Mathematical Models of Cell Migration and Proliferation in Scratch Assays

Scratch assays are standard in-vitro experimental procedures for studying cell migration. In these experiments, a scratch is made on a cell monolayer. By imaging the recolonisation process of the scratched region, we are able to quantify cell migration rates. This experimental technique is commonly used in the pharmaceutical industry to identify new compounds that target cell migration, and to evaluate the efficacy of potential drugs that inhibit cancer invasion. Given the key role this method plays in assessing the potential of new compounds for clinical use, it is important to develop robust quantification frameworks that accurately describe the movement of the front of migrating cells. We develop a migration quantification method that fits experimental data more closely than existing methods, provides a more accurate statistical classification of the migration rate between different assays and is able to cope with experimental data of lower quality than the classic quantification methods can handle. The robustness of our new method is validated using both in-vitro and in-silico data. Developing robust quantification methods allows the validation of mathematical models that can be used to test hypotheses about the physical and biological mechanisms that govern cell migration. Typically scratch assays are modelled by continuum reaction-diffusion equations depicting cell migration by diffusion and carrying capacity-limited proliferation by a logistic source term. An age-structured population model is presented that aims to explain the two phases of proliferation in scratch assays previously observed experimentally: where an initial phase is observed where proliferation is not logistic, followed by a second phase where proliferation appears to be logistic. The cell population is modelled by a McKendrick-von Foerster partial differential equation. The conditions under which the model captures this two-phase behaviour are presented. Finally, an important aspect of modelling biological systems is the development of efficient algorithms. The scratch assay is a classical example of a system in which there is low cell number in some regions of the spatial domain and high cell number in others. When the cell number is sufficiently high, mean-field models, like partial differential equations, can capture the relevant dynamics. However, when the cell number is low, such models are not appropriate and stochastic representations must be employed. Hybrid algorithms allow multiple modelling frameworks for the same species in different parts of the spatial domain. Typically hybrid algorithms consider heuristic methods based on the cell density for determining which compartments will be updated deterministically or stochastically. We introduce a hybrid algorithm that couples the mesoscopic description of a reaction-diffusion system with its mean-field analogue. We consider a natural indicator of when the mean-field approximation is valid: the system variance. We estimate the system variance using the intrinsic noise approximation and use this estimate to determine the regions in which the system is updated stochastically or deterministically over time. We apply the hybrid algorithm to the stochastic Fisher-Kolmogorov-Petrovsky-Piscounov model, a typical model of scratch assays. We analyse systematically how good is the approximation to the stochastic process and compare its performance to another hybrid algorithm