Mathematical Modelling of Arteries – How Can Biomechanics ... · Outline Introduction Mechanical...

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Mechanical modelling of soft tissueApplication

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Mathematical Modelling of Arteries –How Can Biomechanics Predict Arterial Diseases?

Guest Lecture by Jonas Stalhand

Division of MechanicsLinkoping Institute of Technology

Linkoping, Sweden

26 November 2009

Guest Lecture by Jonas Stalhand Mathematical Modelling of Arteries

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1 Introduction

2 Mechanical modelling of soft tissue

3 Application

4 Closure


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Definition

Biomechanics is mechanics applied to a biological system.


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Biomechanical examples

Figure: Kim Howe (left), knee replacement joint (centre), graft (right)


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Motivation for cardiovascular research

Figure: Demography for the EU1960 to 2050

Cost for cardiovasculardiseases (CD) in EU 2003:169 billion euro (Eur Heart J).

Age is a risk factor formany CDs

Ageing population

CDs are often linked tobiomechanics


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Ageing population



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Ageing population



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Ageing population



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Physiology

Figure: The arterial (systemic)system.

Arteries carry blood fromthe heart

Elastic and musculararteries

Three layers: intima,media, adventitia

Adventitia and media arethe main passive layers

Intima is important for theactive properties


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Physiology

Figure: The arterial (systemic)system.







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Physiology

Figure: The three arterial layers(Stalhand, 2005).







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Physiology








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Physiology








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Physiology (contd.)

Figure: Collagen (above) andelastin (below)

Arterial wall is a compositestructure

Collagen and elastin arestiff and resilient proteins,respectively

Collagen is structuredwhile elastin is random.

Unloaded collagen isundulated (wavy)

Collagen and elastinhalf-life are 15–90 days and70 years(!), respectively


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Physiology (contd.)








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Physiology (contd.)








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Physiology (contd.)








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Physiology (contd.)








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Physiology (contd.)

Figure: The circumferential (above) andaxial (below) stress-stretch response.

Large nonlineardeformations

‘Linear region’ =elastin, exponentialregion = collagen

Waviness gives thetransition region

Anisotropic due tocollagen fibres

Incompressible


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Physiology (contd.)






Incompressible


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Physiology (contd.)






Incompressible


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Physiology (contd.)






Incompressible


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Physiology (contd.)






Incompressible


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One-dimensional example: Chordae tendinae


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One-dimensional balance law

L0

A0

F

F

Reference

L

A

F

F

Deformed

Principle of virtual power:

F

A0v∣∣∣L0

0=

∫ L0

0Pd

dv

dXdX , ∀v ∈ V

(1)


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One-dimensional balance law

L0

A0

F

F

Reference

L

A

F

F

Deformed

Principle of virtual power:

F

A0v∣∣∣L0

0=

∫ L0

0Pd

dv

dXdX , ∀v ∈ V

(1)

Equilibrium equations:

dPd

dX= 0, 0 < X < L0

Pd =F

A0, X = 0, L0

(2)


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One-dimensional constitutive equation

Stress and force are coupled by the balance law. How is stresscoupled to deformation?

A constitutive equation couples stress and deformation

The classical Hooke’s law (σ = Eε) is linear

From the introduction we know that soft tissues are nonlinear

How can we find nonlinear constitutive equations?


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One-dimensional constitutive equation (contd.)

A0

L0

ρ0

X

F

Pd

Reference

A

ρ

L

x

F

σd

Deformed

Deformation:

x = λX , v = x = λ (3)

Dissipation inequality:

W ≤ Pddv

dX(4)

where W = W (λ) is the strain energy.

Pd =dW

dλ(5)


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One-dimensional constitutive equation (contd.)

A0

ρ0

L0

F

Pd

Reference

A

ρ

L

F

σd

Deformed

Mass conservation:

ρ0A0dX = ρAdx (6)

Axial force:

PdA0 = σdA (7)

Constitutive equation (true stress):

σd =ρ

ρ0λ

dW

dλ(8)


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One-dimensional model

Figure: Stress-strainresponse. Measurements(circles) and model response(solid line).

Kunzelman and Cochran, TransAm Soc Artif Int Org, 1990

Incompressible deformation: ρ = ρ0

W = dW /dλ = 0 for λ = 1

W (λ) = c( 1k ek(λ−1) − λ)

Least-squares-fitting givesc = 4644 Pa and k = 28.7


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Three-dimensional modelling of arteries

θ

z

r

Aorta is approximated by a thinwalled homogeneous cylinder

Rotationally symmetric strainfield and material properties

Pressurised inner and free outerboundary

Two collagen fibre familiesdirections symmetrically aroundthe tangential direction in theθ − z plane

Incompressibility


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θ

z

r





Incompressibility


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θ

z

r

P





Incompressibility


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M1

M2

2β

θ

z

r

P





Incompressibility


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M1

M2

2β

θ

z

r

P





Incompressibility


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Three-dimensional kinematics

Membrane stretches (inflation-extension, no shear):

λθ =r

R, λz =

L

L0(9)

Deformation gradient:

Fij =

λr 0 0

0 λθ 0

0 0 λz

, Ckl = FikFil (10)

Incompressibility (constraint):

det Fij = λrλθλz = 1 ⇒ λr =1

λθλz(11)


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Three-dimensional balance law

Equilibrium:

dσij

dxj= 0 in Ω, ti = σijnj on ∂Ω

Equilibrium equation and boundary conditions (cyl.coords.)

dσrr

dr− σθθ − σrr

r= 0

σrr (r0) = −P, σrr (r1) = 0(12)


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Three-dimensional constitutive equation

Constitutive equation for incompressible materials:

σij = −pδij + 2Fik∂ψ

∂CklFjl (13)

Holzapfel-Gasser-Ogden (J. Elasticity, 2000):

ψ = c(Ckj − 3) +k1

k2

(ek2(λ2

f−1)2 − 1)

(14)

where λ2f = cosβλ2

θ + sinβλ2z is the squared collagen fibre stretch


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Three-dimensional constitutive equations (contd.)

Membrane stresses:

σθθ = 2c

(λ2θ −

1

λ2θλ

2z

)+ 4k1

(λ2

f − 1)ek2(λ2

f−1)2λ2θ(cosβ)2,

σzz = 2c

(λ2

z −1

λ2zλ

2θ

)+ 4k1

(λ2

f − 1)ek2(λ2

f−1)2λ2z(sinβ)2

(15)


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An application

Determination of material properties in humansusing clinical data

Jonas Stalhanda, Hakan Astrandb, Jerker Karlssonb, Carl-JohanThorea, Bjorn Sonessonc , Toste Lanneb

a Div. Mechanics/IEI, Linkoping Institute of Technology, Linkopingb Div. Cardiovascular Medicine/pysiology, Dept. Medical and Health Sciences, Linkoping University, Linkopingc Vascular Centre, Malmo University Hospital, Malmo


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Fundamental mechanical principles apply to soft tissues!

Nonlinear continuum mechanics can offer new insights

Can biomechanics predict arterial diseases? Yes!, but furtherresearch is needed.


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Thank you for listening!


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