Dynamics of Cancer: Modeling and Experiment

Mathematical models of this process develop according to two mainstreams. Murray and coworkers (see for instance [1]) focus on the strain process and state that the formation of the lacunae in the cell density is driven by the tension field due to the cells hanging on the matrigel. More recently it has been noticed that pure mechanical effects cannot predict the measured invariance of the chord length: the strain field can only amplifylacunae that are dictated by the initial conditions and no characteristic lengths can arise. One can instead properly account for this experimental fact incorporating into the mathematical description a mechanism largely documentated by the biomedical literature: the chemotactic signalling. This way, it is possible to describe the system of the cells as a fluid, including chemoattraction as a bulk force. A characteristic length naturally arises and can be expressed in terms of physical parameters to be measured independently. Aim of this talk is to illustrate a new model, providing a unified view of the morphogenetic process. When including both chemotactic signalling and matrigel stretch in the description, it is possible to predict both the emerging characteristic length scale and the chords thinning due to the tension field.

[1] Manoussaki, D., Lubkin, S.R., Vernon, R.B., Murray, J.D. A mechanical model for the formation of vascular networks in vitro. Acta Biotheoretica 44:271--282 (1996).

[2] D. Ambrosi, A. Gamba and G. Serini. Cell directional and chemotaxis in vascular morphogenesis, Bulletin of Mathematical Biology, 66 (2004).

Boushaba, K *Mathematical modeling for dormant pre-angiogenic tumors*

Byrne, H *Ductal carcinoma in situ: modelling in the dark*

To date, most mathematical models of solid tumour growth have been developed in response to existing experimental or clinical data, the aim being to explain the mechanisms responsible for phenomena that include the multilayered structure of multicellular spheroids, patterns of angiogenesis and the response of vascular tumours to different combinations of anti-cancer drugs. Unfortunately such information is not always available. The early, pre-invasive stage of ductal carcinoma in situ (DCIS) serves as a good example for two reasons: first, by the time such a tumour is diagnosed it will typically have breached the duct wall and, secondly, any tumour regions that remain confined to the duct wall will be removed by surgery. In this talk I will focus on recent work concerned with modelling DCIS which illustrates the type of insight that can be obtained when relevant data is lacking.

The resulting models involve coupling existing models of avascular tumour growth in a cylindrically-symmetric tube with mechanical models for the finite deformation of the compliant membrane that confines the tumour, the coupling being mediated by interactions between the expansive forces created by tumour cell proliferation and the stresses that develop in the compliant membrane. The models provide insight into DCIS progression and also generate several hypotheses that could be tested experimentally.

Demuth, T, Berens, ME *A Glioma cell’s strategy for malignant invasion*

Gliomas are the most common primary brain tumors, taking the lives of 15,000 patients per year in the US. The early and pervasive tendency of glioma cells to invade into peritumoral normal brain underlies the poor prognosis for patients with glial tumors (median survival <15 months). This malignant dispersion of glioma prevents complete surgical resection, positions tumor cells behind an intact blood-brain barrier, and sequesters tumor cells outside the fields of focal radiation; each of these leads to heightened (almost certain) tumor recurrence.

The invasive phenotype of human gliomas is studied in a three-dimensional invasion assay, designed to faithfully model events inherent to glioma cell egress from a multicellular tumor mass (spheroid) and to mimic the physical and biochemical space into which the cells invade. Kinetic endpoints of glioma cell dispersion are assessed by light-microscopic imaging of spheroid systems over time. Whole genome expression profiling (~35,000 genes) is utilized to help identify drivers for malignant invasion that are subsequently validated by quantitative RT-PCR, immunostaining; candidate genetic drivers are functionally validated using siRNA and small molecule inhibitors (when available). Laboratory measurements of invasion activity, expression microarray data. and immunostaining will be discussed in the context of consolidating these parameters to expedite the mathematical modeling of gene candidates driving glioma invasion.

de Pillis, LG *Modeling cancer treatment: mixed immuno-chemo-vaccine therapy strategies*

Friedman, A *Stability/Instability of dormant tumors*

Jiang, Y *Solid tumor growth: A multiscale cellular model *

We present a new mathematical model for solid tumor growth and development that spans three distinct scales. At the cellular level, a lattice Monte Carlo model describes cellular dynamics (proliferation, adhesion and viability). At the subcellular level, a Boolean network regulates the expression of proteins that control the cell cycle. At the extracellular level, reaction-diffusion equations describe the chemical dynamics (nutrient, waste, growth promoter and inhibitors). Data from experiments with multicellular spheroids were used to determine the parameters of the simulations. Starting with a single tumor cell, this model produces an avascular tumor that quantitatively mimics experimental measurements in multicellular spheroids. The model also accurately ‘postdicts’ spheroid growth curves under different external nutrient supply conditions.

Khain, E *Dynamics and pattern formation in invasive tumor growth*

In this work, we study the in-vitro dynamics of the most malignant form of the primary brain tumor: Glioblastoma Multiforme. Typically, the growing tumor consists of the inner dense proliferating zone and the outer less dense invasive region. Experiments with different types of cells show qualitatively different behavior. Wild type cells invade a spherically symmetric manner, but mutant cells are organized in tenuous branches. We formulate a model for this sort of growth using two coupled reaction-diffusion equations for the cell and nutrient concentrations. When the ratio of the nutrient and cell diffusion coefficients exceeds some critical value, the plane propagating front becomes unstable with respect to transversal perturbations. The instability threshold and the full phase-plane diagram in the parameter space are determined. The results are in a good agreement with experimental findings for the two types of cells.

Lowengrub, J *Nonlinear Simulation of Angiogenesis and Solid Tumor Growth *

In this talk, I will focus on recent efforts to study solid tumor progression. Here we focus on a continuum-scale description and pose the problem in terms of conservation laws for nutrients, chemical factors and tumor cell populations. We focus first on single-phase models. We analyze the equations and develop accurate, adaptive numerical schemes. We will present simulations of the complex nonlinear coupling between the progression of the tumor and neovascularization. We demonstrate the predictive capability of the model through comparisons with in vitro and in vivo experimental studies of tumor growth. We then discuss extensions to multiphase and mixture models and discuss the effects of residual stress and cell-to-cell adhesion.

Maini, PK *Mulitscale modelling in vascular tumours*

The modelling of cancer provides an enormous mathematical challenge because of its inherent multi-scale nature. For example, in vascular tumours, nutrient is transported by the vascular system, which operates on a tissue level. However, it effects processes occuring on a molecular level. Molecular and intra-cellular events in turn effect the vascular network and therefore the nutrient dynamics. Our modelling approach is to model, using partial differential equations, processes on the tissue level and couple these to the intercellular events (modelled by ordinary differential equations) via cells modelled as automaton units. Thusfar, within this framework we have modelled structural adaptation at the vessel level and we have modelled the cell cycle in order to account for the effects of p27 during hypoxia. These results will be presented.

Maley, CC *Differentiation as a Tumor Suppressor Mechanism*

Large metazoans, such as humans, produce so many cells in a lifetime, that there is a real danger of initiating somatic evolution. Any mutant cell that slips the bonds of tissue homeostasis will have a selective advantage within the body. Some of these mutant cells may progress to malignancy. How has organismal evolution, and the selective pressure of cancer, sculpted the processes of differentiation and tissue homeostasis? We hypothesized that if daughter cells can re-enter the cell compartment of their parental cell (“self-renewal”), the tissue will be more susceptible to somatic evolution in comparison to a tissue in which daughter cells are more differentiated than their parental cell (“strict serial differentiation”). To test this hypothesis we developed an agent-based computational model of somatic evolution in a proliferative unit of an adult tissue. The model shows that tissues with self-renewal are more susceptible to somatic evolution, and thus cancer, than tissues with strict serial differentiation. The more transient amplifying cell stages, allowing greater amplification of the stem cells, the lower the risk of somatic evolution, even though adding these stages adds more cells to the system. Asymmetric differentiation in the transient amplifying cell compartments led to a greater risk of somatic evolution than symmetric differentiation. Finally, disruptions in differentiation pathways can generate exponential growth in the cell population due to abrogation of negative feedback loops in tissue homeostasis. Thus, if cancer has been an important selective force in evolution, we predict that tissues should be structured into a series of strictly and symmetrically differentiating cellular stages. These stages must be functionally if not morphologically distinguishable.

Nör, JE *Role of Bcl-2 in tumor angiogenesis: Biological studies*

We believed for many years that the role of the anti-apoptotic protein Bcl-2 in tumor angiogenesis was limited to the enhancement of endothelial cell survival. Recently, we observed that Bcl-2 triggers a pro-angiogenic signaling pathway in endothelial cells that is mediated by NF-kB signaling and results in upregulated expression of CXC chemokines. We hypothesize that Bcl-2 expression in the vascular endothelium has two distinct, and perhaps synergistic, effects on tumor angiogenesis: 1) Bcl-2 enhances the angiogenic phenotype of endothelial cells via an autocrine pathway mediated by the expression of pro-angiogenic CXC chemokines. And, 2) Bcl-2 enhances endothelial cell survival.

Biological systems may offer limited ability to dissect the relative contribution of each individual Bcl-2-mediated signaling pathway in the establishment and maintenance of tumor microvessels. This seminar will involve the presentation and discussion of

in vitroandin vivomodel systems used to evaluate the role of Bcl-2 in tumor angiogenesis in my laboratory. It will be followed by a presentation from our collaborators (Harsh Jain and Trachette Jackson) that have used mathematical models that allowed for better understanding of the complex role of Bcl-2 expression in tumor angiogenesis. We firmly believe that a combination of biological systems and mathematical models is required to enhance our understanding of the complex and inter-related processes involved in tumor angiogenesis.

Perumparani, A *Taking theory to the laboratory: Experiments in cancer metastasis motivated through mathematical modelling*

It is a happy occurrence when predictions made through modelling are borne out in the laboratory.

In my talk today I will discuss three predictions arising from the results of a mathematical model originally designed by Jonathan Sherratt, John Norbury and myself to study invasion by cancer cells.

Extracellular matrix mediated chemotaxis has traditionally been thought to augment malignant invasion. The first prediction suggests that chemotaxis plays a different role – it acts as an inhibitor of invasion. The second prediction suggests that proliferation by malignant cells has a stimulatory effect on invasion, and reciprocally, that invasion acts to promote proliferation. Our third prediction posits that the chemokinetic coefficient of cancer cells can be calculated through simple invasion assays. Each of these predictions, if borne out, has important clinical and therapeutic implications.

The predictions were tested in the laboratory using cancer cells lines and I will discuss the outcome of these experiments and their applications to the management of cancer.

Preziosi, L *Modelling vasculogenesis*

In vertebrates, supply of oxygen and nutrients to tissues is carried out by the blood vascular system through capillary networks. Capillary patterns are closely mimicked by endothelial cells cultured on Matrigel, a preparation of basement membrane proteins. On the Matrigel surface, single randomly dispersed endothelial cells self-organize into vascular networks. The network is characterized by a typical length scale, which is independent on the initial mean density of deposed cells N over a wide range of values of N. We give a detailed description of a mathematical model of the process which has proven able to reproduce several qualitative and quantitative features of in vitro vascularization experiments. The model is basically a multidimensional Burgers' equation coupled to an equation modeling the diffusion of a chemoattractant factor. Starting from sparse initial data, mimicking the initial conditions realized in laboratory experiments, the solutions to the model equations develop characteristic network structures, similar to observed ones, whose average size is related to the finite range of chemoattractant diffusion.

Sleeman, BD *Mathematical Modelling of Cell Motion in Application to Angiogenesis*

The individual and collective or tissue-like movement of cells is a fundamental component of many processes, including angiogenesis and metastasis in cancer. In these processes individual cells detect chemical and mechanical signals in their environment and alter their pattern of movement in response.

Mathematical models will be outlined which attempt to describe both the mechanical dynamics of cell movement as well as a cell’s chemotactic response to environmental signals. In the first case we shall show how ideas and techniques well known in continuum mechanics may be adapted in order to model the motion of eukaryotic cells, in which it is observed that the cells flatten onto surfaces and extend thin sheets of cytosol called lamellipodia. These in turn make attachments to the surface and by the initiation of internal contractors within the cell a forward motion is achieved.

In the second case we shall outline ideas from the theory of reinforced random walks and circular statistics in order to model the response of individual cells to chemical signals. Applications of this idea are used to study angiogenesis.

Swanson, KR *Clinical applications of quantitative modeling for invasive gliomas*

Gliomas account for over half of all primary brain tumors and have been studied extensively for decades. Even with increasingly sophisticated medical imaging technologies, gliomas remain uniformly fatal lesions. A significant gap remains between the goal of designing effective therapy and the present understanding of the dynamics of glioma progression. It has become increasingly clear that, along with the proliferative potential of these neoplasms, it is the subclinically diffuse invasion of gliomas that most contributes to their resistance to treatment. That is, the inevitable recurrence of these tumors is the result of diffusely invaded but practically invisible tumor cells peripheral to the abnormal signal on medical imaging and to the limits of surgical, radiological and chemical treatments.

In this presentation, I will demonstrate how quantitative modeling can not only shed light on the spatio-temporal growth of gliomas but also can have specific clinical application in real patients. Integration of our quantitative model with the T1-weighted and T2-weighted MR imaging characteristics of gliomas can provide estimates of the extent of invasion of glioma cells peripheral to the imaging abnormality. Further model analysis reveals remarkable concordance with patient survival rates. In summary, although current imaging techniques remain woefully inadequate in accurately resolving the true extent of gliomas, quantitative modeling provides a new approach for the dynamic assessment of real patients and helps direct the way to novel therapeutic approaches.