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### Cardiac Muscle andOrgan Mechanics

### Myocyte Connections

### Cardiac Myocytes

### Excitation-Contraction Coupling

OverviewOverview

Roy Kerckhoffs

Dept of Bioengineering,

University of California, San Diego

Tutorial on heart and lungs

Ohio State University,

Columbus, OH, 20 sep 2006

Outline

Ca2+

Ca2+

Ca2+

kn

Roff

Roff

0

*

kb

Ca2+

*

koff

kon

Ron

Ron

0

*

g

g

f

Ca2+

0

A1

A1

*

Ca2+

system

system

organ

organ

tissue

tissue

cell

cell

Overview

- Anatomy and physiology of the heart
- System and organ level
- Resting cardiac tissue
- Active force generation: the sarcomere
- Multi-scale modeling
- Anatomic models
- Models of cardiac mechanics: from cell to system
- Models of cardiac electromechanics

Cardiac Anatomy

Mitral

SVC

Aorta

Pulmonary

artery

Pulmonic

Tricuspid

valves

Base

LA

RA

LV

RV

Septum

Epicardium

Endocardium

Apex

Pericardium

Vaw

Vra

Vla

Vrv

Vlv

Vperi

Vvw

- A sac wherein the heart sits
- Limits sudden increases in volume
- Increases atrio-ventricular and ventricular-ventricular interaction

*Freeman & Little, AJP 1986;251:H421-H427

Physiology of the heart

Conduction System

right atrium

sinusnode

left atrium

AV-node

left bundle branch

right

bundle

branch

left ventricle

Purkinje fibers

right ventricle

The Cardiac Cycle

2

1

3

4b

4a

- Systole:
- 1. Isovolumic contraction
- 2. Ejection
- Diastole:
- 3. Relaxation
- Early
- Isovolumic
- 4. Filling
- a) Early, rapid
- b) Late, diastasis

4a

4b

1

3

2

Pressureand Volume

2

3

4b

1

4a

1

6

AVC

1

4

1

2

Aorta

AVO

1

0

Pressure (kPa)

8

6

4

Left ventricle

2

MVO

MV

C

0

1

5

0

1

2

0

Volume (ml)

9

0

6

0

3

0

0

1

0

0

2

0

0

3

0

0

4

0

0

5

0

0

6

0

0

7

0

0

Time (msec)

The Pressure-Volume Diagram

End-systole (ES)

2

0

SV=EDV-ESV

Ejection Fraction

EF=SV/EDV

Ejection

1

6

AVC

AVO

1

2

Pressure (kPa)

Stroke

volume

(SV)

Isovolumic relaxation

Isovolumic contraction

8

End-diastole (ED)

4

Filling

MVO

MVC

0

5

0

1

0

0

1

5

0

2

0

0

0

Volume (ml)

The Pressure-Volume Diagram

2

0

Ejection

1

6

AVC

AVO

1

2

Pressure (kPa)

Stroke

(external)

work

Isovolumic relaxation

Isovolumic contraction

8

4

Filling

MVO

MVC

0

5

0

1

0

0

1

5

0

2

0

0

0

Volume (ml)

Preload and Afterload

2

0

ESPVR

1

6

control

1

2

Pressure (kPa)

preload

8

EDPVR

afterload

4

0

5

0

1

0

0

1

5

0

2

0

0

0

Volume (ml)

Time-Varying Elastance

E(

20

0)

= E

max

E(1

60

ms

ec)

VR

2

0

E(1

20

ms

ec)

ESP

1

6

1

2

Pressure (kPa)

E(8

0 m

sec

)

8

R

EDPV

4

0

0

50

100

150

200

LV Volume (ml)

P(t) = E(t){V(t) - V0}

Starling’s Law of the Heart(The Frank-Starling Mechanism)

increased contractility (e.g. adrenergic agonist)

decreased contractility

(e.g. heart failure)

Stroke work

“Preload” (EDV or EDP)

Contractility (Inotropic State)

increased contractility (e.g. adrenergic agonist)

2

0

ESPVR

decreased contractility

(e.g. heart failure)

1

6

1

2

Pressure (kPa)

8

EDPVR

4

0

5

0

1

0

0

1

5

0

2

0

0

0

Volume (ml)

Overview

- Anatomy and physiology of the heart
- System and organ level
- Resting cardiac tissue
- Active force generation: the sarcomere
- Multi-scale modeling
- Models of cardiac mechanics: from cell to system
- Models of cardiac electromechanics

Minimizing Stress Gradients

- Residual Stress
- Fiber Angles
- Torsion

Resting Tissue Properties

- Nonlinearity
- Hysteresis
- Creep
- Relaxation
- Preconditioning Behavior
- Strain Softening
- Anisotropy

Passive Biaxial Properties

1

0

Fiber stress

8

6

Stress (kPa)

4

Cross-fiber stress

2

0

0.

0

0

0.

0

5

0.

1

0

0.

1

5

0.

2

0

0.

2

5

Equibiaxial Strain

Measurement of Myocardial strain

- Radiopaque beads and biplane x-ray
- video imaging of markers
- ultrasound
- MRI tagging

- Myocytes connect to an average of 11 other cells (half end-to-end and half side-to-side)
- Myocytes branch (about 12-15º)
- Intercalated disks
- gap junctions

Overview

- Anatomy and physiology of the heart
- System and organ level
- Resting cardiac tissue
- Active force generation: the sarcomere
- Multi-scale modeling
- Models of cardiac mechanics: from cell to system
- Models of cardiac electromechanics

- Rod-shaped
- Striated
- 80-100 m long
- 15-25 m diameter

Striated Muscle Ultrastructure

Electron micrograph of longitudinal section of freeze-substituted, relaxed rabbit psoas muscle. Sarcomere shows A band, I band, H band, M line, and Z line. Scale bar, 100 nm. From Millman BM, Physiol. Rev. 78: 359-391, 1998

- Calcium-induced calcium release
- Calcium current
- Na+/Ca2+ exchange
- Sarcolemmal Ca2+ pump
- SR Ca2+ ATP-dependent pump

http://www.meddean.luc.edu/lumen/DeptWebs/physio/bers.html

Isometric Tension in Skeletal Muscle:Sliding Filament Theory

(a) Tension-length curves for frog sartorius muscle at 0ºC

(b) Developed tension versus length for a single fiber of frog semitendinosus muscle

Isometric Testing

700

500

200

300

400

600

100

Sarcomere

length, mm

Sarcomere isometric

2.1

2.0

Muscle isometric

1.9

Tension, mN

2.0

1.0

time, msec

Length-Dependent Activation

2.2 micrometer

1.6 micrometer

Isometric peak twitch tension in cardiac muscle continues to rise at sarcomere lengths >2 mm due to sarcomere-length dependent increase in myofilament calcium sensitivity

Isotonic Testing

Isovelocity release experiment conducting during a twitch

Cardiac muscle force-velocity relation corrected for viscous forces of passive cardiac muscle which reduce shortening velocity

Ventricular Mechanics: Summary of Key Points

- Ventricular geometryis 3-D and complex
- Fiber anglesvary smoothly across the wall
- Systoleconsists of isovolumic contraction and ejection; diastoleconsists of isovolumic relaxation and filling
- Area of the pressure-volume loop is ventricular stroke workwhich increases with filling (Starling’s Law)
- Ventricles behave like time-varying elastances
- The slope of the end-systolic pressure volume relation is a load-independent measure of contractility or inotropic state.

Ventricular Mechanics: Summary of Key Points (cont’d)

- Collagencontributes to anisotropicresting properties
- Myocardial straincan be measured invasively and non-invasively
- Torsion and residual stress tend to compensate for these gradients in the ventricles to maintain uniform fiber strain

Overview

- Anatomy and physiology of the heart
- System and organ level
- Resting cardiac tissue
- Active force generation: the sarcomere
- Multi-scale modeling
- Models of cardiac mechanics: from cell to system
- Models of cardiac electromechanics

Integrative In-Silico Biology

Functional Integration,

Structural Integration

- Functional integration
- of interacting physiological processes
- Structural integration
- across scales of biological organization

(c) 2004 Andrew McCulloch, UCSD

Why modeling

- hypothesis generation
- clinical applications
- diagnosis
- training platforms for surgeons
- predict outcomes of surgical interventions
- predict outcomes of therapies

Why multiscale modeling

- Cardiac structure and function are heterogeneous: most pathologies are regional and non-homogeneous
- Ca2+ important ion in electrophysiology and responsible for cardiac force generation
- Many interacting subsystems in basic processes: e.g.
- ventricular stress coronary flow
- electrical activation mechanical activation (ECC and MEF)
- feedback of baroreceptors on cardiac contractility and frequency

- Anatomy and physiology of the heart
- System and organ level
- Resting cardiac tissue
- Active force generation: the sarcomere
- Multi-scale modeling
- Models of cardiac mechanics: from cell to system
- Models of cardiac electromechanics

Models of cardiac mechanicscellular

- Development of models of cellular cardiac mechanics have lagged behind models of cellular cardiac electrophysiology, due to
- lack of available solving algorithms (and computer power)
- controversies about basic mechanisms of force generation in myofilaments
- 4 categories:
- phenomenological time-varying elastance models (algebraic)
- phenomenological Hill-models (ODE)
- A.F.Huxley type models of crossbridge formation (PDE)
- Landesberg type myofilament activation model (ODEs)

Modeling Myofilament Force Production

- Ca2+ binding to TnC causes tropomyosin to change to a permissive state
- Force development occurs as actin-myosin crossbridges form
- Crossbridges can ‘hold’ tropomyosin in the permissive state even after Ca2+ has dissociated

Myofilament Activation/Crossbridge Cycling Kinetics

Ca2+

Ca2+

Ca2+

kn

Roff

Roff

0

*

kb

Ca2+

*

koff

kon

Ron

Ron

0

*

g

g

f

Ca2+

0

A1

A1

*

Ca2+

Non-permissive Tropomyosin

koff

Permissive Tropomyosin

f

Permissive Tropomyosin, 1-3 crossbridges attached (force generating states)

Ca2+

not

bound

to TnC

Ca2+

bound

to TnC

This scheme is used to find A(t), the time-course of attached crossbridges for a given input of [Ca2](t)

Myofilament Model Equations

- Total force is the product of the total number of attached crossbridges, average crossbridge distortion, and crossbridge stiffness:

Myofilament model: results

0

*

Noff

Noff

0

*

Roff

Roff

*

0

Non

Non

0

*

Ron

Ron

0

*

A1

A1

µtitin

0

*

A2

A2

Factive

µgel

ηcell

0

*

A3

A3

Myofilament model of active force generation

+

Passive mechanics of single myocyte

Model validation experiments

- Simultaneous measurement of intracellular Ca2+ and shortening in single myocytes

Nonlinear Elasticity of Soft Tissues

- Soft tissues are not elastic — stress depends on strain and the history of strain
- However, the hysteresis loop is only weakly dependent on strain rate
- It may be reasonable to assume that tissues in vivo are preconditioned
- Fung: elasticity may be suitable for soft tissues, if we use a different stress-strain relation for loading and unloading – the pseudoelasticity concept
- a rationale for applying elasticity theory to soft tissues
- Unlike in bone, linear elasticity is inappropriate for soft tissues; we need nonlinear finite elasticity

Models of cardiac mechanicstissue

- Passive
- strain energy functions
- orthotropic (fiber – crossfiber – sheet)
- heterogeneous
- Active
- orthotropic (fiber – crossfiber – sheet?)
- heterogeneous(!) (Cordeiro et al, AJP 286, H1471-H1479, 2004)

Models of cardiac mechanicsorgan

- Solve tissue models on anatomy with e.g. finite element or finite difference method
- Compute part of cardiac cycle, e.g. produce Frank-Starling curves, or
- Compute full cardiac cycle with coupling to circulatory model

Ventricular Geometry

x

1

q

x

b

2

d

a

m

l

Truncated ellipses of revolution

Prolate Spheroidal Coordinates

x1 = dcoshlcosm

x2 = dsinhlsinmcosq

x3 = dsinhlsinmsinq

Anatomic models

Vetter FJ et al (1998) PBMB

Nielsen PMF et al (1991) AJP

Grieshaber J et al (2002)

Smith NP et al (2000) ABME

Stevens C et al (2003) PBMB

LeGrice IJ et al (1997) AJP

Kerckhoffs et al (2003)

Rao J et al

Models of cardiac mechanicsorgan: application

- Diagnostic measures
- Herz et al used finite element model of cardiac ischemia to generate new measures for dyskinetic cardiac tissue

Herz et al. , 2005. Annals Biomed Eng 33: 912-919

Models of cardiac mechanicsorgan: application

- Surgical training platforms
- Predict outcome of surgical interventions

Myosplint reduced fiber stress, but did not affect stroke volume

Guccione et al. 2003. Annals of Thoracic Surgery 76,1171-1180.

Circulatory modelssystem level

- 2-, 3-, 4-element windkessel

Circulatory modelssystem level

- windkessel = air chamber
- used in plunger pumps to ensure a steady flow

Circulatory modelssystem levelPressure serves as hemodynamic boundary condition

Cavity

pressure

Cavity

pressure

Flow Q

Cavity volume from

circulatory model

FE Cavity volume

(c) 2004 Andrew McCulloch, UCSD

Coupling

Estimate LV & RV cavity pressure

FE model

Circulatory model

Circ Cavity volumes

FE Cavity volumes

Calculate difference R

R < criterion?

no

yes

Do not update

Jacobian

Update

Jacobian

next timestep

System compliance matrix

Circ compliance matrix

FE compliance matrix

Update 1: Estimate pressure from history

Update 2: Perturb LV pressure

Update 3: Perturb RV pressure

Updates >3: Update pressures

- Anatomy and physiology of the heart
- System and organ level
- Resting cardiac tissue
- Active force generation: the sarcomere
- Multi-scale modeling
- Models of cardiac mechanics: from cell to system
- Models of cardiac electromechanics

Important components of models of cardiac electromechanics

Anatomic model

Hemodynamic model

Electrophysiology

model

Mechanics

model

time

Models of cardiac electrophysiologytissue

- Couple ionic models or FitzHugh-Nagumo with
- Monodomain Vm(x,t)
- Bidomain Vextr(x,t) and Vintr(x,t)
- Or derive wavefront from bidomain model:
- Eikonal-diffusion tdep(x)

Models of cardiac electromechanicscellular

- Calcium
- Mechano-electric feedback (MEF)
- deformation
- sarcomere length dependent myofilament calcium sensitivity
- stretch-activated ion channels
- Clinic:
- Resynchronization
- asymmetric hypertrophy by chronic pacing
- Couple existing models of electrophysiology to models of cardiac mechanics

Models of cardiac electromechanicstissue

- Electromechanical Coupling:
- Tight (Continuous interplay: EC and MEF)
- Loose (EC only)
- Due to computational demand of tightly coupled electromechanics, solve models on 2D domains

Models of cardiac electromechanicstissue

- Spiral waves with FitzHugh-Nagumo-monodomain model and active contraction
- Mechano-electric feedback through deforming tissue

noncontracting

contracting

Nash and Panfilov. Progr Biophys Mol Biol 85, 501-522, 2004.

Models of cardiac electromechanicsorgan

- Solve tissue models on anatomy with e.g. finite element or finite difference method
- Due to computational demand, either:
- Phenomenological or simple ionic models
- Loosely coupled (ECC only)
- Tightly coupled: parallellization

Models of cardiac electromechanicsorgan

- Loosely coupled model

In a computational model study1:

- A physiological sequence of depolarization results in an unphysiological non-uniform distribution of shortening
- An unphysiological synchronous depolarization results in a physiological homogeneous distribution of shortening

End-systolic

myofiber strain

Experiment (Mazhari et al, Circ. 104, 2001)

0

Simulation with synchronous activation

-0.1

Simulation with normal activation

-0.2

0

0.25

0.5

0.75

1

Epi

Endo

1Kerckhoffs et al, ABME 31, p536-547, 2003

Models of cardiac electromechanicsorgan

- Loosely coupled model

normal dep wave

synchronous

myofiber

strain

1Kerckhoffs et al, ABME 31, p536-547, 2003

Models of cardiac electromechanicsorgan

From an experimental study1:

The latency to onset of contraction in endocardial cells is ~20 ms longer than that of epicardial cells, which allows the impulse to traverse the LV wall and effect a coordinated contraction of the ventricular myocardium.

1Cordeiro et al, AJP 286, H1471-H1479, 2004

Models of cardiac electromechanicsorgan: ventricular pacingElectrical activationin the paced heart

Pacing at the left ventricle

Pacing at the right ventricle

activation completed in: 92 ms

129 ms

Kerckhoffs et al. J Eng Math. 2003;47:201-216.

Models of cardiac electromechanicsorgan: ventricular pacingContraction in the paced heart

Pacing at the left ventricle

Pacing at the right ventricle

strain

Maximum LV

pressure increase:

137 kPa/s

240 kPa/s

Kerckhoffs et al. J Eng Math. 2003;47:201-216.

Models of cardiac electromechanicsorgan

- Tightly coupled model:
- Simple 3-current ionic model
- Model of active contraction

Nickerson et al. Europace 7, 5118-5127, 2005.

Models of cardiac electromechanicsorgan: application

- Tightly coupled model
- Computation time: 3 weeks on 8 parallel processors

Nickerson et al. Europace 7, 5118-5127, 2005.

Computational demand

- With advances in biology, simulations remain large and time-consuming
- Algorithm improvement
- Parallel computing

Computation speed at CMRG

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