Modeling of heart mechanics with adaptation of tissue to load

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Theo Artst.arts@bf.unimaas.nl

Frits Prinzen, Tammo Delhaas*, Peter Bovendeerd**, J. Lumens, W. Kroon (Biophysics, Physiology, Pediatric Cardiology, Engineering)

Maastricht University, *Maastricht University Hospital**University of Technology, Eindhoven

The Netherlands

IPAM Feb 06

Modeling of heart mechanics with adaptation of tissue to load

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Modeling:- Finite Element Model Electro-Mechanics of the heart- CircAdapt = whole circulation model- Incorporation of adaptation with Self-structuring

Models

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left+right ventricle movie:

sarcomere length & pV

FEM of LV+RV- paced, depolarization wave with pacing- full cardiac cycle(R. Kerckhoffs, 2003)

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CircAdapt model of heart and circulation

lungs

RA

RV

LA

LV

aorta

system

Dynamic(t)CompliancesInertiasNon-linear

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Modeling structure of myocardial wall with adaptation:- mechanical load determines

wall massfiber orientationsheet orientation

- size and shape of blood vessels

Enormous reduction of number of parameters

Especially in pathology: Generally 1 basic cause of pathology, followed by physiological adaptation processes.

Parameter reduction: Self-structuring by simulation of adaptation

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gene expressionprotein formation

adaptation

Local tissue adaptation

(pump) function

pressure flow load

tissue propertiesstructuregeometry

tissuestress & strain

Whole organ function

Cardiovascular adaptation to mechanical load

Implication: Cell adaptation determines organ structure

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Myofiber structure and hypertrophy

Result of local adaptation?

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Cardiac fiber structure

F. Torrent-Guasp

LVRV

apical view, upper layers peeled off(randomly?)

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Filling/Rest

Activation

Relaxation

Force/StretchSynchrony

Shortening

What signals to the cells can be used for adaptation?

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fibroblast

extra-cellularmatrix

myocyte

deformation

(filling)

myocyte orientation

in myocyte optimum: - principal stress alingment - sarcomere shortening

(fiber structure)

tissue mass (wall mass)

stretch contractility

extra-cellular matrix strain

softening

Used adaptation rules of cardiac tissue

Arts, 1994, Biophys J. 66:953-961.

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Diffusion Tensor ImagingMyofiber orientation components in cross-section

of the goat heart.

posterior

anterior

perpendicular

in plane Base to apex In plane

1/2

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Transmural course of helix angle (goat)

L. GeertsAJP 2002

model predicted

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15

10

5

0

midwall radius ( )

mm

baseapex-20 20(mm)

equator

0

model prediction

10o

0o

-10o

transverseangle ( )

Transverse angle, model experiment (pig)

leftventricle

L. Geerts, AJP 2002

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Myofiber structure

Proposed adaptation rules for local tissue →

Realistic helical and transverse fiber structure

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Sheet structure

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Fiber-Sheet Structure

endocardium

midwall

epicardium

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Sheets split on plane of maximum shear

Hypothesis

Find plane with maximum shear

45o directions of maximum shear,

2 planes

Fiber direction not in one of those planes

Constraint: Sheet planes also contain the fiber direction

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Sheet Angles6 pooledexperiments BASE

rz rzrc rc

APEX

0o

45o

90o

135o

180o

0o

45o

90o

135o

180o

epi endo Arts, T, Am J Physiol. 280:H2222-H2229 (2001).

model predicted: 2 solutions with different likelyhood

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Simulation of short axis sheet cross-sections

seems realistic

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Sheets facilitate wall thickening and shortening by shear

(and rotation)

(J. Covell)

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Unloading for shear by 90º sheet rotation has same macroscopic effect on mechanics, ab= ba ≈ 0

So, sheet orientation itself is quite irrelevant for modeling, as long as it unloads the major component of shear stress.

2 solutions for sheat orientation

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Cross-sheet slice

Courtesy A. Young

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Sheet structure

- Sheet plane contains myofiber direction

- Designed to facilitate cross fiber deformation, thus minimizing chamber stiffness

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Application of predicted structure in analysis

Non-invasive determination of transmural difference in myofiber shortening with aortic stenosis

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epi

Aortic stenosis

aortic stenosis

endo

region with high intramyocardial pressure,causing coronary flow obstruction

high left ventricular

pressure

Subendocardial dysfunction

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- Torsion tuned to Shortening- IF Subendocardial TSR=( Torsion / Shortening )

Shortening & Torsioninner - - outer - -

outer

inner

Shortening onlyinner - - -outer -

Torsion onlyinner +outer -

Wall segment model (cylindrical)

Determination of transmural differences

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Rotation and deformation of the heart can be quantified

ribs

LV wall

RV

cavity

lung

Magnetic Resonance Tagging (MRT)

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MRTshortening

torsion

torsion

shortening = ln(Cavity Area)

Definitions

Torsion/Shortening measurement

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Mri-Software: Midwall motion and circumferential strain

Normal human left ventricle

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TSR in Control and AVS patients

Inner wall strain ln(L/LVc=Vw)

TSR=slope of torsion versus inner wall strain in systole:

- Dimensionless- Species independent- Expresses transmural difference in contractile function

Control= healthy young

AVSten= Aortic Valve Stenosis

AVRepl= 3 mo after aortic valve replacement

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From TSR to TransDifModel:

Torsion/Shortening (TSR)↓

normalized transmural difference in myofiber shortening (TransDif)

TransDif=Difference/Mean

Van der Toorn A et al. Am J Physiol. 2002;283:H1609-1615

Stunned subendocardial

myocardium regains function

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CircAdapt modela. Modeling of circulation

- Lumped model in modules:chambers, tubes, valves

b. Adaptation of modules to load

Search:

Google + keyword ‘CircAdapt’

hit ‘AJP’: Arts T et al. Am J Physiol. 2005;288:H1943-H1954hit ‘Biophysics’: MatLab source code

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Circulation in modules

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1-fiber model of a thick-walled cavity (chamber or blood vessel)

Vlv

plv

Vw

ventricle (LV)

wrapped in 1 myofiber

wlv3

1f

wlvlvf

VV31ln

VV31p

f= myofiber stressf= myofiber strainplv= LV pressureVlv= LV volumeVw= wall volume

FEM model confirms: Shape is practically irrelevant

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Laws of adaptation

Adaptation of Blood vessel• Shear stress → Diameter ↑

• Wall stress → Wall thickness ↑

• Contractility+diastolic stretch → Hypertrophy

• Deformation → Dilatation

Adaptation of Cavity

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Input to CircAdapt value SI-unit description

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Simulations after adaptation

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Pressure-Volume & Stress-Strain Loops

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Calculated parameters= Result

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Promising applications

1. Boundary conditions for FEM of heart

2. Patient specific modeling

3. “Non-invasive catheterization”Pressure difference over:

- membrane → pressure transducer

- valve with inertiadoppler velocity and accelerationmass ~ dynamic membrane

→ pressure transducer

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Hypertrophy in12 wks myofiber

mechanics in AV-block

Systolic fiber stress is not the stimulus to hypertrophy.

More likely: end-diastolic stress

(Donker et al, Basic Res Cardiol, 2005).

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Adaptation rules with‘deformation intervention’

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fiber direction

torsion

fiber direction

torsion

Situs Solitus(normal)

Situs InversusTotalis (SIT) Hypothesis

SIT heart with mirrored fiber structure is an alternative solution satisfying adaptation rules of cardiac cells.

Test

SIT heart should have equal torsion but in opposite direction.

occurrence

8000 : 1

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Situs Inversus Totalis Normal

(pig SIT heart)

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Situs Solitus (normal) ?? Situs Inversus Totalis ??Torsion (rad)

-0.20 -0.10 0 0.10 0.20

Expected torsion in SIT

NOT TRUE !

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-0.10

0.00

0.10

0.20Rotation

[rad]

-0.20

-0.10

0.00

0.10Torsion

5 base43

2

1 apex

d basecba apex

Situs Solitus(normal)

Situs InversusTotalis (n=14)

0.5 s

Human torsion

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-0.20

Situs Solitus (normal) Situs Inversus Totalis

Torsion (rad)

Torsion in SIT

-0.10 0 0.10 0.20

Delhaas, T. et al. Ann N Y Acad Sci 1015: 190-201.

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Myofiber structure in SITNormal adaptation rules

Start conditions for fiber structure:- normal apex- inverse base

Development to SIT structure

→ RESULT

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Situs Inversus Totalis

A natural in vivo ‘experiment’ to investigate cell’s long term response to a variety of deformations

boundary conditions

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inverse

Dualistic myofiber structure SIT

Helix angleendo - epi

inverse

normal

normal

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Efficency

The structure of the SIT heart is different.

Because each cell optimizes generation of mechanical work (= adaptation rule), all cells work close to their optimum. Since the only escape of mechanical work is pump work, the SIT heart works about just as optimal as the normal heart.

stress-strain work

stress-strain work

stress-strain work

The only escape of work:

pump work (p-V)

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General conclusion: Self-structuring by adaptation

“morphogenetic draft”

result

Self-adaptation until load is proper

growing brick

- works for cardiovascular modeling on all levels- adaptation can be modeled- should be used as prior knowledge in heart modeling

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That ’SIT