Dependence

of

Intramyocardial

Pressure

and

Coronary

Flow

on

Ventricular

Loading

and

Contractility

A

Model

Study

The

phasic

coronary

arterial

inflow

during

the

normal

cardiac

cycle

has

been

explained

with

simple

-LRB-

waterfall

intramyocardial

pump

-RRB-

models

emphasizing

the

role

of

ventricular

pressure

To

explain

changes

in

isovolumic

and

low

afterload

beats

these

models

were

extended

with

the

effect

of

three-dimensional

wall

stress

nonlinear

characteristics

of

the

coronary

bed

and

extravascular

fluid

exchange

With

the

associated

increase

in

the

number

of

model

parameters

a

detailed

parameter

sensitivity

analysis

has

become

difficult

Therefore

we

investigated

the

primary

relations

between

ventricular

pressure

and

volume

wall

stress

intramyocardial

pressure

and

coronary

blood

flow

with

a

mathematical

model

with

a

limited

number

of

parameters

The

model

replicates

several

experimental

observations

the

phasic

character

of

coronary

inflow

is

virtually

independent

of

maximum

ventricular

pressure

the

amplitude

of

the

coronary

flow

signal

varies

about

proportionally

with

cardiac

contractility

and

intramyocardial

pressure

in

the

ventricular

wall

may

exceed

ventricular

pressure

A

parameter

sensitivity

analysis

shows

that

the

normalized

amplitude

of

coronary

inflow

is

mainly

determined

by

contractility

reflected

in

ventricular

pressure

and

at

low

ventricular

volumes

radial

wall

stress

Normalized

flow

amplitude

is

less

sensitive

to

myocardial

coronary

compliance

and

resistance

and

to

the

relation

between

active

fiber

stress

time

and

sarcomere

shortening

velocity

Introduction

Coronary

arterial

inflow

varies

in

time

during

the

cardiac

cycle

Systolic

inflow

is

smaller

than

diastolic

inflow

demonstrating

that

the

pulsatility

of

coronary

flow

is

not

caused

by

variation

of

the

arterial

venous

pressure

difference

Instead

the

phasic

pattern

of

coronary

inflow

has

been

attributed

to

changes

in

cross-sectional

area

of

the

myocardial

vessels

For

example

a

transient

decrease

in

cross-sectional

area

affects

coronary

flow

in

two

ways

First

coronary

inflow

decreases

and

outflow

increases

because

blood

is

squeezed

out

of

the

coronary

bed

Second

both

coronary

inflow

and

outflow

decrease

because

of

the

increase

of

coronary

resistance

Vessel

cross-sectional

area

depends

on

the

local

coronary

pressure

the

vessel

wall

mechanical

properties

and

the

embedment

of

the

vessel

in

the

myocardial

tissue

The

relation

between

vessel

cross-sectional

area

and

transmural

pressure

is

nonlinear

and

changes

due

to

autoregulation

The

embedment

in

the

myocardial

tissue

can

be

represented

by

myocardial

wall

stress

which

consists

of

two

contributions

the

stress

in

the

collagen

matrix

through

which

the

vessel

is

tied

to

the

surrounding

tissue

and

the

intramyocardial

pressure

in

the

interstitial

fluid

In

the

end

the

pattern

of

coronary

flow

is

mainly

determined

by

contraction

of

the

myofibers

in

the

cardiac

wall

since

myofiber

contraction

determines

the

level

of

both

myocardial

wall

stress

and

coronary

pressure

Many

experiments

have

been

performed

to

elucidate

the

interplay

between

the

factors

governing

coronary

flow

It

has

been

observed

that

the

pulsatile

component

of

the

coronary

inflow

signal

-LRB-

-RRB-

is

about

proportionally

related

to

left

ventricle

-LRB-

LV

-RRB-

contractility

flow

amplitude

being

about

zero

when

contractility

is

about

zero

-LRB-

-RRB-

is

virtually

independent

of

systolic

LV

pressure

in

isovolumic

beats

executed

at

various

LV

volumes

for

pressures

up

to

about

kPa

and

-LRB-

-RRB-

is

about

the

same

in

isobaric

beats

at

low

LV

pressure

as

in

isovolumic

beats

at

LV

pressures

up

to

about

kPa

It

was

observed

also

that

minimum

systolic

flow

-LRB-

-RRB-

is

virtually

independent

of

LV

pressure

for

pressures

below

kPa

but

-LRB-

-RRB-

decreases

with

increasing

LV

pressure

for

pressures

above

about

kPa

In

addition

it

has

been

found

that

-LRB-

-RRB-

systolic

intramyocardial

pressure

may

exceed

left

ventricular

pressure

in

subendocardial

layers

in

low

afterload

beats

and

-LRB-

-RRB-

that

an

increase

of

coronary

perfusion

pressure

leads

to

an

increase

of

intramyocardial

pressure

Mathematical

models

have

been

proposed

as

a

tool

to

interpret

the

experimental

data

In

early

models

the

interaction

between

the

coronary

vessel

and

the

myocardium

was

modeled

through

the

intramyocardial

pressure

only

In

the

waterfall

and

the

intramyocardial

pump

model

this

pressure

was

assumed

to

be

determined

completely

by

left

ventricular

pressure

with

intramyocardial

pressure

decreasing

linearly

from

left

ventricular

pressure

at

the

endocardial

surface

to

zero

at

the

epicardial

surface

These

models

can

explain

the

pulsatility

of

coronary

inflow

under

normal

physiological

conditions

and

the

experimental

observations

-LRB-

-RRB-

and

-LRB-

-RRB-

listed

above

In

later

models

the

interaction

between

vessel

and

tissue

was

still

modeled

through

intramyocardial

pressure

only

but

intramyocardial

pressure

was

assumed

to

depend

both

on

left

ventricular

pressure

and

on

transverse

tissue

stress

i.e.

stress

perpendicular

to

the

muscle

fiber

direction

In

a

finite

element

model

of

the

beating

heart

incorporating

transverse

tissue

stress

observation

-LRB-

-RRB-

was

replicated

Transverse

stress

is

also

included

in

models

by

Beyar

et

al.

together

with

nonlinear

characteristics

of

the

properties

of

the

coronary

bed

and

exchange

of

fluid

between

the

coronary

vessels

and

the

myocardial

interstitium

In

these

models

experimental

observations

-LRB-

-RRB-

through

-LRB-

-RRB-

are

replicated

Vis

et

al.

extended

the

model

for

interaction

between

the

vessel

and

the

cardiac

wall

by

assuming

that

the

coronary

vessel

is

subject

to

an

extravascular

pressure

that

depends

both

on

intramyocardial

pressure

and

local

tissue

stress

So

the

effective

compliance

of

the

coronary

vessel

depends

both

on

the

compliance

of

the

vessel

wall

and

the

stiffness

of

the

myocardium

thus

implementing

an

idea

introduced

before

as

the

time-varying

elastance

concept

In

a

finite

element

model

of

one

vessel

in

the

myocardial

wall

Vis

et

al.

investigated

the

influence

of

contractility

pressure

and

circumferential

wall

stretch

on

vessel

area

for

static

diastole

and

systole

With

the

model

observations

-LRB-

-RRB-

and

-LRB-

-RRB-

were

reproduced

With

the

increasing

complexity

of

the

models

more

experimental

observations

have

been

replicated

Simultaneously

the

number

of

model

parameters

has

increased

making

it

difficult

to

perform

a

detailed

parameter

sensitivity

analysis

and

identify

the

critical

model

parameters

Therefore

our

aim

was

to

study

the

primary

relations

between

left

ventricular

pressure

and

volume

wall

stress

in

fiber

and

transverse

direction

intramyocardial

pressure

and

the

coronary

blood

flow

with

a

mathematical

model

with

a

limited

number

of

parameters

and

to

assess

the

sensitivity

of

the

model

results

to

the

model

parameter

settings

Material

and

methods

The

complete

model

consists

of

four

parts

describing

ventricular

wall

mechanics

myocardial

constitutive

properties

intramyocardial

pressure

and

the

coronary

and

systemic

circulation.Figure

Passive

material

behavior

Top

passive

stress

as

a

function

of

stretch

ratio

λ

for

uniaxial

stretch

along

fiber

direction

-LRB-

σm

f

-RRB-

and

radial

direction

-LRB-

σm

r

-RRB-

Bottom

passive

left

ventricular

pressure

volume

relation

according

to

model

-LRB-

solid

line

-RRB-

and

experimental

data

experimental

data

given

for

average

minimum

and

maximum

volume

-LRB-

dashed

line

-RRB-

and

minimum

and

maximum

volume

SD

-LRB-

dash-dot

lines

-RRB-

Ventricular

Wall

Mechanics

The

model

of

ventricular

wall

mechanics

describes

how

left

ventricular

pressure

and

volume

are

related

to

local

tissue

properties

i.e.

fiber

stress

and

strain

and

radial

wall

stress

and

strain

The

model

is

based

on

a

previously

published

model

which

is

extended

to

describe

the

influence

of

radial

wall

stress

While

the

original

model

is

derived

for

arbitrary

ventricular

geometries

with

rotational

symmetry

here

we

will

derive

the

equations

for

the

special

case

of

a

thick

walled

sphere

We

consider

the

sphere

to

consist

of

a

set

of

nested

thin

spherical

shells

In

each

shell

stresses

satisfy

the

condition

of

force

equilibrium

in

radial

direction

where

σr

σc

and

σl

denote

the

radial

circumferential

and

longitudinal

component

of

the

tissue

stress

tensor

respectively

and

r

indicates

the

radial

position

in

the

wall

We

neglected

shear

stress

components

in

view

of

the

low

shear

stiffness

of

the

tissue

The

myocardial

tissue

was

assumed

to

be

incompressible

and

to

consist

of

myofibers

embedded

in

a

collagen

matrix

The

Cauchy

stress

tensor

in

tissue

is

written

as

where

pim

denotes

the

intramyocardial

pressure

the

stress

in

the

collagen

matrix

and

σa

the

stress

generated

in

the

myofibers

along

the

myofiber

direction

We

assume

that

the

fibers

are

located

in

spherical

shells

at

an

angle

α

with

the

circumferential

direction

and

adopt

the

assumption

by

Arts

that

the

stress

in

the

collagen

matrix

in

the

plane

of

a

shell

is

completely

determined

by

the

passive

stress

σm

f

in

the

fibers

Then

it

holds

where

σm

r

represents

the

radial

wall

stress

generated

in

the

collagen

matrix

and

the

total

fiber

stress

σf

is

introduced

as

the

sum

of

passive

and

active

fiber

stress

Substitution

in

-LRB-

-RRB-

yields

This

equation

describes

how

in

each

shell

fiber

stress

σf

and

radial

wall

stress

in

the

collagen

matrix

σm

r

contribute

to

the

variation

of

total

radial

wall

stress

σr

in

radial

direction

Together

all

shells

increase

radial

wall

stress

from

zero

stress

at

the

outer

surface

-LRB-

r

ro

-RRB-

to

minus

left

ventricular

pressure

at

the

inner

surface

-LRB-

r

ri

-RRB-

The

relation

between

wall

stress

and

left

ventricular

pressure

is

found

by

integrating

Eq

-LRB-

-RRB-

from

the

endocardial

to

the

epicardial

surface

To

evaluate

the

integral

we

first

adopt

the

assumption

by

that

σf

is

constant

across

the

wall

Secondly

we

assume

that

a

representative

position

can

be

found

such

that

Evaluation

of

the

integral

and

substitution

of

the

boundary

conditions

-LRB-

-RRB-

yields

where

ri

and

ro

are

rewritten

in

terms

of

the

cavity

volume

Vlv

and

wall

volume

Vw

and

is

introduced

as

a

short

notation

of

Apart

from

the

term

this

relation

is

identical

to

that

derived

in

Arts

et

al.

To

complete

the

model

relating

wall

mechanics

to

cavity

mechanics

Eq

-LRB-

-RRB-

is

complemented

by

a

relation

between

ventricular

volume

and

tissue

strain

We

choose

the

passive

ventricle

at

zero

transmural

pressure

as

a

reference

state

In

this

state

we

assume

a

sarcomere

length

and

a

cavity

volume

It

has

been

shown

that

the

fiber

stretch

ratio

λf

can

be

approximated

by

This

ratio

corresponds

to

the

circumferential

stretch

ratio

at

the

outer

surface

of

a

shell

that

contains

the

left

ventricular

cavity

volume

and

one

third

of

the

wall

volume

At

the

same

location

the

radial

stretch

ratio

λr

equals

where

we

assumed

incompressibility

of

the

myocardial

tissue

Equations

-LRB-

-RRB-

-LRB-

-RRB-

describe

how

global

ventricular

properties

Vlv

and

plv

are

related

to

local

tissue

properties

λf

λr

σf

and

Myocardial

Constitutive

Properties

The

model

of

ventricular

mechanics

is

completed

with

constitutive

laws

for

fiber

stress

and

radial

stress

The

active

fiber

stress

σa

was

modeled

to

depend

on

contractility

c

sarcomere

length

ls

time

elapsed

since

activation

ta

and

sarcomere

shortening

velocity

vs

as

with

contractility

c

-LRB-

c

-RRB-

scaling

constant

σar

and

Here

ls

denotes

the

sarcomere

length

below

which

active

stress

becomes

zero

and

ls

ar

represents

the

sarcomere

length

to

which

the

reference

stress

σar

is

referred

to

Times

ta

and

tmax

denote

the

time

elapsed

since

activation

and

the

duration

of

the

twitch

respectively

Velocity

represents

the

unloaded

sarcomere

shortening

velocity

while

cv

governs

the

shape

of

the

stress

velocity

relation

Parameter

values

in

the

active

stress

law

-LRB-

Table

-RRB-

are

chosen

in

agreement

with

experimental

data

Table

Reference

settings

for

parameters

in

the

model

of

the

left

PaVlv

ls

mls

Pals

m

s

The

passive

stress

along

the

fiber

direction

is

modeled

as

where

it

was

assumed

that

no

stress

can

be

transmitted

in

compression

Passive

transverse

stress

is

modeled

similarly

Since

the

radial

direction

at

ventricular

level

coincides

with

the

transverse

direction

at

the

tissue

level

we

write

the

transverse

stress

model

in

terms

of

radial

stress

and

stretch

ratio

Settings

of

parameter

values

cf

and

cr

are

based

on

experimental

data

on

the

pressure

volume

relation

of

the

passive

left

ventricle

In

the

latter

study

the

pressure

volume

relation

was

described

by

a

logarithmic

relation

involving

chamber

stiffness

at

positive

and

negative

pressures

and

maximum

and

minimum

attainable

volume

The

bottom

panel

of

Fig.

shows

the

experimental

data

and

our

model

prediction

for

parameter

settings

in

Table

Uniaxial

stress

strain

responses

are

shown

in

the

top

panel

of

Fig.

Figure

Schematic

representation

of

the

left

ventricle

-LRB-

LV

-RRB-

in

the

systemic

circulation

and

coronary

circulation

Rart

Rper

and

Rven

represent

the

systemic

arterial

peripheral

and

venous

resistance

respectively

MV

and

AV

represent

the

mitral

and

aortic

valve

respectively

Cart

and

Cven

the

systemic

arterial

and

venous

compliance

Lven

and

Lart

the

inertia

of

the

blood

in

the

venous

and

arterial

system

Rart

c

Rmyo

Rmyo

and

Rven

c

represent

the

coronary

arterial

the

two

intramyocardial

and

venous

resistances

Cart

c

Cmyo

c

and

Cven

c

the

coronary

arterial

intramyocardial

and

venous

compliance

part

c

pmyo

c

and

pven

c

the

pressure

in

the

coronary

arteries

myocardium

and

veins

respectively

pao

and

pla

the

aortic

and

left

atrial

pressure

respectively

qao

qm

represent

the

AV

flow

and

MV

flow

qart

c

and

qven

c

the

coronary

arterial

inflow

and

venous

outflow

represents

the

intramyocardial

pressure

at

the

representative

radial

position

When

simulating

isovolumic

and

isobaric

contractions

the

pressure

drop

across

the

coronary

circulation

is

switched

from

pao

pla

to

a

constant

perfusion

pressure

pper.Figure

Hemodynamics

in

the

beating

heart

Left

simulation

of

the

physiological

state

with

aortic

pressure

driving

the

coronary

circulation

Right

simulation

of

isolated

heart

experiment

with

a

constant

perfusion

pressure

and

maximum

vasodilation

From

top

to

bottom

ventricular

pressure

plv

intramyocardial

pressure

-LRB-

bold

-RRB-

and

radial

wall

stress

aortic

valve

-LRB-

AV

-RRB-

flow

qao

-LRB-

bold

-RRB-

and

mitral

valve

-LRB-

MV

-RRB-

flow

qm

left

ventricular

cavity

volume

Vlv

coronary

perfusion

pressure

pper

myocardial

coronary

pressure

pmyo

c

-LRB-

bold

-RRB-

and

left

atrial

pressure

pla

coronary

inflow

-LRB-

bold

-RRB-

and

outflow

per

g

wall

volume

volume

of

myocardial

coronary

bed

Vmyo

c.Figure

Hemodynamics

at

maximal

vasodilatation

and

constant

perfusion

pressure

expressed

in

time

courses

of

ventricular

pressure

plv

ventricular

volume

Vlv

radial

wall

stress

intramyocardial

pressure

and

coronary

arterial

inflow

Left

isovolumic

beats

for

ventricular

volumes

ranging

from

to

ml

Middle

isovolumic

beats

at

ml

for

contractility

parameter

c

ranging

from

to

Left

isobaric

beats

for

systolic

pressures

ranging

from

to

kPa

Intramyocardial

Pressure

To

derive

the

expression

for

the

intramyocardial

pressure

again

we

consider

the

ventricular

wall

to

be

composed

of

a

number

of

nested

shells

The

pressure

in

between

two

shells

is

assumed

to

be

a

fraction

β

of

the

left

ventricular

pressure

In

analogy

to

Eq

-LRB-

-RRB-

this

pressure

is

in

equilibrium

with

radial

tissue

stress

σr

In

the

model

we

assume

a

linear

variation

of

β

with

the

transmural

position

in

the

wall

with

β

at

the

endocardial

surface

and

β

at

the

epicardial

surface

Again

stress

at

the

shell

containing

the

LV

cavity

volume

and

one

third

of

LV

wall

volume

is

considered

representative

For

this

location

at

a

radius

in

the

wall

we

find

where

we

introduced

the

notation

and

the

radial

positions

ri

ro

and

are

defined

as

Equation

-LRB-

-RRB-

describes

intramyocardial

pressure

as

a

function

of

LV

pressure

and

volume

the

volume

dependency

enters

the

equation

through

the

radial

stress

σm

r

-LSB-

Eq

-LRB-

-RRB-

-RSB-

which

depends

the

radial

stretch

ratio

λr

-LSB-

Eq

-LRB-

-RRB-

-RSB-

which

in

turn

depends

on

LV

volume

-LSB-

Eq

-LRB-

-RRB-

-RSB-

Systemic

and

Coronary

Circulation

The

model

of

left

ventricular

wall

mechanics

is

incorporated

in

lumped

parameter

models

for

the

coronary

and

systemic

circulation

-LRB-

Fig.

-RRB-

The

aortic

-LRB-

AV

-RRB-

and

mitral

valve

-LRB-

MV

-RRB-

are

modeled

as

an

ideal

diode

Vessels

are

modeled

with

constant

resistances

R

inertances

L

and

capacitances

C

The

pressure

drop

Δp

across

each

of

these

components

is

given

by

with

V

the

volume

in

the

capacitance

and

q

the

flow

through

a

resistance

or

an

inertance

The

pressure

volume

relation

of

the

capacitance

represents

a

linearization

around

the

physiologic

working

point

representing

the

volume

at

zero

pressure

Values

of

parameters

in

the

circulation

model

were

based

on

literature

-LRB-

Table

-RRB-

Table

Reference

settings

for

parameters

in

the

circulation

model

coronary

resistance

values

in

parentheses

represent

maximum

vasodilation.Systemic

circulationCoronary

Pa

s

m

-LRB-

-RRB-

Pa

s

m

Pa

s

m

-LRB-

-RRB-

Pa

s

m

Pa

s

m

-LRB-

-RRB-

Pa

s

m

Pa

Pa

s

m

Pa

Pa

Pa

Pa

Pa

s

m

Pa

s

m

The

connection

between

the

model

of

LV

mechanics

and

the

coronary

circulation

model

is

made

through

the

intramyocardial

pressure

that

acts

on

the

myocardial

capacitance

Cmyo

c

The

values

of

the

coronary

capacitances

were

based

on

measurements

by

Spaan

et

al.

and

ml

mm

Hg

g

LV

in

large

coronary

arteries

myocardial

coronary

bed

and

coronary

veins

respectively

Zero

pressure

volumes

were

chosen

such

that

under

normal

physiological

conditions

time-averaged

coronary

volume

was

about

ml

g

of

LV

tissue

distributed

over

arterial

myocardial

and

venous

vessels

in

a

ratio

of

In

the

coronary

circulation

resistance

values

during

normoxia

and

hyperemia

were

derived

from

Chilian

et

al.

In

that

study

total

coronary

resistance

under

normal

and

vasodilated

conditions

was

measured

to

be

and

mm

Hg

min

g

ml

respectively

Distribution

of

resistance

over

the

arterial

myocardial

and

venous

compartment

was

measured

to

be

and

under

normal

conditions

and

and

under

maximal

vasodilation

Systemic

parameters

are

chosen

to

yield

representative

function

curves

for

a

kg

adult

at

a

heart

rate

of

bpm

LV

wall

volume

was

set

to

ml

and

cavity

volume

at

zero

pressure

was

set

to

of

this

volume

The

arterial

load

was

modeled

by

a

three-element

windkessel

model

consisting

of

a

characteristic

aortic

impedance

Rart

an

arterial

compliance

Cart

and

a

peripheral

resistance

Rper

The

peripheral

resistance

was

chosen

to

yield

realistic

time-averaged

values

of

aortic

pressure

and

aortic

flow

Next

arterial

capacitance

was

chosen

to

yield

realistic

values

of

minimum

and

maximum

aortic

pressure

Total

blood

volume

was

set

to

ml

The

blood

volume

at

which

mean

systemic

pressure

is

zero

was

assumed

to

be

equal

of

total

blood

volume

about

of

which

is

contained

in

the

venous

system

Venous

capacitance

was

chosen

such

that

the

additional

of

blood

volume

leads

a

mean

systemic

pressure

of

about

kPa

Simulations

Performed

With

the

model

a

number

of

simulations

were

performed

First

we

considered

the

normal

physiological

situation

with

normal

vessel

tone

and

coronary

flow

driven

by

the

difference

between

aortic

and

left

atrial

pressure

Next

we

investigated

the

changes

induced

in

the

isolated

heart

setup

used

by

Krams

et

al.

by

the

combination

of

maximum

vasodilation

constant

perfusion

pressure

and

zero

coronary

outflow

pressure

Then

under

these

conditions

we

simulated

isovolumic

and

isobaric

beats

for

various

settings

of

LV

volume

pressure

and

contractility

Following

Krams

et

al.

and

Pagliaro

et

al.

results

were

analyzed

in

terms

of

minimal

systolic

coronary

flow

and

the

normalized

coronary

flow

amplitude

-LRB-

NFA

-RRB-

Finally

we

investigated

sensitivity

of

NFA

to

changes

in

myocardial

radial

stiffness

active

material

properties

coronary

myocardial

resistance

and

coronary

capacitance.Figure

Analysis

of

coronary

inflow

data

from

simulations

in

Fig.

expressed

in

normalized

arterial

coronary

flow

amplitude

-LRB-

NFA

-RRB-

defined

in

Eq

-LRB-

-RRB-

and

minimal

coronary

arterial

inflow

Left

NFA

-LRB-

top

-RRB-

and

-LRB-

bottom

-RRB-

in

isovolumic

beats

as

a

function

of

maximum

left

ventricular

pressure

for

contractility

parameter

c

varying

from

to

in

top

panel

experimental

are

given

by

the

dashed

line

-LRB-

reference

-RRB-

dash-dotted

line

-LRB-

high

contractility

-RRB-

and

dotted

line

-LRB-

low

contractility

-RRB-

Middle

NFA

and

in

isovolumic

beats

as

a

function

of

contractility

parameter

c

for

LV

volumes

of

and

ml

Right

NFA

and

in

isobaric

beats

as

a

function

of

maximum

ventricular

pressure

for

contractility

parameter

c

ranging

from

to

Results

The

Normal

Beating

Heart

To

illustrate

the

behavior

of

the

model

first

the

normal

physiological

situation

was

simulated

with

normal

vessel

tone

and

coronary

flow

driven

by

the

difference

between

aortic

and

left

atrial

pressure

Ventricular

and

intramyocardial

pressure

rise

to

and

kPa

respectively

-LRB-

Fig.

left

panel

-RRB-

Intramyocardial

pressure

is

almost

completely

determined

by

ventricular

pressure

since

radial

wall

stress

is

virtually

zero

Maximum

aortic

and

mitral

flow

are

and

ml/s

respectively

Stroke

volume

is

ml

at

an

ejection

fraction

of

Perfusion

pressure

equals

aortic

pressure

and

varies

between

and

kPa

Myocardial

coronary

pressure

varies

between

and

kPa

Left

atrial

pressure

remains

about

constant

at

kPa

Mean

coronary

arterial

inflow

and

venous

outflow

are

ml

s

g

LV

wall

volume

Coronary

arterial

inflow

displays

a

two

peaked

pattern

the

peaks

occurring

in

early

filling

and

in

ejection

Diastolic

inflow

is

larger

than

systolic

inflow

Coronary

arterial

outflow

shows

only

one

peak

per

cycle

occurring

during

ejection

Myocardial

coronary

volume

varies

between

and

ml

Constant

Perfusion

Pressure

and

Maximal

Vasodilation

This

simulation

represents

the

experimental

situation

in

the

isolated

heart

with

a

constant

perfusion

pressure

of

kPa

and

maximal

vasodilation

Myocardial

coronary

pressure

varies

between

and

kPa

-LRB-

Fig.

right

panel

-RRB-

Arterial

inflow

varies

between

and

ml

s

g

LV

wall

volume

with

a

mean

of

ml

s

g

LV

wall

volume

In

the

arterial

inflow

curve

the

positive

flow

peak

in

early

systole

has

disappeared

Myocardial

coronary

volume

varies

between

and

ml

The

Isovolumic

Beating

Heart

The

isovolumic

experiments

by

Krams

et

al.

were

simulated

at

a

constant

perfusion

pressure

of

kPa

and

maximal

vasodilation

Maximum

LV

pressure

decreases

with

decreasing

LV

volume

-LRB-

Fig.

left

panel

-RRB-

As

LV

volume

decreases

below

ml

the

volume

at

which

pressure

in

the

passive

LV

is

zero

diastolic

LV

pressure

becomes

negative

At

these

volumes

radial

wall

stress

becomes

positive

while

being

constant

during

a

beat

With

decreasing

volume

maximum

intramyocardial

pressure

and

coronary

arterial

inflow

decrease

whereas

minimum

coronary

arterial

inflow

increases

The

normalized

coronary

flow

amplitude

-LRB-

NFA

-RRB-

-LSB-

Eq

-LRB-

-RRB-

-RSB-

decreases

with

decreasing

LV

pressure

-LRB-

Fig.

top

left

panel

-RRB-

but

pulsatility

persists

at

zero

ventricular

pressure

NFA

decreases

with

decreasing

contractility

as

well

Minimum

coronary

arterial

inflow

increases

with

decreasing

LV

pressure

-LRB-

Fig.

bottom

left

panel

-RRB-

At

low

pressures

the

slope

of

the

pressure

flow

relation

decreases

For

a

constant

LV

volume

of

ml

developed

LV

pressure

decreases

with

decreasing

contractility

-LRB-

Fig.

middle

panel

-RRB-

At

this

volume

of

ml

radial

wall

stress

is

zero

Maximum

intramyocardial

pressure

decreases

with

maximum

LV

pressure

Maximum

coronary

arterial

inflow

decreases

slightly

whereas

minimum

coronary

arterial

inflow

increases

strongly

With

decreasing

contractility

normalized

coronary

flow

amplitude

-LRB-

NFA

-RRB-

decreases

about

linearly

while

minimal

arterial

inflow

increases

linearly

-LRB-

Fig.

middle

panel

-RRB-

For

other

volumes

a

similar

behavior

is

found

The

Isobaric

Beating

Heart

Simulations

of

hemodynamics

in

the

isobaric

beating

heart

were

performed

at

a

filling

pressure

of

kPa

a

constant

perfusion

pressure

of

kPa

maximal

vasodilation

and

contractility

parameter

c

With

decreasing

LV

pressure

during

ejection

minimal

LV

volume

decreases

and

radial

wall

stress

increases

-LRB-

Fig.

right

panel

-RRB-

Reduction

of

maximum

LV

pressure

from

to

kPa

is

accompanied

by

a

reduction

of

maximum

intramyocardial

pressure

to

kPa

Maximum

arterial

inflow

decreases

whereas

minimum

arterial

inflow

increases

Further

reduction

of

LV

pressure

below

kPa

hardly

affects

maximum

intramyocardial

pressure

due

to

the

increasing

contribution

of

radial

wall

stress

Maximum

and

minimum

arterial

inflow

remain

about

constant

These

changes

are

reflected

in

the

right

panel

of

Fig.

With

decreasing

left

ventricular

pressure

initially

NFA

decreases

and

minimum

arterial

inflow

increases

Below

a

threshold

pressure

of

about

kPa

these

quantities

become

about

constant

A

similar

behavior

is

found

for

lower

contractilities

Sensitivity

Analysis

Results

of

the

sensitivity

analysis

are

shown

in

Fig.

Sensitivity

to

settings

of

parameters

in

the

model

for

radial

wall

stress

was

investigated

since

in

our

model

radial

wall

stress

is

an

important

determinant

of

intramyocardial

pressure

If

the

radial

stress

parameter

-LSB-

Eq

-LRB-

-RRB-

-RSB-

is

decreased

or

increased

by

a

factor

the

pulsatile

character

of

the

coronary

inflow

at

lower

pressures

persists

both

for

isovolumic

and

isobaric

beats

-LRB-

Fig.

top

panel

-RRB-

In

isovolumic

beats

maximum

LV

pressure

increases

with

decreasing

At

the

lowest

LV

volume

simulated

ml

maximum

left

ventricular

pressure

does

not

drop

below

kPa

when

is

set

to

zero

In

isobaric

beats

at

high

pressures

NFA

is

not

affected

by

changes

in

Below

an

ejection

pressure

of

kPa

NFA

increases

with

increasing

radial

stiffness

When

is

set

to

zero

NFA

decreases

proportionally

with

left

ventricular

pressure

until

pulsatility

is

lost

at

zero

left

ventricular

pressure

Sensitivity

to

settings

of

parameters

in

active

stress

model

was

investigated

since

myofiber

contraction

is

the

origin

of

both

left

ventricular

pressure

and

radial

wall

stress

-LSB-

Eq

-LRB-

-RRB-

-RSB-

and

thus

of

intramyocardial

pressure

and

of

coronary

flow

impediment

In

both

isovolumic

and

isobaric

beats

NFA

increases

slightly

if

the

twitch

duration

tmax

-LSB-

Eq

-LRB-

-RRB-

-RSB-

is

reduced

from

to

ms

-LRB-

Fig.

panel

from

top

-RRB-

Changing

the

linear

relation

between

fiber

stress

and

sarcomere

shortening

velocity

into

a

hyperbolic

one

-LRB-

by

setting

cv

in

Eq

-LRB-

-RRB-

does

not

affect

NFA

in

isovolumic

beats

where

shortening

velocity

is

zero

NFA

in

isobaric

beats

is

hardly

affected

Increasing

the

sarcomere

length

ls

below

which

active

stress

is

zero

-LSB-

Eq

-LRB-

-RRB-

-RSB-

from

to

μm

yields

a

reduction

in

NFA

in

isovolumic

and

isobaric

beats

with

a

ventricular

pressure

below

about

kPa

-LRB-

Fig.

-RRB-

Figure

Sensitivity

of

relation

between

maximum

left

ventricular

pressure

and

normalized

flow

amplitude

-LRB-

NFA

-RRB-

to

settings

of

model

parameters

for

isovolumic

beats

-LRB-

left

-RRB-

and

isobaric

beats

-LRB-

right

-RRB-

Results

for

reference

parameter

settings

are

indicated

with

ref

Top

radial

stiffness

parameter

set

to

twice

-LRB-

high

-RRB-

or

half

-LRB-

low

-RRB-

the

reference

value

or

set

to

zero

-LRB-

zero

-RRB-

Next

variation

of

active

stress

model

twitch

duration

reduced

from

to

ms

-LRB-

tim

-RRB-

stress

velocity

relation

changed

from

linear

into

hyperbolic

-LRB-

vel

-RRB-

and

stress-free

sarcomere

length

increased

from

to

μm

-LRB-

len

-RRB-

Next

all

coronary

resistances

Rart

c

Rmyo

Rmyo

and

Rven

c

set

to

twice

-LRB-

high

-RRB-

or

half

-LRB-

low

-RRB-

their

reference

values

Bottom

coronary

myocardial

capacitance

Cmyo

c

set

to

twice

-LRB-

high

-RRB-

or

half

-LRB-

low

-RRB-

the

reference

value

Sensitivity

to

coronary

resistance

was

studied

to

mimic

change

in

resistance

due

to

variation

in

perfusion

pressure

A

two-fold

simultaneous

increase

or

decrease

of

all

resistances

in

the

coronary

circulation

model

-LRB-

Rart

c

Rmyo

Rmyo

and

Rven

c

-RRB-

does

affect

NFA

more

in

isovolumic

beats

than

in

isobaric

beats

-LRB-

Fig.

panel

from

top

-RRB-

Still

changes

in

NFA

are

limited

and

pulsatility

at

low

pressures

is

maintained

Finally

sensitivity

to

myocardial

compliance

Cmyo

c

was

studied

since

pulsatility

of

coronary

inflow

is

closely

related

to

changes

in

coronary

volume

When

myocardial

compliance

Cmyo

c

is

set

to

twice

or

half

its

reference

value

NFA-values

change

quantitatively

but

the

phasic

nature

of

coronary

flow

at

low

left

ventricular

pressure

remains

both

in

isovolumic

and

isobaric

beats

-LRB-

Fig.

bottom

panel

-RRB-

Again

the

effect

is

larger

in

isovolumic

beats

than

in

isobaric

beats

Discussion

Model

Setup

The

aim

of

this

study

was

to

design

a

model

with

a

limited

number

of

parameters

for

investigation

of

the

primary

relations

between

left

ventricular

pressure

and

volume

wall

stress

in

fiber

and

transverse

direction

intramyocardial

pressure

and

the

coronary

blood

flow

Central

to

the

model

are

Eq

-LRB-

-RRB-

which

shows

how

fiber

stress

that

ultimately

drives

the

cardiac

cycle

is

converted

into

both

LV

pressure

and

radial

wall

stress

and

Eq

-LRB-

-RRB-

that

describes

how

LV

pressure

and

radial

wall

stress

contribute

to

intramyocardial

pressure

Finally

it

is

the

variation

of

intramyocardial

pressure

that

causes

a

variation

in

arterial

coronary

inflow

through

a

change

in

coronary

volume

represented

by

intramyocardial

compliance

Since

the

use

of

a

simple

model

is

not

without

danger

we

will

address

the

impact

of

the

main

model

simplifications

In

the

model

of

ventricular

wall

mechanics

spatial

homogeneity

of

fiber

stress

and

strain

was

assumed

The

plausibility

of

this

assumption

has

been

illustrated

in

finite

element

models

of

ventricular

mechanics

The

assumption

that

in

plane

stress

is

dominated

by

fiber

stress

-LSB-

Eq

-LRB-

-RRB-

-RSB-

is

realistic

during

systole

when

myofibers

are

active

and

the

main

flow

impediment

occurs

Obviously

radial

wall

stress

is

spatially

inhomogeneous

since

the

ventricle

is

thick

walled

Yet

we

introduced

a

representative

stress

computed

at

a

representative

radial

position

enclosing

one

third

of

the

wall

volume

in

evaluating

the

integral

of

σm

r

over

the

wall

thickness

-LSB-

Eq

-LRB-

-RRB-

-RSB-

In

principle

since

the

transmural

variation

of

radial

strain

is

known

once

LV

cavity

volume

is

known

using

the

constitutive

Eq

-LRB-

-RRB-

the

integral

in

Eq

-LRB-

-RRB-

can

be

determined

exactly

Due

to

the

strong

increase

of

radial

wall

strain

from

the

outside

to

the

inside

surface

and

the

nonlinearity

of

the

constitutive

law

its

value

would

be

dominated

by

the

inner

layers

However

in

those

layers

the

assumption

of

a

solid

wall

with

a

smooth

endocardial

surface

is

in

error

the

endocardial

surface

shows

many

invaginations

relieving

any

radial

stress

that

would

build

up

in

the

subendocardial

tissue

The

choice

to

use

representative

positions

strains

and

stresses

is

compatible

with

the

aim

of

the

study

to

investigate

determinants

of

the

coronary

flow

signal

Obviously

the

model

can

not

describe

spatial

differences

in

coronary

perfusion

which

are

clinically

relevant

in

relation

to

vulnerability

to

ischemia

In

view

of

the

use

of

a

representative

intramyocardial

pressure

and

the

use

of

average

values

for

the

coronary

compliances

and

resistances

the

flow

in

the

model

should

be

regarded

as

a

mean

flow

over

the

myocardial

wall

In

the

constitutive

model

for

the

cardiac

tissue

the

description

of

the

active

stress

component

is

chosen

as

simple

as

possible

while

maintaining

the

basic

dependence

of

active

stress

on

time

sarcomere

length

and

sarcomere

shortening

velocity

Within

the

limitations

of

the

model

parameter

settings

are

chosen

to

mimic

experimental

data

The

model

could

be

extended

to

incorporate

the

sigmoidal

relation

between

stress

and

sarcomere

length

or

the

increase

of

twitch

duration

with

increasing

sarcomere

length

However

we

prefer

the

current

version

with

a

limited

number

of

parameters

in

view

of

the

sensitivity

analysis

The

model

for

passive

tissue

stress

is

simplified

since

we

do

not

model

a

complete

three-dimensional

state

of

stress

Yet

the

behavior

shown

in

Fig.

agrees

well

with

experimental

data

We

approximate

extravascular

pressure

by

intramyocardial

pressure

thus

neglecting

the

contribution

of

local

tissue

stress

In

fact

this

assumption

is

made

in

many

other

models

including

the

waterfall

model

the

intramyocardial

pump

model

and

the

models

by

Huyghe

et

al.

and

Beyar

et

al.

In

addition

in

our

model

intramyocardial

pressure

is

determined

completely

by

the

model

of

LV

wall

mechanics

Thus

we

can

not

replicate

the

observation

that

intramyocardial

pressure

depends

on

perfusion

pressure

We

feel

this

simplification

is

allowed

in

our

simulations

of

isovolumic

and

isobaric

beats

in

which

we

assumed

a

constant

perfusion

pressure

In

the

coronary

circulation

model

resistances

are

constant

This

is

an

approximation

to

the

real

situation

in

which

vessel

diameter

and

therefore

vessel

resistance

depend

on

vessel

transmural

pressure

The

nature

of

the

approximation

is

two-fold

First

we

neglect

the

chronic

change

in

resistance

due

to

a

chronic

change

in

perfusion

pressure

This

is

not

critical

for

the

value

of

NFA

as

is

apparent

from

our

sensitivity

analysis

where

we

applied

a

two-fold

change

of

coronary

resistance

-LRB-

Fig.

-RRB-

Second

we

neglect

cyclic

changes

in

resistance

during

the

cardiac

cycle

which

seems

incompatible

with

the

ability

in

the

model

to

temporarily

store

blood

in

the

coronary

capacitance

Yet

the

phasic

nature

of

coronary

flow

is

reproduced

in

the

model

We

explain

this

by

noting

that

a

change

in

vessel

diameter

during

the

cardiac

cycle

is

associated

with

change

in

coronary

volume

that

affects

coronary

inflow

irrespective

of

the

location

in

the

coronary

tree

where

the

volume

change

takes

place

However

the

associated

change

in

resistance

is

important

in

the

smallest

vessels

only

due

to

the

nonlinear

relation

between

vessel

diameter

and

resistance

Obviously

the

effect

of

changes

in

coronary

resistance

during

the

cycle

may

show

up

in

aspects

of

the

coronary

flow

signal

that

are

not

captured

by

the

normalized

flow

amplitude

NFA

The

simplifications

in

the

model

may

be

seen

as

limitations

if

one

has

the

goal

to

explain

all

experimental

observations

available

In

this

study

we

consider

it

a

strength

of

the

model

in

view

of

our

aim

to

investigate

primary

interactions

and

dependence

on

parameter

settings

Results

In

contrast

to

the

situation

in

all

other

organs

coronary

inflow

of

the

heart

muscle

occurs

mainly

in

diastole

and

is

significantly

impeded

in

systole

Experimentally

the

amplitude

of

the

coronary

inflow

signal

was

found

to

be

only

weakly

coupled

to

systolic

LV

pressure

in

isovolumic

beats

at

various

LV

volumes

This

behavior

is

illustrated

by

the

three

fits

to

experimental

data

sets

shown

in

top

left

panel

of

Fig.

As

in

the

experiment

in

our

model

pulsatility

of

coronary

inflow

is

maintained

at

low

left

ventricular

pressure

although

dependence

of

NFA

on

pressure

is

slightly

higher

than

in

the

experiment

The

mechanism

by

which

pulsatility

of

coronary

inflow

is

maintained

in

isovolumic

beats

is

explained

as

follows

-LRB-

Fig.

left

panel

-RRB-

Total

LV

pressure

is

the

sum

of

diastolic

pressure

in

the

passive

ventricle

and

the

extra

pressure

generated

by

muscle

contraction

Diastolic

pressure

decreases

with

decreasing

volume

and

becomes

negative

as

volume

decreases

below

the

resting

volume

of

ml

We

note

that

in

the

experiments

by

Krams

et

al.

negative

diastolic

pressure

was

induced

as

well

by

applying

suction

to

the

balloon

inserted

in

the

LV

cavity

Since

LV

volume

is

constant

during

a

beat

radial

wall

stress

and

its

contribution

to

LV

pressure

are

constant

as

well

The

extra

pressure

related

to

muscle

contraction

reduces

with

decreasing

volume

but

at

the

lowest

volume

of

ml

it

still

is

about

kPa

and

maximum

total

LV

pressure

is

still

about

kPa

The

fact

that

the

variation

in

LV

pressure

which

determines

pulsatility

of

coronary

flow

decreases

more

slowly

with

decreasing

volume

than

maximum

LV

pressure

is

reflected

in

the

value

of

NFA

shown

in

Fig.

The

relation

between

maximum

LV

pressure

and

minimum

coronary

inflow

shown

in

the

bottom

left

panel

of

Fig.

qualitatively

corresponds

with

experimental

data

by

Kouwenhoven

et

al.

and

Pagliaro

et

al.

In

the

latter

study

minimum

coronary

inflow

was

found

to

decrease

with

increasing

maximum

LV

pressure

for

pressures

above

kPa

but

was

virtually

independent

of

systolic

LV

pressure

below

kPa

It

was

suggested

that

at

low

pressures

the

shielding

effect

of

the

contracting

ventricle

prevents

ventricular

pressure

from

being

transmitted

in

the

myocardial

wall

In

our

model

this

shielding

effect

is

identified

as

radial

wall

stress

Experimentally

NFA

in

isobaric

beats

at

low

LV

pressure

was

found

to

be

about

equal

to

NFA

in

isovolumic

beats

This

finding

is

replicated

in

our

model

-LRB-

Fig.

right

panel

-RRB-

In

contrast

to

the

isovolumic

case

the

relation

between

maximum

pressure

and

NFA

is

non

monotonous

with

distinct

behavior

at

high

and

low

pressures

At

high

pressures

LV

volume

remains

mostly

above

the

passive

resting

volume

of

ml

and

intramyocardial

pressure

is

dominated

by

LV

pressure

At

low

pressures

the

LV

ejects

to

far

below

the

passive

resting

volume

of

ml

and

the

decreasing

contribution

of

LV

pressure

to

intramyocardial

pressure

is

compensated

by

the

increasing

contribution

of

radial

wall

stress

It

is

to

be

noted

that

in

the

isobaric

beats

both

contributions

to

intramyocardial

pressure

radial

wall

stress

and

left

ventricular

pressure

vary

in

time

while

in

the

isovolumic

beats

radial

wall

stress

is

constant

and

cyclic

flow

impediment

is

due

to

the

varying

contribution

of

left

ventricular

pressure

only

-LRB-

Fig.

-RRB-

This

explains

the

two

peaks

in

the

intramyocardial

pressure-signal

in

Fig.

the

first

one

is

related

to

the

rise

in

left

ventricular

pressure

while

the

second

one

arises

from

the

rise

in

radial

wall

stress

related

to

the

decrease

in

left

ventricular

volume

Another

experimental

observation

is

that

NFA

is

about

proportionally

related

to

LV

contractility

expressed

by

maximum

elastance

NFA

being

about

zero

when

contractility

was

about

zero

This

observation

is

replicated

by

our

model

in

which

NFA

changes

about

proportionally

with

the

contractility

parameter

c

irrespective

of

the

volume

setting

-LRB-

Fig.

-RRB-

For

low

contractilities

minimum

coronary

inflow

approaches

the

theoretical

value

of

ml

s

g

of

tissue

obtained

from

the

coronary

perfusion

pressure

of

kPa

the

total

coronary

resistance

of

Pa

s

m

and

a

wall

volume

of

g.

Finally

it

has

been

observed

that

intramyocardial

pressure

may

exceed

left

ventricular

pressure

in

low

afterload

beats

This

is

also

the

case

in

our

model

-LRB-

Fig.

right

panel

-RRB-

and

explained

again

from

the

contribution

of

radial

wall

stress

at

the

low

ventricular

volumes

associated

with

the

low

LV

ejection

pressures

Sensitivity

Analysis

Sensitivity

of

NFA

to

the

radial

stiffness

parameter

in

isovolumic

beats

is

low

-LRB-

Fig.

top

left

-RRB-

This

can

be

best

understood

from

Eq

-LRB-

-RRB-

which

shows

that

only

the

constant

level

of

radial

wall

stress

is

affected

by

changing

while

the

variation

of

left

ventricular

pressure

and

thus

of

intramyocardial

pressure

and

arterial

inflow

is

unaffected

The

increase

in

left

ventricular

pressure

with

decreasing

stiffness

is

explained

from

Eq

-LRB-

-RRB-

which

shows

that

fiber

stress

is

converted

into

both

LV

pressure

and

radial

wall

stress

With

decreasing

radial

stiffness

an

increasing

part

of

fiber

stress

is

converted

into

LV

pressure

In

the

extreme

case

of

zero

radial

stiffness

minimum

left

ventricular

pressure

is

about

kPa

which

is

higher

than

the

pressure

measured

in

the

experiment

-LRB-

Fig.

-RRB-

Variation

of

NFA

with

radial

stiffness

is

more

prominent

in

isobaric

beats

at

low

LV

pressure

-LRB-

Fig.

top

right

-RRB-

An

increase

of

radial

stiffness

causes

a

more

rapid

increase

of

radial

wall

stress

with

decreasing

volume

below

the

equilibrium

volume

Thus

the

pressure

at

which

the

contribution

of

radial

wall

stress

to

intramyocardial

pressure

takes

over

from

the

contribution

of

ventricular

pressure

is

increased

-LRB-

-RRB-

and

pulsatility

of

coronary

inflow

is

maintained

At

zero

radial

stiffness

intramyocardial

pressure

is

determined

fully

by

left

ventricular

pressure

explaining

loss

of

pulsatility

of

coronary

flow

with

decreasing

ejection

pressure

in

disagreement

with

experimental

findings

Sensitivity

of

NFA

to

settings

of

the

active

stress

model

-LRB-

Fig.

-RRB-

is

low

as

far

as

the

velocity

and

time

dependence

is

concerned

NFA

was

more

affected

by

increasing

the

stress-free

sarcomere

length

from

to

μm

In

fact

this

variation

has

the

same

effect

as

decreasing

contractility

at

low

sarcomere

lengths

Thus

in

this

case

the

decrease

of

NFA

is

explained

from

the

decrease

in

variation

of

left

ventricular

pressure

which

is

reflected

in

a

decreased

variation

of

intramyocardial

pressure

The

increase

of

NFA

with

increasing

myocardial

compliance

-LRB-

Fig.

-RRB-

both

in

isovolumic

and

isobaric

beats

is

explained

from

the

fact

that

at

the

same

variation

of

intramyocardial

pressure

the

volume

change

of

the

myocardial

bed

increases

Hence

the

pulsatility

of

arterial

inflow

is

increased

as

well

A

decrease

of

myocardial

compliance

has

the

opposite

effect

The

first

order

effect

of

a

variation

of

coronary

resistance

is

a

change

in

both

the

average

and

pulsatile

component

of

coronary

flow

as

was

illustrated

already

in

Fig.

In

this

approximation

the

relative

changes

in

average

and

pulsatile

flow

component

are

equal

causing

NFA

to

remain

unaffected

However

since

coronary

compliance

is

kept

constant

the

relative

contribution

of

flow

related

to

volume

change

of

the

coronary

bed

becomes

less

when

decreasing

coronary

resistance

causing

the

decrease

of

NFA

observed

in

Fig.

Increasing

coronary

resistance

has

the

opposite

effect

Relation

with

Other

Models

In

comparison

to

the

waterfall

and

the

intramyocardial

pump

model

our

model

is

more

advanced

in

its

description

of

intramyocardial

pressure

adding

the

effect

of

radial

wall

stress

Radial

wall

stress

becomes

important

only

below

the

zero-pressure

volume

of

the

passive

ventricle

Thus

the

early

models

still

are

applicable

in

normal

physiological

conditions

in

which

ventricular

volume

is

higher

than

this

zero-pressure

volume

during

the

major

part

of

the

cardiac

cycle

In

comparison

to

the

models

by

Beyar

et

al.

our

model

is

simpler

emphasizing

primary

interactions

at

the

cost

of

information

on

transmural

variation

of

coronary

flow

and

long-term

exchange

of

fluid

between

the

coronary

vessels

and

the

interstitium

However

the

reduced

complexity

yields

increased

insight

into

primary

determinants

of

coronary

flow

impediment

Main

limitation

of

our

model

with

respect

to

the

model

presented

by

Vis

et

al.

lies

in

neglecting

the

direct

contribution

of

tissue

stress

to

the

extravascular

pressure

and

effective

compliance

of

the

coronary

vessel

The

approach

of

Vis

et

al.

requires

a

detailed

analysis

of

the

equilibrium

between

the

pressure

in

a

cavity

in

the

LV

wall

and

the

stress

in

the

surrounding

tissue

However

such

an

analysis

would

compromise

the

simplicity

that

we

aim

for

in

this

study

Conclusion

In

conclusion

a

mathematical

model

of

the

interaction

between

coronary

flow

and

cardiac

mechanics

is

presented

with

a

limited

number

of

model

parameters

The

model

replicates

the

experimental

observations

that

the

phasic

character

of

coronary

inflow

is

virtually

independent

of

maximum

left

ventricular

pressure

that

the

amplitude

of

the

coronary

flow

signal

depends

linearly

on

cardiac

contractility

and

that

intramyocardial

pressure

in

the

left

ventricular

wall

may

exceed

left

ventricular

pressure

The

normalized

amplitude

of

coronary

inflow

is

mainly

determined

by

contractility

reflected

in

dependence

of

active

fiber

stress

on

sarcomere

length

and

maintained

at

low

ventricular

volumes

by

radial

wall

stress

The

sensitivity

of

the

NFA

to

myocardial

coronary

compliance

and

resistance

and

to

the

relation

between

active

fiber

stress

time

and

sarcomere

shortening

velocity

is

low

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O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O B I I O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O

ventricular mechanics
radial wall stress
intramyocardial pump

empty beating heart
lumped parameter model