Using mathematical models to simulate dynamic biological processes has a long history. Over the past couple of decades or so, quantitative approaches have also made their way into cancer research. An increasing number of mathematical, physical, computational and engineering techniques have been applied to various aspects of tumor growth, with the ultimate goal of understanding the response of the cancer population to clinical intervention. So-called in silico trials that predict patient-specific response to various dose schedules or treatment combinations and sequencing are on the way to becoming an invaluable tool to optimize patient care. Herein we describe fundamentals of mathematical modeling of tumor growth and tumor-host interactions, and summarize some of the seminal and most prominent approaches.

Ligand-induced activation of peroxisome proliferator-activated receptor γ (PPARγ) inhibits proliferation in cancer cells in vitro and in vivo; however, the downstream targets remain undefined. We report the identification of a peroxisome proliferator response element in the promoter region of the Na⁺/H⁺ transporter gene NHE1, the overexpression of which has been associated with carcinogenesis. Exposure of breast cancer cells expressing high levels of PPARγ to its natural and synthetic agonists resulted in downregulation of NHE1 transcription as well as protein expression. Furthermore, the inhibitory effect of activated PPARγ on tumor colony-forming ability was abrogated on overexpression of NHE1, whereas small interfering RNA–mediated gene silencing of NHE1 significantly increased the sensitivity of cancer cells to growth-inhibitory stimuli. Finally, histopathologic analysis of breast cancer biopsies obtained from patients with type II diabetes treated with the synthetic agonist rosiglitazone showed significant repression of NHE1 in the tumor tissue. These data provide evidence for tumor-selective downregulation of NHE1 by activated PPARγ in vitro and in pathologic specimens from breast cancer patients and could have potential implications for the judicious use of low doses of PPARγ ligands in combination chemotherapy regimens for an effective therapeutic response. [Cancer Res 2009;69(22):8636–44]

The number of publications on mathematical modeling of cancer is growing at an exponential rate, according to PubMed records, provided by the US National Library of Medicine and the National Institutes of Health. Seminal papers have initiated and promoted mathematical modeling of cancer and have helped define the field of mathematical oncology (Norton and Simon in J Natl Cancer Inst 58:1735–1741, 1977; Norton in Can Res 48:7067–7071, 1988; Hahnfeldt et al. in Can Res 59:4770–4775, 1999; Anderson et al. in Comput Math Methods Med 2:129–154, 2000. https://doi.org/10.1080/10273660008833042; Michor et al. in Nature 435:1267–1270, 2005. https://doi.org/10.1038/nature03669; Anderson et al. in Cell 127:905–915, 2006. https://doi.org/10.1016/j.cell.2006.09.042; Benzekry et al. in PLoS Comput Biol 10:e1003800, 2014. https://doi.org/10.1371/journal.pcbi.1003800). Following the introduction of undergraduate and graduate programs in mathematical biology, we have begun to see curricula developing with specific and exclusive focus on mathematical oncology. In 2018, 218 articles on mathematical modeling of cancer were published in various journals, including not only traditional modeling journals like the Bulletin of Mathematical Biology and the Journal of Theoretical Biology, but also publications in renowned science, biology, and cancer journals with tremendous impact in the cancer field (Cell, Cancer Research, Clinical Cancer Research, Cancer Discovery, Scientific Reports, PNAS, PLoS Biology, Nature Communications, eLife, etc). This shows the breadth of cancer models that are being developed for multiple purposes. While some models are phenomenological in nature following a bottom-up approach, other models are more top-down data-driven. Here, we discuss the emerging trend in mathematical oncology publications to predict novel, optimal, sometimes even patient-specific treatments, and propose a convention when to use a model to predict novel treatments and, probably more importantly, when not to.

Tumours are heterogeneous populations composed of different cells types: stem cells with the capacity for self-renewal and more differentiated cells lacking such ability. The overall growth behaviour of a developing neoplasm is determined largely by the combined kinetic interactions of these cells. By tracking the fate of individual cancer cells using agent-based methods in silico, we apply basic rules for cell proliferation, migration and cell death to show how these kinetic parameters interact to control, and perhaps dictate defining spatial and temporal tumour growth dynamics in tumour development. When the migration rate is small, a single cancer stem cell can only generate a small, self-limited clone because of the finite life span of progeny and spatial constraints. By contrast, a high migration rate can break this equilibrium, seeding new clones at sites outside the expanse of older clones. In this manner, the tumour continually ‘self-metastasises’. Counterintuitively, when the proliferation capacity is low and the rate of cell death is high, tumour growth is accelerated because of the freeing up of space for self-metastatic expansion. Changes to proliferation and cell death that increase the rate at which cells migrate benefit tumour growth as a whole. The dominating influence of migration on tumour growth leads to unexpected dependencies of tumour growth on proliferation capacity and cell death. These dependencies stand to inform standard therapeutic approaches, which anticipate a positive response to cell killing and mitotic arrest.

Glioblastoma multiforme (GBM) is one of the most aggressive human malignancies with a poor patient prognosis. Ionizing radiation either alone or adjuvant after surgery is part of standard treatment for GBM but remains primarily noncurative. The mechanisms underlying tumor radioresistance are manifold and, in part, accredited to a special subpopulation of tumorigenic cells. The so-called glioma stem cells (GSC) are bestowed with the exclusive ability to self-renew and repopulate the tumor and have been reported to be less sensitive to radiation-induced damage through preferential activation of DNA damage checkpoint responses and increased capacity for DNA damage repair. During each fraction of radiation, non–stem cancer cells (CC) die and GSCs become enriched and potentially increase in number, which may lead to accelerated repopulation. We propose a cellular Potts model that simulates the kinetics of GSCs and CCs in glioblastoma growth and radiation response. We parameterize and validate this model with experimental data of the U87-MG human glioblastoma cell line. Simulations are conducted to estimate GSC symmetric and asymmetric division rates and explore potential mechanisms for increased GSC fractions after irradiation. Simulations reveal that in addition to their higher radioresistance, a shift from asymmetric to symmetric division or a fast cycle of GSCs following fractionated radiation treatment is required to yield results that match experimental observations. We hypothesize a constitutive activation of stem cell division kinetics signaling pathways during fractionated treatment, which contributes to the frequently observed accelerated repopulation after therapeutic irradiation. Cancer Res; 73(5); 1481–90. ©2012 AACR.