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OutlineIntroduction
Mechanical modelling of soft tissueApplication
Closure
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
OutlineIntroduction
Mechanical modelling of soft tissueApplication
Closure
1 Introduction
2 Mechanical modelling of soft tissue
3 Application
4 Closure
Guest Lecture by Jonas Stalhand Mathematical Modelling of Arteries
OutlineIntroduction
Mechanical modelling of soft tissueApplication
Closure
Definition
Biomechanics is mechanics applied to a biological system.
Guest Lecture by Jonas Stalhand Mathematical Modelling of Arteries
OutlineIntroduction
Mechanical modelling of soft tissueApplication
Closure
Biomechanical examples
Figure: Kim Howe (left), knee replacement joint (centre), graft (right)
Guest Lecture by Jonas Stalhand Mathematical Modelling of Arteries
OutlineIntroduction
Mechanical modelling of soft tissueApplication
Closure
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
Guest Lecture by Jonas Stalhand Mathematical Modelling of Arteries
OutlineIntroduction
Mechanical modelling of soft tissueApplication
Closure
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
Guest Lecture by Jonas Stalhand Mathematical Modelling of Arteries
OutlineIntroduction
Mechanical modelling of soft tissueApplication
Closure
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
Guest Lecture by Jonas Stalhand Mathematical Modelling of Arteries
OutlineIntroduction
Mechanical modelling of soft tissueApplication
Closure
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
Guest Lecture by Jonas Stalhand Mathematical Modelling of Arteries
OutlineIntroduction
Mechanical modelling of soft tissueApplication
Closure
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
Guest Lecture by Jonas Stalhand Mathematical Modelling of Arteries
OutlineIntroduction
Mechanical modelling of soft tissueApplication
Closure
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
Guest Lecture by Jonas Stalhand Mathematical Modelling of Arteries
OutlineIntroduction
Mechanical modelling of soft tissueApplication
Closure
Physiology
Figure: The three arterial layers(Stalhand, 2005).
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
Guest Lecture by Jonas Stalhand Mathematical Modelling of Arteries
OutlineIntroduction
Mechanical modelling of soft tissueApplication
Closure
Physiology
Figure: The three arterial layers(Stalhand, 2005).
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
Guest Lecture by Jonas Stalhand Mathematical Modelling of Arteries
OutlineIntroduction
Mechanical modelling of soft tissueApplication
Closure
Physiology
Figure: The three arterial layers(Stalhand, 2005).
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
Guest Lecture by Jonas Stalhand Mathematical Modelling of Arteries
OutlineIntroduction
Mechanical modelling of soft tissueApplication
Closure
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
Guest Lecture by Jonas Stalhand Mathematical Modelling of Arteries
OutlineIntroduction
Mechanical modelling of soft tissueApplication
Closure
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
Guest Lecture by Jonas Stalhand Mathematical Modelling of Arteries
OutlineIntroduction
Mechanical modelling of soft tissueApplication
Closure
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
Guest Lecture by Jonas Stalhand Mathematical Modelling of Arteries
OutlineIntroduction
Mechanical modelling of soft tissueApplication
Closure
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
Guest Lecture by Jonas Stalhand Mathematical Modelling of Arteries
OutlineIntroduction
Mechanical modelling of soft tissueApplication
Closure
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
Guest Lecture by Jonas Stalhand Mathematical Modelling of Arteries
OutlineIntroduction
Mechanical modelling of soft tissueApplication
Closure
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
Guest Lecture by Jonas Stalhand Mathematical Modelling of Arteries
OutlineIntroduction
Mechanical modelling of soft tissueApplication
Closure
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
Guest Lecture by Jonas Stalhand Mathematical Modelling of Arteries
OutlineIntroduction
Mechanical modelling of soft tissueApplication
Closure
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
Guest Lecture by Jonas Stalhand Mathematical Modelling of Arteries
OutlineIntroduction
Mechanical modelling of soft tissueApplication
Closure
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
Guest Lecture by Jonas Stalhand Mathematical Modelling of Arteries
OutlineIntroduction
Mechanical modelling of soft tissueApplication
Closure
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
Guest Lecture by Jonas Stalhand Mathematical Modelling of Arteries
OutlineIntroduction
Mechanical modelling of soft tissueApplication
Closure
One-dimensional example: Chordae tendinae
Guest Lecture by Jonas Stalhand Mathematical Modelling of Arteries
OutlineIntroduction
Mechanical modelling of soft tissueApplication
Closure
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)
Guest Lecture by Jonas Stalhand Mathematical Modelling of Arteries
OutlineIntroduction
Mechanical modelling of soft tissueApplication
Closure
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)
Guest Lecture by Jonas Stalhand Mathematical Modelling of Arteries
OutlineIntroduction
Mechanical modelling of soft tissueApplication
Closure
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?
Guest Lecture by Jonas Stalhand Mathematical Modelling of Arteries
OutlineIntroduction
Mechanical modelling of soft tissueApplication
Closure
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?
Guest Lecture by Jonas Stalhand Mathematical Modelling of Arteries
OutlineIntroduction
Mechanical modelling of soft tissueApplication
Closure
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?
Guest Lecture by Jonas Stalhand Mathematical Modelling of Arteries
OutlineIntroduction
Mechanical modelling of soft tissueApplication
Closure
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?
Guest Lecture by Jonas Stalhand Mathematical Modelling of Arteries
OutlineIntroduction
Mechanical modelling of soft tissueApplication
Closure
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?
Guest Lecture by Jonas Stalhand Mathematical Modelling of Arteries
OutlineIntroduction
Mechanical modelling of soft tissueApplication
Closure
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)
Guest Lecture by Jonas Stalhand Mathematical Modelling of Arteries
OutlineIntroduction
Mechanical modelling of soft tissueApplication
Closure
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)
Guest Lecture by Jonas Stalhand Mathematical Modelling of Arteries
OutlineIntroduction
Mechanical modelling of soft tissueApplication
Closure
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
Guest Lecture by Jonas Stalhand Mathematical Modelling of Arteries
OutlineIntroduction
Mechanical modelling of soft tissueApplication
Closure
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
Guest Lecture by Jonas Stalhand Mathematical Modelling of Arteries
OutlineIntroduction
Mechanical modelling of soft tissueApplication
Closure
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
Guest Lecture by Jonas Stalhand Mathematical Modelling of Arteries
OutlineIntroduction
Mechanical modelling of soft tissueApplication
Closure
Three-dimensional modelling of arteries
θ
z
r
P
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
Guest Lecture by Jonas Stalhand Mathematical Modelling of Arteries
OutlineIntroduction
Mechanical modelling of soft tissueApplication
Closure
Three-dimensional modelling of arteries
M1
M2
2β
θ
z
r
P
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
Guest Lecture by Jonas Stalhand Mathematical Modelling of Arteries
OutlineIntroduction
Mechanical modelling of soft tissueApplication
Closure
Three-dimensional modelling of arteries
M1
M2
2β
θ
z
r
P
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
Guest Lecture by Jonas Stalhand Mathematical Modelling of Arteries
OutlineIntroduction
Mechanical modelling of soft tissueApplication
Closure
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)
Guest Lecture by Jonas Stalhand Mathematical Modelling of Arteries
OutlineIntroduction
Mechanical modelling of soft tissueApplication
Closure
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)
Guest Lecture by Jonas Stalhand Mathematical Modelling of Arteries
OutlineIntroduction
Mechanical modelling of soft tissueApplication
Closure
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
Guest Lecture by Jonas Stalhand Mathematical Modelling of Arteries
OutlineIntroduction
Mechanical modelling of soft tissueApplication
Closure
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)
Guest Lecture by Jonas Stalhand Mathematical Modelling of Arteries
OutlineIntroduction
Mechanical modelling of soft tissueApplication
Closure
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
Guest Lecture by Jonas Stalhand Mathematical Modelling of Arteries
OutlineIntroduction
Mechanical modelling of soft tissueApplication
Closure
Closure
Fundamental mechanical principles apply to soft tissues!
Nonlinear continuum mechanics can offer new insights
Can biomechanics predict arterial diseases? Yes!, but furtherresearch is needed.
Guest Lecture by Jonas Stalhand Mathematical Modelling of Arteries
OutlineIntroduction
Mechanical modelling of soft tissueApplication
Closure
Closure
Fundamental mechanical principles apply to soft tissues!
Nonlinear continuum mechanics can offer new insights
Can biomechanics predict arterial diseases? Yes!, but furtherresearch is needed.
Guest Lecture by Jonas Stalhand Mathematical Modelling of Arteries
OutlineIntroduction
Mechanical modelling of soft tissueApplication
Closure
Closure
Fundamental mechanical principles apply to soft tissues!
Nonlinear continuum mechanics can offer new insights
Can biomechanics predict arterial diseases? Yes!, but furtherresearch is needed.
Guest Lecture by Jonas Stalhand Mathematical Modelling of Arteries
OutlineIntroduction
Mechanical modelling of soft tissueApplication
Closure
Thank you for listening!
Guest Lecture by Jonas Stalhand Mathematical Modelling of Arteries
OutlineIntroduction
Mechanical modelling of soft tissueApplication
Closure
Guest Lecture by Jonas Stalhand Mathematical Modelling of Arteries