Download Modeling the relation between cardiac pump function and myofiber

Journal of Biomechanics 36 (2003) 731–736
Modeling the relation between cardiac pump function and
myofiber mechanics
T. Artsa,b,*, P. Bovendeerdb, T. Delhaasc, F. Prinzenc
a
Faculty of Biomedical Engineering, Eindhoven University of Technology, Eindhoven, The Netherlands
b
Departments of Biophysics, Maastricht University, Maastricht 6200 MD, The Netherlands
c
Department of physiology, Maastricht University, Maastricht 6200 MD, The Netherlands
Accepted 25 November 2002
Abstract
Complexity of the geometry and structure of the heart hampers easy modeling of cardiac mechanics. The modeling can however
be simplified considerably when using the hypothesis that in the normal heart myofiber structure and geometry adapt, until load is
evenly distributed. A simple and realistic relationship is found between the hemodynamic variables cavity pressure and volume, and
myofiber load parameters stress and strain. The most important geometric parameter in the latter relation is the ratio of cavity
volume to wall volume, while actual geometry appears practically irrelevant. Applying the found relationship, a realistic maximum is
set to left ventricular pressure after chronic pressure load. Pressures exceeding this level are likely to cause decompensation and heart
failure. Furthermore, model is presented to simulate left and right ventricular pump function with left–right interaction.
r 2003 Elsevier Science Ltd. All rights reserved.
1. Introduction
The heart consists of four cavities. For each cavity,
the work generated by the myofibers in the wall is
converted to pump work. Wall mass is about proportional with generated contractile work. Pump work is
closely related to the product of stroke volume and
generated systolic pressure. Consequently, the left
ventricle is the cavity with the largest wall mass,
followed by the right ventricle, the left atrium and the
right atrium, respectively.
Pump function of a cavity can be characterized by the
time-variant relation between cavity pressure pcav and
cavity volume Vcav during the cardiac cycle. Plotting
pressure as a function of volume, a pV-diagram is
obtained with the shape of a loop. The area of the loop
represents stroke work. The shape and position of the
loop in the pV-diagram is characteristic for the quality
of pump function. According to the variable elastance
model (Sagawa, 1978) pressure may be modeled as a
*Corresponding author. Department of Biophysics, Maastricht
University, PO Box 616, 6200 MD Maastricht, The Netherlands,
Tel.: +31-43-3881656.
E-mail address: [email protected] (T. Arts).
time-variant compliance. During activation, the cavity
stiffens considerably, forcing the cavity to a low volume.
In a closer look, active cavity pressure also appeared to
depend on the time derivative of cavity volume (Suga
and Yamakoshi, 1977), implying an inherent timedependent viscous behavior.
Pump work is generated by the summed action of
all separate cardiomyocytes, arranged in a specific
structure of myofiber orientations (Streeter, 1979). Local
myofiber work is quantified by stress and strain as a
function of time. In experiments on isolated myofibers,
under a variety of circumstances myofiber force has
been described as a function of sarcomere length,
velocity of sarcomere shortening and time (De Tombe
and Ter Keurs, 1990; Guccione et al., 1997). In the
literature, stress, force and cavity pressure are closely
related subjects. Similarly, myofiber strain is closely
related to changes in sarcomere length and cavity
volume.
Myofiber stress sf is a function of sarcomere length ls
and time. This stress is a summation of a passive elastic
component (spas ) and an active component (sact ). The
active component is modeled by a contractile element
with length lce in series with a series elastic element. The
velocity of contractile element shortening dlce/dt is a
0021-9290/03/$ - see front matter r 2003 Elsevier Science Ltd. All rights reserved.
doi:10.1016/S0021-9290(02)00451-7
732
T. Arts et al. / Journal of Biomechanics 36 (2003) 731–736
function of time (t) and active stress (sact ) (Hill, 1939).
The following model may be used:
2. Three-dimensional models of cardiac mechanics
sf ¼ spas ðls Þ þ sact ðls ; lce ; tÞ
dlce
¼ functionðsact ðls ; lce ; tÞ; tÞ:
dt
To investigate the relation between left ventricular
pump function and local, time-variant myofiber mechanics, a finite element model of left ventricular
mechanics was developed (Bovendeerd et al., 1992).
The reference shape of the wall was taken to be
rotationally symmetric, having a realistic wall thickness.
The angulation of myofiber orientation with the
circumferential direction was taken from Streeter
(Streeter, 1979). When simulating normal cardiac
mechanics, the distribution of systolic myofiber stress
appeared very sensitive to the chosen distribution of
myofiber orientation. Within the range of accuracy of
the myofiber orientation measurements, systolic myofiber stress could vary locally by a factor of two
(Bovendeerd et al., 1992).
In designing a realistic model, the needed accuracy for
local myofiber orientation appeared much higher than
measurements could provide. The transmural gradient
in myofiber orientation is about 201/mm in humans. So
an error in the location by 0.25 mm causes already a
significant error in the stress calculation. Spatial
inaccuracies due to biological variance are likely to be
much higher. Lack of accuracy forced us to give up
using in vivo measurements on myofiber orientation for
the purpose of stress calculation. In stead we used the
hypothesis that in the normal heart, myofiber orientation adapts locally so that myofiber stress and strain are
about uniform. This hypothesis was investigated in a
model, using a rotationally symmetric ventricle. Local
wall thickness and local myofiber orientation were
optimized to reach a minimum value for the standard
deviation of strain within the wall (Rijcken et al., 1999).
The thus found distribution of myofiber orientation
inside the wall was not significantly different from
reported measurements. Thus it was concluded that the
use of calculated myofiber orientation should be
preferred over measured values. Interestingly, stress
was transmitted predominantly along the direction of
maximum stiffness. In systole, and also in diastole be it
to a lesser degree, this direction practically coincided
with the myofiber direction. We therefore designed the
one-fiber model by simplifying the complex threedimensional material behavior of the tissue to a model
with uniform uniaxial stress.
ð1Þ
Quantification of the relation between pump variables
like pressure and volume to myofiber variables like
stress and strain has been subject of discussion for a long
time. For a given systolic pressure and stroke volume it
has been shown that metabolism is not uniquely
determined. At constant stroke work, oxygen consumption increases with end-diastolic volume (Suga et al.,
1987). Apparently, the starting conditions of a contraction cycle are important to mechanics and metabolism.
To further elucidate the relation between myofiber
mechanics and pump function, numerical models of
cardiac mechanics have been designed (Vetter and
McCulloch, 2000; Bovendeerd et al., 1996). Currently
for many applications, sufficient information is available
to establish this relation. Myocardial material has to be
considered anisotropic (Arts et al., 1979; Hunter et al.,
1998). Passive myocardial material properties are known
from biaxial loading experiments (Humphrey et al.,
1990) on the excised septum. Active myofiber mechanics
is known from experiments on isolated myofibers (De
Tombe and Ter Keurs, 1990; Guccione et al., 1997;
Hunter et al., 1998). Cardiac geometry can be measured
directly by for instance magnetic resonance imaging
(MRI) or post mortem anatomical measurements.
Regional myofiber orientation and collagen lamina
organization may be obtained from anatomical measurements (Streeter, 1979; LeGrice et al., 1995; Costa
et al., 1999; Arts et al., 2001; Bovendeerd et al., 1999).
In the present study, we have investigated what
properties are crucial in relating pump function to
myofiber dynamics, and what properties may be less
relevant under what circumstances. Efficiency in modeling requires customizing the model design. Recent
models of cardiac mechanics most often use finite
element analysis (Vetter and McCulloch, 2000; Bovendeerd et al., 1996). The strong point of these models is
the ability to deal with non-uniformities in material
properties within the heart, occurring for instance in the
paced or in the infarcted heart. From these models, in
the normal heart properties as well as load appeared to
be quite uniform, especially during systole. In the
present study a tremendous simplification was applied
by leaving out spatial variations in modeling the relation
between tissue mechanics and pump function in the
normal heart. The resulting one-fiber model of cardiac
mechanics relates cavity pressure and volume to
myofiber stress and strain. Finally, applications of this
model are shown, concerning 1) adaptation possibilities
and limitations to pressure load, and 2) left and right
ventricular mechanics with their mutual interaction.
3. The one-fiber model of cardiac mechanics
Having myofiber orientation optimized for reaching
uniform stress and strain in the wall, myofiber dynamics
in the normal heart simplifies considerably. In absence
of spatial differences, mean values are good estimates of
local shortening and stress. Following earlier ideas on
the cardiac myofiber structure (Torrent-Guasp, 1959,
T. Arts et al. / Journal of Biomechanics 36 (2003) 731–736
Fig. 1. The wall of a myocardial cavity may be considered as a single
long thin fiber, wrapped around the cavity. Suppose the fiber starts at
midwall equator. After following the circular circumferential pathway,
the fiber end just slightly displaces to a toroidal layer, enclosing the
starting pathway. Following a helical pathway in the toroidal layer,
this layer can be filled completely. Similarly a next toroidal layer is
filled, until the outer toroidal layer, encompassing the epicardium and
endocardium, is filled completely. Thus a whole ventricular wall,
having a realistic distribution of fiber orientations, has been filled with
one single fiber.
Streeter, 1979, p. 856) we consider the cavity to be
wrapped in one single long fiber, filling the whole wall
(Fig. 1), comparable to a rope wrapped on a core,
representing the cavity.
Since mechanical load is transmitted predominantly
along the myofibers, the myocardial tissue was described
by a fluid fiber model, in which total Cauchy stress stotal
is composed of a hydrostatic pressure pwall and fiber
stress sf :
stotal ¼ pwall I þ sf ef ef ;
ð2Þ
where ef represents fiber direction. Assuming the
myofibers to be parallel with the wall surface (ef;r ¼ 0),
using Eq. (2) for myofiber stress sf it is found:
sf ¼ scc þ szz 2srr :
ð3Þ
The indices c, z and r refer to the circumferential, base to
apex and transmural direction. The variables scc ; szz and
srr are components of the stress tensor stotal : In many
model studies of left ventricular wall mechanics, stress in
the wall is estimated as a function of cavity pressure and
various geometric parameters. In performing simulations with many of these models, often having various
wall geometries, fiber structures, and material properties, the following general relation between myofiber
stress sf and cavity pressure pcav has been found having
an accuracy as good as 75% (Arts et al., 1991):
sf
Vcav
E1 þ 3
:
ð4Þ
pcav
Vwall
Eq. (4) shows that for a given pressure wall stress
increases with cavity volume, which is consistent with
common models on wall stress. In relating cavity
733
pressure to myofiber stress, the ratio of cavity volume
to wall volume appears by far the most important
geometric parameter. The actual shape of the wall
appeared of minor importance (Arts et al., 1991). In an
extended trial, the definition of sf according to Eq. (3)
was also applied to models having isotropic wall
material. Unexpectedly, under these conditions, Eq. (4)
appeared to be valid too. So, using Eq. (3) as a definition
of stress, the validity range of the one-fiber model can be
extended to the passive, more isotropic wall material
properties as well.
Having this simple relationship of Eq. (4), the very
robust physical law of conservation of work was applied
to derive a relation for myofiber strain ef : Using that
incremental pump work is equal to incremental mechanical work generated by the myofibers, it follows:
def
pcav
¼
:
pcav dVcav ¼ Vwall sf def dVcav =Vwall
sf
ð5Þ
Note that sf def represents incremental mechanical
myofiber work per unit of tissue volume. Substituting
Eq. (4) into Eq. (5), followed by integration with respect
to normalized cavity volume Vcav /Vwall ; the following
expression was found for the change Def in fiber strain
when cavity volume changes from Vcav1 to Vcav2 :
1
1 þ 3Vcav2 =Vwall
Def ¼ ln
:
ð6Þ
3
1 þ 3Vcav1 =Vwall
Note that myofiber strain ef is a natural strain, linked to
sarcomere length li by:
ls
ef ¼ ln
:
ð7Þ
ls;ref
Parameter ls;ref refers to a reference sarcomere length.
This length becomes irrelevant when considering strain
between two states like it occurs in Eq. (6). Inversion of
Eq. (6) provides expressions for cavity volume Vcav as a
function of myofiber strain ef :
Vcav
1
Vcav;ref 3ðef ef;ref Þ
1þ3
¼
1 :
ð8Þ
e
Vwall 3
Vwall
Eq. (8) requires knowledge of Vwall and Vcav;ref at some
reference state, which may be used as the zero reference
of ef :
In Fig. 2 at known wall volume, for a given time
course of cavity volume, myofiber strain was calculated
using Eq. (6). For a given time course of left ventricular
pressure, using the cavity volume data, myofiber stress
was calculated using Eq. (4). Myofiber stress was thus
determined as a function of myofiber strain by using
measurements that can be performed non-invasively in
the clinic.
During ejection the level of left ventricular pressure is
relatively constant, causing the pressure volume loop to
be about symmetric. In contrast, at the beginning of
ejection, fiber stress is considerably higher than at the
734
T. Arts et al. / Journal of Biomechanics 36 (2003) 731–736
about the stress-strain relation of myofibers are
obtained only in experiments on isolated myofibers.
4. Adaptation of cavity and wall volume to load
The circulation requires a given amount of stroke
volume Vstroke at a given amount of stroke work Estroke :
We define ejection pressure peject by:
Estroke
peject ¼
:
ð9Þ
Vstroke
Fig. 2. Simulation of left ventricular cavity mechanics during a cardiac
cycle. The left panel shows the myofiber parameters fiber strain ef and
myofiber stress sf ; and the pump parameters cavity volume Vlv and
cavity pressure plv ; as a function of time. Cavity volume is normalized
to wall volume Vwall : In the right panel the high narrow loop shows
myofiber stress as a function of myofiber strain. The area of the sf 2Def
loop represents myofiber stroke work per unit of tissue volume.
Similarly, the wide low curve represents the plv 2Vlv loop. The area of
the loop represents pump stroke work, normalized to wall volume.
Note: Both loops have the same physical dimension along their axes
and the same area, showing that pump work is equal to the mechanical
work as generated by the tissue.
end of ejection, causing the stress–strain loop to be
clearly asymmetric.
The presented model provides an instantaneous
relation between cardiac pump function parameters
and myofiber function parameters. Despite its simplicity, realistic time dependent myofiber stress results in
realistic time courses of left ventricular pressure (Fig. 2).
For instance an isometric myofiber stress of 120 kPa in
an end-diastolic left ventricle with a cavity to wall
volume ratio of 0.6 results in an isovolumic pressure of
43 kPa (=300 mmHg), which is a very realistic value for
a healthy, intact left ventricle (Arts et al., 1982). A
thorough test of the model under many circumstances
was considered beyond the scope of this article.
Eqs. (4)–(8) form the backbone of the one-fiber
model. In models it is often convenient to express
pressure as a function of volume, applying the
constitutive behavior of the myofibers. Then sarcomere
length ls can be calculated from volume by Eqs. (8) and
(7). Stress sf is calculated from ls through the
constitutive equations of the myocardial material.
Finally, Eq. (4) is used to calculate cavity pressure. In
another application, shown in relation to Fig. (2), from
in vivo measured time courses of left ventricular volume
and pressure, the stress–strain relation of the myocardial
tissue can be estimated as a function of time during the
cardiac cycle (Schreuder et al., 2000). Thus in patients
the constitutive behavior of the myofiber may be
obtained from non-invasive or moderately invasive
hemodynamic measurements. Currently, information
The variable peject represents afterload. Normally,
healthy cardiac tissue adapts mass and myofiber
direction such that regular mechanical load reaches a
given, biologically determined level. Normal load levels
of myocardial tissue may be characterized by mechanical stroke work per unit of tissue volume wstroke and
myofiber strain Def during ejection. Wall volume after
adaptation is proportional to stroke work
Estroke peject Vstroke
Vwall ¼
¼
:
ð10Þ
wstroke
wstroke
Using Eq. (6), substituting for cavity volumes Vcav1 and
Vcav2 the volumes at the beginning (Vbe ) and end of
ejection (Vee ), while applying Vbe ¼ Vee þ Vstroke and
Eq. (10), it is found:
Vstroke
Vwall
1
peject
¼ Vstroke 3De
Vee ¼ 3De
:
e f 1 3wstroke
e f 1
3
ð11Þ
According to Eqs. (11) after complete adaptation to
load, ventricular cavity and wall volume are determined
by the tissue properties of adaptation, wstroke and Def ;
and required afterload peject and stroke volume Vstroke :
We found the following tissue parameter values to be
reasonable: wstroke E6000 Jm3 and Def E0:16: Note that
the physical dimension of wstroke is the same as that of
pressure, i.e. [Pa]=[N m2]=[J m3].
4.1. Maximum pressure load
Interestingly, in adapting geometry to pressure load
using Eq. (11), an upper limit is set to systolic pressure
peject ; because end-systolic cavity volume has to be
positive. The latter condition is equivalent to: ejection
fraction o100%. Using Eqs. (11) it is found:
3wstroke
peject opeject;max ¼ 3De
:
ð12Þ
e f 1
For the abovementioned normal values of wstroke and
Def ; maximum pressure with full adaptive compensation
appears peject;max ¼ 29:2 kPa (E218 mmHg). Normally,
left ventricular pressure during ejection is readily below
this upper limit. However, the pressure safety margin
may decrease considerably by: (1) increase of afterload,
caused by aortic stenosis or by stiffening of the blood
T. Arts et al. / Journal of Biomechanics 36 (2003) 731–736
735
Table 1
Predicted wall and cavity volumes
Cavity
a
Left ventricle
Right ventriclea
Outer wall
Inner wall
Left ventricleb
Right ventricleb
Peject (kPa)
Vstroke (ml)
Vwall (ml)
Vcav;be (ml)
Vcav;ee (ml)
15
3.75
3.75
11.25
15
3.75
80
80
160
80
80
80
200
50
100
150
143
193
386
160
147
117
63
113
226
80
67
37
Stroke work per tissue volume: wstroke ¼ 6000 N m2 :
Systolic myofiber shortening: Def ¼ 0:16:
a
Separate ventricular walls.
b
Overlapping ventricular walls, overlap volume: Voverlap ¼ 40 ml:
vessels due to aging, or (2) deterioration of myocardial
tissue performance, to be expressed by a decreased value
of wstroke : Qualitatively as well as quantitatively, the
latter conditions are in remarkable agreement with
known clinical risks of heart failure. Limitation of
afterload by reducing load pressure and/or limiting
systolic myofiber shortening by for instance negative
inotropic drugs, vasodilators or b-blockade may also
help to increase the pressure safety margin. The
maximum pressure set by Eq. (12) is likely to be an
overestimation, because maximum ejection fraction is
always less than the assumed 100%.
4.2. Left and right ventricular wall volume and cavity size
Knowing pressure load peject during ejection and
stroke volume Vstroke ; wall volume and begin- and endsystolic cavity volumes can be predicted by using
Eqs. (11). For the left and right ventricle, stroke volume
is about 80 ml for both, and systolic pressure load is
about 15 and 3.75 kPa, respectively. In the upper two
lines of Table 1, the thus calculated values for wall
volume, and cavity volume at the beginning and end of
ejection are shown.
In modeling the left and right ventricle one may
consider a different set up (Fig. 3, middle panel) (Arts
and Reneman, 1988). A common outer wall, having a
transmural pressure equal to right ventricular pressure,
and a stroke volume of the two cavities together,
encapsulates the right and left ventricular cavities. The
inner wall encapsulates the left ventricular cavity only,
having a transmural pressure equal to the difference
between left and right ventricular pressure, and a stroke
volume of one cavity only. Using Eqs. (10)–(11) inner
and outer wall volumes and cavity volumes are
calculated (Table 1). In this composite, even at the end
of ejection, the inner side of the outer shell is slightly
wider than the outer side of the inner shell. To free some
extra space for the right cavity, the shells are assembled
slightly asymmetrically (Fig. 3, right panel). As a result,
the outer and inner wall partially overlap inside the free
wall of the left ventricle. The left ventricular free wall
appears somewhat thicker than the septum. From a
realistic Fig. 3, the overlap volume is estimated to be
about a fraction of 0.4 of the outer wall. The left
ventricular cavity is estimated to be smaller by about
one third of the overlap volume due to the increase of
free wall thickness. The thus found composite appears
very realistic.
Representing the heart (Fig. 3) by an outer wall
(index o), encapsulating the related cavity, the inner wall
and cavity (index i), inner and outer cavity volumes,
Vcav;i and Vcav;o ; are determined from left and right
ventricular cavity volumes, Vlv and Vrv ; by:
1
Vcav;i ¼ Vlv Voverlap ;
3
Vcav;o ¼ Vlv þ Vrv þ ðVwall;i Voverlap Þ:
Fig. 3. Composing left (LV) and right (RV) ventricle to a heart in endsystole. Left panel: Separate left and right ventricular walls. Middle
panel: In a more realistic configuration an outer wall (o) encapsulates
an inner (i) wall. Right panel: The inner wall encapsulates the LV only.
Overlap of the outer wall with the inner wall enlarges the RV cavity.
The crosshatched area represents the extra volume of the LV free wall
due to this overlap.
ð13Þ
The volume Voverlap is considered a constant, representing the overlap of the inner wall and outer wall in the
composite of left and right ventricle. Eq. (10) is used to
estimate outer and inner wall volume from the related
ejection pressures and stroke volumes. Next, Eq. (6) is
used to express inner and outer shell myofiber strain
(Def;i and Def;o ; respectively) as a function of inner and
outer cavity (Vcav;i and Vcav;o ; respectively) and wall
T. Arts et al. / Journal of Biomechanics 36 (2003) 731–736
736
(Vwall;i and Vwall;o ; respectively) volume:
1
1 þ 3Vcav;i;be =Vwall;i
Def;i ¼ ln
and
3
1 þ 3Vcav;i =Vwall;i
1
1 þ 3Vcav;o;be =Vwall;o
:
Def;o ¼ ln
3
1 þ 3Vcav;o =Vwall;o
ð14Þ
Begin of ejection (be) has been used as strain reference
state. Myofiber stresses are determined from myofiber
strains, using the constitutive equations of myofibers
(Eq. (1)). From these stresses, applying Eq. (4), pressures in the inner and outer cavities are calculated.
Finally, these pressures are converted to left and right
cavity pressures by
plv ¼ pcav;i þ pcav;o
and
prv ¼ pcav;o :
ð15Þ
Thus, a model of the composite of right and left
ventricle is designed, estimating cavity pressures as a
function of cavity volumes, and presenting left right
interaction. For example, imagine the presence of a
pulmonary artery hypertension. Then the elevated
pressure in the right ventricular cavity is added to this
pressure in the left ventricular cavity (Eq. (15)), thus
influencing both filling and ejection.
5. Summary
For the non-uniform heart, a three-dimensional finite
element model is needed to relate the hemodynamic
parameters cavity pressure and volume to the local
myocardial parameters myofiber strain and myofiber stress.
In the normal heart, we use the hypothesis that myofiber
structure and geometry adapt so that the myofiber
shortening and tissue workload are constant. Using this
hypothesis: (1) a realistic relationship is found between
myofiber mechanics and hemodynamic parameters, (2) the
ratio of cavity volume to wall volume is the most important
geometric parameter relating cavity pressure and volume to
myofiber stress and strain, (3) a realistic maximum is set to
chronic systolic pressure load, and (4) left and right
ventricular interaction can be simulated.
References
Arts, T., Reneman, R.S., 1988. The importance of the geometry of the heart
to the pump. In: ter Keurs, H., Noble, M. (Eds.), Starling’s Law of the
Heart Revisited. Kluwer Acad Publishers, Dordrecht, pp. 94–111.
Arts, T., Veenstra, P.C., Reneman, R.S., 1979. A model of the
mechanics of the left ventricle. Annals of Biomedical Engineering 7,
299–318.
Arts, T., Veenstra, P.C., Reneman, R.S., 1982. Epicardial deformation
and left ventricular wall mechanics during ejection in the dog.
American Journal of Physiology 243, H379–H390.
Arts, T., Bovendeerd, P.H.M., Prinzen, F.W., Reneman, R.S., 1991.
Relation between left ventricular cavity pressure and volume and
systolic fiber stress and strain in the wall. Biophyscial Journal 59,
93–103.
Arts, T., Costa, K.D., Covell, J.W., McCulloch, A.D., 2001. Relating
myocardial laminar architecture to shear strain and muscle fiber
orientation. American Journal of Physiology 280, H2222–H2229.
Bovendeerd, P.H.M., Arts, T., Delhaas, T., Huyghe, J.M., Van
Campen, D.H., Reneman, R.S., 1996. Regional wall mechanics in
the ischemic left ventricle: numerical models and dog experiments.
American Journal of Physiology 270, H398–H410.
Bovendeerd, P.H.M., Arts, T., Huyghe, J.M., Van Campen, D.H.,
Reneman, R.S., 1992. Dependence of left ventricular wall
mechanics on myocardial fiber orientation: a model study. Journal
of Biomechanics 25, 1129–1140.
Bovendeerd, P.H.M., Rijcken, J., Van Campen, D.H., Schoofs,
A.J.G., Nicolay, K., Arts, T., 1999. Optimization of left ventricular
muscle fiber orientation. In: Pederson, M.P.B.P. (Ed.), IUTAM
Symposium on Synthesis in Bio Solid Mechanics. Kluwer
Academic Publishers, Netherlands, pp. 285–296.
Costa, K.D., Takayama, Y., McCulloch, A.D., Covell, J.W., 1999.
Laminar fiber architecture and three-dimensional systolic mechanics in canine ventricular myocardium. American Journal of
Physiology. 276, H595–607.
De Tombe, P.P., Ter Keurs, H.E.D.J., 1990. Force and velocity of
sarcomere shortening in trabeculae from rat heart. Circulation
Research 66, 1239–1254.
Guccione, J.M., Le Prell, G.S., de Tombe, P.P., Hunter, W.C., 1997.
Measurements of active myocardial tension under a wide range
of physiological loading conditions. Journal Biomechanics 30,
189–192.
Hill, A.V., 1939. The transformation of energy and mechanical work
of muscles. Proceedings of Physiological Society 51, 1–18.
Humphrey, J.D., Strumpf, R.K., Yin, F.C.P., 1990. Biaxial mechanical
behavior of excised ventricular epicardium. American Journal of
Physiology 259, H101–H108.
Hunter, P.J., McCulloch, A.D., ter Keurs, H.E., 1998. Modelling the
mechanical properties of cardiac muscle. Progression Biophysics
Molecular Biology 69, 289–331.
LeGrice, I.J., Takayama, Y., Covell, J.W., 1995. Transverse shear
along myocardial cleavage planes provides a mechanism for
normal systolic wall thickening. Circulation Research 77, 182–193.
Rijcken, J., Bovendeerd, P.H.M., Schoofs, A.J.G., van Campen, D.H.,
Arts, T., 1999. Optimization of cardiac fiber orientation for
homogeneous fiber strain during ejection. Annals of Biomedical
Engineering 27, 289–297.
Sagawa, K., 1978. The ventricular pressure volume diagram revisited.
Circ Research 43, 677–687.
Schreuder, J.J., Steendijk, P., van der Veen, F.H., Alfieri, O., van der
Nagel, T., Lorusso, R., van Dantzig, J.M., Prenger, K.B., Baan, J.,
Wellens, H.J., Batista, R.J., 2000. Acute and short-term effects of
partial left ventriculectomy in dilated cardiomyopathy: assessment
by pressure–volume loops. Journal of the American College of
Cardiololgy 36, 2104–2114.
Streeter, D.D., 1979. Gross morphology and fiber geometry of the
heart. In: Berne, R.M. (Ed.), The Cardiovascular System, The
Heart. American Physiological Society, Bethesda, Maryland, USA,
pp. 61–112.
Suga, H., Yamakoshi, K.I., 1977. Effect of stroke volume and velocity
of ejection on end-systolic pressure of canine left ventricle.
Circulation Research 40, 445–450.
Suga, H., Yasumara, Y., Nozawa, T., Futaki, S., Igarashi, Y., Goto,
Y., 1987. Prospective prediction of O2 consumption from pressurevolume area in dog hearts. American Journal of Physiology 252,
H1258–H1264.
Torrent-Guasp, F., 1959. An Experimental Approach on Heart
Dynamics. Aguirre Torre, Madrid.
Vetter, F.J., McCulloch, A.D., 2000. Three-dimensional stress and
strain in passive rabbit left ventricle: a model study. Annals of
Biomedical Engineering 28, 781–792.