A Modified Stochastic Gompertz Model for Tumour Cell Growth

Based upon the deterministic Gompertz law of cell growth, we have proposed a stochastic model of tumour cell growth, in which the size of the tumour cells is bounded. The model takes account of both cell fission (which is an ‘action at a distance’ effect) and mortality too. Accordingly, the density function of the size of the tumour cells obeys a functional Fokker–Planck Equation (FPE) associated with the bounded stochastic process. We apply the Lie-algebraic method to derive the exact analytical solution via an iterative approach. It is found that the density function exhibits an interesting kink-like structure generated by cell fission as time evolves

Comparing System Dynamics and Agent-Based Simulation for Tumour Growth and its Interactions with Effector Cells

There is little research concerning comparisons and combination of System Dynamics Simulation (SDS) and Agent Based Simulation (ABS). ABS is a paradigm used in many levels of abstraction, including those levels covered by SDS. We believe that the establishment of frameworks for the choice between these two simulation approaches would contribute to the simulation research. Hence, our work aims for the establishment of directions for the choice between SDS and ABS approaches for immune system-related problems. Previously, we compared the use of ABS and SDS for modelling agents' behaviour in an environment with nomovement or interactions between these agents. We concluded that for these types of agents it is preferable to use SDS, as it takes up less computational resources and produces the same results as those obtained by the ABS model. In order to move this research forward, our next research question is: if we introduce interactions between these agents will SDS still be the most appropriate paradigm to be used? To answer this question for immune system simulation problems, we will use, as case studies, models involving interactions between tumour cells and immune effector cells. Experiments show that there are cases where SDS and ABS can not be used interchangeably, and therefore, their comparison is not straightforward.Comment: 8 pages, 8 figures, 2 tables, International Summer Computer Simulation Conference 201

Study of Birth-Death Processes with Immigration

Birth-death processes are applied in the modelling of many biological populations, such as tumour cells and viruses. Various studies have established that birth-death processes, which occur when the population size is zero, are not in-line with reality in many situations. Therefore, in this study, the birth-death processes with immigration were investigated. We considered two immigration policies. First, immigration is allowed if and only if the population size is zero. Second, immigration at a constant rate is allowed irrespective of the population size. Birth and death rates were chosen such that the mean population size is a Gompertz function when the immigration rate is zero. The transient population size probability was obtained for both cases. Several tumour growth datasets were fitted using the mean population size of the above models and standard birth-death model without immigration. The two models with immigration provided entirely different probabilities of the population size being zero at an arbitrary epoch when compared with the model without immigration. Moreover, all three models provided a similar fit to the data. For each of the datasets studied, the models that allowed immigration produced less variance than the non-immigration model

Modelling The Cancer Growth Process By Stochastic Delay Diffferential Equations Under Verhults And Gompertz's Law

In this paper, the uncontrolled environmental factors are perturbed into the intrinsic growth rate factor of deterministic equations of the growth process. The growth process under two different laws which are Verhults and Gompertz’s law are considered, thus leading to stochastic delay differential equations (SDDEs) of logistic and Gompertzian, respectively. Gompertzian deterministic model has been proved to fit well the clinical data of cancerous growth, however the performance of stochastic model towards clinical data is yet to be confirmed. The prediction quality of logistic and Gompertzian SDDEs are evaluating by comparing the simulated results with the clinical data of cervical cancer growth. The parameter estimation of stochastic models is computed by using simulated maximum likelihood method. We adopt 4-stage stochastic Runge-Kutta to simulate the solution of stochastic models

When the optimal is not the best: parameter estimation in complex biological models

Background: The vast computational resources that became available during the past decade enabled the development and simulation of increasingly complex mathematical models of cancer growth. These models typically involve many free parameters whose determination is a substantial obstacle to model development. Direct measurement of biochemical parameters in vivo is often difficult and sometimes impracticable, while fitting them under data-poor conditions may result in biologically implausible values. Results: We discuss different methodological approaches to estimate parameters in complex biological models. We make use of the high computational power of the Blue Gene technology to perform an extensive study of the parameter space in a model of avascular tumor growth. We explicitly show that the landscape of the cost function used to optimize the model to the data has a very rugged surface in parameter space. This cost function has many local minima with unrealistic solutions, including the global minimum corresponding to the best fit. Conclusions: The case studied in this paper shows one example in which model parameters that optimally fit the data are not necessarily the best ones from a biological point of view. To avoid force-fitting a model to a dataset, we propose that the best model parameters should be found by choosing, among suboptimal parameters, those that match criteria other than the ones used to fit the model. We also conclude that the model, data and optimization approach form a new complex system, and point to the need of a theory that addresses this problem more generally

The roles of T cell competition and stochastic extinction events in chimeric antigen receptor T cell therapy

Chimeric antigen receptor (CAR) T cell therapy is a remarkably effective immunotherapy that relies on in vivo expansion of engineered CAR T cells, after lymphodepletion (LD) by chemotherapy. The quantitative laws underlying this expansion and subsequent tumour eradication remain unknown. We develop a mathematical model of T cell–tumour cell interactions and demonstrate that expansion can be explained by immune reconstitution dynamics after LD and competition among T cells. CAR T cells rapidly grow and engage tumour cells but experience an emerging growth rate disadvantage compared to normal T cells. Since tumour eradication is deterministically unstable in our model, we define cure as a stochastic event, which, even when likely, can occur at variable times. However, we show that variability in timing is largely determined by patient variability. While cure events impacted by these fluctuations occur early and are narrowly distributed, progression events occur late and are more widely distributed in time. We parameterized our model using population-level CAR T cell and tumour data over time and compare our predictions with progression-free survival rates. We find that therapy could be improved by optimizing the tumour-killing rate and the CAR T cells' ability to adapt, as quantified by their carrying capacity. Our tumour extinction model can be leveraged to examine why therapy works in some patients but not others, and to better understand the interplay of deterministic and stochastic effects on outcomes. For example, our model implies that LD before a second CAR T injection is necessary

Mathematical models of avascular cancer

This review will outline a number of illustrative mathematical models describing the growth of avascular tumours. The aim of the review is to provide a relatively comprehensive list of existing models in this area and discuss several representative models in greater detail. In the latter part of the review, some possible future avenues of mathematical modelling of avascular tumour development are outlined together with a list of key questions