On the mathematical modelling of tumor-induced angiogenesis

Abstract

An angiogenic system is taken as an example of extremely complex ones in the field of Life Sciences, from both analytical and computational points of view, due to the strong coupling between the kinetic parameters of the relevant branching - growth - anastomosis stochastic processes of the capillary network, at the microscale, and the family of interacting underlying biochemical fields, at the macroscale. To reduce this complexity, for a conceptual stochastic model we have explored how to take advantage of the system intrinsic multiscale structure: one might describe the stochastic dynamics of the cells at the vessel tip at their natural microscale, whereas the dynamics of the underlying fields is given by a deterministic mean field approximation obtained by an averaging at a suitable mesoscale. But the outcomes of relevant numerical simulations show that the proposed model, in presence of anastomosis, is not self-averaging, so that the ``propagation of chaos" assumption cannot be applied to obtain a deterministic mean field approximation. On the other hand we have shown that ensemble averages over many realizations of the stochastic system may better correspond to a deterministic reaction-diffusion system.

Simulation analysis for tumor radiotherapy based on three‐component mathematical models

“…In other papers, mathematical model was combined with the technique of medical image processing for evaluation and prediction of tumor multidisciplinary treatment . Although there has not been a comprehensive mathematical model for clinical application until now, some positive improvements have been achieved in serials of fields . It is assured that the tumor mathematical models will reach an excellent level with the rapid development of computers in the near future…”

Optimal duration and dosage of BCG intravesical immunotherapy: A free final time optimal control approach

“…Some research papers on mathematical medicine have developed mathematical models which describe the evolution of tumors and analyze the different interactions between the tumor, immune‐system, and treatment…”

“…Many optimal control strategies have been proposed for highlighting the advantage of optimization of some parameters in some biomathematical models. [9][10][11] In this context, some applied mathematicians advise to rely dosage problems; also, on their optimal control approaches, they see beneficial for developing therapeutic strategies that aim to reduce tumor volume with lesser side-effects, see, for instance, in the case of chemotherapy, the work in Yegorov and Todorov, 12 and in the case of immunotherapy, papers in Hamdache et al 13,14 Some research papers on mathematical medicine have developed mathematical models which describe the evolution of tumors 15,16 and analyze the different interactions between the tumor, immune-system, and treatment. 17,18 In this paper, we formulate a minimization problem from which we aim to find the optimal dose and duration of the BCG immunotherapy procedure in the case of superficial bladder cancer, based on a prevalidated mathematical model 19 in the form of 4 ordinary differential equations which describe the superficial bladder tumor-immune interactions after the injection of BCG in the bladder of a hypothetical patient.…”

Well posedness of an angiogenesis related integrodifferential diffusion model

“…Positivity is therefore a key property of the solutions. For positive p the sign of the source term in (5) cannot be controlled. A possibility to generate approximate solutions with a controlled sign is to freeze the integral coefficient and to include the linearized integral source in a linearized diffusion operator.…”