Discrete and Continuous Dynamical Systems - B
February 2004 , Volume 4 , Issue 1
Mathematical Models in Cancer
A special issue based on the Cancer Workshop at Vanderbilt University 2002
Guest Editors: Mary Ann Horn and Glenn Webb
This special issue is featured in Economist, one of the most influential publications
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"As in many hierarchies of scientific models, the virtues of a simpler theory can, under the right circumstances, outweigh its vices." This comment concerning modeling, expressed by Raymond Lee and Alistair Fraser (see bibliography), is illustrated in the context of wound healing by means of a series of increasingly sophisticated analytical models. The motivation for these models is based on experimental studies of the critical size defect (CSD) in animal models which has been defined as the smallest intraosseous wound that does not heal by bone formation during the lifetime of the animal. For practical purposes this timescale can usually be taken as one year (later, the definition was further extended to a defect which has less than ten percent bony regeneration during the lifetime of the animal). CSDs can "heal" by fibrous connective tissue formation, but since this is not bone it does not have the properties (strength, etc.) that a completely healed defect would. A sequence of increasingly sophisticated models is developed and their biological implications are discussed in some detail.
Analytical arguments are used to enhance findings related to the Gompertzian growth kinetics of disseminated cancer cells. It is shown that such cells could also obey kinetics described by a modified Gompertz representation arising from looking at the bone marrow as a porous medium.
In this paper we present several ODE systems encoding the most essential observations and assumptions about the complex hierarchical interactive processes of tumor neo-vascularization (angiogenesis). From experimental results we infer that a significant marker of tumor aggressiveness is the oscillatory behavior of tumor size, which is driven by its vascularization dynamics. To study the forces underlying these oscillations we perform a Hopf point analysis of the angiogenesis models. In the analyzed models Hopf points appear if and only if a nontrivial set of time-delays is introduced into the tumor proliferation or the neo-vascularization process. We suggest to examine in laboratory experiments how to employ these results for containing cancer growth.
In this paper a mathematical model is presented that describes growth, immune escape, and siRNA treatment of tumors. The model consists of a system of nonlinear, ordinary differential equations describing tumor cells and immune effectors, as well as the immuno-stimulatory and suppressive cytokines IL-2 and TGF-$\beta$. TGF-$\beta$ suppresses the immune system by inhibiting the activation of effector cells and reducing tumor antigen expression. It also stimulates tumor growth by promoting angiogenesis, explaining the inclusion of an angiogenic switch mechanism for TGF-$\beta$ activity. The model predicts that increasing the rate of TGF-$\beta$ production for reasonable values of tumor antigenicity enhances tumor growth and its ability to escape host detection. The model is then extended to include siRNA treatment which suppresses TGF-$\beta$ production by targeting the mRNA that codes for TGF-$\beta$, thereby reducing the presence and effect of TGF-$\beta$ in tumor cells. Comparison of tumor response to multiple injections of siRNA with behavior of untreated tumors demonstrates the effectiveness of this proposed treatment strategy. A second administration method, continuous infusion, is included to contrast the ideal outcome of siRNA treatment. The model's results predict conditions under which siRNA treatment can be successful in returning an aggressive, TGF-$\beta$ producing tumor to its passive, non-immune evading state.
This paper deals with the development of an asymptotic theory for large systems of interacting cells in a vertebrate. Macroscopic diffusion and evolution equations are derived from the microscopic behavior represented by a class of nonlinear kinetic equations obtained as a generalization of the Boltzmann equation in mathematical biology. The analysis shows how the time-scaling plays a crucial role in the derivation of different type of equations. The application, developed in the second part of the paper refers to a model of progressing tumor cells in competition with the immune system. The asymptotic analysis is addressed to derive the mathematical framework of macroscopic equations to describe the evolution of solid tumors in "vivo" or in "vitro" environments.
Experimentalists are developing new therapies that exploit the tendency of macrophages, a type of white blood cell, to localise within solid tumours. The therapy studied here involves engineering macrophages to produce chemicals that kill tumour cells. Accordingly, a simple mathematical model is developed that describes interactions between normal cells, tumour cells and infiltrating macrophages. Numerical and analytical techniques show how the ability of the engineered macrophages to eliminate the tumour changes as model parameters vary. The key parameters are $m^*$, the concentration of engineered macrophages injected into the vasculature, and $k_1$, the rate at which they lyse tumour cells. As $k_1$ or $m^*$ increases, the average tumour burden decreases although the tumour is never completely eliminated by the macrophages. Also, the stable solutions are oscillatory when $k_1$ and $m^*$ increase through well-defined bifurcation values. The physical implications of our results and directions for future research are also discussed.
A mathematical model is presented that describes several signaling events that occur in cells from patients with chronic myeloid leukemia, i.e. autophosphorylation of the Bcr-Abl oncoprotein and subsequent signaling through the Crkl pathway. Dynamical effects of the drug STI-571 (Gleevec) on these events are examined, and a minimal concentration for drug effectiveness is predicted by simulation. Most importantly, the model suggests that, for cells in blast crisis, cellular drug clearance mechanisms such as drug efflux pumps dramatically reduce the effectiveness of Gleevec. This is a new prediction regarding the efficacy of Gleevec. In addition, it is speculated that these resistance mechanisms might be present from the onset of disease.
We consider a mathematical model of a proliferating tumor cell population forming a cylindrical structure around an interior blood vessel. We analyse the equilibrium problem when cells are distinguished by maturity, radial distance from the interior vessel, and proliferative or quiescent state. We give sufficient conditions to assure the existence of a unique steady state.
We investigate a mathematical model for the dynamics between tumor cells, immune-effector cells, and the cytokine interleukin-2 (IL-2). In order to better determine under what circumstances the tumor can be eliminated, we implement optimal control theory. We design the control functional to maximize the effector cells and interleukin-2 concentration and to minimize the tumor cells. Next, we show that an optimal control exists for this problem. After which, we characterize our unique optimal control in terms of the solutions to the optimality system, which is the state system coupled with the adjoint system. Finally, we analyze the optimal control and optimality system using numerical techniques.
A variety of PDE models for tumor growth have been developed in the last three decades. These models are based on mass conservation laws and on reaction-diffusion processes within the tumor.
Tumour cells growing around blood vessels form structures called tumour cords. We review some mathematical models that have been proposed to describe the stationary state of the cord and the cord evolution after single-dose cell killing treatment. Whereas the cell population has been represented with age or maturity structure to describe the cord stationary state, for the response to treatment a simpler approach was followed, by representing the cell population by means of the cell volume fractions. In this latter model, where transport of oxygen is included and its concentration is critical for cell viability, some constraints to be imposed on the interface separating the tumour from the necrotic region have a crucial role. An analysis of experimental data from untreated tumour cords, which involves modelling by cell age and by volume fractions, and some results about the cord response to impulsive cell killing, are also presented.
A mathematical model is developed that investigates polyclonality and decreased apoptosis as mechanisms for the androgen-independent relapse of human prostate cancer. The tumor is treated as a continuum of two types of cells (androgen-dependent and androgen- independent) whose proliferation and apoptotic death rates differ in response to androgen rich and androgen poor conditions. Insight into the tumor's response to therapies which both partially and completely block androgen production is gained by applying a combination of analytical and numerical techniques to the model equations. The analysis predicts that androgen deprivation therapy can only be successful for a small range of the biological parameters no matter how completely androgen production is blocked. Numerical simulations show that the model captures all three experimentally observed phases of human prostate cancer progression including exponential growth prior to treatment, androgen sensitivity immediately following therapy, and the eventual androgen-independent relapse of the tumor. Simulations also agree with experimental evidence that androgen-independent relapse is associated with a decrease in apoptosis without an increase in proliferation.
Vasculogenesis, i.e. self-assembly of endothelial cells leading to capillary network formation, has been the object of many experimental investigations in recent years, due to its relevance both in physiological and in pathological conditions. We performed a detailed linear stability analysis of two models of in vitro vasculogenesis, with the aim of checking their potential for structure formation starting from initial data representing a continuum cell monolayer. The first model turns out to be unstable at low cell densities, while pressure stabilizes it at high densities. The second model is instead stable at low cell densities. Detailed information about the instability regions and the structure of the critical wave numbers are obtained in several interesting limiting cases. We expect that altogether, this information will be useful for further comparison of the two models with experiments.
Many lines of evidence lead to the conclusion that ribosomes, and therefore phosphorus, are potentially important commodities in cancer cells. Also, the population of cancer cells within a given tumor tends to be highly genetically and physiologically varied. Our objective here is to integrate these elements, namely natural selection driven by competition for resources, especially phosphorus, into mathematical models consisting of delay differential equations. These models track mass of healthy cells within a host organ, mass of parenchyma (cancer) cells of various types and the number of blood vessels within the tumor. In some of these models, we allow possible mechanisms that may reduce tumor phosphorous uptake or allow the total phosphorus in the organ to vary. Mathematical and numerical analyses of these models show that tumor population growth and ultimate size are more sensitive to total phosphorus amount than their growth rates are. In particular, our simulation results show that if an artificial mechanism (treatment) can cut the phosphorus uptake of tumor cells in half, then it may lead to a three quarter reduction in ultimate tumor size, indicating an excellent potential of such a treatment. Also, in general we find that tumors with a relatively high cell death rate are more susceptible to treatments that block phosphorus uptake by tumor cells. Similarly, tumors with a large phosphorus requirement and (or) low cell reproductive rates are also strongly affected by phosphorus limitation.
The interactions between a solid tumor and the immune system are described both prior to and after neovascularization by a predator-prey model, and predictions about tumor behavior in a host are made. Trajectory analysis of phase-plane portraits as well as standard perturbation analysis show that most system steady states are unstable but that stability is theoretically possible. Reasonable parameter value estimation enables meaningful analysis of system behavior, and Mathematica is used to simulate model dynamics. The model accounts for many observed tumor behaviors, and regions of uncontrolled tumor growth, tumor extinction in finite time, and irreversible lymphocyte decline are found either analytically or numerically. A better understanding of tumor-immune dynamics is obtained, allowing for improved research on treatment specifically in the area of immunotherapy.
The effects of angiogenesis on oxygenation of an epidermal wound are described using a mathematical model. Diffusion equations are used to characterize the dependence of the wounded tissue regeneration on oxygen availability, which in turn affects the production of the Macrophage Derived Growth Factors (MDGFs) and as a result the growth of capillary density. When the capillaries grow beyond a certain point, they contribute to their own growth retardation, and as a result, a negative feedback mechanism is build into the system. The results of this model suggest that in order for an epidermal wound to be healed successfully the levels of oxygen concentration within the wounded area must be low. This process parallels an earlier mathematical model developed to describe the capillary growth in the retina, and demonstrates the generality and application of such a modeling approach to various biological phenomena involving growth factors.
A central goal of human genetics is the identification of combinations of DNA sequence variations that increase susceptibility to common, complex human diseases. Our ability to use genetic information to improve public health efforts to diagnose, prevent, and treat common human diseases will depend on our ability to understand the hierarchical relationship between complex biological systems at the genetic, cellular, biochemical, physiological, anatomical, and clinical endpoint levels. We have previously demonstrated that Petri nets are useful for building discrete dynamic systems models of biochemical networks that are consistent with nonlinear gene-gene interactions observed in epidemiological studies. Further, we have developed a machine learning approach that facilitates the automatic discovery of Petri net models thus eliminating the need for human-based trial and error approaches. In the present study, we evaluate this automated model discovery approach using four different nonlinear gene-gene interaction models. The results indicate that our model-building approach routinely identifies accurate Petri net models in a human-competitive manner. We anticipate that this general modeling strategy will be useful for generating hypotheses about the hierarchical relationship between genes, biochemistry, and measures of human health.
Glioblastomas are the most malignant and most common glioma, a type of primary brain tumor with the unfortunate ability to recur despite extensive treatment. Even with the advent of medical imaging technology during the last two decades, successful treatment of glioblastomas has remained elusive. It has become increasingly clear that, along with the proliferative potential of these neoplasms, it is the subclinically diffuse invasion of glioblastomas that primarily contributes to their resistance to treatment. In other words, the inevitable recurrence of these tumors is the result of diffusely invaded but invisible tumor cells peripheral to the abnormal signal on medical imaging and to the current limits of surgical, radiological and chemical treatments.
Mathematical modeling has presented itself as a viable tool for studying complex biological processes (Murray, 1993, 2002). We have developed a mathematical model that portrays the growth and extension of theoretical glioblastoma cells in a matrix that accurately describes the brain's anatomy to a resolution of 1 cu mm (Swanson, et al, 1999, 2000, 2002, 2003a, 2003b). The model assumes that only two factors need be considered for such predictions: net growth rate and infiltrative ability. The model has already provided illustrations of theoretical glioblastomas that not only closely resemble the MRIs (magnetic resonance imaging) of actual patients, but also show the distribution of the diffusely infiltrating cells.
In this paper we have developed a state space model for carcinogenesis. By using this state space model we have also developed statistical procedures to estimate the unknown parameters via multi-level Gibbs sampling method. We have applied this model and the methods to the British physician data on lung cancer with smoking. Our results indicate that the tobacco nicotine is an initiator. If $t > 60$ years old, then the tobacco nicotine is also a promoter.
Cell cycle perturbations occur after treatment with all anticancer drugs. The perturbations are usually classified as cytostatic (cell cycle arrest) or cytotoxic (cell killing). Our approach for analysis of cell cycle perturbations in vitro was to consider all the data provided by different experimental tests and interpret them through a mathematical formulation of the problem.
The model adopted for data analysis and interpretation is the result of merging two models, one for the cell cycle and the other for the drug effects. The first exploits the results of the theory of age-structured cell population dynamics while the second is based on distinct parameters ("effect descriptors") directly linked to cell cycle arrest, damage repair or cell death in $G_1$ and $G_2M$ and to inhibition of DNA synthesis and death in $S$. The set of values of the effect descriptors which are coherent with all experimental data are used to estimate the cytostatic and cytotoxic effects separately.
Applying the procedure to data from in vitro experiments, we found complex but biologically consistent patterns of time and dose dependence for each cell cycle effect descriptor, opening the way for a link to the parallel changes in the molecular pathways associated with each effect.
Ductal carcinoma in situ (DCIS) refers to a specific diagnosis of cancer that is isolated within the breast duct, and has not spread to other parts of the breast. We modify a model proposed by Byrne and Chaplain for the growth of a tumour consisting of live cells (nonnecrotic tumour) to describe the tumour growth inside a cylinder, a model mimicking the growth of a ductal carcinoma. The model is in the form of a free boundary problem. The analysis of stationary solutions of the problem shows interesting results that are similar to the patterns found in DCIS.
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