.png" /></figure>

<figure><img src="image_87.png" /></figure>

<figure><img src="image_88.png" /></figure>

Mathematical models

$$
\begin{cases}
\partial_t \rho(t, v) + \partial_v (g(v)\rho(t, v)) = 0 \\
g(V_0)\rho(t, V_0) = d(V_p(t)) \left( + \int_{V_0}^{+\infty} d(v)\rho(t, v)dv \right) \\
\rho(0, v) = \rho^0
\end{cases}
$$

<figure><img src="image_89.png" /></figure>

Data

<figure><img src="image_90.png" /></figure>
# Data

<figure><img src="image_91.png" /></figure>

## Experimental

<figure><img src="image_92.png" /></figure>

n > 400 animals with different treatments and schedules

SMARTc wet lab
collaboration with J. Ebos, Roswell Park Cancer Institute, Buffalo, NY

## Clinical

* **Number and sizes** of brain metastasis in individual NSCLC patients (n=31)

<figure><img src="image_93.png" /></figure>

<figure><img src="image_94.png" /></figure>

* 3 clinical databases of metastatic relapse in breast cancer

**Bergonié (N = 591):**

* Tumor size

* age

* Nodes status

* grade

* ER

* PR

* HER2

* Ki67

**IPC (N = 676):**

* Tumor size

* age

* Nodes status

* grade

* ER

* PR

* HER2

* Ki67

* molecular subtype

**AP-HM (N = 167):**

* Tumor size

* age

* Nodes status

* grade

* ER

* PR

* TK

* UPA

* PAI-1
Clinical problem

<figure><img src="image_95.png" /></figure>

<figure><img src="image_96.png" /></figure>

<figure><img src="image_97.png" /></figure>

Mathematical models

$$
\begin{cases}
\partial_t \rho(t, v) + \partial_v (g(v)\rho(t, v)) = 0 \\
g(V_0)\rho(t, V_0) = d(V_p(t)) \left( + \int_{V_0}^{+\infty} d(v)\rho(t, v)dv \right) \\
\rho(0, v) = \rho^0
\end{cases}
$$

<figure><img src="image_98.png" /></figure>

Data

<figure><img src="image_99.png" /></figure>
Mathematical models

<figure><img src="image_100.png" /></figure>

Metastatic burden (total number of metastatic cells)

$$M(t) = \int_{V_0}^{+\infty} v \rho(t, v) dv = \int_{0}^{t} d(V_p(t-s)) V(s) ds$$

Number of metastases with size larger than the visible size $V_{vis}$

$$N_{vis}(t) = \int_{V_{vis}}^{+\infty} \rho(t, v) dv = \int_{0}^{t-\tau_{vis}} d(V_p(t)) dt$$

Growth rates of primary and secondary tumors $g_p$ and $g$

$$\frac{dV_p}{dt} = g_p(t, V_p) \quad \text{ex: Gompertz}$$

Dissemination rate $d(V_p) = \mu V_p^\gamma$

Size distribution of the metastases $\rho(t,v)$ [Iwata et al., 2000]

$$\begin{cases} \partial_t \rho(t, v) + \partial_v(g(v)\rho(t, v)) = 0 \\ g(V_0)\rho(t, V_0) = d(V_p(t)) \left( + \int_{V_0}^{+\infty} d(v)\rho(t, v)dv \right) \\ \rho(0, v) = \rho^0 \end{cases}$$

<figure><img src="image_101.png" /></figure>

Time to relapse (TTR) = time from diagnosis to first visible met

$$TTR = \inf \{t > 0 : N_{vis}(t_{diag} + t) \ge 1\}$$
<figure><img src="image_102.png" /></figure>

# Results: experimental data

**Methodology:** mixed-effects modeling

Individual *i*

Time *t*<sub>j</sub>

$$y_j^i = M(t_j^i, \theta^i) + \varepsilon_j^i, \quad \varepsilon_j^i \sim \mathcal{N}(0, \sigma_j^i)$$

$$\theta^i = \theta_{\text{pop}} + \eta^i, \quad \eta^i \sim \mathcal{N}(0, \omega^2)$$

* Likelihood maximization performed using the SAEM algorithm

* Main difficulty: PDE model

* Solution: fast Fourier transform for computation of metastatic burden

## Results:

* *g* = Gompertz for breast, exponential for kidney, *γ* = 1

* same growth for PT and mets

* good practical identifiability (all rse < 30%)

* parameter *μ* is critical for inter-animal variability (CV = 176%)

* insights on quantitative link between PT size and proba of relapse

## Fit

<figure><img src="image_103.png" /></figure>

## Prediction

<figure><img src="image_104.png" /></figure>

<figure><img src="image_105.png" /></figure>

<figure><img src="image_106.png" /></figure>

<figure><img src="image_107.png" /></figure>
# Results: individual clinical data

## Methodology

### Primary tumor

$$ \frac{dS_p}{dt} = \alpha_1 S_p - \kappa e^{-\frac{t-T_d}{t_{res}}} \ln^2 S_p, \quad \forall t \ge T_d $$

<figure><img src="image_108.png" /></figure>

### Metastases

$$ f(t, v) = \int_{v}^{+\infty} \rho(t, u) du = N(t - \tau(v)) $$

<figure><img src="image_109.png" /></figure>

* Main difficulties:

no data on PT pre-treatment + fit PDE to discrete data

## Results

* Dormancy, estimated to 133 days ± 4.2%

* Good fit of the entire longitudinal data (47 BM sizes over 6 time points) with only 3 parameters.

* Onset of BM 14-19 months before diagnosis

<figure><img src="image_110.png" /></figure>

<figure><img src="image_111.png" /></figure>

## Ongoing (collab. TU München)

* Extend to larger cohort: complex treatment histories

* Include effect of treatment ⇒ g(t, v) ⇒ nontrivial difficulties for param estimation
# Results: prediction of clinical metastatic relapse

## Methodology

* Reduce to two parameters: growth ($\alpha$) and dissemination ($\mu$)

$$ \ln(T^i) = \ln(TTR(V_{diag}^i; \alpha^i, \mu^i)) + \varepsilon^i, \quad \varepsilon^i \sim \mathcal{N}(0, \sigma^2) \quad S(t|\alpha^i, \mu^i) = \mathbb{P}(T^i > t|\alpha^i, \mu^i) $$

$$ \ln(\mu^i) = \ln(\mu_{pop}) + \beta_{\mu}^{T} \mathbf{x}_{\mu}^{i} + \eta_{\mu}^{i}, \quad \eta_{\mu}^{i} \sim \mathcal{N}(0, \omega_{\mu}^{2}) $$

$$ \ln(\alpha^i) = \ln(\alpha_{pop}) + \beta_{\alpha}^{T} \mathbf{x}_{\alpha}^{i} + \eta_{\alpha}^{i}, \quad \eta_{\alpha}^{i} \sim \mathcal{N}(0, \omega_{\alpha}^{2}) $$

* Main difficulties: selection of covariates + survival data

* Solution: random survival forest + adapt for censoring

## Results

* Similar predictive power for mechanistic VS classical Cox or survival random forest (c-index = 0.63-0.70 vs 0.67-0.71 vs 0.67-0.72) or other ML

* Gives mechanistic insights on metastatic biology

<figure><img src="image_112.png" /></figure>

<figure><img src="image_113.png" /></figure>

<figure><img src="image_114.png" /></figure>

<table><thead><tr><th>Parameter</th><th>Estimate</th><th>r.s.e. (%)</th><th>p-value</th></tr></thead><tbody><tr><td>log α<sub>non</sub></td><td>-8.883</td><td>10.151</td><td></td></tr><tr><td>β<sub>Ki67,α</sub></td><td>0.086</td><td>27.376</td><td>2.59 · 10<sup>-4</sup></td></tr><tr><td>β<sub>HER2,α</sub></td><td>0.029</td><td>42.833</td><td>0.020</td></tr><tr><td>β<sub>CD44,α</sub></td><td>0.011</td><td>60.816</td><td>0.1</td></tr><tr><td>β<sub>TRIO,α</sub></td><td>0.016</td><td>58.119</td><td>0.085</td></tr><tr><td>log μ<sub>pop</sub></td><td>-26.342</td><td>3.696</td><td></td></tr><tr><td>β<sub>EGFR,μ</sub></td><td>0.039</td><td>47.527</td><td>0.035</td></tr><tr><td>σ</td><td>0.606</td><td>23.104</td><td></td></tr><tr><td>ω<sub>α</sub></td><td>2.062</td><td>22.715</td><td></td></tr><tr><td>ω<sub>μ</sub></td><td>3.563</td><td>16.759</td><td></td></tr></tbody></table>

<figure><img src="image_115.png" /></figure>

<figure><img src="image_116.png" /></figure>

<figure><img src="image_117.png" /></figure>
TAKE HOME MESSAGE

Dynamics matter! Modeling can help :)

Big data? Statistics can help :)

Combining both can bring relevant answers to biological and clinical research
# Acknowledgements

<figure><img src="image_118.png" /></figure>

* René Bruno

* Candice Jamois

* Suresh Vatakuti

* François Mercier

<figure><img src="image_119.png" /></figure>

* Célestin Bigarré

* Mélanie Karlsen

* Linh Nguyen

MDs: F. Barlesi, L. Greillier, S. Salas, X. Muracciole, F. Bertucci, L. Padovani, E. Simon, P. Tomasini,...

<figure><img src="image_120.png" /></figure>

Thank you for your attention!

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