Modelling recurrence and second cancer risks induced by proton therapy.

In the past few years, proton therapy has taken the centre stage in treating various tumour types. The primary contribution of this study is to investigate the tumour control probability (TCP), relapse time and the corresponding secondary cancer risks induced by proton beam radiation therapy. We incorporate tumour relapse kinetics into the TCP framework and calculate the associated second cancer risks. To calculate proton therapy-induced secondary cancer induction, we used the well-known biologically motivated mathematical model, initiation-inactivation-proliferation formalism. We used the available in vitro data for the linear energy transfer (LET) dependence of cell killing and mutation induction parameters. We evaluated the TCP and radiation-induced second cancer risks for protons in the clinical range of LETs, i.e. approximately 8 $\mathrm{keV/\mu m}$ for the tumour volume and 1-3 $\mathrm{keV/\mu m}$ for the organs at risk. This study may serve as a framework for further work in this field and elucidates proton-induced TCP and the associated secondary cancer risks, not previously reported in the literature. Although studies with a greater number of cell lines would reduce uncertainties within the model parameters, we argue that the theoretical framework presented within is a sufficient rationale to assess proton radiation TCP, relapse and carcinogenic effects in various treatment plans. We show that compared with photon therapy, proton therapy markedly reduces the risk of secondary malignancies and for equivalent dosing regimens achieves better tumour control as well as a reduced primary recurrence outcome, especially within a hypo-fractionated regimen.

Modeling Invasion Dynamics with Spatial Random-Fitness Due to Micro-Environment.

Numerous experimental studies have demonstrated that the microenvironment is a key regulator influencing the proliferative and migrative potentials of species. Spatial and temporal disturbances lead to adverse and hazardous microenvironments for cellular systems that is reflected in the phenotypic heterogeneity within the system. In this paper, we study the effect of microenvironment on the invasive capability of species, or mutants, on structured grids (in particular, square lattices) under the influence of site-dependent random proliferation in addition to a migration potential. We discuss both continuous and discrete fitness distributions. Our results suggest that the invasion probability is negatively correlated with the variance of fitness distribution of mutants (for both advantageous and neutral mutants) in the absence of migration of both types of cells. A similar behaviour is observed even in the presence of a random fitness distribution of host cells in the system with neutral fitness rate. In the case of a bimodal distribution, we observe zero invasion probability until the system reaches a (specific) proportion of advantageous phenotypes. Also, we find that the migrative potential amplifies the invasion probability as the variance of fitness of mutants increases in the system, which is the exact opposite in the absence of migration. Our computational framework captures the harsh microenvironmental conditions through quenched random fitness distributions and migration of cells, and our analysis shows that they play an important role in the invasion dynamics of several biological systems such as bacterial micro-habitats, epithelial dysplasia, and metastasis. We believe that our results may lead to more experimental studies, which can in turn provide further insights into the role and impact of heterogeneous environments on invasion dynamics.

Efficacy of dose escalation on TCP, recurrence and second cancer risks: a mathematical study.

OBJECTIVE: We investigated the effects of conventional and hypofractionation protocols by modelling tumour control probability (TCP) and tumour recurrence time, and examined their impact on second cancer risks. The main objectives of this study include the following: (a) incorporate tumour recurrence time and second cancer risks into the TCP framework and analyse the effects of variable doses and (b) investigate an efficient protocol to reduce the risk of a secondary malignancy while maximizing disease-free survival and tumour control. METHODS: A generalized mathematical formalism was developed that incorporated recurrence and second cancer risk models into the TCP dynamics. RESULTS: Our results suggest that TCP and relapse time are almost identical for conventional and hypofractionated regimens; however, second cancer risks resulting from hypofractionation were reduced by 22% when compared with the second cancer risk associated with a conventional protocol. The hypofractionated regimen appears to be sensitive to dose escalation and the corresponding impact on tumour recurrence time and reduction in second cancer risks. The reduction in second cancer risks is approximately 20% when the dose is increased from 60 to 72 Gy in a hypofractionated protocol. CONCLUSION: Our results suggest that hypofractionation may be a more efficient regimen in the context of TCP, relapse time and second cancer risks. Overall, our study demonstrates the importance of including a second cancer risk model in designing an efficient radiation regimen. ADVANCES IN KNOWLEDGE: The impact of various fractionation protocols on TCP and relapse in conjunction with second cancer risks is an important clinical question that is as yet unexplored.

Spatial invasion dynamics on random and unstructured meshes: implications for heterogeneous tumor populations.

In this work we discuss a spatial evolutionary model for a heterogeneous cancer cell population. We consider the gain-of-function mutations that not only change the fitness potential of the mutant phenotypes against normal background cells but may also increase the relative motility of the mutant cells. The spatial modeling is implemented as a stochastic evolutionary system on a structured grid (a lattice, with random neighborhoods, which is not necessarily bi-directional) or on a two-dimensional unstructured mesh, i.e. a bi-directional graph with random numbers of neighbors. We present a computational approach to investigate the fixation probability of mutants in these spatial models. Additionally, we examine the effect of the migration potential on the spatial dynamics of mutants on unstructured meshes. Our results suggest that the probability of fixation is negatively correlated with the width of the distribution of the neighborhood size. Also, the fixation probability increases given a migration potential for mutants. We find that the fixation probability (of advantaged, disadvantaged and neutral mutants) on unstructured meshes is relatively smaller than the corresponding results on regular grids. More importantly, in the case of neutral mutants the introduction of a migration potential has a critical effect on the fixation probability and increases this by orders of magnitude. Further, we examine the effect of boundaries and as intuitively expected, the fixation probability is smaller on the boundary of regular grids when compared to its value in the bulk. Based on these computational results, we speculate on possible better therapeutic strategies that may delay tumor progression to some extent.