Let X(t) be the number of tumor cells at time t, and Pr{X(t) = n} = pn(t) is the density of X. A “birth”, i.e., an increase of one of the total population of cancer cells, can occur either by mutation of a normal cell caused by the action of the carcinogen, consisting of randomly (Poisson) distributed hits, or by reproduction of existing cancer cells. A death of a tumor cell occurs as an additive combination of non-immunological and immunological elements. Once a tumor is initiated by carcinogenic action, it undergoes a birth and death process with infinitesimal birth rate linear and infinitesimal death rate composed of a linear and a nonlinear term, the former due to non-immunological deaths, the latter to immunological feedback. The death rate per tumor cell due to immunological response is proportional to the total number of antigen-producing (tumor) cells; thus, the total death rate is quadratic. Although this assumes a very simple mechanism for the action of immunological feedback, it is nevertheless a first step.

The Mathematical Model

Cancer modeling: From mechanistic to data-driven approaches, and from fundamental insights to clinical applications

Prostate cancer can be lethal in advanced stages, for which chemotherapy may become the only viable therapeutic option. While there is no clear clinical management strategy fitting all patients, cy...

Optimal control of cytotoxic and antiangiogenic therapies on prostate cancer growth

Model-informed drug development (MIDD) approaches have rapidly advanced in drug development in recent years Additionally, the Prescription Drug User Fee Act (PDUFA) VI has specific commitments to further enhance MIDD Tumor growth dynamic (TGD) modeling, as one of the commonly utilized MIDD approaches in oncology, fulfills the purposes to accelerate the drug development, to support new drug and biologics license applications, and to guide the market access Increasing knowledge of TGD modeling methodologies, encouraging applications in clinical setting for patients' survival, and complementing assessment of regulatory review for submissions, together fueled promising potentials for imminent enhancement of TGD in oncology This review is to comprehensively summarize the history of TGD, and present case examples of the recent advance of TGD modeling (mixture model and joint model), as well as the TGD impact on regulatory decisions, thus illustrating challenges and opportunities Additionally, this review presents the future perspectives for TGD approach

Tumor Growth Dynamic Modeling in Oncology Drug Development and Regulatory Approval: Past, Present, and Future Opportunities.

Treatment-driven tumour heterogeneity and drug resistance: Lessons from solid tumours.

Increasing knowledge of intertumor heterogeneity, intratumor heterogeneity, and cancer evolution has improved the understanding of anticancer treatment resistance. A better characterization of cancer evolution and subsequent use of this knowledge for personalized treatment would increase the chance to overcome cancer treatment resistance. Model-based approaches may help achieve this goal. In this review, we comprehensively summarized mathematical models of tumor dynamics for solid tumors and of drug resistance evolution. Models displayed by ordinary differential equations, algebraic equations, and partial differential equations for characterizing tumor burden dynamics are introduced and discussed. As for tumor resistance evolution, stochastic and deterministic models are introduced and discussed. The results may facilitate a novel model-based analysis on anticancer treatment response and the occurrence of resistance, which incorporates both tumor dynamics and resistance evolution. The opportunities of a model-based approach as discussed in this review can be of great benefit for future optimizing and personalizing anticancer treatment.

https://ascpt.onlinelibrary.wiley.com/doi/pdf/10.1002/psp4.12450

Johan G.C. van Hasselt

Papers

A Review of Mathematical Models for Tumor Dynamics and Treatment Resistance Evolution of Solid Tumors