Mathematical modelling of drug transport in tumour multicell spheroids and monolayer cultures

The minimal, phase-transition model for the cell-number maintenance by the hyperplasia-extended homeorhesis

Oncogenic hyperplasia is the first and inevitable stage of formation of a (solid) tumor. This stage is also the core of many other proliferative diseases. The present work proposes the first minimal model that combines homeorhesis with oncogenic hyperplasia where the latter is regarded as a genotoxically activated homeorhetic dysfunction. This dysfunction is specified as the transitions of the fluid of cells from a fluid, homeorhetic state to a solid, hyperplastic-tumor state, and back. The key part of the model is a nonlinear reaction-diffusion equation (RDE) where the biochemical-reaction rate is generalized to the one in the well-known Schlögl physical theory of the non-equilibrium phase transitions. A rigorous analysis of the stability and qualitative aspects of the model, where possible, are presented in detail. This is related to the spatially homogeneous case, i.e. when the above RDE is reduced to a nonlinear ordinary differential equation. The mentioned genotoxic activation is treated as a prevention of the quiescent G0-stage of the cell cycle implemented with the threshold mechanism that employs the critical concentration of the cellular fluid and the nonquiescent-cell-duplication time. The continuous tumor morphogeny is described by a time-space-dependent cellular-fluid concentration. There are no sharp boundaries (i.e. no concentration jumps exist) between the domains of the homeorhesis- and tumor-cell populations. No presumption on the shape of a tumor is used. To estimate a tumor in specific quantities, the model provides the time-dependent tumor locus, volume, and boundary that also points out the tumor shape and size. The above features are indispensable in the quantitative development of antiproliferative drugs or therapies and strategies to prevent oncogenic hyperplasia in cancer and other proliferative diseases. The work proposes an analytical-numerical method for solving the aforementioned RDE. A few topics for future research are suggested.

A spatiotemporal, patient individualized simulation model of solid tumor response to chemotherapy in vivo: the paradigm of glioblastoma multiforme treated by temozolomide

A novel four-dimensional, patient-specific Monte Carlo simulation model of solid tumor response to chemotherapeutic treatment in vivo is presented. The special case of glioblastoma multiforme treated by temozolomide is addressed as a simulation paradigm. Nevertheless, a considerable number of the involved algorithms are generally applicable. The model is based on the patient's imaging, histopathologic and genetic data. For a given drug administration schedule lying within acceptable toxicity boundaries, the concentration of the prodrug and its metabolites within the tumor is calculated as a function of time based on the drug pharamacokinetics. A discretization mesh is superimposed upon the anatomical region of interest and within each geometrical cell of the mesh the most prominent biological "laws" (cell cycling, necrosis, apoptosis, mechanical restictions, etc.) are applied. The biological cell fates are predicted based on the drug pharmacodynamics. The outcome of the simulation is a prediction of the spatiotemporal activity of the entire tumor and is virtual reality visualized. A good qualitative agreement of the model's predictions with clinical experience supports the applicability of the approach. The proposed model primarily aims at providing a platform for performing patient individualized in silico experiments as a means of chemotherapeutic treatment optimization.

A multi-phase mathematical model of quorum sensing in a maturing Pseudomonas aeruginosa biofilm

It is well known that sessile bacteria have a strong tendency to exist in a biofilm phenotype, whereby bacterial cells aggregate and produce a gel-like extracellular matrix, which, in an infection scenario, offers a significant barrier to attack by conventional antibiotics and the immune system. In this paper we develop a multi-phase model of a maturing Pseudomonas aeruginosa biofilm, allowing for the production and secretion of exopolysaccharide (EPS). The primary quorum-sensing system of P. aeruginosa (namely the lasR system) is believed to be required for full biofilm development, and we thus take the synthesis of EPS to be regulated by the cognate signal molecule, 3-oxo-C12-HSL. We also take EPS and signal production, along with bacterial growth, to be limited by oxygen availability, thus factoring in the nutrient poor conditions deep inside the biofilm. We use simulations to examine the role played by quorum sensing in the biofilm maturation process, and to investigate the effect of anti-quorum sensing and antibiotic treatments on EPS concentration, signal level, bacterial numbers and biofilm growth rate. In addition, we undertake analysis of the associated travelling-wave behaviour.

Continuum versus discrete model: a comparison for multicellular tumour spheroids

We study multicellular tumour spheroids with a continuum model based on partial differential equations (PDEs). The model includes viable and necrotic cell densities, as well as oxygen and glucose concentrations. Viable cells consume nutrients and become necrotic below critical nutrient concentrations. Proliferation of viable cells is contact-inhibited if the total cellular density locally exceeds volume carrying capacity. The model is discussed under the assumption of spherical symmetry. Unknown model parameters are determined by simultaneously fitting the cell number to several experimental growth curves for different nutrient concentrations. The outcome of the PDE model is compared with an analogous off-lattice agent-based model for tumour growth. It turns out that the numerically more efficient PDE model suffices to explain the macroscopic growth data. As in the agent-based model, we find that the experimental growth curves are only reproduced when a necrotic core develops. However, evaluation of morphometric properties yields differences between the models and the experiment.

Incorporating energy metabolism into a growth model of multicellular tumor spheroids

Diffusion limitations in tumors create regions that are deficient in essential nutrients and contain a large number of quiescent and dying cells. Chemotherapeutic compounds are not effective against quiescent cells and therefore have reduced efficacy against tumors with extensive quiescence. We have formulated a mathematical model that predicts the extent and location of quiescence in multicellular spheroids. Multicellular spheroids are in vitro models of in vivo tumor growth that have proven to be useful experimental systems for studying radiation therapy, drug penetration, and novel chemotherapeutic strategies. Our model incorporates a realistic description of primary energy metabolism within reaction-diffusion equations to predict local glucose, oxygen, and lactate concentrations and an overall spheroid growth rate. The model development is based on the assumption that local cellular growth and death rates are determined by local ATP production generated by intracellular energy metabolism. Dynamic simulation and parametric sensitivity studies are used to evaluate model behavior, including the spatial distribution of proliferating, quiescent, and dead cells for different cellular characteristics. Using this model we have determined the critical cell survival parameters that have the greatest impact on overall spheroid physiology, and we have found that oxygen transport has a greater effect than glucose transport on the distribution of quiescent cells. By predicting the extent of quiescence based on individual cellular characteristic alone this model has the potential to predict therapeutic efficiency and can be used to design effective chemotherapeutic strategies.

A multiscale mathematical model of cancer, and its use in analyzing irradiation therapies

Background

Radiotherapy outcomes are usually predicted using the Linear Quadratic model. However, this model does not integrate complex features of tumor growth, in particular cell cycle regulation.

Methods

In this paper, we propose a multiscale model of cancer growth based on the genetic and molecular features of the evolution of colorectal cancer. The model includes key genes, cellular kinetics, tissue dynamics, macroscopic tumor evolution and radiosensitivity dependence on the cell cycle phase. We investigate the role of gene-dependent cell cycle regulation in the response of tumors to therapeutic irradiation protocols.

Results

Simulation results emphasize the importance of tumor tissue features and the need to consider regulating factors such as hypoxia, as well as tumor geometry and tissue dynamics, in predicting and improving radiotherapeutic efficacy.

Conclusion

This model provides insight into the coupling of complex biological processes, which leads to a better understanding of oncogenesis. This will hopefully lead to improved irradiation therapy.

Simulating chemotherapeutic schemes in the individualized treatment context: the paradigm of glioblastoma multiforme treated by temozolomide in vivo

A novel patient individualized, spatiotemporal Monte Carlo simulation model of tumor response to chemotherapeutic schemes in vivo is presented. Treatment of glioblastoma multiforme by temozolomide is considered as a paradigm. The model is based on the patient's imaging, histopathologic and genetic data. A discretization mesh is superimposed upon the anatomical region of interest and within each geometrical cell of the mesh the most prominent biological "laws" (cell cycling, apoptosis, etc.) in conjunction with pharmacokinetics and pharmacodynamics information are applied. A good qualitative agreement of the model's predictions with clinical experience supports the applicability of the approach to chemotherapy optimization.

The competitive dynamics between tumor cells, a replication-competent virus and an immune response

Replication-competent viruses have been used as an alternative therapeutic approach for cancer treatment. However, new clinical data revealed an innate immune response to virus that may mitigate the effects of treatment. Recently, Wein, Wu and Kirn have established a model which describes the interaction between tumor cells, a replication-competent virus and an immune response (Cancer Research 63 (2003):1317-1324). The purpose of this paper is to extend their model from the viewpoints of mathematics and biology and then prove global existence and uniqueness of solution to this new model, to study the dynamics of this novel therapy for cancers, and to explore a explicit threshold of the intensity of the immune response for controlling the tumor. We also study a time-delayed version of the model. We analytically prove that there exists a critical value tau0 of the time-delay tau such that the system has a periodic solution if tau > tau0. Numerical simulations are given to verify the analytical results. Furthermore, we numerically study the spatio-temporal dynamics of the model. The effects of the diffusivity of the immune response on the tumor growth are also discussed.

The use of 3-D cultures for high-throughput screening: the multicellular spheroid model

Over the past few years, establishment and adaptation of cell-based assays for drug development and testing has become an important topic in high-throughput screening (HTS). Most new assays are designed to rapidly detect specific cellular effects reflecting action at various targets. However, although more complex than cell-free biochemical test systems, HTS assays using monolayer or suspension cultures still reflect a highly artificial cellular environment and may thus have limited predictive value for the clinical efficacy of a compound. Today's strategies for drug discovery and development, be they hypothesis free or mechanism based, require facile, HTS-amenable test systems that mimic the human tissue environment with increasing accuracy in order to optimize preclinical and preanimal selection of the most active molecules from a large pool of potential effectors, for example, against solid tumors. Indeed, it is recognized that 3-dimensional cell culture systems better reflect the in vivo behavior of most cell types. However, these 3-D test systems have not yet been incorporated into mainstream drug development operations. This article addresses the relevance and potential of 3-D in vitro systems for drug development, with a focus on screening for novel antitumor drugs. Examples of 3-D cell models used in cancer research are given, and the advantages and limitations of these systems of intermediate complexity are discussed in comparison with both 2-D culture and in vivo models. The most commonly used 3-D cell culture systems, multicellular spheroids, are emphasized due to their advantages and potential for rapid development as HTS systems. Thus, multicellular tumor spheroids are an ideal basis for the next step in creating HTS assays, which are predictive of in vivo antitumor efficacy.

Tumour eradication by antiangiogenic therapy: analysis and extensions of the model by Hahnfeldt et al. (1999)

The model proposed by Hahnfeldt et al. (1999) describes the growth of a tumour assuming that tumour growth is strictly controlled by the evolution of the vascular network that supplies oxygen and nutrients to tumour cells. Consequently, it provides a framework to represent the effects of antiangiogenic therapies. In this paper, some possible modifications of that model are proposed, and conditions that guarantee the eradication of the tumour under a regimen of periodic antiangiogenic therapy are derived. The model variants considered assume the potential doubling time of the vasculature to be constant, and subdivide the endothelial cell pool, which is involved in angiogenesis, in resting and proliferating cells allowing for a more detailed description of drug effects.