Monika Joanna Piotrowska

A quantitative cellular automaton model of in vitro multicellular spheroid tumour growth

We report numerical results from a 2D cellular automaton (CA) model describing the dynamics of the in vitro cultivated multicellular spheroid obtained from EMT6/Ro (mammary carcinoma) cell line. Significantly, the CA model relaxes the often assumed one-to-one correspondence between cells and CA sites so as to correctly model the peripheral mitotic boundary region, and to enable the study of necrosis in large avascular tumours. By full calibration and scaling to available experimental data, the model produces with good accuracy experimentally comparable data on a range of bulk tumour kinetics and necrosis measures. Our main finding is that the metabolic production of H(+) ions is not sufficient to cause central necrosis prior to the sub-viable nutrient-deficient stage of tumour development being reached. Thus, the model suggests that an additional process is required to explain the experimentally observable onset of necrosis prior to the non-viable nutrient-deficient point being reached.

Hopf bifurcation in a solid avascular tumour growth model with two discrete delays

In this paper the model of solid, avascular, uniformly proliferating tumour growth with two independent time delays is presented and analysed. The model considers two main cellular processes: proliferation and apoptosis. The aim of this paper is to investigate the influence of time delays on the Hopf bifurcation when one of delays is used as a bifurcation parameter.

The nature of Hopf bifurcation for the Gompertz model with delays

Abstract In this paper, we study the influence of time delays on the dynamics of the classical Gompertz model. We consider the models with one discrete delay introduced in two different ways and the model with two delays which generalise those with one delay. We study asymptotic behaviour and bifurcations with respect to the ratio of delays τ = τ 1 / τ 2 . Our results show that in such model with two delays there is only one stability switch and for a threshold value of bifurcation parameter, Hopf bifurcation (HB) occurs. However, the type of HB, and therefore its stability (i.e. stability of periodic orbits arising due to it), strongly depends on the magnitude of τ . The function describing stability of HB is periodic with respect to τ . Within one period of length 4 five changes of HB stability are observed. We also introduce the second model with two delays which has a better biological interpretation than the first one. In that model several stability switches can occur, depending on the model parameters. We illustrate analytical results on the example of tumour growth model with parameters estimated on the basis of experimental data.

Activator-inhibitor system with delay and pattern formation

In the present paper, a description and mathematical analysis of a simple model of nonlinear pattern formation is given. The model is based on the so-called activator-inhibitor system proposed by Thomas. We introduce time delay into the reaction term and focus on its influence on morphogenesis and pattern formation. Numerical simulations are presented and compared for both cases without and with delay.

Gompertz model with delays and treatment: Mathematical analysis

In this paper we study the delayed Gompertz model, as a typical model of tumor growth, with a term describing external interference that can reflect a treatment, e.g. chemotherapy. We mainly consider two types of delayed models, the one with the delay introduced in the per capita growth rate (we call it the single delayed model) and the other with the delay introduced in the net growth rate (the double delayed model). We focus on stability and possible stability switches with increasing delay for the positive steady state. Moreover, we study a Hopf bifurcation, including stability of arising periodic solutions for a constant treatment. The analytical results are extended by numerical simulations for a pharmacokinetic treatment function.

A simple model of carcinogenic mutations with time delay and diffusion

In the paper we consider a system of delay differential equations (DDEs) of Lotka-Volterra type with diffusion reflecting mutations from normal to malignant cells. The model essentially follows the idea of Ahangar and Lin (2003) where mutations in three different environmental conditions, namely favorable, competitive and unfavorable, were considered. We focus on the unfavorable conditions that can result from a given treatment, e.g. chemotherapy. Included delay stands for the interactions between benign and other cells. We compare the dynamics of ODEs system, the system with delay and the system with delay and diffusion. We mainly focus on the dynamics when a positive steady state exists. The system which is globally stable in the case without the delay and diffusion is destabilized by increasing delay, and therefore the underlying kinetic dynamics becomes oscillatory due to a Hopf bifurcation for appropriate values of the delay. This suggests the occurrence of spatially non-homogeneous periodic solutions for the system with the delay and diffusion.

Model of tumour angiogenesis -- analysis of stability with respect to delays

In the paper we consider the model of tumour angiogenesis process proposed by Bodnar and Fory (2009). The model combines ideas of Hahnfeldt et al. (1999) and Agur et al. (2004) describing the dynamics of tumour, angiogenic proteins and effective vessels density. Presented analysis is focused on the dependance of the model dynamics on delays introduced to the system. These delays reflect time lags in the proliferation/death term and the vessel formation/regression response to stimuli. It occurs that the dynamics strongly depends on the model parameters and the behaviour independent of the delays magnitude as well as multiple stability switches with increasing delay can be obtained.

LOGISTIC TYPE EQUATIONS WITH DISCRETE DELAY AND QUASI-PERIODIC SUPPRESSION RATE

Abstract In this paper a single species dynamics governed by the logistic type delayed equation with an external influence (more precisely, suppression) on the population size is studied. Conditions guaranteeing global stability of the zero steady state are proved. These conditions are necessary and sufficient for a periodic or quasi-periodic suppression rate. Moreover, if the external influence is periodic and the zero steady state is repulsive, the existence of a periodic solution is shown. From the applications point of view, the logistic type equations are often used in the description of underlying tumour dynamics. In this case the external influence reflects the tumour treatment and the global stability of the zero steady state can be interpreted as the cure.

Mathematical model for NF-kappaB-driven proliferation of adult neural stem cells.

Abastract.  Neural stem cells (NSCs) are early precursors of neuronal and glial cells. NSCs are capable of generating identical progeny through virtually unlimited numbers of cell divisions (cell proliferation), producing daughter cells committed to differentiation. Nuclear factor kappa B (NF‐κB) is an inducible, ubiquitous transcription factor also expressed in neurones, glia and neural stem cells. Recently, several pieces of evidence have been provided for a central role of NF‐κB in NSC proliferation control. Here, we propose a novel mathematical model for NF‐κB–driven proliferation of NSCs. We have been able to reconstruct the molecular pathway of activation and inactivation of NF‐κB and its influence on cell proliferation by a system of nonlinear ordinary differential equations. Then we use a combination of analytical and numerical techniques to study the model dynamics. The results obtained are illustrated by computer simulations and are, in general, in accordance with biological findings reported by several independent laboratories. The model is able to both explain and predict experimental data. Understanding of proliferation mechanisms in NSCs may provide a novel outlook in both potential use in therapeutic approaches, and basic research as well.

A Matter of Timing: Identifying Significant Multi-Dose Radiotherapy Improvements by Numerical Simulation and Genetic Algorithm Search

Multi-dose radiotherapy protocols (fraction dose and timing) currently used in the clinic are the product of human selection based on habit, received wisdom, physician experience and intra-day patient timetabling. However, due to combinatorial considerations, the potential treatment protocol space for a given total dose or treatment length is enormous, even for relatively coarse search; well beyond the capacity of traditional in-vitro methods. In constrast, high fidelity numerical simulation of tumor development is well suited to the challenge. Building on our previous single-dose numerical simulation model of EMT6/Ro spheroids, a multi-dose irradiation response module is added and calibrated to the effective dose arising from 18 independent multi-dose treatment programs available in the experimental literature. With the developed model a constrained, non-linear, search for better performing cadidate protocols is conducted within the vicinity of two benchmarks by genetic algorithm (GA) techniques. After evaluating less than 0.01% of the potential benchmark protocol space, candidate protocols were identified by the GA which conferred an average of 9.4% (max benefit 16.5%) and 7.1% (13.3%) improvement (reduction) on tumour cell count compared to the two benchmarks, respectively. Noticing that a convergent phenomenon of the top performing protocols was their temporal synchronicity, a further series of numerical experiments was conducted with periodic time-gap protocols (10 h to 23 h), leading to the discovery that the performance of the GA search candidates could be replicated by 17–18 h periodic candidates. Further dynamic irradiation-response cell-phase analysis revealed that such periodicity cohered with latent EMT6/Ro cell-phase temporal patterning. Taken together, this study provides powerful evidence towards the hypothesis that even simple inter-fraction timing variations for a given fractional dose program may present a facile, and highly cost-effecitive means of significantly improving clinical efficacy.