Mathematical and numerical analysis of a model for anti-angiogenic therapy in metastatic cancers

Benzekry, Sébastien

doi:10.1051/m2an/2011041

Cited by 20 publications

(28 citation statements)

References 23 publications

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“…This includes optimizing dosing regimen through better understanding of PK/PD relationships (35). Several extensions of the metastatic model including a treatment have been proposed (36)(37)(38). The next step of this work should be to confront model predictions with data in a therapeutic framework.…”

Section: Resultsmentioning

confidence: 99%

Mathematical Modeling of Tumor Growth and Metastatic Spreading: Validation in Tumor-Bearing Mice

et al. 2014

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Defining tumor stage at diagnosis is a pivotal point for clinical decisions about patient treatment strategies. In this respect, early detection of occult metastasis invisible to current imaging methods would have a major impact on best care and long-term survival. Mathematical models that describe metastatic spreading might estimate the risk of metastasis when no clinical evidence is available. In this study, we adapted a top-down model to make such estimates. The model was constituted by a transport equation describing metastatic growth and endowed with a boundary condition for metastatic emission. Model predictions were compared with experimental results from orthotopic breast tumor xenograft experiments conducted in Nod/Scidg mice. Primary tumor growth, metastatic spread and growth were monitored by 3D bioluminescence tomography. A tailored computational approach allowed the use of Monolix software for mixed-effects modeling with a partial differential equation model. Primary tumor growth was described best by Bertalanffy, West, and Gompertz models, which involve an initial exponential growth phase. All other tested models were rejected. The best metastatic model involved two parameters describing metastatic spreading and growth, respectively. Visual predictive check, analysis of residuals, and a bootstrap study validated the model. Coefficients of determination were R 2 ¼ 0:94 for primary tumor growth and R 2 ¼ 0:57 for metastatic growth. The data-based model development revealed several biologically significant findings. First, information on both growth and spreading can be obtained from measures of total metastatic burden. Second, the postulated link between primary tumor size and emission rate is validated. Finally, fast growing peritoneal metastases can only be described by such a complex partial differential equation model and not by ordinary differential equation models. This work advances efforts to predict metastatic spreading during the earliest stages of cancer. Cancer Res; 74(22); 6397-407. Ó2014 AACR.

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Section: Resultsmentioning

confidence: 99%

Mathematical Modeling of Tumor Growth and Metastatic Spreading: Validation in Tumor-Bearing Mice

et al. 2014

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“…We use this parameter set as a basis point for exploration of the parameter space. Numerical simulations were performed adapting a previously developed discretization scheme based on the straightening of the characteristics [4]. …”

Section: Simulation Resultsmentioning

confidence: 99%

“…The approach we use is based on previous work [4,5] on modeling of metastatic development, in the framework of structured population dynamics.…”

Section: Introductionmentioning

confidence: 99%

A mathematical model of systemic inhibition of angiogenesis in metastatic development

2014

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Abstract. We present a mathematical model describing the time development of a population of tumors subject to mutual angiogenic inhibitory signaling. Based on biophysical derivations, it describes organism-scale population dynamics under the influence of three processes: birth (dissemination of secondary tumors), growth and inhibition (through angiogenesis). The resulting model is a nonlinear partial differential transport equation with nonlocal boundary condition. The nonlinearity stands in the velocity through a nonlocal quantity of the model (the total metastatic volume). The asymptotic behavior of the model is numerically investigated and reveals interesting dynamics ranging from convergence to a steady state to bounded non-periodic or periodic behaviors, possibly with complex repeated patterns. Numerical simulations are performed with the intent to theoretically study the relative impact of potentiation or impairment of each process of the birth/growth/inhibition balance. Biological insights on possible implications for the phenomenon of "cancer without disease" are also discussed.Résumé. Nous présentons un modèle mathématique décrivant le développement temporel d'une population de tumeurs en interactions mutuelles via des signaux d'inhibition de l'angiogénèse. Basé sur une dérivation biophysique, il décrit la dynamique,à l'échelle de l'organisme, qui résulte de l'influence relative de trois processus: naissance (dissémination de tumeurs secondaires), croissance et inhibition (de l'angiogénèse). Le modèle résultant est uneéquation aux dérivées partielles de transport non-linéaire avec condition aux limites non-locale. La non-linéarité réside dans une quantité non-locale (le volume métastatique global) présente dans la vitesse de croissance. Le comportement asymptotique du modèle estétudié numériquement et révèle des dynamiques intéressantes, allant de la convergence vers un point d'équilibreà des solutions périodiques exhibant des comportements non-triviaux. Des solutions bornées et non-périodiques sont aussi observées. Nous présentons une exploration numérique de l'espace des paramètres dans le but d'étudier théoriquement l'effet de la balance naissance/croissance/inhibition. De potentielles implications biologiques concernant le phénomène de "cancer sans maladie" sont aussi discutées.

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“…In this model, developed in [7,8,9], the tumor growth rate is bi-dimensional and integrates dynamics of the vascular capacity following the model of [26]. We first study and compare effect of the scheduling of anti-angiogenic drugs on both primary tumor growth and metastases.…”

Section: Xρ(t X)dxmentioning

confidence: 99%

“…Considering that all metastases are born with size x min = 1 cell = 10 −6 mm 3 and vascular capacity θ 0 we take N (σ) = δ σ=(1,θ 0 ) , the Dirac mass at point (1, θ 0 ). We refer to [7,8,9] for mathematical and numerical analysis of the resulting two-dimensional, measure-valued renewal equation. [26].…”

Section: Modelmentioning

confidence: 99%

Modeling the Impact of Anticancer Agents on Metastatic Spreading

Benzekry

André

Benabdallah

et al. 2012

Math. Model. Nat. Phenom.

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Abstract. Treating cancer patients with metastatic disease remains an ultimate challenge in clinical oncology. Because invasive cancer precludes or limits the use of surgery, metastatic setting is often associated with (poor) survival, rather than sustained remission, in patients with common cancers like lung, digestive or breast carcinomas. Mathematical modeling may help us better identify non detectable metastatic status to in turn optimize treatment for patients with metastatic disease. In this paper we present a family of models for the metastatic growth. They are based on four principles : to be as simple as possible, involving the least possible number of parameters, the main informations are obtained from the primary tumor and being able to recover the variety of phenomena observed by the clinicians. Several simulations of therapeutic strategies are presented illustrating possible applications of modeling to the clinic.

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Mathematical and numerical analysis of a model for anti-angiogenic therapy in metastatic cancers

Cited by 20 publications

References 23 publications

Mathematical Modeling of Tumor Growth and Metastatic Spreading: Validation in Tumor-Bearing Mice

Mathematical Modeling of Tumor Growth and Metastatic Spreading: Validation in Tumor-Bearing Mice

A mathematical model of systemic inhibition of angiogenesis in metastatic development

Modeling the Impact of Anticancer Agents on Metastatic Spreading

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