Case Studies in Cancer Modelling: Connecting with …Case Studies in Cancer Modelling: Connecting with the Clinic Mark Chaplain School of Mathematics and Statistics Mark Chaplain Multiscale

Case Studies in Cancer Modelling:Connecting with the Clinic

Mark ChaplainSchool of Mathematics and Statistics

Mark Chaplain Multiscale cancer modelling INI 9th December 2015 1 / 54

Talk Overview

Biology of cancer

Intracellular modelling

Cell-scale modelling

Tissue-scale modelling

Applications and Impact


Cancer: A Nonlinear Dynamical System


The Hallmarks of Cancer


The Hallmarks of Cancer


Cancer: A Multiscale System






Intracellular modelling: Gene regulatory networks


Gene regulatory networks: Transcription Factors andOscillations

Spatio-temporal oscillations in the NF-κB system


Gene regulatory networks: Negative feedback loops

A generic negative feedback loop: species x produces y which then inhibitsx, in turn reducing levels of y. . .


The Hes1 Transcription Factor


Hes1 Spatial Stochastic Model

µp

!m

Pf Po

k1

k2

!m/"

mRNA

!p

#

#protein

µm


Hes1 Spatial Stochastic Model

Pf + proteink1−−k2

Po, (promoter, xm, nucleus)

Pfαm−−−→ mRNA, (promoter, xm, nucleus)

Poαm/γ−−−−−→ mRNA, (promoter, xm, nucleus)

mRNAαp−−→ mRNA + protein, (cytoplasm,Ωc)

mRNAµm−−−→ φ, (entire cell,Ω)

proteinµp−−→ φ, (entire cell,Ω)

proteiniD/h2

−−−−→ proteini+1, (entire cell,Ω)

mRNAiD/h2

−−−−→ mRNAi+1, (entire cell,Ω)

proteiniD/h2

−−−−→ proteini−1, (entire cell,Ω)

mRNAiD/h2

−−−−→ mRNAi−1, (entire cell,Ω)


Hes1: Experimental Data/Simulation Results

Experimental data from Kobayashi et al.1 showing Hes1 protein levels in murineembryonic stem cells.

1Kobayashi et al. (2009) The cyclic gene Hes1 contributes to diverse differentiationresponses of embryonic stem cells Genes Dev. 23, 1870 - 1875


Hes1: Experimental Data/Simulation Results

0 200 400 600 800 10000

100

200

300

400

time (min)

copy

num

ber

mRNAprotein

0 200 400 600 800 10000

100

200

300

400

500

time (min)

copy

num

ber

mRNAprotein

0 200 400 600 800 10000

20

40

60

80

time (min)

copy

num

ber

mRNAprotein

0 200 400 600 800 10000

20

40

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time (min)

copy

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ber

mRNAprotein

Corresponding simulation results from the spatial stochastic model1.

1Sturrock, Hellander, Matzavinos, Chaplain (2013) Spatial stochastic modelling ofthe Hes1 gene regulatory network: intrinsic noise can explain heterogeneity in embryonicstem cell differentiation. J. R. Soc. Interface 10, 20120988


Period Estimation


Intracellular modelling: p53-Mdm2 System





Experimental data from Lahav et al.2 showing p53 and Mdm2 protein levels inindividual cells.

2Lahav et al. (2004) Dynamics of the p53-Mdm2 feedback loop in individual cells.Nature Genetics 36, 147 - 150



Corresponding simulation results from a spatial stochastic p53 model.Mark Chaplain Multiscale cancer modelling INI 9th December 2015 19 / 54

Diffusion Causes Oscillations: Rigorous Proof


A Force-based, Individual-based Model of Tumour Growth

cells modelled as elastic spheres

maximum radius of the cells: R= 5 µm

cells divide into two spheres of radius R

213

cell cycle length are uniformly distributed between 17h and 27h



Cell-Cell Interaction

cell-cell repulsion calculated by the Hertz model

cell-cell adhesion calibrated with separation force measurements



The potential Vij between two cells of radius Ri and Rj is given by

Vij = (Ri +Rj − dij)5/21

5Eij

√RiRj

Ri +Rj︸︷︷︸repulsive forces

+ εs︸︷︷︸adhesive forces

.

Eij is defined by

Eij =3

4

(1− σ2iEi

+1− σ2jEj

).

Ei, Ej are the elastic moduli of the cells i, j, σi, σj the Poisson ratios ofthe spheres.


Intercellular Adhesion: E-cadherin and β-catenin







The interaction force results from deriving the potential function

F ij = −(∂Vij/∂dij)(d(dij)/dx, d(dij)/dy, d(dij)/dz)


Individual-based, Force-based Model

Cell movement:

Γfisvi︸︷︷︸

s-friction

+∑j nn i

Γfij

(vi − vj

)︸︷︷︸cell-cell friction

=∑i nn j

F ij︸︷︷︸forces

+ fi(t)︸︷︷︸

noise

+ χ∇Q(t)︸︷︷︸chemotaxis

+ ρ∇H(t)︸︷︷︸hapotaxis


Cell-Matrix Interactions

Elements of the model:

cell modelled as flat hemispherical object

I base radius : 15µmI height : 2.6µm

extracellular matrix fibres modelled as rigid cylinders

I length ∼N(75µm, 5µm)I width 200nmI uniformly distributed in 2D space

Cell-Fibre Interaction:

cell polarisation and migration

contact inhibition of locomotion

matrix remodelling


Cell-Matrix InteractionsMatrix Remodelling - fibre realignment due to adhesion and forcegeneration

a polarised cell pulls fibres towards itself whereby:

fibre is lever that is rotated about the moment of force

end of fibre that is furthest away from cell acts as fulcrum

realignment of fibre dependent on integrin expression and matrixstiffness


Summary


Single Cell Simulation


Multicell Simulation


Further Single Cell Simulations

Investigation of changes in:

fibre length

fibre density

matrix stiffness

matrix architecture

and the effect on:

cell speed

persistence time

Results:

matrix stiffness has a large influence on cell speed and persistence→ matrix reorientation seems to be a very important process in cellmigration

decrease in matrix isotropy leads to increase in persistence

nonlinear relationships between persistence times and fibre length anddensity


Further Single Cell Simulations

Investigation of changes in:

fibre length

fibre density

matrix stiffness

matrix architecture

and the effect on:

cell speed

persistence time

Results:

matrix stiffness has a large influence on cell speed and persistence→ matrix reorientation seems to be a very important process in cellmigration

decrease in matrix isotropy leads to increase in persistence

nonlinear relationships between persistence times and fibre length anddensity


Metastasis


Chemotherapy and Radiotherapy Treatment Modelling

hybrid cellular automaton - PDE approach

individual cancer cells

cell-cycle

blood vessels

oxygen, chemotherapy drug

radiation





Cell-cycle - ODEs in each cancer cell



Blood vessel distribution - 2D cross-section of tissue



Oxygen and chemotherapy drug:



Radiation:





Radiation:



Radiation:


Chemotherapy and Radiotherapy Treatment ModellingRadiation:


Cancer Invasion Modelling

Thursday 10th December:15:30 - 16:15 Pia Domschke (Technische Universitat Darmstadt)Mathematical Modelling of cancer invasion: The role of cell adhesionvariability

Friday 11th December11:30 - 12:30 Alf Gerisch (Technische Universitat Darmstadt)Nonlocal models for interaction drive cell movement











“The REF”



“The REF”



“The REF”(Unfortunately NOT) “The Last Judgement”



“The REF”(Unfortunately NOT) “The Last Judgement”


Applications and Impacthttp://www.ref.ac.uk








Summary

Mathematical models of cancer growth/treatment at multiple scales

Quantitative, predictive

Applications to clinical practice, patient-specific therapy


Collaborators

Marc Sturrock (ICL)

Ignacio Ramis-Conde (UCLM, Spain), Daniela Schluter (Lancaster)

Gibin Powathil (Swansea)


Case Studies in Cancer Modelling: Connecting with …Case Studies in Cancer Modelling: Connecting with the Clinic Mark Chaplain School of Mathematics and Statistics Mark Chaplain Multiscale

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