Modelling the response of spatially structured tumours to chemotherapy: Drug kinetics

In this paper we consider the effects of a single anticancer agent on the growth of a solid tumour in the context of a simple mathematical model for the latter. The tumour is assumed to comprise a single cell population which reproduces and dies at a rate dependent on the local drug concentration. This causes cell movement and so establishes a velocity field within the tumour. We investigate the action of a single chemotherapeutic drug on the tumour and explore how different drug kinetics and treatment regimes may affect the final treatment outcome. A single infusion of drug is shown to be more effective than repeated short applications. We are able to construct asymptotic solutions to the model in the limit of a small drug degradation rate; these closely match solutions obtained numerically and provide additional insight into the behaviour of the tumour, in particular allowing the prediction of the strength of drug required to achieve tumour regression.     

Cross-linked actin networks: Micro- and macroscopic effects

Actin plays a crucial role in the mechanical response of cells. Together with other proteins, it also drives protrusion, motility and cell division. Two important aspects of the mechanical modeling of this kind of protein are considered: its microscopic and macroscopic behavior. At the microscopic level, we start with a model proposed by Holzapfel and Ogden [1] providing a relationship between the stretch of a single polymer chain and the applied tension force. The model is advantageous as it simulates the so-called ‘exceptional normal stresses’. This effect is typical for biopolymers and contradicts with the Poynting effect typically observed in rubber-like polymers. The multiscale finite element method (FEM) is applied to simulate the effective mechanical behavior of cell cytoplasm. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A mathematical analysis of a model for capillary network formation in the absence of endothelial cell proliferation

An analysis of a parabolic partial differential equation modelling capillary network formation is presented. The model includes terms representing cell random motility, chemotaxis, and haptotaxis due to the presence of chemical stimuli: tumour angiogenic factors and fibronectin. The analysis provides an underlying insight into mechanisms of cell migration which are crucial for tumour angiogenesis. Specific 1 and 2D examples are discussed in detail.     

A mathematical model of vascular tumour growth and invasion

In this paper, we develop a simple mathematical model of the vascularization and subsequent growth of a solid spherical tumour. The key elements that are encapsulated in this model are the development of a central necrotic core due to the collapse of blood vessels at the centre of the tumour and a peak of tumour cells advancing towards the main blood vessels together with the regression of newly-formed capillaries. Diffusion alone cannot account for all observed behaviour, and hence, we include ‘taxis’ in our model, whereby the movement of the tumour cells is directed towards high blood vessel densities. This means that the growth of the tumour is accompanied by the invasion of the surrounding tissue. Invasion is closely linked to metastasis, whereby tumour cells enter the blood or lymph system and hence secondary tumours or metastases may arise. In the second part of the paper, we conduct a travelling wave analysis on a simplified version of the model and obtain bounds on the parameters such that the solutions are nonnegative and hence biologically relevant and also an estimate for the rate of invasion.     

On a phase field model of Cahn–Hilliard type for tumour growth with mechanical effects

Mechanical effects have mostly been neglected so far in phase field tumour models that are based on a Cahn–Hilliard approach. In this paper we study a macroscopic mechanical model for tumour growth in which cell–cell adhesion effects are taken into account with the help of a Ginzburg–Landau type energy. In the overall model an equation of Cahn–Hilliard type is coupled to the system of linear elasticity and a reaction–diffusion equation for a nutrient concentration. The highly non-linear coupling between a fourth-order Cahn–Hilliard equation and the quasi-static elasticity system lead to new challenges which cannot be dealt within a gradient flow setting which was the method of choice for other elastic Cahn–Hilliard systems. We show existence, uniqueness and regularity results. In addition, several continuous dependence results with respect to different topologies are shown. Some of these results give uniqueness for weak solutions and other results will be helpful for optimal control problems.     

Modelling of glioblastoma growth by linking a molecular interaction network with an agent-based model

In this work, a mathematical model of malignant brain tumour growth is presented. In particular, the growth of glioblastoma is investigated on the intracellular and intercellular scale.

The Go or Grow principle of tumour cells states that tumour cells either migrate or proliferate. For glioblastoma, microRNA-451 has been shown to be an energy dependent key regulator of the LKB1 (liver kinase B1) and AMPK (AMP-activated protein kinase) pathway that influences the signalling for migration or cell division.

We introduce a mathematical model that reproduces these biological processes. The intracellular molecular interaction network is represented by a system of nine ordinary differential equations. This is put into a multiscale context by applying an agent-based approach: each cell is equipped with this interaction network and additional rules to determine its new phenotype as either migrating, proliferating or quiescent.

The evaluation of the proposed model by comparison of the results with in vitro experiments indicates its validity.     

Therapy burden, drug resistance, and optimal treatment regimen for cancer chemotherapy

Email: boldrini{at}ime.unicamp.brAuthor to whom correspondence should be addressed. Email: michel{at}Incc.br Three nonlinear models of tumour cell growth under continuousdelivery of cycle nonspecific anticancer agents are studied.A dynamical optimization problem with the objective of minimizingthe final level of tumour cells is posed for these mathematicalsetups. The simplest setup does not possess toxicity constraints,whereas the other setups contain a dynamical equation describingthe therapy burden as a toxicity criterion. In addition, thethird setting contains the dynamics of drug resistant cells.A discussion conceming the optimal strategies of the respectivemodels is performed.     

Discontinuous cellular automaton method for crack growth analysis without remeshing

A numerical technical of discontinuous cellular automaton method for crack growth analysis without remeshing is developed. In this method, the level set method is employed to track the crack location and its growth path, where the level set functions and calculation grids are independent, so no explicit meshing for crack surface and no remeshing for crack growth are needed. Then, the discontinuous enrichment shape functions which are enriched by the Heaviside function and the exact near-tip asymptotic field functions are constructed to model the discontinuity of cracks. Finally, a discontinuous cellular automaton theory is proposed, which are composed of cell, neighborhood and updating rules for discontinuous case. There is an advantage that the calculation is only applied on local cell, so no assembled stiffness matrix but only cell stiffness is needed, which can overcome the stiffness matrix assembling difficulty caused by unequal degrees of nodal freedom for different cells, and much easier to consider the local properties of cells. Besides, the present method requires much less computer memory than that of XFEM because of it local property.     

Numerical optimisation of chemotherapy dosage under antiangiogenic treatment in the presence of drug resistance

We consider a two-compartment model of chemotherapy resistant tumour growth under angiogenic signalling. Our model is based on the one proposed by Hahnfeldt et al. (1999), but we divide tumour cells into sensitive and resistant subpopulations. We study the influence of antiangiogenic treatment in combination with chemotherapy. The main goal is to investigate how sensitive are the theoretically optimal protocols to changes in parameters quantifying the interactions between tumour cells in the sensitive and resistant compartments, that is, the competition coefficients and mutation rates, and whether inclusion of an antiangiogenic treatment affects these results. Global existence and positivity of solutions and bifurcations (including bistability and hysteresis) with respect to the chemotherapy dose are studied. We assume that the antiangiogenic agents are supplied indefinitely and at a constant rate. Two optimisation problems are then considered. In the first problem a constant, indefinite chemotherapy dose is optimised to maximise the time needed for the tumour to reach a critical (fatal) volume. It is shown that maximum survival time is generally obtained for intermediate drug dose. Moreover, the competition coefficients have a more visible influence on survival time than the mutation rates. In the second problem, an optimal dosage over a short, 30-day time period, is found. A novel, explicit running penalty for drug resistance is included in the objective functional. It is concluded that, after an initial full-dose interval, an administration of intermediate dose is optimal over a broad range of parameters. Moreover, mutation rates play an important role in deciding which short-term protocol is optimal. These results are independent of whether antiangiogenic treatment is applied or not.     

Optimal Bang-Bang Controls for a Two-Compartment Model in Cancer Chemotherapy

A class of mathematical models for cancer chemotherapy which have been described in the literature take the form of an optimal control problem over a finite horizon with control constraints and dynamics given by a bilinear system. In this paper, we analyze a two-dimensional model in which the cell cycle is broken into two compartments. The cytostatic agent used as control to kill the cancer cells is active only in the second compartment where cell division occurs and the cumulative effect of the drug is used to model the negative effect of the treatment on healthy cells. It is shown that singular controls are not optimal for this model and the optimality properties of bang-bang controls are established. Specifically, transversality conditions at the switching surfaces are derived. In a nondegenerate setting, these conditions guarantee the local optimality of the flow if satisfied, while trajectories will be no longer optimal if they are violated.     

Mathematical modelling of immune reaction against gliomas: Sensitivity analysis and influence of delays

In the paper we considered a model of immune reaction against malignant glioma. The model proposed by Kronik et al. (Cancer Immunol. Immunother., 2008) describes simplified interactions between tumour cells and five components of the immune system. We studied the effects of uncertainties of the parameters values to the system behaviour. We showed that the tumour growth rate is one of the most important parameters only in case of fast growing tumours, that is for GBM in our case.On the basis of the performed sensitivity analysis we proposed a reduced model in which the role of time delays in loops appearing in the described interactions is considered. The proposed model includes only two main components of the reaction, that is tumour cells and cytotoxic T-lymphocytes. It occurs that although the reduced system is described by several non-linear terms with three time delays, its dynamics is simple and time delays have hardly any influence on it.Both considered models confirmed that the non-linearities present in interactions between tumour cells and CTLs play a major role in the system dynamics, while other components or delays can be taken into account as supplementary elements only.     

Modelling the role of cell-cell adhesion in the growth and development of carcinomas

In this paper, a mathematical model is presented to describe the evolution of an avascular solid tumour in response to an externally-supplied nutrient. The growth of the tumour depends on the balance between expansive forces caused by cell proliferation and cell-cell adhesion forces which exist to maintain the tumour's compactness. Cell-cell adhesion is incorporated into the model using the Gibbs-Thomson relation which relates the change in nutrient concentration across the tumour boundary to the local curvature, this energy being used to preserve the cell-cell adhesion forces.

Our analysis focuses on the existence and uniqueness of steady, radially-symmetric solutions to the model, and also their stability to time-dependent and asymmetric perturbations. In particular, our analysis suggests that if the energy needed to preserve the bonds of adhesion is large then the radially-symmetric configuration is stable with respect to all asymmetric perturbations, and the tumour maintains a radially-symmetric structure—this corresponds to the growth of a benign tumour. As the energy needed to maintain the tumour's compactness diminishes so the number of modes to which the underlying radially-symmetric solution is unstable increases—this corresponds to the invasive growth of a carcinoma. The strength of the cell-cell bonds of adhesion may at some stage provide clinicians with a useful index of the invasive potential of a tumour.     

Modelling and remodelling of biological tissue in the framework of continuum biomechanics

A biological tissue in general is formed by cells, extracellular matrix (ECM) and fluids. Consequently, its overall material behaviour results from its components and their interaction among each other. Furthermore, in case of living tissues, the material properties do not remain constant but naturally change due to adaptation processes or diseases. In the context of the Theory of Porous Media (TPM), a continuum-mechanical model is introduced to describe the complex fluid-structure interaction in biological tissue on a macroscopic scale. The tissue is treated as an aggregate of two immiscible constituents, where the cells and the ECM are summarised to a solid phase, whereas the fluid phase represents the extracellular and interstitial liquids as well as necrotic debris and cell or matrix precursors in solution. The growth and remodelling processes are described by a distinct mass exchange between the fluid and solid phase, which also results in a change of the constituent material behaviour. To furthermore guarantee the compliance with the entropy principle, the growth energy is introduced as an additional quantity. It measures the average of chemical energy available for cell metabolism, and thus, controls the growth and remodelling processes. To set an example, the presented model is applied for the simulation of the early stages of avascular tumour growth in the framework of the finite element method (FEM). (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Mathematical model of optimal chemotherapy strategy with allowance for cell population dynamics in a heterogeneous tumor

A mathematical model of tumor cell population dynamics is considered. The tumor is assumed to consist of cells of two types: amenable and resistant to chemotherapeutic treatment. It is assumed that the growth of the cell populations of both types is governed by logistic equations. The effect of a chemotherapeutic drug on the tumor is specified by a therapy function. Two types of therapy functions are considered: a monotonically increasing function and a nonmonotone one with a threshold. In the former case, the effect of a drug on the tumor is stronger at a higher drug concentration. In the latter case, a threshold drug concentration exists above which the effect of the therapy reduces. The case when the total drug amount is subject to an integral constraint is also studied. A similar problem was previously studied in the case of a linear therapy function with no constraint imposed on the drug amount. By applying the Pontryagin maximum principle, necessary optimality conditions are found, which are used to draw important conclusions about the character of the optimal therapy strategy. The optimal control problem of minimizing the total number of tumor cells is solved numerically in the case of a monotone or threshold therapy function with allowance for the integral constraint on the drug amount.     

On strategies on a mathematical model for leukemia therapy

A mathematical model for leukemia therapy based on the Gompertzian law of cell growth is studied. It is assumed that the chemotherapeutic agents kill leukemic as well as normal cells.Effectiveness of the medicine is described in terms of a therapy function. Two types of therapy functions are considered: monotonic and non-monotonic. In the former case the level of the effect of the chemotherapy directly depends on the quantity of the chemotherapeutic agent. In the latter case the therapy function achieves its peak at a threshold value and then the effect of the therapy decreases. At any given moment the amount of the applied chemotherapeutic is regulated by a control function with a bounded maximum. Additionally, the total quantity of chemotherapeutic agent which can be used during the treatment process is bounded too.The problem is to find an optimal strategy of treatment to minimize the number of leukemic cells while at the same time retaining as many normal cells as possible.With the help of Pontryagin’s Maximum Principle it was proved that the optimal control function has at most one switch point in both monotonic and non-monotonic cases for most relevant parameter values.A control strategy called alternative is suggested. This strategy involves increasing the amount of the chemotherapeutical medicine up to a certain value within the shortest possible period of time, and holding this level until the end of the treatment.The comparison of the results from the numerical calculation using the Pontryagin’s Maximum Principle with the alternative control strategy shows that the difference between the values of cost functions is negligibly small.     

A new modelling approach for the description of tissues at cell level

The understanding of tissues from a biomechanical perspective requires a deeper knowledge about cell mechanics. A lot of experimental and theoretical investigation has already been done to consider cells' behaviour under various conditions. Several types of cells exist, each with specific properties. In this work, the cell microstructure is characterized and explained via tensegrity systems. In doing so, we consider in a first step a simple cubic shaped cell built up by trusses and ropes that are discretized by finite 1D elements. This simplified microstructure represents the micro/cell level on the integration point of the finite element discretization at macro level. By using an appropriate homogenization technique for the micro level, it is assumed to gain a more detailed view on deformations occur on cell level. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A continuum-mechanical analysis of the influence of mechanical stimuli on biological tissue

Mechanical stimuli play a crucial role in the differentiation process of mesenchymal stem cells (MSC). The resulting mechanical signals are important in the regulation of various cell functions and maintenance of many tissues. The underlying molecular and biophysical mechanisms of the differentiation process are poorly understood. Present remodelling and growth models are purely phenomenological without linkage to cell mechanisms. The presented macroscopic model of MSC mechanics is based on a multiphasic-multicomponent formulation within the framework of Theory of Porous Media (TPM), where a single cell is considered as a mixture of interacting constituents. In particular, the constituents are the solid cytoskeleton saturated by a fluid phase (cytoplasm), which itself consists of a liquid solvent and mobile components, e. g., chemical messengers, proteins, etc. To demonstrate the capabilities of the developed model, first qualitative numerical simulations of the impact of external forces on MSC are presented. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Case Studies in cancer and its treatment by radiotherapy

Mathematical modelling has played a major role in certain aspects of the wide area of study of cancer and its treatment. Relevant background material is followed by mathematical modelling case studies in tumour growth and the response of tumour cells to irradiation. These components are synthesized in Case Study 4 where the problem of the derivation of optimal radiotherapy treatment schedules is discussed. The case studies are presented in a form suitable for classroom development.