Tumour Growth Models

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Mechanical Aspects of

Tumour Growth

Aim: Are tumors solids or liquids?

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Multiphase model Multiphase model tumour cells host cells deformable and degradable ECM extracellular liquid DDDD _DDDD

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Multiphase models Multiphase models

with

volume ratios

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Interaction of the constituents with the liquid Interaction of the constituents with the liquid

Observation:

The interactions involving the liquid phase are much

smaller than those between cells and ECM

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Interaction of the constituents with the liquid Interaction of the constituents with the liquid

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Adhesion forces Adhesion forces

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Human Brain Tumor

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Interfering with the adhesion mechanisms Interfering with the adhesion mechanisms

Disrupts actin cytoskeleton removes hyaluronan backbone from membrane

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Modelling the interaction between cells and ECM Modelling the interaction between cells and ECM

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Using force balance

i.e., if cells are pulled strong enough they move relative to the ECM if not they stick to the ECM

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Summary Summary

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− Adhesion depends on the amount of ECM, e.g.

moves slows down stops

− Different clones have different σ

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• Non-remodelling ECM • Growth by loss of contact inhibition mechanisms

• Cell ensembles as elastic fluids

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•_{Non-remodelling EC}

• Growth by loss of contact inhibition mechanisms

• Cell ensembles as elastic fluids • Vascularised case

Islets of Langerhans _{Cell volume ratio}

Nutrient

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Cell-cell interactions Cell-cell interactions E-cadherin β-catenin α-catenin α-actinin actin P120 cytoplasm

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Stress relaxation in a liquid Stress relaxation in a liquid

s λ s str ai n str ess G(s) = relaxation kernel If

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Stress relaxation in a solid Stress relaxation in a solid

s λ s str ai n str ess G(s) = relaxation kernel } elasticity

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Creep Creep s Λ s str ess str ai n

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Plastic re-organisation during growth Plastic re-organisation during growth

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Cell re-organisation Cell re-organisation (

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Stress relaxation Stress relaxation

G(s) = relaxation kernel

λ₂ λ₁

• Winters et al., Int. J. Cancer,

114: 371-379,(2005) • Forgacs et al., Biophys. J. 74: 2227–2234 .(1998)

(

}

surface tension

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Cell re-organisation Cell re-organisation

Fig. 5. A spherical heart aggregate on the lower compression plate

(A), compressed for 3.5 hours

(B), and then released (C), temporarily retains ﬂat upper and lower surfaces. Behaving as a Fig. 4. A spherical heart aggregate on the

• Foty et al., PRL 72: 2298-2301 (1996)

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Stress relaxation Stress relaxation

Forgacz et al. (1998)

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Spheroids are Viscoplastic Spheroids are Viscoplastic

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Cell re-organisation Cell re-organisation fluid-like region σ₁ solid-like region

• some bonds break • cells re-organise • stress relaxes

yield surface • bonds don’t break

• cells deform • elastic recovery

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Multiple Natural Configurations Multiple Natural Configurations

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Small elastic deformations

Limit cases: small elastic deformation Limit cases: small elastic deformation

= characteristic time to relax the stress to the yield value

t str ess str ai n t τ str ess str ai n τ _Unrecovered ce ll -orga ni sa ti on

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Standard tests Standard tests t st rai n str ess t } yield stress = • Stress relaxation T(t) g τ + η γ

• Steady shear rate

.

regardless of imposed strain

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Foty's cell reorganisation test Foty's cell reorganisation test

• Imposing a deformation and then releasing the stress

t t str ai n str ess str ai n st re ss

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Small elastic deformations

Limit cases: small elastic deformation Limit cases: small elastic deformation

Maxwell fluid Elastic solid Recall

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Limit cases: comparison with previous models Limit cases: comparison with previous models

τ₀ 0 (negligible yield stress)

viscous models (e.g., Byrne-Preziosi, Chaplain et al., King et al.) fluid-like region σ₁ solid-like

region _surfaceyield

σ₂

fluid-like region

σ₁ σ₂

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Limit cases: comparison with previous models Limit cases: comparison with previous models

“large yield stress”

(till yield surface is reached)

“elastic models” (e.g., Araujo et al.)

σ₁

yield surface

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Example: Growth in a Duct Example: Growth in a Duct

cells tumour

host

ECM

• K = K₀ /K=0.1, ratio between the interaction force between tumour cells and the liquid ﬂowing around it and the one between tumour cells and the ECM.

• µ = µ₀ /µ_t =0.1, ratio between the elastic moduli

• τ = τ /µ_t =0.01, ratio between yield stress and Young modulus

• η_p = νη_p /µ_t =0.01, ratio between the characteristic stress relaxation time due to cell sation and the duplication time.

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growth plastic deformatio

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stress ECM axial stress wall stress Example: Growth in a Duct Example: Growth in a Duct