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Tumour Growth Modelling and Computational Simulation Ana Leal 2 , Joana Louren co 2 , Tom as Cruz 1 1 Mestrado Integrado em Engenharia F sica Tecnol ogica 2 Mestrado Integrado em Engenharia Biom edica Instituto Superior T ecnico

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Tumour Growth Modelling and Computational Simulation

Ana Leal2, Joana Louren¸ co2, Tom´ as Cruz1

1Mestrado Integrado em Engenharia F´

ısica Tecnol´

  • gica

2Mestrado Integrado em Engenharia Biom´

edica Instituto Superior T´ ecnico

May 30, 2012

Ana Leal, Joana Louren¸ co, Tom´ as Cruz Tumour Growth Modelling and Computational Simulation

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Index

  • I. State of the art.
  • II. Principles of tumour growth simulation

Biological principles concerning tumour growth Biophysical principles concerning tumour growth Mathematical models for tumour growth

  • III. Case Study
  • IV. Final Remarks.
  • V. References.

Ana Leal, Joana Louren¸ co, Tom´ as Cruz Tumour Growth Modelling and Computational Simulation

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State of the art

To better understand tumour growth mechanisms, researchers are applying mathematical models to predict tumour behaviour, for example, it’s aggressiveness or it’s susceptibility to chemotherapy.

Ana Leal, Joana Louren¸ co, Tom´ as Cruz Tumour Growth Modelling and Computational Simulation

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State of the art

To better understand tumour growth mechanisms, researchers are applying mathematical models to predict tumour behaviour, for example, it’s aggressiveness or it’s susceptibility to chemotherapy. In Cancer Research (2009), Bearer E. L. et al., created a model that links the behaviour of cancer cells and their surrounding to tumour growth, shape and treatment response.

They conclude that tumour growth and invasion are not only explained by genomic and molecular events, but can be predictable processes that obey physical laws.

Ana Leal, Joana Louren¸ co, Tom´ as Cruz Tumour Growth Modelling and Computational Simulation

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State of the art

Frieboes, H. B. et al. (2009), created a model to predict a patient’s response to a particular drug with a basic model representing the tumour as a sphere-like structure. Sophisticated multiphase tumour simulators, capable of simulating vascularized tumour growth in 3D, have the potential to predict cancer behaviour.

Ana Leal, Joana Louren¸ co, Tom´ as Cruz Tumour Growth Modelling and Computational Simulation

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Biological principles concerning tumour growth

Tumour is characterized by an abnormal cell proliferation.

Ana Leal, Joana Louren¸ co, Tom´ as Cruz Tumour Growth Modelling and Computational Simulation

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Biological principles concerning tumour growth

Tumour is characterized by an abnormal cell proliferation. There are two types of tumours:

Benign Malignant - Cancer

Ana Leal, Joana Louren¸ co, Tom´ as Cruz Tumour Growth Modelling and Computational Simulation

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Biological principles concerning tumour growth

Tumour is characterized by an abnormal cell proliferation. There are two types of tumours:

Benign Malignant - Cancer

Invasion of other tissues - METASTASIS

Ana Leal, Joana Louren¸ co, Tom´ as Cruz Tumour Growth Modelling and Computational Simulation

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Biological principles concerning tumour growth

Tumour is characterized by an abnormal cell proliferation. There are two types of tumours:

Benign Malignant - Cancer

Invasion of other tissues - METASTASIS Multicellular spheroid

Ana Leal, Joana Louren¸ co, Tom´ as Cruz Tumour Growth Modelling and Computational Simulation

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Biological principles concerning tumour growth

Carcinogenesis: Process through which cancer is generated.

Ana Leal, Joana Louren¸ co, Tom´ as Cruz Tumour Growth Modelling and Computational Simulation

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Biological principles concerning tumour growth

Carcinogenesis: Process through which cancer is generated. Phases Initiation Promotion Progression

Ana Leal, Joana Louren¸ co, Tom´ as Cruz Tumour Growth Modelling and Computational Simulation

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Biological principles concerning tumour growth

Carcinogenesis: Process through which cancer is generated. Phases Initiation Promotion Progression Activation of Oncogene

Ana Leal, Joana Louren¸ co, Tom´ as Cruz Tumour Growth Modelling and Computational Simulation

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Biophysical principles concerning tumour growth

The mechanical properties of the cell can be altered by biochemical processes, invasion of foreign organisms or disease development.

Ana Leal, Joana Louren¸ co, Tom´ as Cruz Tumour Growth Modelling and Computational Simulation

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Biophysical principles concerning tumour growth

The mechanical properties of the cell can be altered by biochemical processes, invasion of foreign organisms or disease development. Mechanic deformation characteristics are determined by the cytoskeleton.

Ana Leal, Joana Louren¸ co, Tom´ as Cruz Tumour Growth Modelling and Computational Simulation

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Biophysical principles concerning tumour growth

The mechanical properties of the cell can be altered by biochemical processes, invasion of foreign organisms or disease development. Mechanic deformation characteristics are determined by the cytoskeleton. The physics of adhesion between biological cells influences cancer cell motility, invasion and metastasis.

Ana Leal, Joana Louren¸ co, Tom´ as Cruz Tumour Growth Modelling and Computational Simulation

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Biophysical principles concerning tumour growth

The mechanical properties of the cell can be altered by biochemical processes, invasion of foreign organisms or disease development. Mechanic deformation characteristics are determined by the cytoskeleton. The physics of adhesion between biological cells influences cancer cell motility, invasion and metastasis. The ECM plays an important role in tumour growth and shape.

Ana Leal, Joana Louren¸ co, Tom´ as Cruz Tumour Growth Modelling and Computational Simulation

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Mathematical models for tumour growth

Types of models

Discrete Models

Figure: Example discrete model.

Continuum Models

Figure: Example continuum model.

Ana Leal, Joana Louren¸ co, Tom´ as Cruz Tumour Growth Modelling and Computational Simulation

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Mathematical models for tumour growth

Oxygen/nutrient consumption

Continuity Equation ∂ni ∂t +∇Γi +Si( r, t)−Li( r, t) = 0

Ana Leal, Joana Louren¸ co, Tom´ as Cruz Tumour Growth Modelling and Computational Simulation

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Mathematical models for tumour growth

Oxygen/nutrient consumption

Continuity Equation ∂ni ∂t +∇Γi +Si( r, t)−Li( r, t) = 0 Fick’s Law Γi = −Di∇ni Γi = ±µini E − Di∇ni

Ana Leal, Joana Louren¸ co, Tom´ as Cruz Tumour Growth Modelling and Computational Simulation

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Mathematical models for tumour growth

Oxygen/nutrient consumption

Continuity Equation ∂ni ∂t +∇Γi +Si( r, t)−Li( r, t) = 0 Fick’s Law Γi = −Di∇ni Γi = ±µini E − Di∇ni General equation that describe uncharged substances behaviour ∂ni ∂t − Di∇2ni + Si( r, t) − Li( r, t) = 0

Ana Leal, Joana Louren¸ co, Tom´ as Cruz Tumour Growth Modelling and Computational Simulation

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Mathematical models for tumour growth

Cell behaviour as a function of the diffusion of substances

Each substance in the extracellular medium has its movement described by one of the equations previously shown;

Ana Leal, Joana Louren¸ co, Tom´ as Cruz Tumour Growth Modelling and Computational Simulation

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Mathematical models for tumour growth

Cell behaviour as a function of the diffusion of substances

Each substance in the extracellular medium has its movement described by one of the equations previously shown; Function that outputs the probability of the behaviour depending on the concentration of the relevant substances;

Ana Leal, Joana Louren¸ co, Tom´ as Cruz Tumour Growth Modelling and Computational Simulation

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Mathematical models for tumour growth

Cell behaviour as a function of the diffusion of substances

Each substance in the extracellular medium has its movement described by one of the equations previously shown; Function that outputs the probability of the behaviour depending on the concentration of the relevant substances; These relation functions are very dependent on experimental parameters

Ana Leal, Joana Louren¸ co, Tom´ as Cruz Tumour Growth Modelling and Computational Simulation

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Mathematical models for tumour growth

Cell behaviour as a function of the diffusion of substances

Each substance in the extracellular medium has its movement described by one of the equations previously shown; Function that outputs the probability of the behaviour depending on the concentration of the relevant substances; These relation functions are very dependent on experimental parameters Example Pdeath = e−

  • n

σnd

2

Ana Leal, Joana Louren¸ co, Tom´ as Cruz Tumour Growth Modelling and Computational Simulation

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Mathematical models for tumour growth

Cell movement and proliferation

Cell population as an incompressible fluid ∂Φi ∂t + ∇ (uiΦi) = ∇ (Di∇Φi) + λi − µi

  • ni d

ri dt = ui ni

Figure: Schematic representation of an invading

carcinoma.

Single cell movement

Figure: Illustration based

  • n a discrete model.

Ana Leal, Joana Louren¸ co, Tom´ as Cruz Tumour Growth Modelling and Computational Simulation

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Mathematical models for tumour growth

Tissue mechanical properties

Pressure can be related to the effective tissue stress and strain: σij = 2Gεij +

  • K − 2

3G

  • εkkδij − piδij − Kηδij

σij is the effective stress, εij is the strain tensor of the tumour tissue, δij is the Kronecker delta, K is the drained bulk modulus, G is the shear modulus of the tissue and lastly η is the volume growth of the tissue.

Figure: ECM rigidity influences

proliferation velocity.

Ana Leal, Joana Louren¸ co, Tom´ as Cruz Tumour Growth Modelling and Computational Simulation

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Mathematical models for tumour growth

Tissue mechanical properties

Pressure can be related to the effective tissue stress and strain: σij = 2Gεij +

  • K − 2

3G

  • εkkδij − piδij − Kηδij

σij is the effective stress, εij is the strain tensor of the tumour tissue, δij is the Kronecker delta, K is the drained bulk modulus, G is the shear modulus of the tissue and lastly η is the volume growth of the tissue. Darcy’s law Darcy’s law relates proliferation velocity and pressure gradient. vi = µi∇pi µi is the motility.

Figure: ECM rigidity influences

proliferation velocity.

Ana Leal, Joana Louren¸ co, Tom´ as Cruz Tumour Growth Modelling and Computational Simulation

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Mathematical models for tumour growth

Other mathematical models

  • J. Sherratt and M. Chaplain, 2001

Travelling wave analysis of a non linear diffusion to introduce cell movement into the growth

  • f tumour spheroids.

Ana Leal, Joana Louren¸ co, Tom´ as Cruz Tumour Growth Modelling and Computational Simulation

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Mathematical models for tumour growth

Other mathematical models

  • J. Sherratt and M. Chaplain, 2001

Travelling wave analysis of a non linear diffusion to introduce cell movement into the growth

  • f tumour spheroids.

Antonio Br´ u et al., 2003

Characterization of tumour growth dynamics was based on fractality and scale invariance

  • f the colony contour.

Ana Leal, Joana Louren¸ co, Tom´ as Cruz Tumour Growth Modelling and Computational Simulation

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Mathematical models for tumour growth

Other mathematical models

  • S. C. Ferreira et al., 2002

Discrete nutrient-limited model for avascular tumour growth including cell proliferation, motility and death.

Ana Leal, Joana Louren¸ co, Tom´ as Cruz Tumour Growth Modelling and Computational Simulation

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Mathematical models for tumour growth

Other mathematical models

  • S. C. Ferreira et al., 2002

Discrete nutrient-limited model for avascular tumour growth including cell proliferation, motility and death.

Caroline Rosello, 2004

Model for quantification random motility coefficient, migration speed and trajectory persistence time.

Ana Leal, Joana Louren¸ co, Tom´ as Cruz Tumour Growth Modelling and Computational Simulation

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Case Study

Kansal et al. (2000)

Discrete model to simulate brain cancer growth dynamics.

Figure: Gompertzian growth.

Ana Leal, Joana Louren¸ co, Tom´ as Cruz Tumour Growth Modelling and Computational Simulation

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Case Study

Kansal et al. (2000)

Discrete model to simulate brain cancer growth dynamics.

Figure: Gompertzian growth.

Voronoi tesselation to create cells

Ana Leal, Joana Louren¸ co, Tom´ as Cruz Tumour Growth Modelling and Computational Simulation

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Case Study

Kansal et al. (2000)

Parameters compared between experimental data and the simulation results.

Ana Leal, Joana Louren¸ co, Tom´ as Cruz Tumour Growth Modelling and Computational Simulation

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Case Study

Kansal et al. (2000)

Figure: Cross section of a tumour

showing the emergence and eventual dominance of a more rapidly growing secondary strain. Black zones are the necrotic regions of both strains.

Figure: Cross-section of fully

developed tumour in different positions. Dark - necrotic cells, Light-gray - non-proliferative cells, Drak-gray - proliferating cells.

Ana Leal, Joana Louren¸ co, Tom´ as Cruz Tumour Growth Modelling and Computational Simulation

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Final Remarks

Cancer is one of the biggest causes of death in the world.

Ana Leal, Joana Louren¸ co, Tom´ as Cruz Tumour Growth Modelling and Computational Simulation

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Final Remarks

Cancer is one of the biggest causes of death in the world. Models can predict tumour growth and important features about specific tumours like growth fractions, necrotic fractions and volume doubling time.

Ana Leal, Joana Louren¸ co, Tom´ as Cruz Tumour Growth Modelling and Computational Simulation

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Final Remarks

Cancer is one of the biggest causes of death in the world. Models can predict tumour growth and important features about specific tumours like growth fractions, necrotic fractions and volume doubling time. Prospects to the future Increase in complexity of the models. Models applied to each type of tumour. Models will be a great help in diagnosis and treatment in real cases.

Ana Leal, Joana Louren¸ co, Tom´ as Cruz Tumour Growth Modelling and Computational Simulation

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References

Devi, P Uma, “Basics of carcinogenesis”, Health Administrator, Vol: XVII, No. 1, 16-24, (2005). Byrne, H. M. and Chaplain, M. A. J., “Modelling the role of cell-cell adhesion in the growth and development of carcinomas”, Math. Comput. Modelling, Vol. 24, No. 12, 1-17, (1996). Roose, T. et al., “Mathematical models of avascular tumour growth”, Siam Review, Vol. 49, No. 2, 179-208, (2007). Sherrart, J. A. and Chaplain, M. A. J., “A new mathematical model for avascular tumour growth”, J. Math. Biol., Vol. 43, 291-312, (2001). Br´ u, A. et al., “The universal dynamics of tumour growth”, Biophysical Journal, Vol. 85, 2948-2961, (2003). Delsanto P. P. et al., “Growth model for multicellular tumour spheroids”,

  • Appl. Phys. Lett., Vol. 85, 4225-4227, (2004).

Kansal, A. R. et al.,“Simulated brain tumour growth dynamics using a three-dimensional cellular automaton”, J. Theor. Biol., Vol. 203, 367-382, (2000). To see more consult our Report on Tumour Growth Modelling and Tumour Computational Simulation

Ana Leal, Joana Louren¸ co, Tom´ as Cruz Tumour Growth Modelling and Computational Simulation