Mathematical Modelling of the Spatio-temporal Response of Cytotoxic T-Lymphocytes to a Solid Tumour Mark A.J. Chaplain Anastasios Matzavinos Vladimir A. Kuznetsov Mathematical Medicine and Biology 21, 1-34 (2004) C. R. Biologies 327, (2004)

Talk Overview Biological (pathological) background The immune system Mathematical model of immune-tumour interactions Numerical analysis and simulations Model analysis Discussion and conclusions

Chemical ChemicalCarcinogen Radiation Viruses Carcinogens interact with cell components (nucleus) Genetic mutations result (e.g. p53) Normal cell becomes a transformed cell Key difference from normal cell: uncontrolled proliferation Small cluster of malignant cells may still be destroyed Normal/Transformed Cell

The Individual Cancer Cell: “A Nonlinear Dynamical System”

Solid Tumour Growth Avascular growth phase (no blood supply) Angiogenesis (blood vessel network) Vascular growth Invasion and metastasis

~ 10 6 cells maximum diameter ~ 2mm Necrotic core Quiescent region Thin proliferating rim Avascular Growth: The Multicellular Spheroid

Tumour-induced angiogenesis

Invasive Growth Generic name for a malignant epithelial (solid) tumour is a CARCINOMA (Greek: Karkinos, a crab). Irregular, jagged shape often assumed due to local spread of carcinoma. Cancer cells break through basement membrane Basement membrane

Metastasis: “A Multistep Process”

Chemical ChemicalCarcinogen Radiation Viruses Carcinogens interact with cell components (nucleus) Genetic mutations result (e.g. p53) Normal cell becomes a transformed cell Key difference from normal cell: uncontrolled proliferation Small cluster of malignant cells may still be destroyed The Transformed Cell

The Immune System The immune system is a complex system of cells and molecules distributed throughout our bodies that provides us with a basic defence against bacteria, viruses, fungi, and other pathogenic agents.

The Immune System Neutrophil “attacking” a bacterium

Cytotoxic T-Lymphocytes

 One of the most important cell types of the immune system is a class of white blood cells known as lymphocytes.  These cells are created in the bone marrow (B), along with all of the other blood cells, and the thymus (T) and are transported throughout the body via the blood stream.  They can leave the blood through capillaries, explore tissues for foreign molecules or cells (antigens), and then return to the blood through the lymph system. Lymphocytes

Cytotoxic T-Lymphocytes  A particular sub-population of lymphocytes called cytotoxic T cells (CTLs) are responsible for killing virally infected cells and cells that appear abnormal, such as some tumour cells.  Tumour-specific CTLs can be isolated from animals and humans with established tumours, such as melanomas.  Tumours express antigens that are recognized as “foreign” by the immune system of the tumour-bearing host.

CTL – Tumour Cell Complexes  The process of killing a tumour cell by a CTL consists of two main stages: (I) CTL binding to the membrane of the tumour cell and (II) the delivery of apoptotic biochemical signals from the CTL to the tumour cell.  During the formation of tumour cell–CTL complexes, the CTLs secrete certain soluble diffusible chemicals (chemokines), which recruit more effector cells to the immediate neighbourhood of the tumour.

Cancer Dormancy  In some cases, relatively small tumours are in cell-cycle arrest or there is a balance between cell proliferation and cell death.  This “dynamic” steady state of a fully malignant, but regulated-through-growth-control, tumour, could continue many months or years.  In many (but not all) cases such a latent form of small numbers of malignant tumours is mediated by cellular immunity and in particular by CTLs.  Clinically, such latent forms of tumours have been referred to as cancer dormancy.

Clinical Implications of Cancer Dormancy  Patients with breast cancers have recurrences at a steady rate 10 to 20 years after mastectomy.  Both melanoma and renal carcinoma can have recurrences a decade or two after removal of the primary tumour.  There is a need for controlling biological processes such as micrometastases and cancer dormancy.  Mathematical modelling can be a powerful tool in predicting therapeutic spatial and temporal regimes for the application of various immunotherapies.

Mathematical Model Our mathematical model will be based around the key interactions between the CTLs, the tumour cells and the secretion of chemokine. Initially 6 dependent variables in a 1-dimensional domain.

Basic Kinetic Scheme of the Model Our mathematical model will be based around the key interactions between the CTLs and the tumour cells.

Basic Kinetic Scheme (cont.) Applying the law of mass action:

Equation Governing the CTLs Term derived through fitting to experimental data.

The Role of the Heaviside Function The Heaviside function h separates the domain of interest into two subregions, an epidermis-like one and a dermis-like one.

Equations Governing Remaining Cell Types and Chemokine

Parameter Estimation  The murine B cell lymphoma is used as an experimental model of cancer dormancy in mice (Uhr & Marches 2001).  The kinetic parameters of our model were determined to have the following values:

Initial Conditions

Travelling Wave Solutions

“Cancer Dormancy” Simulation

Temporal Dynamics of the CTL Overall Population

Temporal Dynamics of the Tumour Cell Overall Population

Temporal Dynamics of the Cell Complex Overall Population

Early Oscillations in the Total Number of Tumour Cells

Reaction Kinetics ODE System We consider the following autonomous system of ODEs that describes the underlying spatially homogeneous kinetics of our system (with the Heaviside function omitted):

Linear Stability Analysis The “healthy” steady state (unstable) The “tumour dormancy” steady state (unstable)

Limit Cycle

Hopf Bifurcation

Coexistence of Limit Cycles

Spatio-temporal Chaos ?  The evolution of the kinetics of our system appears to have some similarities with the evolution of the ODE kinetics of the predator- prey ecological models presented in Sherratt et al. (1995).  The systems presented there were able to depict an invasive wave of predators with irregular spatio-temporal oscillations behind the wave front.  Sherratt et al. (1995) undertook a detailed investigation of that particular behaviour in the framework of simplified reaction- diffusion systems of λ–ω type.  They were able to relate the appearance of these irregularities with periodic doubling and bifurcations to tori, which are well known routes to chaos.

Bifurcation Diagrams With Respect to Parameter p

Travelling Wave Analysis

5-dimensional 1 st order ODE system….

Steady states => Existence of heteroclinic connection…. (Guckenheimer & Holmes)

Travelling Wave Analysis

Travelling Wave Approximation

Radially Symmetric Solid Tumour Growth

“Three-dimensional” Simulation

Two-dimensional Simulation

Conclusions  The model is able to reproduce the experimentally observed phenomenon of tumour dormancy by depicting a quasi-stationary evolution in time.  The spatial distributions that arise from the dynamics of the model are unstable and highly heterogeneous.  The existence of heterogeneities in solid tumours infiltrated by CTLs has been reported in numerous immunomorphological investigations.  Potential application to treatment of patients via immunotherapy i.e. “cancer vaccination”.

Future Work  An explicitly two dimensional (space) model could be investigated, enabling us to study the effect of asymmetry.  Explicit interactions between the cancer cells and the host tissue could be incorporated into the model.  Explicit interactions between the lymphocytes and other immune cell types (mainly macrophages and neutrophils) could be considered.  The concept of a central necrotic core (and explicit oxygen distribution/uptake) could also be incorporated.