A mixture-like model for tumor-immune system interactions.

We introduce a mathematical model based on mixture theory intended to describe the tumor-immune system interactions within the tumor microenvironment. The equations account for the geometry of the tumor expansion, and the displacement of the immune cells, driven by diffusion and chemotactic mechanisms. They also take into account the constraints in terms of nutrient and oxygen supply. The numerical investigations analyze the impact of the different modeling assumptions and parameters. Depending on the parameters, the model can reproduce elimination, equilibrium or escape phases and it identifies a critical role of oxygen/nutrient supply in shaping the tumor growth. In addition, antitumor immune cells are key factors in controlling tumor growth, maintaining an equilibrium while protumor cells favor escape and tumor expansion.

Tumorigenesis and axons regulation for the pancreatic cancer: A mathematical approach.

The nervous system is today recognized to play an important role in the development of cancer. Indeed, neurons extend long processes (axons) that grow and infiltrate tumors in order to regulate the progression of the disease in a positive or negative way, depending on the type of neuron considered. Mathematical modeling of this biological process allows to formalize the nerve-tumor interactions and to test hypotheses in silico to better understand this phenomenon. In this work, we introduce a system of differential equations modeling the progression of pancreatic ductal adenocarcinoma (PDAC) coupled with associated changes in axonal innervation. The study of the asymptotic behavior of the model confirms the experimental observations that PDAC development is correlated with the type and densities of axons in the tissue. We study then the identifiability and the sensitivity of the model parameters. The identifiability analysis informs on the adequacy between the parameters of the model and the experimental data and the sensitivity analysis on the most contributing factors on the development of cancer. It leads to significant insights on the main neural checkpoints and mechanisms controlling the progression of pancreatic cancer. Finally, we give an example of a simulation of the effects of partial or complete denervation that sheds lights on complex correlation between the healthy, pre-cancerous and cancerous cell densities and axons with opposite functions.

Sympathetic axonal sprouting induces changes in macrophage populations and protects against pancreatic cancer.

Neuronal nerve processes in the tumor microenvironment were highlighted recently. However, the origin of intra-tumoral nerves remains poorly known, in part because of technical difficulties in tracing nerve fibers via conventional histological preparations. Here, we employ three-dimensional (3D) imaging of cleared tissues for a comprehensive analysis of sympathetic innervation in a murine model of pancreatic ductal adenocarcinoma (PDAC). Our results support two independent, but coexisting, mechanisms: passive engulfment of pre-existing sympathetic nerves within tumors plus an active, localized sprouting of axon terminals into non-neoplastic lesions and tumor periphery. Ablation of the innervating sympathetic nerves increases tumor growth and spread. This effect is explained by the observation that sympathectomy increases intratumoral CD163+ macrophage numbers, which contribute to the worse outcome. Altogether, our findings provide insights into the mechanisms by which the sympathetic nervous system exerts cancer-protective properties in a mouse model of PDAC.

A Growth-Fragmentation Approach for Modeling Microtubule Dynamic Instability.

Microtubules (MTs) are protein filaments found in all eukaryotic cells which are crucial for many cellular processes including cell movement, cell differentiation, and cell division. Due to their role in cell division, they are often used as targets for chemotherapy drugs used in cancer treatment. Experimental studies of MT dynamics have played an important role in the development and administration of many novel cancer drugs; however, a complete description of MT dynamics is lacking. Here, we propose a new mathematical model for MT dynamics, that can be used to study the effects of chemotherapy drugs on MT dynamics. Our model consists of a growth-fragmentation equation describing the dynamics of a length distribution of MTs, coupled with two ODEs that describe the dynamics of free GTP- and GDP-tubulin concentrations (the individual dimers that comprise of MTs). Here, we prove the well-posedness of our system and perform a numerical exploration of the influence of certain model parameters on the systems dynamics. In particular, we focus on a qualitative description for how a certain class of destabilizing drugs, the vinca alkaloids, alter MT dynamics. Through variation of certain model parameters which we know are altered by these drugs, we make comparisons between simulation results and what is observed in in vitro studies.

Exploring the effect of end-binding proteins and microtubule targeting chemotherapy drugs on microtubule dynamic instability.

Microtubules (MTs) play a key role in normal cell development and are a primary target for many cancer chemotherapy MT targeting agents (MTAs). As such, understanding MT dynamics in the presence of such agents, as well as other proteins that alter MT dynamics, is extremely important. In general, MTs grow relatively slowly and shorten very fast (almost instantaneously), an event referred to as a catastrophe. These dynamics, referred to as dynamic instability, have been studied in both experimental and theoretical settings. In the presence of MTAs, it is well known that such agents work by suppressing MT dynamics, either by promoting MT polymerization or promoting MT depolymerization. However, recent in vitro experiments show that in the presence of end-binding proteins (EBs), low doses of MTAs can increase MT dynamic instability, rather than suppress it. Here, we develop a novel mathematical model, to describe MT and EB dynamics, something which has not been done in a theoretical setting. Our MT model is based on previous modeling efforts, and consists of a pair of partial differential equations to describe length distributions for growing and shortening MT populations, and an ordinary differential equation (ODE) system to describe the time evolution for concentrations of GTP- and GDP-bound tubulin. A new extension of our approach is the use of an integral term, rather than an advection term, to describe very fast MT shortening events. Further, we introduce an ODE system to describe the binding and unbinding of EBs with MTs. To compare simulation results with experiment, we define novel mathematical expressions for time- and distance-based catastrophe frequencies. These quantities help to define MT dynamics in in vivo and in vitro settings. Simulation results show that increasing concentrations of EBs work to increase time-based catastrophe while distance-based catastrophe is less affected by changes in EB concentration, a result that is consistent with experiment. We further describe how EBs and MTAs alter MT dynamics. In the context of this modeling framework, we show that it is likely that MTAs and EBs do not work independently from one another. Thus, we propose a mechanism for how EBs can work synergistically with MTAs to promote MT dynamic instability at low MTA dose.

Study of metastatic kinetics in metastatic melanoma treated with B-RAF inhibitors: Introducing mathematical modelling of kinetics into the therapeutic decision.

BACKGROUND: Evolution of metastatic melanoma (MM) under B-RAF inhibitors (BRAFi) is unpredictable, but anticipation is crucial for therapeutic decision. Kinetics changes in metastatic growth are driven by molecular and immune events, and thus we hypothesized that they convey relevant information for decision making. PATIENTS AND METHODS: We used a retrospective cohort of 37 MM patients treated by BRAFi only with at least 2 close CT-scans available before BRAFi, as a model to study kinetics of metastatic growth before, under and after BRAFi. All metastases (mets) were individually measured at each CT-scan. From these measurements, different measures of growth kinetics of each met and total tumor volume were computed at different time points. A historical cohort permitted to build a reference model for the expected spontaneous disease kinetics without BRAFi. All variables were included in Cox and multistate regression models for survival, to select best candidates for predicting overall survival. RESULTS: Before starting BRAFi, fast kinetics and moreover a wide range of kinetics (fast and slow growing mets in a same patient) were pejorative markers. At the first assessment after BRAFi introduction, high heterogeneity of kinetics predicted short survival, and added independent information over RECIST progression in multivariate analysis. Metastatic growth rates after BRAFi discontinuation was usually not faster than before BRAFi introduction, but they were often more heterogeneous than before. CONCLUSIONS: Monitoring kinetics of different mets before and under BRAFi by repeated CT-scan provides information for predictive mathematical modelling. Disease kinetics deserves more interest.

Mathematical modeling of tumor growth and metastatic spreading: validation in tumor-bearing mice.

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/Scidγ 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 [Formula: see text] for primary tumor growth and [Formula: see text] 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.

Mathematical and numerical analysis for a model of growing metastatic tumors.

In cancer diseases, the appearance of metastases is a very pejorative forecast. Chemotherapies are systemic treatments which aim at the elimination of the micrometastases produced by a primitive tumour. The efficiency of chemotherapies closely depends on the protocols of administration. Mathematical modeling is an invaluable tool to help in evaluating the best treatment strategy. Iwata et al. [K. Iwata, K. Kawasaki, N. Shigesad, A dynamical model for the growth and size distribution of multiple metastatic tumors, J. Theor. Biol. 203 (2000) 177.] proposed a partial differential equation (PDE) that describes the metastatic evolution of an untreated tumour. In this article, we conducted a thorough mathematical analysis of this model. Particularly, we provide an explicit formula for the growth rate parameter, as well as a numerical resolution of this PDE. By increasing our understanding of the existing model, this work is crucial for further extension and refinement of the model. It settles down the framework necessary for the consideration of drugs administration effects on tumour development.