245
Special Article
Calculation of Left Ventricular Wall Stress
David M. Regen
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Chamber-stress equations relate wall stresses to pressure and wall dimensions. Such equations
play a central role in the analysis and understanding of heart-chamber function. Over the past
three decades, several stress equations giving radically different results have been derived,
used, and/or espoused. They can be classified into two categories, according to the definition of
stress underlying the equation. The stresses in one class of equations are total forces per unit
normal area, excluding ambient pressure but including pressure in the wall exerted by more
external elements of the wall. The stresses in the other class of equations are fiber-pulling forces
per unit normal area, that is, total forces per unit normal area excluding all pressure. The
validity of stress equations can be tested at least three ways: 1) Do they predict that the pressure
inside a small chamber nested in a larger chamber would be the sum of transmural pressures
of the two chambers? 2) Do they satisfy the expectation from Laplace's law that a sphere with
a given circular stress and thickness/radius ratio would exert twice the pressure of a cylinder
with the same circular stress and thickness/radius ratio? 3) Do they predict that the ratio of
principle stresses depends on chamber shape but not on wall/cavity ratio, with the circular/
longitudinal stress ratio of a cylinder being 2 and that of a prolate spheroid being between 1 and
2? Stress equations of the first class fail all of these tests by large margins, whereas those of the
second class pass all of these tests exactly. When selecting a stress equation, one should
consider these distinctions as they might affect 1) evaluation of stress afterload and stress
preload, 2) evaluation of contractility, 3) understanding the roles of contractility and wall/cavity
ratio as determinants of pressure-making ability, 4) understanding the role of fiber orientation
as a determinant of chamber shape, and 5) translations between pressure-volume relations and
stress-length relations. (Circulation Research 1990;67:245-252)
C hamber function depends on dimensions and
mechanical properties of the chamber wall.
Goals of the heart-mechanics discipline
include the following: understanding this dependence, accounting for normal and abnormal function
in terms of chamber dimensions and wall properties,
understanding the biological determinants of chamber dimensions and wall properties, and understanding the biological significance of normal and abnormal chamber function.
Mechanical properties are characteristics of relations between stresses and either normalized dimensions or their rates of change. This simple sentence
becomes quite complicated when applied to thickwalled biological chambers. How should one define
and calculate stress, what dimensions best express
extension or distension of the wall, changes of what
dimensions best express systolic performance, what is
the best reference state for normalizing dimensions
and defining characteristics, and what features of the
From the Department of Molecular Physiology and Biophysics,
Vanderbilt University Medical School, Nashville, Tenn.
Address for correspondence: David M. Regen, Department of
Molecular Physiology and Biophysics, Vanderbilt University Medical School, Nashville, TN 37232-0615.
Received May 11, 1989; accepted March 30, 1990.
stress-dimension relation best characterize the relation? My efforts to examine these issues have brought
me to the following positions: 1) The germane stresses
of a heart-chamber wall should be defined as fiberpulling force per unit normal area1 2 and should not be
defined as total force per unit area (as in classical
mechanics) or total force per unit area excluding
ambient pressure (as in cardiology). 2) Wall extension
or distension should be defined as normalized dimensions of a specific midwall element,3-6 not as normalized cavity dimensions. 3) Systolic performance should
be defined as fractional displacement of that midwall
element,4'6 not fractional cavity-surface displacement.
4) A definable state of positive strain, not the unloaded
state, should be the reference state for normalizing
dimensions and defining wall properties.2'4-7 5) Elastic
properties are amplitude, spread, and shape of the
isometric relation between stress and extension3-5;
stiffness is not a valid or informative characteristic of
contracting muscle.
My positions on the aforementioned items 1-4
have been held by a few authorities for more than
two decades, and my position on item 5 is not without
precedence: 1) The stress of Badeer's equation8 and
that of McHale and Greenfield's equation9 is fiber-
246
Circulation Research Vol 67, No 2, August 1990
pulling force per unit area, as is Mirsky and Rankin's
"incremental stress."10 2) Glantz and Kernoffl1 and
Weber et al'2 expressed wall stretch in terms of midwall
dimensions. 3) It is not uncommon for investigators to
measure midwall displacements and their rates.13 4)
Spann et al14 and Lakatta and Jewell15 normalized
muscle lengths to optimal length. 5) Amplitude and
spread of the stress-extension relation are analogous to
the parameters (PO, Vma,) with which Sonnenblick'6
characterized the velocity-afterload relation. My current article contains a candid discussion of chamberstress equations (item 1 in the previous paragraph).
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Stresses of Classical Elastic Theory
In classical mechanics, stress is defined as the total
force in a given direction per unit area normal to that
direction.17 Stress is positive for a pulling force and
negative for a pushing force. Thus, a rod with 1-mm2
cross-sectional area, being stretched longitudinally
with a force of 2,000 dynes in outer space, would have
a longitudinal stress of 2,000 dynes/mm2 and a radial
stress of 0 dynes/mm2, but that same rod under the
same stretch force at sea level (atmospheric pressure
of 10,133 dynes/mm2) would have a longitudinal
stress of -8,133 dynes/mm2 and a radial stress of
-10,133 dynes/mm2, with pressure being a negative
component of stress in all directions. The rod's
longitudinal stress as defined in classical mechanics
bears no relation to the rod's stretch or recoil action
and is uninformative in itself. In this classical system,
the force-intensity response to deformation is represented in the difference between orthogonal stresses,
in this case longitudinal stress minus radial stress:
-8,133-(-10,133)=2,000 dynes/mm2. The above
convention of defining stress is one of several
adopted by mechanical engineers. It may be the most
systematic convention for dealing with materials
undergoing significant volume changes in response to
their external forces.
Class I Chamber-Stress Equations
A common convention is to exclude ambient pressure as a component of stress, in which case the
stretched rod mentioned above has virtually the same
stresses at sea level and in outer space. This is the
convention under which several chamber-stress equations were derived. Most such equations are essentially the same as the equations of Falsetti et al18 for
average meridional and hoop stresses at the equator
of a prolate spheroid:
(1a)
P(C,0(2h/bc +h bc)
P= oQ0(2h/bc+h2/ac)/(2-bc2/a~c)
(1b)
where P is transmural pressure, ocfi is average meridional stress at the equator, oco is average hoop stress
at the equator, h is equatorial thickness, bc is cavity
minor semiaxis, and ac is cavity major semiaxis.
Similar equations in the literature are akin to the
Falsetti equations,12 slightly erroneous forms of the
Falsetti equations,19 analogues of the Falsetti equations
with possibly better representation of chamber
shape,20-22 or erroneous analogues of the Falsetti
equations.'0,23 The following pair of equations was
attributed to Mirsky and Rankin's 1979 treatment'0 by
Yin24:
P = 2o-Ck,(h / bm)/(1 -h / 2bm)2
(2a)
P=cuc(h / bm)/(1 -b2/2a2-h/2bm+h2/8a2) (2b)
where bm is midwall minor semiaxis and am is midwall
major semiaxis. The last term of Equation 2b differs
from that presented by Mirsky and Rankin.10 Equations la and 2a are identities, as will be seen in the
tables.
Recently stress equations resembling Equation 2b in
form and symbolism have been attributed to Mirsky.20
One of them is a correct analogue.22 Another is
erroneous,23 and I cannot tell whether the large error
was in the equation as used or occurred in publication.
Recently Mirsky and collaborators25 presented an
equation for hoop stress averaged over the whole
meridional section, which was derived with the assumption that apical thickness is the same as equatorial
thickness. As applied to a sphere or cylinder, it is
identical to Equation lb, so I will pair it with Equation
la (a similarly derived equation for meridional stress
averaged over the equatorial section).
P = oC+5(2h / bC + h / bc)
(3a)
P= ouco,(h / bc+h/ac+h2/ bcac)
(3b)
The subscript v means averaged over a cross section.
By definition, or0 and uo, are total forces per unit
area in the respective directions excluding that due to
ambient pressure. Specifically, pressure in the wall
exerted by more external fibers (alternatively, transmitted from the cavity) is included as a negative
component of uo and o,. This mural pressure is no
more relevant to fiber action than was atmospheric
pressure to the rod described above. Thus, the
stresses of the commonly used chamber-stress equations are less than fiber-pulling force per unit area to
the extent that more external fibers exert pressure.
The discrepancy between uo- or uo, and fiberpulling force per unit area increases with wall thickness, so there is no consistent relation between o,,0 or
uo, and fiber-pulling force per unit area. As distension
decreases, wall/cavity ratio increases, so the discrepancy
increases. Therefore, the relation between oc- or 'co
and any expression of stretch or distension is not an
expression of mechanical properties. This can be
proven mathematically with simple chamber models.
For example, one can predict the relation between
pressure and cavity volume (or diameter) of a thickwalled sphere with any stress-stretch relation,2,7,26,27
then calculate oco (same as or6 in the sphere) by
Equation 1, 2, or 3, and show that the relation between
oc, and any expression of stretch is not the stressstretch relation of the wall material. Such a complicated
proof might not be very convincing. The following
paragraphs present three simple, direct, and intuitive
Regen Left Ventricular Stress
247
TABLE 1. Transmural Pressures Exerted by Spheres Having Unity Wall Stress (1 dyne/cm2) According to
Various Formulations
Falsetti
et al18:
Equation 1
Mirsky and
Rankin'0:
Equation 2
(ao=1)
3
3
Pressure (dynes/cm')
Mirsky
et al25:
Regenl:
Equation 3
Equation 4
(OjC=1)
(up =1)
3
1.386
3
1.386
Regenl:
Equation 5
Sphere dimensions (cm)
(o-c=1)
(op,=1)
1.386
3
Small sphere: rc=1, h=1
1.386
3
Large sphere: rc=2, h=2
Nested small and large:
15
15
15
2.773
2.773
rc=1, h=3
h,
thickness.
fiber
stress;
cavity
radius;
rc,
ac, average stress at equator; o-,, stress averaged over a cross section; cup,
In a sphere, bc=ac=rc, bm=am=rm, acr=oco=o7c, and apj=opp==op, where bc is cavity minor semiaxis, a, is cavity major
semiaxis, bm is midwall minor semiaxis, am is midwall major semiaxis, rm is midwall radius, rco is average hoop stress at
the equator, aco is average meridional stress at the equator, ojpe is apparent average hoop fiber stress at the equator,
and opb is apparent average meridional fiber stress at the equator. For Equation 2, rm=rc+h/2. For Equation 4,
rm=(ro-rc)/(ln r0-ln rc), where r. is the outer radius.
tests by which one can prove that Equations 1-3 do not
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express
rational physical relations.
Validity Tests for Stress Equations
The first test is the nested-chamber test. According
to Equations 1, 2, or 3, the pressure exerted by a
sphere of o, = 1 dyne/cm2, cavity radius (re) = 1 cm, and
h=1 cm would be 3 dynes/cm2, and the pressure
exerted by a sphere of o, = 1 dyne/cm2, rc = 2 cm, and
h=2 cm would also be 3 dynes/cm2. These two spheres
exert the same pressure because they have the same
stress and the same wall/cavity ratio (Table 1). The
pressure exerted by a chamber is its transmural pressure. Transmural pressures of nested chambers are
additive. That is, pressure outside the inner chamber
will exceed ambient pressure by the outer chamber's
transmural pressure, and pressure inside the inner
chamber will exceed that pressure by the inner chamber's transmural pressure. Thus, if the above two
spheres in the same state of stretch were nested, cavity
pressure of the inner sphere would exceed ambient
pressure by 6 dynes/cm2. The nested spheres together
would have an r, of 1 cm and an h of 3 cm. According
to Equations 1, 2, or 3, a sphere of uc= 1 dyne/cm2,
rc=1 cm, and h=3 cm would exert a pressure of 15
dynes/cm2 (Table 1), in defiance of common sense and
basic energetics.
This same test can be applied to cylinders. In a
cylinder, only circular stress (oo) exerts pressure
(Laplace's law, see below); longitudinal stress (o,p)
balances the longitudinal force of pressure, but does
not exert pressure. The b,/ac, bm/am, h/ac, and h/am
ratios of a cylinder are zero, because a cylinder is a
prolate spheroid of infinite long axis. With a given
circular stress (ac,c=l dyne/cm2), a cylinder of r,=2
cm and h=2 cm exerts the same pressure (1 dyne/
cm2) as does a cylinder of rc=1 cm and h=1 cm
(Table 2). Nested, these cylinders in this state of
stretch should exert the sum of the pressures they
would exert in isolation, that is, 2 dynes/cm2. The
nested cylinders would have an r, of 1 cm and an h of
3 cm. According to Equation lb, 2b, or 3b, a cylinder
of these dimensions and the same stress would exert
a pressure of 3 dynes/cm2 (Table 2), again defying
common sense.
The second test requires one to consider Laplace's
law: P=T /r1+TJr2, where T1 and T2 are principal
tensions (force per unit normal circumferential length)
and r1 and r2 are the corresponding principal radii of
curvature. Tension is the product of stress and wall
thickness. All we need to see in this relation is that a
sphere, with two equal orthogonal curvatures, should
exert twice the pressure of a cylinder with only one
curvature if the sphere and cylinder have the same
TABLE 2. Transmural Pressures Exerted by Cylinders Having Unity Circular Stress (1 dyne/cm2) According to
Various Formulations
Pressure (dynes/cm')
Falsetti
Mirsky
Mirsky and
et al25:
Rankin10:
et all8:
Regenl:
Regenl:
Equation 5
Equation 3
Equation 4
Equation 2
Equation 1
(ap0=l)
Cylinder dimensions (cm)
(aco,=1)
(a-C=1)
(aoU.=1)
(o-po=1)
0.693
0.693
1
1
1
Small cylinder: rc=1, h=1
0.693
1
0.693
1
1
Large cylinder: rc=2, h=2
Nested small and large:
1.386
3
1.386
3
3
rc=1, h=3
stress
over
circular
averaged
longitudinal section; opg, apparent average circular
crco, average circular stress; o-,r,
fiber stress; rc, cavity radius; h, thickness. In a cylinder, bc=rc, bm=rm, and am=ac= x, where bc is cavity minor semiaxis,
bm is midwall minor semiaxis, rm is midwall radius, am is midwall major semiaxis, and ak is cavity major semiaxis. For
Equation 2, rm=rc+h/2. For Equation 4, rm=(ro-rc)/(ln r0-ln rJ), where r. is the outer radius.
248
Circulation Research Vol 67, No 2, August 1990
TABLE 3. Ratios of Circular Stress to Longitudinal Stress in Cylinders of Various Wall/Cavity Ratios
Falsetti
Mirsky and
Mirsky
et al'8:
Rankin'0:
et al25:
Regend:
Equation 1
Equation 2
Equation 3
Equation 4
Cylinder dimensions (cm)
(
(c-Iovc/)
)
(Q/O.c5)
(O'eI/opo)
Very thin: rc=1, h=0.01
2.01
2.01
2.01
2
Thin: r,=1, h=0.1
2.1
2.1
2.1
2
Thick: r,=1, h=1
3
3
3
2
Thick: rc=2, h=2
3
3
3
2
Nested thick walls: r =1, h=3
5
5
5
2
ace, average circular stress; om,g,, average longitudinal stress; o, 0v, circular stress averaged over longitudinal section;
oco, longitudinal stress averaged over cross section; 0ap9, apparent average circular fiber stress; orp, apparent average
longitudinal fiber stress; rc, cavity radius; h, thickness. In a cylinder, bc=rc, bm=rm, and am=ac= o, where bc is cavity
minor semiaxis, bm is midwall minor semiaxis, rm is midwall radius, am is midwall major semiaxis, and ac is cavity major
semiaxis. For Equation 2, rm=rc+h/2. For Equation 4, rm=(ro-rc)/(ln r0-ln rc), where r. is outer radius.
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circular stresses and thickness/radius ratios. Comparing
Tables 1 and 2, we see that Equations 1-3 all say that
a sphere will exert more than twice the pressure that a
cylinder exerts with the same circular stress and thick-
ness/radius ratio, in defiance of Laplace's law.
The third test is to calculate the oO/Ota, ratios of
nonspherical chambers of various wall/cavity ratios.
It is well known that the circular/longitudinal tension
ratio of a cylinder is 2. According to Equation 1, 2, or
3, if r,= 1 cm and h=0.01 cm, then the ocl/co- ratio is
2.01, already a slight error (Table 3). If r,=1 cm and
h= 1 cm, then the oc/aol0 ratio is 3. Since every
element of the thicker cylinder is cylindrical, the
stress ratio of the thicker cylinder should not differ
from that of the thinner cylinder. If rc=2 cm and h=2
cm, Equation 1, 2, or 3 again yields the aolo/rc, ratio
of 3. Nesting the two thick-walled cylinders in their
states of stretch would not change the tension ratio.
The nested cylinders would have an rc of 1 cm and an
h of 3 cm. According to Equation 1, 2, or 3, the
cro/ac,o ratio of such a cylinder would be 5 (Table 3),
in defiance of common sense.
This same test can be applied to a prolate spheroid
of any length/diameter (L/D) ratio and h/b ratio.
According to Equation 1 or 2, if the L/D ratio is 2,
then the or/olom¢4 ratio of a thin-walled prolate spheroid is 1.76 (Table 4). With each increase of thickness,
the oC/oJc4,> ratio increases, exceeding 2 if the wall is
quite thick. Since the thicker prolate spheroids can
be shaped like the thinner prolate spheroids, there is
no reason for the equatorial/meridional tension ratio
to increase with thickness. Moreover, the average
equatorial/meridional tension ratio of a prolate
spheroid must be between 1 and 2 and is never 2.33,
2.80, 3.25, or 3,000. According to a numerically
similar analogue of Equation 1,19 the aola/oc-5 ratio of
the normal human left ventricle is 2.6-2.7.28,29
In a prolate spheroid of "uniform thickness,"
global average hoop stress (Equation 3b) is less than
equatorial stress (Equation lb or 2b). As seen in
Table 4, the ratio of global average hoop stress to
meridional stress (according to Equation 3) increases
with thickness. In a chamber with a prolate-spheroidal
cavity and uniform thickness, the outer elements and
the average element are more spherical (less prolate)
than the cavity, the more so if the wall is thicker.
Therefore, the average hoop/meridional tension ratio
TABLE 4. Ratios of Hoop Stress to Meridional Stress in Prolate Spheroids of Various Wall/Cavity Ratios
Falsetti et a118:
Equation 1
(o'CO/-o if
Mirsky and
Rankin10:
Equation 2
Mirsky
et a125:
Equation 3
(
if
Regen':
Equation 4
(ap/Oap, if
Lm/Dm=2)
Prolate-spheroid
(oro/o-,ac if
dimensions (cm)
Lm/Dm=2)
Lc/Dc=2)
L[/Dc=2)
...
1.76
bc=1, h=0.01
1.33
bm=l, h=0.01
*
1.76
*
1.75
1.81
bc=l, h=0.1
1.35
...
...
bm=l, h=0.1
1.83
1.75
...
2.33
bc=1, h=1
1.50
bm=l, h=1
3.25
*
1.75
2.80
bc=1, h=2
*
1.60
bm= 1, h= 1.999
*
3000
*
1.75
oa, average hoop stress at equator; ao-p average meridional stress at equator; Lc, cavity long axis, Dc, cavity diameter;
L,., midwall long axis; D., midwall diameter; ac0v, hoop stress averaged over meridional section; crc,, meridional stress
averaged over equatorial section; a-p,, apparent average hoop fiber stress at equator; opo, apparent average meridional
fiber stress at equator; bc, cavity minor semiaxis; h, equatorial thickness; bi, midwall minor semiaxis. The semiaxis
length ratio is the same as the axis length ratio: Lc/D,=ac/bc and L,/Dm=am/bm, where a, is cavity major semiaxis and
am is midwall major semiaxis.
...
...
...
...
...
Regen Left Ventricular Stress
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declines toward 1 as wall/cavity ratio increases, opposite the result from Equation 3.
Class I equations fail these tests because their
stresses (o,) are defined as pulling-action stresses
minus mural pressure, pulling-action stress being the
response to circumferential stretch, which is the
potential for shortening circumferentially and for
exerting pressure according to Laplace's law, and
mural pressure being irrelevant with respect to these
relations and increasing relative to pulling-action
stress as wall/cavity ratio increases. Since wall/cavity
ratio is inversely dependent on distension (wall/cavity
ratio decreasing with filling and increasing with ejection) and since the wall/cavity ratio at any point in the
cardiac cycle varies from chamber to chamber and
from heart to heart: class I equations do not give valid
expressions of stress preload, stress afterload, or stressdeveloping ability (contractility), they do not account
properly for the roles of contractility and wall/cavity
ratio as determinants of pressure-making ability (peak
isovolumic pressure at reference distension), they cannot be used to account for chamber shape in terms of
fiber orientations, and they cannot be used to predict
chamber hydraulic properties from fiber mechanical
properties or to estimate fiber mechanical properties
from pressure-volume or pressure-diameter relations.
Class II Chamber-Stress Equations
For more than 2 decades, a few investigators have
espoused chamber-stress equations in which the
stresses were defined as fiber-pulling force per unit
normal area. This pulling-action stress, fiber stress,
is the difference between circumferential stress and
radial stress. The fiber-stress analogues of Equations 1 and 2 are as follows':
(4a)
P=2Opj,h/bm
(4b)
P=2crp,(h/bm)/(2-bml/a2)
where opo is apparent average meridional fiber stress
at the equator and apo is apparent average hoop fiber
stress at the equator. We see that Equation 4 is
simpler than Equation 1, 2, or 3. Given the fact that
fibers near the cavity have more hydraulic advantage
than do those near the outer surface and the fact that
stresses in these two locations may differ, apparent
average fiber stresses calculated by Equation 4 are
approximate average fiber stresses, and they are
approximate but accurate midwall fiber stresses.5
Moreover, if bm and am are defined not in terms of the
halfway point between surfaces but as logarithmic
means of cavity and outer semiaxes,1'2 the equations
can be almost exact as average-fiber-stress or as
midwall-fiber-stress equations. The logarithmic am is
(a.-a )/(ln a.-ln aj) and the logarithmic bm is (b0b,)/(ln b.-ln bc), where a. and b. are outer semiaxes.
Fiber-stress equations have a long history but have
not gained wide acceptance. In 1963, Badeer8 espoused
the meridional-fiber-stress equation (Equation 4a) as a
rational expression of the relation between tension and
pressure; and in 1966, Ross et aP30 used that equation in
249
an analysis of left ventricular function. In 1973, McHale
and Greenfield9 demonstrated that an equation numerically similar to Equation 4b expressed the relation
between circumferential tension and pressure better
than did Equation lb. Unfortunately, their equation
was derived from the Sandler and Dodge version19 of
Equation lb, and it retained a minor mathematical
error in the Sandler and Dodge equation, which had
been pointed out by Falsetti et al18 in 1970. Had they
derived their equation from Falsetti's hoop-stress equation, the result would have been Equation 4b.
One possible reason that Equation 4 has not gained
wide acceptance is the fact that Mirsky20 derived class I
equations in 1969. In 1969 and 1974, he presented
Equation 4 (Laplace's law with midwall geometry) but
recommended against this formulation because it did
not agree with his 1969 formulation.2021 He stated
explicitly that the disagreement was due to errors in
Equation 4 and that Equation 4 should not be used. In
1981, Yin24 published an influential review of stress
equations in which he compared Equations 1 and 2 but
did not consider Equation 4 in any version and did not
address the issue of stress definition. However, by 1976,
Mirsky et aP26 recognized that the stresses of Equations
1 and 2 contain pressure as a negative component and
do not express the recoil response to stretch. Mirsky et
alP' and Mirsky and Rankin10 proposed calculating the
stresses that represent fiber action by subtracting midwall radial stress from the stresses of Equation 2, which
is the same as adding midwall pressure or average wall
pressure (about half of cavity pressure) to the stresses
of Equation 2. They correctly pointed out that the
resulting "incremental stress" expresses fiber action
and that its dependence on stretch expresses myocardial elastic properties. They did not derive or present
an equation directly relating cavity pressure and incremental stress (Equation 4), and Mirsky et a125 still
espouse the procedure of calculating the kind of stress
in Equations 1-3 (with wall pressure as a negative
component) and then adding average wall pressure
(half of cavity pressure) to get incremental stress, which
they now call "stress difference." Incremental stress or
stress difference is the same as fiber stress, which is
given directly, simply, and accurately by Equation 4.
Incremental stress calculated from the stress of Equation lb, 2b, or 3b does not agree exactly with the fiber
stress of Equation 4b, but the incremental stress calculated from Wisenbaugh's version22 of Equation 2b
(lacking the h2/8al term) agrees with the fiber stress of
Equation 4b exactly.
Apparent Average Fiber Stress
In 1982, Arts et a131 analyzed a cylindrical model
with fiber orientations like those of the left ventricle.
They presented an equation (based on energetic
considerations) relating pressure to the sum of
orthogonal fiber stresses (opo,+ apo) and volume
dimensions. The study made several useful contributions, among them the fiber-stress concept and the
fact that pressure depends on wall/cavity volume
ratio and volume-averaged fiber stress. This depen-
250
Circulation Research Vol 67, No 2, August 1990
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dence is shape indifferent1l32 and is expressed most
exactly as follows:
(5a)
P=(2/3)up (ln Vo-ln Vj)
where V, is cavity volume, VO is chamber volume
(cavity+wall), and ap is average of orthogonal fiber
stresses, (op,+ o-(p)/2. This equation can also be
written2:
P= (2/3) opVw/Vm,,
(Sb)
where Vw is wall volume and Vmu is volume enclosed
by a midwall element. This "midwall volume" is the
logarithmic mean of cavity volume and outer volume,
(Vo-V )/(ln VO-ln Vj). In Equation Sb, one can see
the analogy with Equation 4. Equation 5 is an exact
average-fiber-stress equation at one distension, a
distension where stresses near the cavity are comparable with stresses near the outer surface. At other
distensions (where stresses near the inner and outer
surfaces differ substantially), it is an approximate
average-fiber-stress equation and an accurate averagemidwall-fiber-stress equation. There is now an averagemidwall-fiber-stress equation,5 which is more accurate
over a wider range of distensions (Equation 7 below).
Fiber-Stress Equations Tested
We can now examine Equations 4 and 5 by the
commonsense tests. First is the nested-chamber test. A
sphere of r,=1 cm and h=1 cm would have a logarithmic rm of 1.4427 cm. With a orp of 1 dyne/cm2, it
would exert a pressure of 1.3863 dynes/cm2 according to
Equation 4 (Table 1). A sphere of rc=2 cm and h=2 cm
would have a logarithmic rm of 2.8854 cm. With a op of
1 dynes/cm2, it would exert a pressure of 1.3863 dynes/
cm2 according to Equation 4. These spheres when
nested would exert the sum of their pressures, or 2.7726
dynes/cm2. Overall dimensions of the nested spheres
would be rc= 1 cm and h=3 cm, so logarithmic rm would
be 2.164 cm. With these dimensions and a cp of 1
dyne/cm2, Equation 4 predicts a pressure of 2.7726
dynes/cm2, which is exactly the sum of pressures exerted
by the small and large spheres in isolation. In this test,
Equation 5 is also exact (Table 1). For a sphere,
Equation 5 can take the following form: P=2orp ln
(r0/r,), where r. is the outer radius. This is an identity of
Equation 4 with logarithmic rm.
Since Equation 4 passes the nested-sphere test
exactly, it will pass the nested-cylinder test exactly,
because all factors of Equation 4 are the same for a
sphere and cylinder except the denominator of Equation 4b, which for a sphere is 1 and for a cylinder is
2. As seen in Table 2, Equation 4 with logarithmic rm
passes the nested-cylinder test exactly. In a cylinder,
the -p/o-po ratio is 2; so o-po is 4/3 of the average of
orthogonal stresses (uJp) in Equation 5, and Equation 5
can take the following form: P=uop, ln (r0/rc). Again,
this is an identity of Equation 4b with logarithmic rm,
and it passes the nested-cylinder test exactly (Table 2).
Comparing Tables 1 and 2, we see that Equations 4
and 5 pass the second test exactly, because both say
that a sphere exerts exactly twice the pressure exerted
by a cylinder of the same circular stress and thickness/
radius ratio, in agreement with Laplace's law.
We can see by inspection of Equation 4 that it
passes the third test exactly; the Oap/ap, ratio of a
cylinder is 2 regardless of thickness (Table 3), as
expected intuitively. According to Equation 4, the
o./Jpo,p ratio of a prolate spheroid is between 1 and 2
and depends on the bm/am ratio, not on wall/cavity
ratio (Table 4). The op//op¢, ratio of the normal
human left ventricle is about 1.7; its shape is determined in part by its fiber orientations.33
Thus, Equations 4 and 5 pass all three commonsense tests exactly. This is because the stresses in these
equations are defined as pulling force per unit area. In
their derivations, pressure in the wall was treated as a
contributor to the distending force borne by wall
stress, not as a negative component of wall stress.
Midwall Dimensions and Stresses
Recently there have been some refinements in the
calculation of midwall dimensions and midwall fiber
stresses. I have shown that there is a midwall element
or isobar whose reference dimensions best express
chamber size, whose reference-normalized dimensions
best express extension or distension, and whose fractional displacements best express systolic performance.4-6,32 Its fractional displacements from reference dimensions depend according to wall properties
on fractional changes of isovolumic pressure from its
value at reference distension. The theoretical reference
distension is the one where inner and outer elements
are comparably stretched,'-7 but this distension is not
readily identified. For characterizing muscle dynamics,
a good reference distension is the one that is optimal
for stress development,14,15 but this reference distension may be inaccessible in vivo or hard to identify. For
characterizing pump function a good reference distension is the average basal end-diastolic distension to
which the chamber is accustomed.34,6,7 Owing to myocardial remodeling,34-36 sarcomere lengths at average
end-diastolic distension tend to be "normal."7 This is
obviously an accessible reference state, but it remains
to be seen whether it can be identified. In fact, inner
and outer elements are comparably stretched at the
average end-diastolic distension.731,37 Volume enclosed
by the midwall element can be easily calculated by the
following formulas4-6:
Vmu=VcuVou(ln Vou-ln Vcj/(V0u-Vcj (6a)
Vm=Vmu +Vc -VCU
(6b)
where Vcu, Vou, and Vmu are cavity, chamber, and
midwall volumes at reference distension and V, and
Vm are cavity and midwall volumes at any distension.
Chamber hydraulic properties are characteristics of
the relation between transmural pressure and midwall volume, which depend consistently on characteristics of the stress-length relation of the wall
material.4-6 The average of orthogonal fiber stresses
in the midwall element (Cm) is calculated very accurately by the following formula5:
Regen Left Ventricular Stress
Oxm=(3/2)(Vm/Vmu)P/(ln VOU.-In V,,)
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(7)
This average-midwall fiber-stress formula is exact for
a homogenous sphere that has a straight pressurevolume relation and is uniformly stretched at reference distension.5 In this simple model, the midwall
fiber-stress equation takes into account the stress
gradients at all distensions. It is very accurate for
chambers of any shape and fiber orientation and for
curved pressure-volume relations.4'32 It is numerically
similar to the apparent-average-fiber-stress equation
(Equation 5) at all distensions and identical to it at
reference distension (where Vm=Vmu). Average wall
properties are seen directly in relations between am
and (Vm/Vmu)/3, the latter being apparent average
midwall extension.
One can also calculate apparent axes of the midwall element (am, bm) and apparent principal fiber
stresses of the midwall element (ot.,k, arn) based on a
model of isobar shapes in a prolate spheroid.1,3'7
Elastic properties should be reflected in the relation
between equatorial fiber stress (ap- or Umr) and equatorial extension (bm normalized to its reference value,
bmu). Elastic properties should also be reflected in the
relation between meridional fiber stress (apo, or amj)
and meridional extension, but meridional extension is
not easily estimated.
Concluding Remarks
This essay demonstrates some of the paradoxes
that result from defining wall stress as a mixture of
pulling-force intensity and pressure transmitted from
the cavity (Equations 1-3). Pulling-force intensity is
the response to deformation; it is the potential for
approaching an undeformed shape. Pressure is the
response to compaction; it is the potential for occupying uncompacted volume. Though these two
potentials have the same dimensions (force per unit
area), they are entirely different quantities.
These two quantities are mixed in the stresses of
classical elastic theory, because those equations are
designed to account for the relations between a
body's external forces and departures of its external
dimensions from unstrained dimensions, in terms of
material properties, internal forces, and internal
dimension changes - recognizing that dimension
changes might be mixtures of deformation and compaction. It is possible to treat this kind of problem
with compaction and deformation separated, in
terms of deviatoric stresses and strains, but deviatoric
stresses are not the pulling-action stresses that
account for pressure according to Laplace's law.
However, when dealing with materials that are not
significantly compacted or rarified by their deforming
forces, a simpler treatment of elastic behavior is
possible, with negative radial stress of elastic theory
being called pressure and the difference between radial
stress and its orthogonal stresses being called fiber
stresses (the shape-changing potentials). This treatment provides a simple and enlightening description of
equilibrium states regardless of material compactabil-
251
ity. That is, one can account for external forces in terms
of dimensions and fiber-stress field without concern for
the deformations and elastic properties that gave rise to
them. Biological chambers qualify for this treatment on
both counts. First, their walls are not significantly
compacted or rarified by their deformating forces, so
one can describe viscoelastic behavior in terms of
fiber stresses as affected by deformations and deformation rates. Second, with biological chambers, one
is concerned with forces at prevailing distensions, not
with the path of deformation from an unstrained
state that the wall never experienced. The forces are
in fact exerted by fibers deposited with the chambers
distended (like the rubber band in a golf ball).
As seen in this essay, chamber-fiber-stress equations exhibit no paradoxes in the commonsense validity tests. This is because fiber stress is a valid expression
of pulling action, and it is this pulling action that exerts
pressure (Laplace's law). I have shown elsewhere5 that
the relation between apparent average fiber stress
(Equation 5) or apparent average midwall fiber stress
(Equation 7) and normalized midwall volume (Equation 6) expresses wall properties faithfully in the
range of dimensions found in the left ventricle.4'5,32
The relations between circumferential fiber stresses
(Equation 4) and corresponding midwall extensions
would also express wall properties faithfully.
I espoused Equation 5 in 1983 as an energetically
sound expression of the relation between wall dimensions and pressure,2 and I espoused Equations 4 and
5 in 1984 for the same reasons.' However, there must
have been a time before 1969 when Equation 4 was
common knowledge, because Mirsky20'21 referred to
Equation 4 without citation and without derivation as
"Laplace's law with midwall geometry," and he
stated that Equation 4a was "often observed in the
literature on heart mechanics." Perhaps the latter
was a reference to Badeer8 and Ross et al.30
As is evident from this essay, I consider the mixing
of pressure and pulling-action stress into a single
quantity to be a conceptual error resulting in erroneous accounting for the roles that wall dimensions and
wall properties play as determinants of hydrodynamic
characteristics. In the cases of Sandler and Dodge19
and of Falsetti et al,18 the mixing occurred simply
because pressure was assumed to act only on the cavity
silhouette, so all forces in the wall were attributed to
wall stress. In the cases of Mirsky20'21 and possibly
Streeter et al,38 it appears that fiber stress was conceived and formulated, and then wall pressure transmitted from the cavity was subtracted from the fiber
stress to obtain the mixed quantities. This was presumably because the stresses of classical infinitesimal-displacement theory are total forces per unit
area. However, the common practice of omitting
ambient pressure but including internal pressure as a
component of stress is not a feature of infinitesimaldisplacement theory. In that theory, it is recognized
that pulling-action stress is not expressed in the total
stresses but in the differences among orthogonal total
stresses. For a chamber, pulling-action stress is cir-
252
Circulation Research Vol 67, No 2, August 1990
cumferential stress minus radial stress, that is, fiber
stress. Fiber stress is the intensity of force tending to
shorten circumferences, the intensity of force exerting pressure, and the intensity of force corresponding
to deformation state.
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KEY WORDS * chamber theory * wall stress * heart mechanics
Calculation of left ventricular wall stress.
D M Regen
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Circ Res. 1990;67:245-252
doi: 10.1161/01.RES.67.2.245
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