Stochastic Modeling of Cancer Dynamics

Cancer is a highly complex disease with wide range of heterogenous dynamics. Some major hallmarks of cancer include excessive and sustained proliferation, evasion of growth inhibitors and apoptosis, inducing angiogenesis, invasion and metastasis. Furthermore, other emerging characteristics include deregulation of cellular energetics, genomic instability and phenotypic plasticity. Understanding these interconnected behaviors is key to controlling or mitigating their progression.

One on the major reason for the lethality of cancer is its ability to aggressively invade the tissue and spread to distal regions of the organ and also the body. Interestingly, there seems to be a high correlation of upregulated glucose metabolism and proteolytic tissue degradation with the invasive potential of the cancer. Indeed, wet lab experiments have reinforced that glycolysis, reversed pH-gradient (intracellular pH higher than extracellular pH), and acid resistance subsequently developed by tumor cells confers substantial survival advantage, thereby boosting both proliferation and invasion.

Our current and past research activities involves studying the migratory and invasive patterns of cancer cells especially under the influence of micro-environmental factors such as cellular acidity and tissue structure. As a primary tool I have mainly relied on the techniques of multi-scale mathematical modelling. This consequently results in a nonlinear system of partial-differential equations coupled with stochastic/random ordinary differential equations. The PDEs represent the averaged dynamics at the tissue level while the latter represents cell level dynamics occurring at a much faster scale. The presence of different kids of uncertainties, heterogeneities and variations in the cellular processes, are abstractly captured via random nonlinear disturbance term in the cell level model. Below you may find an example of one such model that studies the emergence pseudopalisade (and alike) patterns during the development of a malignant gliobasltoma cancer.

Data driven modeling of pseudopalisade pattern formation.
Journal of Mathematical Biology 87(4), 2023. DOI
S. Hiremath, C. Surulescu.

Modeling of pH regulation in tumor cells: Direct interaction between proton-coupled lactate transporters and cancer-associated carbonic anhydrase.
MBE, 2019, 16(1), 320-337.
S. Hiremath, C. Surulescu, H. Becker.

On a coupled SDE-PDE system modeling acid-mediated tumor invasion.
DCDS-B, 2018, 23(6), 2339-2369.
S. Hiremath, S. Sonner, C. Surulescu, A. Zhigun.

Mathematical models for acid-mediated tumor invasion: deterministic and stochastic approaches.
In A. Gerisch, R. Penta, J. Lang (eds): Multiscale Models in Mechano and Tumor Biology: Modeling, Homogenization, and Applications, Lecture Notes in Computational Science and Engineering, Springer Verlag Heidelberg, (2017), ISBN 978-3-319-73371-5
S. Hiremath, C. Surulescu.

A stochastic model featuring acid induced gaps during tumor progression.
Nonlinearity 29 (2016) 851-914.
S. Hiremath, C. Surulescu.

A stochastic multiscale model for acid mediated cancer invasion.
Nonlin. Analysis B: Real World Appl. 22 (2015) 17-205.
S. Hiremath, C. Surulescu.