Mathematical Modelling of the Spatio-temporal Response of Cytotoxic T-Lymphocytes to a Solid Tumour Mark A.J. Chaplain Anastasios Matzavinos Vladimir A.

Mathematical Modelling of the Mathematical Modelling of the Spatio-temporal Response of Cytotoxic Spatio-temporal Response of Cytotoxic

T-Lymphocytes to a Solid TumourT-Lymphocytes to a Solid Tumour

Mark A.J. ChaplainMark A.J. Chaplain

Anastasios Matzavinos Anastasios Matzavinos

Vladimir A. KuznetsovVladimir A. Kuznetsov

Mathematical Medicine and Biology 21, 1-34 (2004)

C. R. Biologies 327, 995-1008 (2004)

Talk Overview

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

ChemicalChemicalCarcinogenCarcinogen RadiationRadiation

VirusesViruses

• 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) tumouris a CARCINOMA (Greek: Karkinos, a crab). Irregular, jagged shape often assumed due to localspread of carcinoma.

Cancer cells break through basement membrane Basement membrane

Metastasis:“A Multistep Process”

ChemicalChemicalCarcinogenCarcinogen RadiationRadiation

VirusesViruses

• 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 SystemThe 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 SystemThe Immune System

Neutrophil “attacking” a bacterium

Cytotoxic T-LymphocytesCytotoxic 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.

LymphocytesLymphocytes

Cytotoxic T-LymphocytesCytotoxic 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 ComplexesCTL – 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 DormancyCancer 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 Clinical Implications of Cancer DormancyDormancy

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 ModelMathematical ModelOur 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.

complexesby secreted chemokine,:

cells tumour dead :~

CTLs d/deadinactivate :~

complexes CTL-cell tumour :

cells tumour :

(CTLs)slymphocyte T cytotoxic :

T

E

C

T

E

complexesby secreted chemokine,:

cells tumour dead :~

CTLs d/deadinactivate :~

complexes CTL-cell tumour :

cells tumour :

(CTLs)slymphocyte T cytotoxic :

T

E

C

T

E

Basic Kinetic Scheme of the ModelBasic Kinetic Scheme of the Model

Our mathematical model will be based around the key interactions between the CTLs and the tumour cells.

TE1k

1k

pk 12

pk2 TE~

TE~

C

TE1k

1k

pk 12

pk2 TE~

TE~

C

cells tumour dead :~

CTLs d/deadinactivate :~

complexes CTL-cell tumour :

cells tumour :

slymphocyte T cytotoxic :

T

E

C

T

E

cells tumour dead :~

CTLs d/deadinactivate :~

complexes CTL-cell tumour :

cells tumour :

slymphocyte T cytotoxic :

T

E

C

T

E

Basic Kinetic Scheme (cont.)Basic Kinetic Scheme (cont.)

Applying the law of mass action:

. ~

,1~

,

, 1

,

kinetics local

2

kinetics local

2

kinetics local

211

kinetics local

211

kinetics local

211

pCkdt

Td

Cpkdt

Ed

CkkETkdt

dC

CpkkETkdt

dT

CpkkETkdt

dE

. ~

,1~

,

, 1

,

kinetics local

2

kinetics local

2

kinetics local

211

kinetics local

211

kinetics local

211

pCkdt

Td

Cpkdt

Ed

CkkETkdt

dC

CpkkETkdt

dT

CpkkETkdt

dE

TE1k

1k

pk 12

pk2 TE~

TE~

C

TE1k

1k

pk 12

pk2 TE~

TE~

C

Equation Governing the CTLsEquation Governing the CTLs

kinetics local

211

decay

1

ionproliferatsupplychemotaxismotility random

21

CpkkETkEd

Tg

fCxshEED

t

E

kinetics local

211

decay

1

ionproliferatsupplychemotaxismotility random

21

CpkkETkEd

Tg

fCxshEED

t

E

Term derived through fitting to experimental data.

The Role of the Heaviside FunctionThe 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 Equations Governing Remaining Cell Types and ChemokineCell Types and Chemokine

kinetics local

211

kinetics local

211

growth logistic

21

motility random

23

decay

4

production

3

diffusion

22

11

CkkETktC

CpkkETkTTbbTDtT

dCkDt

kinetics local

211

kinetics local

211

growth logistic

21

motility random

23

decay

4

production

3

diffusion

22

11

CkkETktC

CpkkETkTTbbTDtT

dCkDt

~~

~

1~

decay

3

kinetics local

2

decay

2

kinetics local

2

TdpCkt

T

EdCpkt

E

~~

~

1~

decay

3

kinetics local

2

decay

2

kinetics local

2

TdpCkt

T

EdCpkt

E

Parameter EstimationParameter Estimation

cmcellsday 1036.1cmcells 1002.2

cmcellsday 102988.0day 0412.0

9997.0day 2.7

day 0.24cmcellsday 103.1

cmcells 100.2day 18.0

11417

11811

12

11

1171

192

11

sg

fd

pk

kk

bb

cmcellsday 1036.1cmcells 1002.2

cmcellsday 102988.0day 0412.0

9997.0day 2.7

day 0.24cmcellsday 103.1

cmcells 100.2day 18.0

11417

11811

12

11

1171

192

11

sg

fd

pk

kk

bb

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 ConditionsInitial Conditions

1,0 ,00,

, if 1000exp

, if 00,

if 0

0 if 1000exp10,

if 1000exp1

0 if 00,

20

0

20

02

0

xx

llxlxC

llxxC

xxl

lxlxTxT

xxllxE

lxxE

1,0 ,00,

, if 1000exp

, if 00,

if 0

0 if 1000exp10,

if 1000exp1

0 if 00,

20

0

20

02

0

xx

llxlxC

llxxC

xxl

lxlxTxT

xxllxE

lxxE

Travelling Wave SolutionsTravelling Wave Solutions

““Cancer Dormancy” Simulation Cancer Dormancy” Simulation

Temporal Dynamics of theTemporal Dynamics of theCTL Overall PopulationCTL Overall Population

Temporal Dynamics of the Temporal Dynamics of the Tumour Cell Overall PopulationTumour Cell Overall Population

Temporal Dynamics of theTemporal Dynamics of theCell Complex Overall Population Cell Complex Overall Population

Early Oscillations in theEarly Oscillations in theTotal Number of Tumour CellsTotal Number of Tumour Cells

Reaction Kinetics ODE SystemReaction Kinetics ODE System

CkkETkdtdC

CpkkETkTTbbdtdT

CpkkETkEdTg

fCs

dtdE

211

21121

2111

11

CkkETkdtdC

CpkkETkTTbbdtdT

CpkkETkEdTg

fCs

dtdE

211

21121

2111

11

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 AnalysisLinear Stability Analysis

12157,4.18,317,,

2.1,03.1,56.0,,

68.0,02.0,85.17,,

0,0,1,,

444

333

222

111

CTE

CTE

CTE

CTE

12157,4.18,317,,

2.1,03.1,56.0,,

68.0,02.0,85.17,,

0,0,1,,

444

333

222

111

CTE

CTE

CTE

CTE

The “healthy” steady state (unstable)

The “tumour dormancy” steady state (unstable)

Limit CycleLimit Cycle

Limit CycleLimit Cycle

Hopf BifurcationHopf Bifurcation

Coexistence of Limit CyclesCoexistence of Limit Cycles

Spatio-temporal Chaos ?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 WithBifurcation Diagrams WithRespect toRespect to Parameter Parameter pp

Travelling Wave AnalysisTravelling Wave Analysis

1

212

2

2

2

CETt

C

CETTTr

T

t

T

CETET

C

r

E

t

E

1

212

2

2

2

CETt

C

CETTTr

T

t

T

CETET

C

r

E

t

E

.....z ctx .....z ctx

Travelling Wave AnalysisTravelling Wave Analysis

c-

1c-

c-

212

2

2

2

CETdz

dC

CETTTdz

Td

dz

dT

CETET

C

dz

Ed

dz

dE

c-

1c-

c-

212

2

2

2

CETdz

dC

CETTTdz

Td

dz

dT

CETET

C

dz

Ed

dz

dE

5-dimensional 1st order ODE system….

Steady statesSteady states

0

0

0

1

0

,

24.1

97.0

0

62.0

0

10 xx

0

0

0

1

0

,

24.1

97.0

0

62.0

0

10 xx

1dim))(Wdim())(dim(W 5su R10 xx 1dim))(Wdim())(dim(W 5su R10 xx

)x0(W manifold unstable ldimensiona three u )x0(W manifold unstable ldimensiona three u

)(W manifold stable ldimensiona three s 1x )(W manifold stable ldimensiona three s 1x

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

Travelling Wave AnalysisTravelling Wave Analysis

c-

1c-

c-

21

CETdz

dC

CETTTdz

dT

CETET

C

dz

dE

c-

1c-

c-

21

CETdz

dC

CETTTdz

dT

CETET

C

dz

dE

Travelling Wave ApproximationTravelling Wave Approximation

Travelling Wave ApproximationTravelling Wave Approximation

Travelling Wave ApproximationTravelling Wave Approximation

Radially Symmetric SolidRadially Symmetric SolidTumour GrowthTumour Growth

11

1

212

2

22

CETtC

CETTTrT

rrrt

T

CETET

Crh

rE

rrrt

E

11

1

212

2

22

CETtC

CETTTrT

rrrt

T

CETET

Crh

rE

rrrt

E

““Three-dimensional” SimulationThree-dimensional” Simulation

Two-dimensional SimulationTwo-dimensional Simulation

Two-dimensional SimulationTwo-dimensional Simulation

Two-dimensional SimulationTwo-dimensional Simulation

Two-dimensional SimulationTwo-dimensional Simulation

Two-dimensional SimulationTwo-dimensional Simulation

Two-dimensional SimulationTwo-dimensional Simulation

Two-dimensional SimulationTwo-dimensional Simulation

ConclusionsConclusions

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 WorkFuture 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.