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# Mechanistic modeling of the combination between cytotoxics and anti-angiogenics for sequence optimization

Sébastien Benzekry

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Sébastien Benzekry. Mechanistic modeling of the combination between cytotoxics and anti-angiogenics for sequence optimization. Quantitative Systems Pharmacology, Leiden University, Apr 2022, Leiden (Netherlands), Netherlands. hal-05239911

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https://inria.hal.science/hal-05239911v1

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Distributed under a Creative Commons Attribution 4.0 International License
Mechanistic modeling of the combination between
cytotoxics and anti-angiogenics for sequence optimization

Sébastien Benzekry

QSPC 2022, Leiden, April 21st

Inria – Inserm team COMPO

COMPutational pharmacology and clinical Oncology

Centre Inria Méditerranée

Centre de Recherches en Cancérologie de Marseille
Outline

1. Modeling at the tumor level

2. Modeling at the organism level : metastasis

1. Ortho-surgical models of post-surgical metastatic development

2. Modeling neo-adjuvant treatment
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# Vascular normalization: a time window for improved pharmacokinetics?

* Bevacizumab = anti-VEGF monoclonal antibody ⇒ **anti-angiogenic** action (first approved in 2004)

* Only proved clinical efficacy when **combined (concomitantly) with cytotoxics**

* Possible explanation: transient **normalization** of the otherwise abnormal (leaky, tortuous) vascular architecture

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

Vakoc et al., Jain, 2009, Nat Med

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Jain, Nat Med, 2001

## Question

What is the **optimal time gap** between administration of bevacizumab and cytotoxic chemotherapy?

How to capture **inter-individual variability** for designing **personalized therapies**?
# Hypothesis: sequential use of bevacizumab with chemotherapy would achieve better efficacy

Current beva-chemo regimen are underpowered

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## Breast cancer model

Mollard et al. (Benzekry), Oncotarget 2017

## Lung cancer model

Experiment 1

Experiment 2

Calibration

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Prediction

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# Experiment-1: 4 days gap

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* Bevacizumab 20 mg/kg

* Pemetrexed 100 mg/kg

* Cisplatin 3 mg/kg

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3 cycles

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* Human NSCLC H460-Luc+ xenograft

* Subcutaneous graft

* Matrigel support

* Follow-up

* Bioluminescence imaging

* Weight monitoring

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Sequential administration Beva then Chemo improves response and survival

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-71.2% tumor size at study conclusion (day 60)

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⇒ Sequential use increases survival by 44%
# Semi-mechanistic mathematical model

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Simeoni et al., Rocchetti, Cancer Res, 2004

Benzekry et al., CRAS, 2012

Mollard et al., Benzekry, Oncotarget, 2017

Imbs et al., Benzekry, CPT: Pharmacometrics Syst Pharmacol, 2018

+ PK models for beva A(t) and CT C(t) concentrations
# Model fits: individual + population level (NLME)

## Control

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## Simultaneous

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## Median growth curves

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## Sequential C/B

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## Sequential B/C

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Parameters identifiability

<table><thead><tr><td>Parameters</td><td>Description</td><td>Units</td><td>Estimates</td><td>RSE<sup>a</sup> (%)</td><td>IAV<sup>b</sup> %</td></tr></thead><tbody><tr><td>α</td><td>Proliferation rate</td><td>Day<sup>-1</sup></td><td>0.767</td><td>8</td><td>8.62</td></tr><tr><td>β</td><td>Exponential decay rate of the relative tumor growth rate (Gompertz model)</td><td>Day<sup>-1</sup></td><td>0.037</td><td>10</td><td>57.3</td></tr><tr><td>γ</td><td>Baseline effect of the chemotherapy</td><td>(mg/g)<sup>-1</sup>.day<sup>-1</sup></td><td>1</td><td>—</td><td>—</td></tr><tr><td>δ</td><td>Cytotoxics efficacy improvement following vascular normalization</td><td>(mg/mL)<sup>-1</sup></td><td>1,200</td><td>36</td><td>0</td></tr><tr><td>τ</td><td>Delay parameter for dynamics of Q</td><td>Day</td><td>2</td><td>20</td><td>10</td></tr><tr><td>k</td><td>Delay of the tumor cell loss following chemotherapy</td><td>Day<sup>-1</sup></td><td>0.3</td><td>—</td><td>—</td></tr><tr><td>Σ</td><td>Exponential error parameter</td><td>—</td><td>0.951</td><td>4</td><td>—</td></tr></tbody></table>
Prediction of the optimal delay

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⇒ to be **tested experimentally**

⇒ **personalized scheduling**
Model fit to the second experimental dataset

<figure><img src="image_31.png" /><figcaption>Log Tumor Size (RFU*)</figcaption></figure>

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Time (days)
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Now, in real life, cancer patients treated
with systemic therapy are usually in
advanced stages

*What about the effect on metastases?*
# A mathematical modeling framework for metastases

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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]

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

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

$$TTR = \inf \{t > 0 : N_{vis}(t_{diag} + t) \ge 1\}$$
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# Experimental data and questions

## Data

* Clinically relevant ortho-surgical **animal models** of metastasis

* **Longitudinal** measurements of primary tumor size and total **metastatic burden**

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## Questions

* Minimal model of metastatic development. **Dissemination** law? Differences between mets and primary tumor **growth**?

* Quantify the **inter-subject variability** of metastatic development

* What is the impact of the primary tumor size at surgery on metastatic development and survival?

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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, r = 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

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Prediction

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* Data primary tumor
  Median model primary tumor
  10th and 90th percentiles model primary tumor
* Data metastatic burden
  Median model metastatic burden
  10th and 90th percentiles model metastatic burden
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# Differential effects of anti-angiogenic therapies between primary tumor and metastases

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Cancer Cell
Report

Published online: October 31, 2014

Research Article

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TRANSPARENT PROCESS

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OPEN ACCESS

EMBO
Molecular Medicine

## Accelerated Metastasis after Short-Term Treatment with a Potent Inhibitor of Tumor Angiogenesis

John M.L. Ebos¹,² Christina R. Lee,¹ William Cruz-Munoz,¹ Georg A. Bjarnason,³ James G. Christensen,⁴ and Robert S. Kerbel¹,²,*

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## Neoadjuvant antiangiogenic therapy reveals contrasts in primary and metastatic tumor efficacy

John M L Ebos¹,*, Michalis Mastri¹, Christina R Lee², Amanda Tracz², John M Hudson², Kristopher Attwood³, William R Cruz-Munoz², Christopher Jedeszko², Peter Burns²,⁴ & Robert S Kerbel²,⁴

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

## Aims

Test possible hypotheses for the **differential effects** of sunitinib on primary tumor and metastases

Build a **minimal kinetics-pharmacodynamics (K-PD) model** of neoadjuvant sunitinib treatment

Assess the predictive power of biomarkers on the model parameters using **machine learning** algorithms
Modeling schematic

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# Simulations of the effect of neoadjuvant sunitinib treatment on metastases suggest no effect on growth of metastases

* Parameter values from the previous study on control groups ⇒ simulations are pure mechanistic predictions

* In first approximation, the effect of the drug was modeled by setting the tumor growth rate to zero during the phase of treatment

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A K-PD model for the effect of Sunitinib

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$$g_p^{treat}(t, v) = g_p(t, v)(1 - kC(t))$$

Primary tumor

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Metastatic burden

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<table><thead><tr><th>Parameter (Unit)</th><th>Meaning</th><th>Median</th><th>CV (%)</th><th>r.s.e. (%)</th></tr></thead><tbody><tr><td>μ (cell<sup>-1</sup>·day<sup>-1</sup>)</td><td>dissemination coefficient</td><td>2.12·10<sup>-11</sup></td><td>1.48·10<sup>3</sup></td><td>17.3</td></tr><tr><td>α (day<sup>-1</sup>)</td><td>gompertzian growth parameter</td><td>1.94</td><td>18.1</td><td>2</td></tr><tr><td>β (day<sup>-1</sup>)</td><td>gompertzian growth parameter</td><td>0.0911</td><td>19.7</td><td>2.21</td></tr><tr><td>λ (day<sup>-1</sup>)</td><td>in vitro proliferation rate</td><td>0.837 (fixed)</td><td>-</td><td>-</td></tr><tr><td>k (L/mg)</td><td>drug efficacy</td><td>0.446</td><td>32.1</td><td>6.34</td></tr><tr><td>k<sub>e</sub> (day<sup>-1</sup>)</td><td>drug elimination rate</td><td>3.26 (fixed)</td><td>-</td><td>-</td></tr><tr><td>V<sub>d</sub> (L/kg)</td><td>volume of distribution</td><td>12 (fixed)</td><td>-</td><td>-</td></tr></tbody></table>

η-shrinkage < 20% for each parameter ⇒ the individual parameter estimates can be considered reliable
# Increased duration of neo-adjuvant treatment could substantially increase probability of metastatic relapse

D = 60 mg/kg

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A

D = 120 mg/kg

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D = 240 mg/kg

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B

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<figure><img src="image_60.png" /></figure>
The biomarkers considered in this study have limited predictive power for metastatic potential as defined by μ

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Theoretical simulations of CT/AA combination in the metastatic setting

<figure><img src="image_67.png" /></figure>

A

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

B

⇒ Impact of the CT/AA sequence on outcome
might differ between the primary tumor and the
metastases

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

C

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

D
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Discussion

* Construction of a semi-mechanistic model for the **combination of CT and AA** drugs in oncology

* Based on a biological mechanism of transient **vascular normalization** following bevacizumab treatment

* The model was able to quantify experimentally-observed differences between CT/AA and AA/CT sequential administrations

* **Predictions** of the model in terms of optimal sequence were confirmed in a second set of experiments

* **Systemic therapies** should be further modeled at the **systemic (organism) scale**

* Include the **dissemination** process in addition to growth

* Consider not just one tumor but a **population of tumors** (cf individual lesion TGI models, e.g. *Mercier et al.*, J PK PD, 2020)

* In addition, there are **tumor-tumor systemic interactions** (cf *Benzekry et al.*, *Cancer Res*, 2017)

* A modeling framework for populations of tumors was proposed and compared to experimental data

* **Post-surgical** metastatic development in untreated animals

* Extension to the **neoadjuvant** setting
# Acknowledgments

## Experimental therapeutics

**J. Ciccolini**

S. Mollard

A. Boyer

S. Giacommetti

COMPO

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

## Preclinical data of ortho-surgical animal models of metastasis

**J. Ebos**

M. Mastri

A. Tracz

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

## Modeling

D.C. Imbs

R. El Cheikh

J. Mochel

C. Nicolò

B. Schneider

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

Multiple **open positions** in COMPO !!

* Tenure researcher

* Postdoc

* PhD

* Engineer

<figure><img src="image_75.png" /></figure>

sebastien.benzekry@inria.fr
@SBenzekry

Thank you for your attention !
# Results: individual clinical data

## Methodology

### Primary tumor

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

<figure><img src="image_76.png" /></figure>

### Metastases

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

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* 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

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<figure><img src="image_79.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 ($\mu$)

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

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

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

* 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

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<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>pop</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_83.png" /><figcaption>Mechanistic</figcaption></figure>

<figure><img src="image_84.png" /><figcaption>Random survival forest</figcaption></figure>

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