Download A mathematical model of left ventricular contraction and

A mathematical model of left ventricular contraction
and its application in heart disease
D.H. MacIver
Department of Cardiology, Taunton and Somerset Hospital, UK.
Abstract
Conflicting clinical data have led to a number of theories to explain both normal and abnormal
cardiac function. The pathophysiology of heart failure is controversial although it is known to
be related somehow to reduced cardiac performance. Following myocardial damage, a number
of homeostatic mechanisms attempt to repair the physiological abnormalities by, for example,
ventricular remodelling.
A simple new method for mathematically modelling left ventricular contraction is described
and the potential implications in clinical practice are discussed. The modelling helps in the
understanding of the relationship between left ventricular structure and function in health and
disease. It improves comprehension of how myocardial fibre shortening may be translated into
changes in whole organ physiology and supports the displacement pump hypothesis to describe
left ventricular function.
The model can use either the area-length method or the hemi-cylinder-hemi-ellipsoidal
volumetric shape to calculate both the internal and external left ventricular volumes. The model
allows adjustment of left ventricular function, for example, left ventricular long- and short-axis
shortening, and the effects of left ventricular hypertrophy on end-diastolic and end-systolic
volume, stroke volume, systolic wall thickening and ejection fraction. A new definition of heart
failure is proposed.
1
Introduction
1.1 Left ventricular structure and function
In fluid mechanics there are different types of pump described, for example displacement,
centrifugal and kinetic. Most authors accept that the heart is probably a dynamic positive displacement pump but other theories include the proposal that the heart works as a perpetual
vortex [1].
The heart comprises of three main but continuous layers of muscle bundles [2]. These are
illustrated in Figure 1 (reproduced from Anderson [2]). At the endocardium and epicardium
the fibres run relatively longitudinally from the base of the heart to loop around the apex and
then return towards the base forming a helical structure. Circumferential fibres tend to run in
the mid-wall of the left ventricle. The left ventricle ejects its stroke volume during systole by a
combination of long- and short-axis shortening associated with twisting of the ventricle. In systole, myocardial shortening and thickening causes inward movement of the endocardium and a
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66 R EPAIR AND R EDESIGN OF PHYSIOLOGICAL SYSTEMS
Epicardial longitudinal fibres
Endocardial longitudinal fibres
Mid-wall circumferential fibres
Figure 1: Myocardial fibre architecture.
reduction in left ventricular cavity volume. This results in an increase in pressure and expulsion
of blood into the aorta (the pressure-propulsion hypothesis). During diastole there is myocardial
fibre lengthening, ventricular wall thinning, untwisting and refilling of the ventricle.
It is generally agreed that the total heart volume does not change during the cardiac cycle
and there is alternate emptying and filling of the left ventricle and atrium. A change in the total
external volume in systole would require energy, physical interaction with the surrounding tissues and can be wasteful. There is only a small fall in the outer circumference of the mid-left
ventricle with little change towards the apex [3–6]. As the long-axis of the ventricle shortens
the wall thickens with an inward movement of the endocardium and significant endocardial
short-axis shortening but a lesser degree of mid-wall circumferential shortening. Studies suggest that at rest long-axis shortening contributes to 75–82% of the stroke volume, suggesting
that circumferential shortening is less important [4, 5].
1.2
Current concepts of the pathophysiology of heart failure
Heart failure has been characterised as due to ‘systolic’ dysfunction when the left ventricular ejection
fraction is reduced to 50% or ‘diastolic’ dysfunction when the left ventricular ejection fraction
is 50% [7]. This nomenclature implies that the mechanisms of heart failure are understood
and heterogeneous. ‘Diastolic heart failure’ has been described as ‘non-systolic’ heart failure
or heart failure with preserved (or normal) ejection fraction (HFpEF) and is associated with a
relatively normal left ventricular end-diastolic volume. The term HFpEF is preferable as it does
not assume the pathophysiology is wholly established.
Many individuals with HFpEF have a sub-normal ejection fraction [8]. Yip et al. showed
that the mean left ventricular ejection fraction in HFpEF was 55% whereas in controls it was
68% [9]. Therefore, the term ‘heart failure with a normal ejection fraction’ is unsound. Often
the term ‘heart failure with a normal or preserved function’ is used but this assumes that the
ejection fraction equates to ventricular function. This does not appear to be true as explained
below. A better term for ‘systolic’ heart failure without presumptions regarding the mechanism
is heart failure with reduced ejection fraction (HFrEF). This phenotype typically has significant
left ventricular remodelling or dilation.
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A MATHEMATICAL MODEL OF LEFT VENTRICULAR CONTRACTION AND ITS APPLICATION IN HEART DISEASE
67
It is important to note that, although the causes differ, the clinical features and neurohumoral changes are indistinguishable between HFpEF and HFrEF, suggesting a mechanistic
association. The processes involved on heart failure have been investigated for many decades
but despite this the underlying pathophysiology remains poorly understood [10]. This has led some
authorities to circumvent defining heart failure by using a collection of signs, symptoms and the
results of investigations to describe the condition [11]. Heart failure is most commonly defined as
the condition in which the heart is unable to pump blood at a rate commensurate with the requirements of the metabolising tissues or can do so only from an elevated filling pressure [11].
The Frank-Starling mechanism is the observation that the more the ventricle is filled with
blood during diastole, the greater the volume of ejected blood will be during the resulting systolic contraction. This mechanism is exhausted in chronic heart failure and so elevated filling
pressures cannot maintain stroke volume alone [12]. Further, patients may have normal filling
pressures following treatment with diuretics yet still maintain a normal cardiac output. As there
is no proven increase in demand of metabolising tissues (perhaps with the exception of thyrotoxicosis) this defi nition implies that a reduction in cardiac output is the cause of heart failure.
However, stroke volume and cardiac output are not used to help in the diagnosis of heart
failure despite the ability to make these measurements non-invasively. Patients with chronic
heart failure in fact have a relatively normal stroke volume and cardiac output [13, 14]. A left
ventricle that is unable to maintain cardiac output causes hypotension or shock rather than the
heart failure syndrome.
Since no current definitions of heart failure are satisfactory, it is not surprising that the
diagnosis of heart failure is difficult and treatment challenging. An erroneous diagnosis of
heart failure is often made because the condition is ill-defi ned. Not only is this unfortunate for
the individual but has also resulted in under-powering of heart failure trials that then become
difficult to extrapolate to clinical practice.
Only in acute deterioration of heart failure is stroke volume reduced and cardiac output
maintained by tachycardia. A reduction in stroke volume has also been demonstrated in HFpEF
initially in some patients following admission with an acute deterioration [15].
It is unlikely that neuro-humoral activation has a sustained inotropic effect because of the
down-regulation of 1-adrenergic receptors and uncoupling of 2-receptors to second messengers [16]. Further, cardiac output is maintained despite neuro-humoral antagonists used to treat
heart failure. Therefore, in chronic heart failure the most important compensatory change is
likely to be ventricular dilation or remodelling.
1.3 End-diastolic volume and remodelling in HFpEF
Left ventricular end-diastolic volume in HFpEF ranges from 19% to 17% of control groups
(Figure 2). There is no consistent increase or decrease in end-diastolic volumes. This may
partially reflect the severity, aetiology and patient selection in the different patient cohorts.
Further, note the lack of correlation with end-diastolic volume and degree of left ventricular
hypertrophy (Figure 2). What is clear is that some patients with HFpEF have an end-diastolic
volume greater than controls. As will be demonstrated below most, if not all, patients also have
greater than expected end-diastolic volumes, suggesting remodelling has taken place.
1.4 Left ventricular hypertrophy in HFpEF
Epidemiological studies suggest HFpEF is strongly associated with concentric left ventricular
hypertrophy [17–19]. A left ventricular mass index of more than 122 g/m2 in women and 149 g/m2
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68 R EPAIR AND R EDESIGN OF PHYSIOLOGICAL SYSTEMS
Observational trial data comparing LV mass and EDV in
different cohorts
20
Hypertension only
HFpEF °Ht
15
EDV (% of control)
HPpEF & Ht
10
Combined HFpEF
5
0
-5
-10
-15
-20
0
10
20
30
40
LV mass (% of control)
Figure 2: A graph showing observational data comparing left ventricular mass (LV mass as %
of controls) and end-diastolic volumes (EDV as % of controls) in different cohorts of
HFpEF and hypertension (data taken from Maurer [13], Maurer [20], Lam 2007 [21]
and Baicu [22]).
in men is sufficient evidence for the diagnosis of HFpEF when tissue Doppler yields inconclusive results or when plasma levels of natriuretic peptides are elevated [7].
Observational clinical trials in HFpEF [13, 20–22] that have data corrected for body surface
area are shown in Figure 2. Note how HFpEF patients in these trials have consistently increased
left ventricular mass (1540%) greater than normals.
This raises the possibility that left ventricular hypertrophy might be causally linked to the
preservation of the ejection fraction in HFpEF.
1.5 Ejection fraction as a measure of systolic function
Ejection fraction has long been regarded as the principal measure of systolic function. It is important to consider that ejection fraction is merely a ratio of left ventricular stroke volume and enddiastolic volume and is only one of many indicators of systolic function. If resting stroke volume
is relatively fixed at the lowest level that can allow tissue perfusion and maintain blood pressure
then it is not surprising that stroke volume in heart failure remains relatively normal. The only
way this can be achieved is by increasing end-diastolic volume. This is referred to as ‘adverse’
remodelling and has been thought of as an undesirable event to be avoided at all costs.
The ejection fraction in heart failure has a unimodal distribution showing that it is not a useful ‘rule out’ measurement [23]. Additional evidence that the ejection fraction is a sub-optimal
measure of function include the following: there is poor concordance between the left ventricular
ejection fraction and severity of heart failure symptoms in clinical practice [24] and the ejection
fraction is a suboptimal predictor of mortality [25]. Furthermore, given a similar severity of
symptoms, the prognosis is equally poor in both HFpEF and HFrEF [26, 27]. Therefore, ejection
fraction cannot be regarded as the gold standard for measuring left ventricular performance.
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A MATHEMATICAL MODEL OF LEFT VENTRICULAR CONTRACTION AND ITS APPLICATION IN HEART DISEASE
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69
Systolic Performance in HFpEF
It is commonly thought that individuals with a normal ejection fraction and heart failure have
normal systolic performance and, therefore, that the heart failure must be due to isolated diastolic dysfunction. Recent echocardiographic methods using tissue Doppler or myocardial velocity
imaging have shown that HFpEF is associated with significant systolic abnormalities. These
abnormalities include an important decline in long-axis displacement, reduced systolic velocities of basal myocardial and mitral annular motion, and decrease of both longitudinal strain and
strain rate [28–33].
In left ventricular hypertrophic conditions such as hypertension [34], hypertrophic cardiomyopathy [35], Fabry disease [36], diabetes [37], haemochromatosis [38], and amyloid [39]
there are major systolic long-axis abnormalities.
Brain natriuretic peptide (BNP) is a biochemical marker which is used in the diagnosis of
heart failure and closely correlated with severity and prognosis. There is a better correlation
of BNP with long-axis systolic function than with ejection fraction in HFrEF patients [40].
This suggests that long-axis shortening may better reflect ventricular performance than ejection
fraction.
Additional evidence of systolic function abnormalities in hypertrophic left ventricular
disease are supported by the finding of depressed longitudinal myocardial shortening in nonhypertensive [41] and hypertensive left ventricular hypertrophy in both echocardiographic [57]
and magnetic resonance imaging studies [42] even in the absence of heart failure.
In addition to the many human studies discussed confirming overwhelming evidence of
systolic abnormalities in HFpEF and hypertensive heart disease, there are numerous experimental small and large animal studies verifying reduced contractile function in hypertensivehypertrophy ventricular disease [43–53].
The long-axis systolic abnormalities in HFpEF have been explained by suggesting
(a) the abnormalities are subtle, [54] which they are not,
(b) differentiating ‘regional’ from ‘global’ function, [55] which is erroneous. Tissue Doppler studies show that diastolic and systolic shortening velocities are reduced in all left ventricular
segments in HFpEF. The presence of abnormalities in all parts of the myocardium indicates
a widespread (i.e. global) abnormality,
(c) by distinguishing contractile from pump function, [56] which assumes the heart is not a
displacement pump,
(d) citing a compensatory increase in circumferential function, [28] which is challenged by many
reports [57– 60]. Indeed, there is some evidence that mid-wall circumferential shortening is
in fact reduced even though endocardial circumferential shortening is increased in hypertensive left ventricular hypertrophy [42]. This apparent paradox can only be explained by the
presence of hypertrophy.
1.7 Diastolic function in heart failure
The documentation of diastolic dysfunction in HFrEF indicates that its mere presence in HFpEF
does not prove it is the cause. Patients with HFrEF and HFpEF have similar degrees of diastolic
dysfunction when assessed using tissue velocity imaging [61]. In fact, Bursi et al. showed that
diastolic function is actually worse in HFrEF compared with HFpEF [62]. These findings probably represent reduced annular motion and/or elevated filling pressures and probably correspond
to the severity of heart failure rather than the phenotype.
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LV pressure
70 R EPAIR AND R EDESIGN OF PHYSIOLOGICAL SYSTEMS
HFpEF
Normal control
EDPVR
LV volume
LV pressure
Figure 3: Expected pressure-volume loops in HFpEF and controls.
3
HFpEF (3-6)
4
1
6
Normal controls (1,2)
2
5
LV volume
Figure 4: Actual EDPVR in HFpEF and controls.
The end-diastolic pressure-volume relationships (EDPVR) have long been regarded as the
method of choice for diagnosing HFpEF and assessing diastolic function as there is believed to
be a shift in the curves upward and to the left in pure ‘diastolic’ heart failure.
Figure 3 (reproduced from Burkhoff [63]) shows the expected pressure-volume loops seen in
normal controls and patients with heart failure and normal ejection fraction. However, Figure 4
shows the EDPVR of the HFpEF (HFNEF) patients can be shifted leftward (curve 3), rightward
(curves 5 & 6) or are no different (curve 4) to the normal controls (1 & 2). This implies the heart
failure seen in HFpEF is not due to isolated diastolic dysfunction [63, 64].
Furthermore, the interpretation of pressure-volume loops depends on the assumption that
the left ventricular end-diastolic volumes are compared with the controls. However, as will be
seen later, the predicted pre-compensatory (unremodelled) end-diastolic volumes are actually
less than controls. Following remodelling and normalisation of stroke volume the end-diastolic
volumes increase towards control values. Therefore, the reference volumes should be related to
the predicted end-diastolic volume for that degree of hypertrophy rather than the normal control
group. This explains the apparent incongruous findings in HFpEF and, potentially, invalidates a
primary assumption of pressure-volume loops in the diagnosis of HFpEF.
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A MATHEMATICAL MODEL OF LEFT VENTRICULAR CONTRACTION AND ITS APPLICATION IN HEART DISEASE
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Other determinants of left ventricular performance
The left ventricular volumes and geometry are influenced by multiple considerations such as
body mass index, body fat, degree of fitness, gender, age, ethnicity, valvular disease, inotropic
and chronotropic effects, neuro-humoral factors, inter- and intra-ventricular dyssynchrony,
ventricular–ventricular interaction, filling pressure, blood pressure, ventricular compliance,
ventricular torsion/rotation, ventricular ‘suction’, abnormal ventricular–arterial interaction, blood
pressure, reflected waves and peripheral vascular resistance. This makes in vivo assessment difficult if not impossible. In addition, any perturbation of function in vivo is associated with both
acute and chronic physiological responses which may conceal or confound any changes.
2 The hypotheses
2.1
Hypothesis 1
The heart failure syndrome is caused initially by a cardiac disorder resulting in a fall in stroke
volume (pre-compensatory or non-remodelled stage) with subsequent compensatory ventricular dilation (post-compensatory or remodelled stage) to normalise net stroke volume in both
HFrEF and HFpEF.
2.2
Hypothesis 2
The presence of an increased muscle mass in the non-dilated heart (i.e. concentric left ventricular
hypertrophy) would explain the preserved ejection fraction and the relatively normal end-diastolic
volumes seen in HFpEF through augmented radial thickening. More specifically, HFpEF may be
caused by systolic dysfunction when concentric left ventricular hypertrophy is present.
3
3.1
Methods
Role of mathematical modelling
Current methods for modelling left ventricular contraction are complex and so difficult to
employ. Therefore, there is a need for a simpler mathematical method.
To remove both confounding factors and the effects of any confusing or opposing physiological
responses a mathematical model has been developed and recently published [65]. We described
the early, pre-compensatory stage (or non-remodelled stage) of heart failure [65]. The model
has been extended to assess the effect of ventricular remodelling following normalisation stroke
volume to mimic the chronic phase of heart failure.
3.2
Basic model
The method employs one or other of two well-established three-dimensional internal (I ) left
ventricular geometric shapes using a combination of the area perpendicular to the long-axis at
the level of the mitral valve and length (L) of long-axis to determine internal left ventricular
volume (see Figure 5).
If one assumes that the external (E) left ventricular volume is a similar shape to the internal
volume then total or external left ventricular volume can be calculated using the same formula.
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72 R EPAIR AND R EDESIGN OF PHYSIOLOGICAL SYSTEMS
End-diastole
ELd
EWd
IWd
ILd
End-systole
ELs
EWs
Radial
thickness
Apex
Endocardium
Epicardium
Base
Figure 5: Schematic diagram of the left ventricle used in the model to calculate internal (I )
and external (E ) volumes during diastole from the short-axis diameter/width (IWd
& EWd) and long-axis length (ILd & ELd ). The apex, base, endocardium and epicardium are labelled. The diastolic external and internal volumes are calculated and the
myocardial volume obtained from the difference. The external short-axis width and
length are reduced to simulate systole (EWs & ELs, respectively) and the new external
volume calculated. The internal end-systolic volume is calculated by subtracting the
muscle volume from the external end-systolic volume.
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A MATHEMATICAL MODEL OF LEFT VENTRICULAR CONTRACTION AND ITS APPLICATION IN HEART DISEASE
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The total myocardial volume is derived from the difference in total (external) left ventricular
volume and the internal volume (Figure 5).
Two different formulae [66, 67] are available and each assumes a slightly different geometry
of the left ventricle:
(a) Area-length method of Dodge
V W 2.L/6
where V volume, W width (i.e. short-axis diameter), L long-axis length.
So
External volume in diastole EWd2 .ELd /6
Internal volume IWd2 .ILd /6
where I internal, E external, ddiastole
Myocardial volume External volume Internal volume
(b) Hemi-cylinder-hemi-ellipsoidal method
V 5(W/2) 2.L/6
3.3
Simulating systole
In order to simulate systole (s) the external short- and long-axis lengths are reduced and the new
systolic external left ventricular volume recalculated (see Figure 5). The new systolic internal
volume is calculated from the difference in external end-systolic volume and the muscle volume.
It is assumed that there is no significant muscle compression and the muscle acts as a simple
elastomer.
External volume in systole EWs2.ELs/6
Internal volume in systole external volume in systole muscle volume.
3.4
Physiological and pathophysiological changes
Adjustments for varying degrees of long- and short-axis dysfunction can then be mimicked by
changing the long- and short-axis shortening. The degree of left ventricular hypertrophy can
be adjusted by altering the difference between the end-diastolic internal (IDd ) and external
diameters (EDd ).
Multiple measurement can then be derived including ejection fraction, stroke volume,
endocardial/mid-wall/epicardial circumferential shortening, radial thickening, ventricular wall
stress and other parameters.
4
4.1
Results
Results pre-compensation (un-remodelled or before remodelling)
The initial results of changing long-axis shortening and the effect of left ventricular hypertrophy
on ventricular volumes have been described recently [65] using the area-length method of
Dodge. We modelled an increase in left ventricular mass of between 13 and 35% of the control
value. These results show that internal left ventricular hypertrophy resulted in augmentation of
systolic wall thickening, a reduction in end-diastolic volume and an increased ejection fraction
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74 R EPAIR AND R EDESIGN OF PHYSIOLOGICAL SYSTEMS
Long-axis
displacement
1.6 cm
1.6 cm
80
1.0 cm
0.5 cm
60
0.2 cm
40
Long-axis
displacement
Post-remodelling
80
Stroke volume (ml)
Stroke volume (ml)
Pre-remodelling
1.0 cm
0.5 cm
60
0.2 cm
40
20
20
159
179
191
203
159
214
179
191
203
214
LV mass (g)
LV mass (g)
Long-axis
displacement
100%
100%
0.5 cm
60%
0.2 cm
40%
20%
Ejection fraction
1.0 cm
80%
Ejection fraction
Long-axis
displacement
1.6 cm
1.0 cm
60%
0.5 cm
0.2 cm
40%
20%
0%
159
179
203
191
LV mass (g)
0%
214
159
179
203
191
LV mass (g)
214
Long-axis
displacement
Long-axis
displacement
3
1.6 cm
80%
3
1.6 cm
1.0 cm
2
0.5 cm
0.2 cm
1
End-systolic wall
thickness (cm)
End-systolic wall
thickness (cm)
1.6 cm
0
1.0 cm
2
0.5 cm
0.2 cm
1
0
179
203
191
LV mass (g)
159
214
159
179
203
191
LV mass (g)
214
Long-axis
displacement
LV end-diastolic
volume (ml)
250
200
1.6 cm
300
1.0 cm
250
0.5 cm
150
0.2 cm
100
50
Long-axis
displacement
1.6 cm
LV end-diastolic
volume (ml)
300
1.0 cm
200
0.5 cm
150
0.2 cm
100
50
0
0
159
179
191
203
LV mass (g)
214
159
179
191
203
214
LV mass (g)
Figure 6: Effect of long-axis shortening and left ventricular mass (up to a 35% increase) on
stroke volume, ejection fraction, end-systolic wall thickness and end-diastolic volume
(Left: before remodelling; Right: after remodelling).
without a change in stroke volume. Abnormal long-axis function resulted in a reduction in
stroke volume, ejection fraction and a small fall in end-systolic wall thickness (Figure 6).
We showed that the ejection fraction could remain >50% together with reduced long axis
shortening and a reduced stroke volume if there was significant left ventricular hypertrophy
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A MATHEMATICAL MODEL OF LEFT VENTRICULAR CONTRACTION AND ITS APPLICATION IN HEART DISEASE
75
present. This implied that HFpEF may be explained by the presence of left ventricular
hypertrophy. The resulting amplified radial thickening in the setting of reduced long-axis shortening can explain the preservation of ejection fraction. We suggested that the reduced stroke
volume in the pre-compensated state rather than diastolic dysfunction may be the cause of heart
failure in both HFrEF and HFpEF.
4.2
Results post-compensation (following remodelling to normalise stroke volume)
As patients with chronic heart failure generally do not have reduced stroke volumes, the effect
of left ventricular dilation or remodelling to normalise stroke volume on ejection fraction, enddiastolic volume and end-systolic wall thickness has been assessed.
This unpublished data used the hemi-cylinder-hemi-ellipsoidal method and has produced
some interesting results (Figure 6). The previous findings using the Dodge formula in the precompensatory stage were also confirmed.
Following remodelling, when there is no left ventricular hypertrophy, significant ventricular
dilation is necessary to correct the stroke volume. The presence of left ventricular hypertrophy causes the end-diastolic volume to fall compared with normal. In the presence of reduced
long-axis shortening, the end-diastolic volume increases. Importantly, when both long-axis
shortening and left ventricular hypertrophy are present remodelling results in the end-diastolic
volume returning towards normal. This is the typical pattern seen in HFpEF in clinical trials
and explains the apparent divergent results seen in Figure 4.
4.3
Summary of results
Figure 7 shows the scaled diagrams of left ventricular contraction. At the top of the figure is
the control situation with relatively normal internal dimensions, an ejection fraction of 58%
and stroke volume of 83 ml. On the left is the effect in a non-hypertrophic heart. Following
acute damage, as may occur in acute fulminant myocarditis, there is a reduction in myocardial
shortening which results in a fall in ejection fraction and stroke volume. This would normally
be compensated for by a tachycardia resulting in normalisation of cardiac output and the syndrome of acute heart failure. If the tachycardia were insufficient to improve the cardiac output
then the patient would have a low cardiac output and develop a shock/hypotensive syndrome.
Should the patient survive and go on to the chronic phase then renal under-perfusion would lead
to an increase in intra-vascular volume, an increase in right heart filling pressures, a mismatch
between right and left ventricular stroke volumes, a rise in left ventricular end-diastolic pressure and finally a significant increase in end-diastolic volume, a normalisation of stroke volume
with an associated reduction in ejection fraction. Some patients with dilated cardiomyopathy
have a more gradual onset of dysfunction and will never suffer from acute heart failure. In this
case the left ventricle will gradually dilate (remodel) in parallel with the fall in myocardial
shortening to normalise the stroke volume.
The situation in pathological hypertrophic left ventricular disease, such as amyloid or hypertrophic-hypertensive left ventricular disease, is different and is shown on the right in Figure 7.
Cardiac hypertrophy is relatively slow to develop so that one does not see a similar acute dysfunction phase as is observed in (non-hypertrophic) acute myocarditis. Therefore, patients will
gradually remodel as the hypertrophic process progresses concurrently with the myocardial
dysfunction as demonstrated with the solid arrow. The uncompensated stage is the hypothetical
predicted situation following hypertrophy but in the absence of any remodelling (shown with
the dashed arrow). Although this will not be seen in clinical practice it does not diminish its
importance as a reference point prior to remodelling.
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76 R EPAIR AND R EDESIGN OF PHYSIOLOGICAL SYSTEMS
No LVH.
No long-axis
disfunction.
83 ml
58%
Normal
Systole
61 ml
Diastole
143 ml
No LVH.
Long-axis
dysfunction.
Acute
dysfunction
55 ml
38%
88 ml
Chronic
onset
dysfunction
Not
remodelled
55ml
60%
143 ml
Moderate
concentric LVH
(+35%).
Long-axis
dysfunction.
36 ml
Systole
91 ml
Diastole
75 ml
Systole
159 ml
Diastole
Chronic compensatory
dilation to normalise
stroke volume.
83 ml
35%
154 ml
237 ml
83 ml
52%
2° eccentric left ventricular hypertrophy
possibly due to wall stress/stretch
H
HFFrrE
EFF
H
HFFppE
EFF
Figure 7: Schematic illustration to scale showing the normal situation (upper part) with normal
long-axis displacement (1.6 cm). The effects of reduced myocardial shortening (longaxis displacement 0.5 cm) without left ventricular hypertrophy (on left) versus moderate (35% increase in mass) left ventricular hypertrophy (on right) before (middle) and
after compensatory dilation (lower) to normalise stroke volume are shown. The dashed
arrow represents predicted hypothetical changes probably not occurring in vivo. Also
shown are the intra-ventricular volumes (ml) in systole and diastole, stroke volume
(ml) and ejection fraction (shown within the cylinders).
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5.1
77
Conclusions
Interpretation
A simple and yet flexible mathematical model has been described that can determine the effect
of left ventricular contraction. The model assumes the heart is a positive displacement pump
with a known geometric shape. It removes the confounding effects of other known as well as
non-predictable physiological and pathological responses. The model can determine the effect
of left ventricular hypertrophy and myocardial shortening and other physiological changes on
stroke volume, ejection fraction, end-systolic wall thickness and end-diastolic volume.
The study has determined that, in the absence of remodelling, stroke volume can be predicted
to fall following reduced myocardial shortening. This effect is independent of the presence
of left ventricular hypertrophy. The changes necessary to normalise stroke volume through
ventricular dilation were calculated. Remodelling without hypertrophy results in a high enddiastolic volume and low ejection fraction (typical features of HFrEF). Following remodelling,
left ventricular hypertrophy resulted in a near-normal end-diastolic volume and preserved ejection fraction (as seen in HFpEF).
These fi ndings are consistent with the hypothesis that heart failure is caused initially by a
reduced stroke volume compensated by tachycardia (i.e. acute HF). This is followed by gradual
remodelling (i.e. compensated by dilation) to normalise stroke volume in both phenotypes.
HFpEF can be explained fully by reduced systolic myocardial shortening due to a fall in contractility in presence of concentric left ventricular hypertrophy rather than isolated diastolic
dysfunction. A central observation is that the amount of ventricular dilation is highly dependent on the degree of left ventricular hypertrophy. These fi ndings are validated by published
observational data.
5.2 Advantages of the model
The model allows the assessment of changes in shape, size and shortening without confounding
effects of body size etc. It also removes the physiological or pathological responses which could
modify the consequences of these changes. A similar clinical or in vivo experimental study
would be impossible to perform because of all the many homeostatic changes.
Additional modifications to the model can be made such as measuring the effect of changes in
mid-wall circumferential shortening or the effect of thinner apical wall and differing shaped ventricles (e.g. sphericity). The model has allowed the facility to determine the outcome of modification
in long-axis, short-axis as well as the degree of left ventricular hypertrophy on ejection fraction,
stroke volume, wall stress etc. whilst keeping other factors constant and avoiding problems of
interpretation following physiological feedback or secondary pathological mechanisms.
5.3 Limitations of the model
The main limitation of the model is that it is based on certain geometric assumptions although
these do appear to correlate with direct observation and have been well validated in the clinical
setting [66, 67]. Furthermore, using either the hemi-cylinder-hemi-ellipsoidal or the area-length
method gave similar results.
The model presupposes that the shapes of both the external and internal left ventricular walls
are similar and remain the same in systole and diastole. Moreover, it was assumed that internal concentric hypertrophy occurs in the pre-compensation (unremodelled) state. This appears
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78 R EPAIR AND R EDESIGN OF PHYSIOLOGICAL SYSTEMS
to be appropriate as the model resulted in a constant stroke volume that was independent of
the amount of left ventricular hypertrophy. This was employed because maintaining a normal
stroke volume was felt to be the most likely physiological response following the development
of hypertrophy.
The simulation presupposes that there is no loss in myocardial muscle volume (that it is
non-compressible) during contraction. This appears reasonable as the myocardium comprises
mainly of water; however, there may be a small loss due to vascular compression.
Although the calculation of ventricular volumes may be approximate, the concepts proposed
are likely to be sound. A normalisation of stroke volume in the post-compensation stage was
used rather than a partially corrected stroke volume to simulate the study findings; it may be
assumed that the resting stroke volume in normal controls is likely to be near the lowest level
that physiologically maintains sufficient tissue (including renal) perfusion.
The model will produce some examples that may not occur in clinical practice such as
pathological hypertrophy with normal long-axis function. With these exceptions, the model
results approximate satisfactorily well to those found in clinical practice.
5.4
Correlation of model predictions with observational studies
The model predicts that reduced long-axis shortening results in a decreased stroke volume in
the pre-compensation stage in heart failure. Acute myocardial damage, say following acute
myocarditis or a myocardial infarction, results in a reduced stroke volume and normalisation of
cardiac output is brought about by a tachycardia. This is regularly seen in HFrEF and may be
occasionally seen in acute exacerbations of HFpEF [13].
In chronic heart failure due to systolic dysfunction, the ejection fraction is reduced (HFrEF)
and ventricular dilation (remodelling) results in a high end-diastolic volume with normalisation of
stroke volume. The data presented predicts that HFpEF would be more frequent with increasing
concentric left ventricular hypertrophy and HFrEF would occur in the absence of hypertrophy.
HFpEF is thought by many to be due to diastolic dysfunction. The end-diastolic volume is
usually relatively normal and so it has been assumed that ventricular remodelling has not taken
place. Moreover, it is often thought that ejection fraction equates to ventricular performance;
however, there is overwhelming evidence of major systolic dysfunction in human and animal
studies both in HFpEF and in hypertensive heart disease as outlined above.
The germane finding of the model in both phenotypes of heart failure is the predicted
reduced stroke volume in the pre-compensation stage. The study also shows the effect on left
ventricular end-diastolic volume with the predicted volumes being significantly greater when
there is no left ventricular hypertrophy and lower left ventricular end-diastolic volumes when
there is hypertrophy present despite remodelling occurring in both phenotypes. This is compatible with the typical findings in HFrEF and HFpEF and is supported by a substantial body of
data (also see Figure 2).
The finding of an increase in ejection fraction with hypertrophy demonstrated by the model
is supported by an observational study in hypertension which confi rmed a higher ejection fraction (69% vs. 63%) in patients with hypertensive left ventricular hypertrophy than in controls
despite having reduced long-axis shortening (1.8 vs. 2.1 cm) [68].
5.5
Implications in heart failure
This study suggests that the development of concentric left ventricular hypertrophy, occurring
concurrently or before the perturbation causing contractile dysfunction may be the cardinal
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feature that determines whether HFrEF or HFpEF develops. This would be entirely consistent
with observational studies showing a strong association of left ventricular hypertrophy with
HFpEF.
The term ‘global’ function when referring to ejection fraction and ‘regional’ when referring
to myocardial velocities is flawed. Heart failure can occur when the ejection fraction is preserved
if sufficient left ventricular hypertrophy takes place to result in increased radial thickening
to maintain ejection fraction but not stroke volume in the presence of reduced long-axis systolic shortening. The model shows how patients with HFpEF can have significant reductions in
long-axis shortening and reduced stroke volume yet maintain their ejection fraction in the presence of hypertrophy. In fact the maintained radial function seen in HFpEF is explained by the
additional radial thickening and augmented inward endocardial displacement secondary to left
ventricular hypertrophy rather than by increased contractility itself.
The poor relationship seen between ejection fraction and symptoms in clinical practice may
be explained by the lack of association between the ejection fraction and stroke volume demonstrated by the model. The main association is with long-axis shortening and stroke volume
in the pre-compensation stage. Assuming a worse stroke volume indicates a more severe heart
failure syndrome then these fi ndings would imply that measures of long-axis function should
be even better predictors of morbidity and mortality than ejection fraction. Certainly, longaxis function is good at assessing prognosis [69–72]. The preserved ejection fraction seen in
HFpEF reflects the relative normal end-diastolic volumes and reduced stroke volume observed
in the pre-compensation stage. ‘Global’ left ventricular performance is impaired as evident by
the reduced stroke volume and long-axis dysfunction. The reduced stroke volume in the precompensatory stage may be brought about by either reduced contractile performance, restricted
filling secondary to poor compliance or, in all probability, a combination of both. The preserved
ejection fraction in HFpEF gives the false impression of normal systolic myocardial function
and, in effect, conceals the adaptive changes that normalise the reduced stroke volume.
As stroke volume is usually not reduced in chronic heart failure, it is understandable that
ejection fraction became a commonly used alternative measure of ventricular function. However, the teleological argument is that it does not matter to the organs what the ejection fraction
is. As far as tissue perfusion is concerned, the ejection fraction is irrelevant; it is the net stroke
volume, rather than ejection fraction, that is most pertinent in enabling tissue perfusion. Since
it is long-axis shortening that mainly determines the stroke volume it probably also determines
the severity of the heart failure syndrome. Furthermore, it may be speculated that although
stroke volume may be normal or near normal at rest there may be blunting of the expected
increase in stroke volume with exercise. This might explain why most treated and euvolaemic
patients only have symptoms on exertion.
It is likely that in normal individuals the resting stroke volume is maintained near the minimum necessary to allow adequate tissue perfusion. Any greater stroke volume would be both
physiologically unnecessary and inefficient. The increase in stroke volume seen in some studies
[13] in heart failure may be the result of valvular regurgitation where the gross stroke volume is
increased but the net stroke volume, the ejected stroke volume and tissue perfusion, is normal. If
the compensatory increase in net stroke volume were inadequate, it would lead to hypotension
or shock rather than the syndrome of heart failure. Presumably, some patients with Class IV
heart failure are also in this category.
The mechanism by which end-diastolic volume increases is unknown. It may be speculated
that an elevated end-diastolic pressure may arise from increased blood volume secondary to
reduced tissue/renal perfusion, neuro-humoral activation, fluid retention and a right ventricular
stroke volume transiently greater than the left ventricular stroke volume. It is suggested that
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the mechanisms involved in chronic heart failure would be as follows: myocardial injury or disease, reduced stroke volume, reduced renal perfusion, fluid retention, increased blood volume,
increased right ventricular filling pressures, mismatch between right and left ventricular stroke
volumes, increased left ventricular end-diastolic pressure, increased left ventricular volumes
and finally normalisation of stroke volume. In heart failure, the stroke volume returns towards
normal by increasing the left ventricular end-diastolic volume. This would predict that additional left ventricular dilation (remodelling) would occur only if systolic function deteriorates
further. It is likely that intra-vascular volume expansion occurs until left ventricular dilation
causes a normalisation of stroke volume.
Only cases of severe hypertrophic disease (such as amyloid) at one end of the spectrum
and dilated cardiomyopathy at the other with moderate hypertensive-hypertrophic disease in
between have been modelled. More complex situations, such as a combination of hypertensive
and ischaemic heart disease, have not been specifically modelled. However, this may explain
some of the intermediate cases when markedly reduced long-axis shortening and moderate
hypertrophy result in only a mildly reduced ejection fraction. It has been proposed that HFrEF
and HFpEF are either two separate disorders or part of a continuum [7]. This study would suggest potentially three different phenotypes of myocardial (i.e. excluding valvular etc.) heart failure. Firstly, pure HFrEF as seen in non-hypertensive, young dilated cardiomyopathy patients at
one end of a spectrum, secondly, pure HFpEF of hypertensive-hypertrophic disease or amyloid
at the other. A third more complex group would have a combination of hypertension with left
ventricular hypertrophy and myocardial infarction. This third group may be the most common
and would explain the disparity in ejection fraction and symptom severity commonly seen following a myocardial infarction. Each type would not only have a continuum of the amount of
hypertrophy but also a spectrum of severity from mild to severe. Furthermore, the third type
would have a proportional impact of each cause. This would satisfactorily explain both the
apparent continuum and the unimodal distribution of ejection fraction in heart failure as well as
the morphological difference in the diverse cohorts.
Further, this study would predict that the presence of reduced ejection fraction confi rms
systolic dysfunction but conversely the presence of a normal ejection fraction does not rule
out significant systolic dysfunction. Left ventricular dysfunction accordingly becomes easier
to understand. If the left ventricle is dilated with reduced ejection fraction both systolic and
diastolic function are likely to be equally impaired. If the ejection fraction is normal then the
systolic and diastolic function may be either normal or abnormal. This study would predict that
measures of systolic and diastolic function are likely to be closely related and that consequently
either diastolic dysfunction or long-axis systolic function can be measured to support the diagnosis of HFpEF. Measuring endocardial circumferential shortening or radial thickening may
be misleading particularly in the presence of concentric hypertrophy because of augmented
thickening.
The term ‘left ventricular function’ when referring to ejection fraction should probably be
avoided as it is imprecise and confusing. As patients with ‘systolic’ heart failure have significant diastolic dysfunction and HFpEF patients have significant systolic dysfunction the use of
heart failure with reduced ejection fraction is more precise. The term ‘diastolic heart failure’
oversimplifies the pathophysiology and the phrase ‘heart failure with normal ejection fraction’
is inconsistent as many patients have a subnormal ejection fraction. At present, the term ‘heart
failure with preserved ejection fraction’ is preferred.
The isolated diastolic dysfunction hypothesis has been accepted as fact by some authors and
has formed the basis of research and clinical trials for more than two decades. The diastolic dysfunction premise may appear outwardly plausible but the mechanisms of heart failure remain
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contentious. Any erroneous theory would have serious consequences. The misunderstanding in
HFpEF appears to arise from the belief that ejection fraction is a suitable surrogate for systolic
function.
In hypertensive-hypertrophic myocardial disease and HFpEF, myocardial dysfunction occurs
gradually and concurrently with concentric left ventricular hypertrophy; the reduced stroke and
end-diastolic volumes are compensated simultaneously so that only the post-compensated state
is seen in practice. A sudden further small fall in contractility for any reason, such as arrhythmia
or ischaemia, could explain the acute exacerbations observed in HFpEF.
Accordingly, all the apparent paradoxes of heart failure can be explained: important systolic
dysfunction and yet preserved ejection fraction in HFpEF, the remodelling in HFpEF yet ‘normal’
end-diastolic volume, the maintained cardiac output in chronic heart failure, ‘regional’ vs. ‘global’
function and the poor correlation of symptoms with ejection fraction. The disparate results of
pressure-volume loops are also elucidated as they have assumed that the control end-diastolic
volume is equivalent to the un-remodelled situation. Remodelling can be seen as a positive
adaptive physiological process to correct a reduction in stroke volume. Thus a controversial
unifying hypothesis of chronic heart failure is proposed as a syndrome where reduced net resting stroke volume is normalised through remodelling in both phenotypes. The isolated diastolic
dysfunction hypothesis is redundant.
5.6
Implications for repair and redesign
Left ventricular remodelling has been thought of as an adverse event in heart failure and so methods to repair and prevent the dilation have been proposed. These techniques include ventricular
reduction surgery (Batista operation), to reduce the left ventricular end-diastolic volume and
the use of an epicardial mesh (CorCap®, HeartNet®) to restrain the heart and prevent ventricular
dilation. It would be predicted from this work that both these procedures would result in a reduction in stroke volume with additional neuro-humoral activation associated with an improvement
in ejection fraction but not myocardial function. It comes as little surprise therefore that these
methods have had disappointing outcomes. This may have been predicted before patients had
been subjected to these procedures if a better comprehension of the pathophysiology of heart
failure had been available.
A greater understanding of the mechanisms of both normal and abnormal left ventricular function will help the development and redesign of artificial hearts and ventricular assist devices. It
will also facilitate understanding of ventricular dysfunction and so allow a better assessment
of left ventricular performance than ejection fraction. An improved appreciation of myocardial
function will enhance the design and evaluation of the effects of drugs and stem cell research to
aid repair of myocardium and help our patients.
5.7
Summary
The mathematical modelling demonstrated supports the view that the heart is a positive displacement pump. Ockham’s razor would suggest a unifying theory would be preferable to
describe the pathophysiology of heart failure with preserved and reduced ejection fraction than
the current heterogeneous mechanisms proposed.
A tenable hypothesis for both HFrEF and HFpEF is that reduced stroke volume in the preor un-remodelled state is the initial event. Compensatory mechanism, such as intra-vascular
volume expansion, occurs which leads to ventricular dilation and results in a normalisation of
stroke volume.
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One of the essential conclusions is that, in the presence of reduced myocardial shortening, a
normal ejection fraction can occur when concentric hypertrophy is present and equally reduced
ejection fraction occurs in the absence of hypertrophy. Therefore, the model predicts that to
diagnose HFpEF concentric hypertrophy must be present.
The study demonstrates the mechanism by which patients could have significant contractile
dysfunction yet have a normal ejection fraction. Conversely, if a reduced ejection fraction is
present it does suggest significant systolic dysfunction. The model explains the paradox of heart
failure with reduced long-axis function and a normal ejection fraction by demonstrating the disparity between the ejection fraction and stroke volume. It is suggested that the terms ‘ejection
fraction’ and ‘function’ are not interchangeable and therefore, should not be used synonymously.
The term ‘left ventricular function’ should be abandoned as it causes misunderstanding.
The mathematical model supports a new paradigm of chronic heart failure. Specifically,
heart failure secondary to myocardial disease is a syndrome where reduced net resting stroke
volume is normalised through left ventricular remodelling.
References
[1] Marinelli, R., Fuerst, B., van der Zee, H., McGinn, A. & William, M., The heart is not a
pump: a refutation of the pressure propulsion hypothesis. Frontier perspectives, 5, 1995.
(http://www.rsarchive.org/RelArtic/Marinelli/).
[2] Robert, H., Anderson, R.H., Yen Ho, S., Redmann, K., Sanchez-Quintana, D. &
Lunkenheimer, P.P., The anatomical arrangement of the myocardial cells making up the
ventricular mass. European Journal of Cardio-thoracic Surgery, 28, pp. 517–525, 2005.
[3] Assman, P.E., Slager, C.J., Dreysse, S.T., Van der Borden, S.G., Oomen, J.A. &
Roelandt, J.R., Two-dimensional echocardiographic analysis of the dynamic geometry of
the left ventricle: the basis for an improved model of wall motion. Journal of the American
Society of Echocardiography, 1, pp. 393–405, 1988.
[4] Carlsson, M., Cain, P., Holmqvist, C., Ståhlberg, F., Lundbäck, S. & Arheden, H., Total
heart volume variation throughout the cardiac cycle in humans. American Journal of
Physiology. Heart and Circulatory Physiology, 287, pp. H243–H250, 2004.
[5] Emilsson, K., Brudin, L. & Wandt, B., The mode of left ventricular pumping: is there
an outer contour change in addition to the atrioventricular plane displacement? Clinical
Physiology, 21, pp. 437–446, 2001.
[6] Lundbäck, S., Cardiac pumping and function of the ventricular septum. Acta Physiologica
Scandinavica Supplementum, 50, 1–101, 1986.
[7] Paulus, W.J., Tschöpe, C., Sanderson, J.E., Rusconi, C., Flachskampf, F., Rademakers, F.E.,
Marino, P., Smiseth, O.A., De Keulenaer, G., Leite-Moreira, A.F., Borbély, A., Édes,
I., Handoko, M.L., Heymans, S., Pezzali, N., Pieske, B., Dickstein, K., Fraser, A.G. &
Brutsaert, D.L., How to diagnose diastolic heart failure: a consensus statement on the
diagnosis of heart failure with normal left ventricular ejection fraction by the Heart
Failure and Echocardiography. European Heart Journal, 28, pp. 2539–2550, 2007.
[8] Kitzman, D.W., Gardin, J.M., Gottdiener, J.S., Arnold, A., Boineau, R. & Aurigemma, G.,
Importance of heart failure with preserved systolic function in patients or 65 years
of age. CHS research Group. Cardiovascular Health Study. American Journal of Cardiology,
87, pp. 413–419, 2001.
[9] Yip, G., Wang, M., Zhang, Y., Fung, J.W.H., Ho, P.Y. & Sanderson, J.E., Left ventricular
long axis function in diastolic heart failure is reduced in both diastole and systole: time
for a redefinition? Heart, 87, pp. 121–125, 2002.
WIT Transactions on State of the Art in Science and Engineering, Vol 35, © 2008 WIT Press
www.witpress.com, ISSN 1755-8336 (on-line)
WITPress_RRPS_ch004.indd 82
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A MATHEMATICAL MODEL OF LEFT VENTRICULAR CONTRACTION AND ITS APPLICATION IN HEART DISEASE
83
[10] Guidelines for the diagnosis and treatment of chronic heart failure: executive summary.
European Heart Journal, 26, pp. 1115–1140, 2005.
[11] Braunwald’s Heart Disease: A Textbook of Cardiovascular Medicine. 7th Ed., Saunders,
2004.
[12] Gill, R.M., Jones, B.D., Corbly, A.K., Ohad, D.G., Smith, G.D., Sandusky, G.E.,
Christe, M.E., Wang, J. & Shen, W., Exhaustion of the Frank–Starling mechanism in
conscious dogs with heart failure induced by chronic coronary microembolization. Life
Sciences, 79, pp. 536–544, 2006.
[13] Maurer, M.D., Burkhoff, D., Fried, L.P., Gottdiener, J., King, D.L. & Kitzman, D.W.,
Ventricular structure and function in hypertensive participants with heart failure and a
normal ejection fraction. Journal of American College of Cardiology, 49, pp. 972–981,
2007.
[14] Garcia, E.H., Perna, E.R., Farias, E.F., Obregon, R.O., Macin, S.M., Parras, J.I.,
Aguero, M.A., Moratio, D.A., Pitzus, A.E., Tassano, E.A. & Rodrriguez, L., Reduced
systolic performance by tissue Doppler in patients with preserved and abnormal ejection
fraction: new insights in chronic heart failure. International Journal of Cardiology, 108,
pp. 181–188, 2006.
[15] Andrew, P., Diastolic heart failure demystified. Chest, 124, pp. 744–753, 2003.
[16] Insel, P.A., –Adrenergic receptors in heart failure. The Journal of Clinical Investigation,
92, p. 2564, 1993.
[17] Hogg, K., Swedberg, K. & McMurray, J.J.V., Heart failure with preserved left ventricular
systolic function: epidemiology, clinical characteristics, and prognosis. Journal of the
American College of Cardiology, 43, pp. 317–327, 2004.
[18] Yip, G.W.K., Ho, P.P.Y., Woo, K.S. & Sanderson, J.E., Comparison of frequencies of left
ventricular systolic and diastolic heart failure in Chinese living in Hong Kong. American
Journal of Cardiology, 84, pp. 563–567, 1999.
[19] Abhayaratna, W.P., Marwick, T.H., Smith, W.T. & Becker, N.G. Characteristics of left
ventricular diastolic dysfunction in the community: an echocardiographic survey. Heart,
93, pp. 1259–1264, 2006.
[20] Maurer, M.S., King, D.L., El-Khoury Rumbarger, L., Packer, M. & Burkhoff, D., Left
heart failure with a normal ejection fraction: identification of different pathophysiological mechanisms. Journal of Cardiac Failure, 11, pp. 177–187, 2005.
[21] Lam, C.S.P., Roger, V.L., Rodeheffer, R.J., Bursi, F., Borlaug, B.A., Ommen, S.R.,
Kass, D.A. & Redfield, M.M., Cardiac structure and ventricular–vascular function in persons with heart failure and preserved ejection fraction from Olmsted County, Minnesota.
Circulation, 115, pp. 1982–1990, 2007.
[22] Baicu, C.F., Zile, M.R., Aurigemma, G.P. & Gaasch, W.H., Left ventricular systolic performance, function, and contractility in patients with diastolic heart failure. Cirulation,
111, pp. 2306–2312, 2005.
[23] Lenzen, M.J., Scholte, O.P., Reimer, W.J.M., Boersma, W.J.E., Vantrimpont, P.J.M.J.,
Follath, F., Swedberg, K., Cleland, J. & Komajda, M., Differences between patients with
a preserved and a depressed left ventricular function: a report from the EuroHeart Failure
Survey. European Heart Journal, 25, pp. 1214–1220, 2004.
[24] Harrington, D.S., Anker, S.D. & Coats, A.J.S., Preservation of exercise capacity and lack
of peripheral changes in asymptomatic patients with severely impaired left ventricular
function. European Heart Journal, 22, pp. 392–399, 2001.
[25] Muntwyler, J., Abetel, G., Gruner, C. & Follath, F., One-year mortality among unselected
outpatients with heart failure. European Heart Journal, 23, pp. 861–1866, 2002.
WIT Transactions on State of the Art in Science and Engineering, Vol 35, © 2008 WIT Press
www.witpress.com, ISSN 1755-8336 (on-line)
WITPress_RRPS_ch004.indd 83
4/29/2008 5:09:36 PM
84 R EPAIR AND R EDESIGN OF PHYSIOLOGICAL SYSTEMS
[26] Bhatia, R.S., Tu, J.V., Lee, D.S., Austin, P.C., Fang, J., Haouzi, A., Gong, Y. & Liu, P.P.,
Outcome of heart failure with preserved ejection fraction in a population-based study.
The New England Journal of Medicine, 355, pp. 260–269, 2006.
[27] Owan, T.E., Hodge, D.O., Herges, R.M., Jacobsen, S.J., Roger, V.L. & Redfield, M.M.,
Trends in prevalence and outcome of heart failure with preserved ejection fraction. The
New England Journal of Medicine, 355, pp. 251–259, 2006.
[28] García, E.H., Perna, E.R., Farías, E.F., Obregón, R.O., Macin, S.M., Parras, J.I.,
Agüero, M.A., Moratorio, D.A., Pitzus, A.E., Tassano, E.A. & Rodriguez, L., Reduced
systolic performance by tissue Doppler in patients with preserved and abnormal ejection
fraction: new insights in chronic heart failure. International Journal of Cardiology, 108,
pp. 181–188, 2006.
[29] Yu, C.M., Lin, H., Yang, H., Kong, S.L., Zhang, Q. & Lee, S.W.L., Progression of systolic
abnormalities in patients with ‘isolated’ diastolic heart failure and diastolic dysfunction.
Circulation, 105, pp. 1195–1201, 2002.
[30] Nikitin, N.P., Witte, K.K.A., Clark, A.L. & Cleland, J.G.F., Color tissue Doppler-derived
long-axis left ventricular function in heart failure with preserved global systolic function.
American Journal of Cardiology, 90, pp. 1174–1177, 2002.
[31] Bruch, C., Gradaus, R., Gunia, S., Breithardt, G. & Wichter, T., Doppler tissue analysis
of mitral annular velocities: evidence for systolic abnormalities in patients with diastolic
heart failure. Journal of the American Society of Echocardiography, 16, pp. 1031–1036,
2003.
[32] Pellerin, D., Sharma, R., Elliott, P. & Veyrat, C., Tissue Doppler, strain, and strain rate
echocardiography for the assessment of left and right systolic ventricular function. Heart,
89, pp. iii9–17, 2003.
[33] Vinereanu, D., Nicolaides, E., Tweddel, A.C. & Fraser, A.G., “Pure” diastolic dysfunction
is associated with long-axis systolic dysfunction. Implications for the diagnosis and classification of heart failure. European Journal of Heart Failure,7, pp. 820–828, 2005.
[34] Miyajima, E., Shintani, Y. & Yonezawa, H., A new technique for quantitative assessment
of the heart in hypertension. American Journal of Hypertension, 18(5)(Suppl 1), p. A3,
2005.
[35] Nagueh, S., Bachinski, L., Meyer, D., Hill, R., Zoghbi, W.A., Tam, J.W., Quiñones, M.A.,
Roberts, R. & Marian, A.J., Tissue Doppler imaging consistently detects myocardial abnormalities in patients with hypertrophic cardiomyopathy and provides a novel
means for an early diagnosis before and independently of hypertrophy. Circulation, 104,
pp. 128–130, 2001.
[36] Weidemann, F., Breunig, F., Beer, M., Sandstede, J., Turschner, O., Voelker, W., Ertl, G.,
Knoll, A., Wanner, C. & Strotmann, J.M., Improvement of cardiac function during
enzyme replacement therapy in patients with Fabry disease. a prospective strain rate
imaging study. Circulation, 108, pp. 1299–1301, 2003.
[37] Galderisi, M., Diastolic dysfunction and diabetic cardiomyopathy. Journal of the American
College of Cardiology, 48(8), pp. 1548–1551, 2006.
[38] Palka, P., Macdonald, G., Lange, A. & Burstow, D.J.J., The role of Doppler left ventricular filling indexes and Doppler tissue echocardiography in the assessment of cardiac involvement in hereditary hemochromatosis. Journal of the American Society of
Echocardiography, 15, pp. 884–890, 2002.
[39] Koyama, J., Ray-Sequin, P.A. & Falk, R., Longitudinal myocardial function assessed by
tissue velocity, strain and strain rate in patients with AL (primary) amyloid. Circulation,
107, pp. 2446–2452, 2003.
WIT Transactions on State of the Art in Science and Engineering, Vol 35, © 2008 WIT Press
www.witpress.com, ISSN 1755-8336 (on-line)
WITPress_RRPS_ch004.indd 84
4/29/2008 5:09:37 PM
A MATHEMATICAL MODEL OF LEFT VENTRICULAR CONTRACTION AND ITS APPLICATION IN HEART DISEASE
85
[40] Vinereanu, D., Lim, P.O., Frenneaux, M.P. & Frazer, A.G., Reduced myocardial velocities
of left ventricular long-axis contraction identify both systolic and diastolic heart failure—
a comparison with brain natriuretic peptide. European Journal of Heart Failure, 5,
pp. 512–509, 2005.
[41] Wandt, B., Bojö, L., Tolagen, K. & Wranne, B., Echocardiographic assessment of ejection
fraction in left ventricular hypertrophy. Heart, 82, pp. 192–198, 1999.
[42] Palmon, L.C., Reichek, N., Yeon, S.B., Clark, N.R., Brownson, D., Hoffman, E. & Axel, L.,
Intramural myocardial shortening in hypertensive left ventricular hypertrophy with normal pump function. Circulation, 89, pp. 122–131, 1994.
[43] Alpert, N.R., Hamrell, B.B. & Halpern, W., Mechanical and biochemical correlates of
cardiac hypertrophy. Circulation Research, 34, pp. II71–II81, 1974.
[44] Bing, O.H., Matsushita, S., Fonburg, B.L. & Levine, H.J., Mechanical properties of rat cardiac muscle during experimental hypertrophy. Circulation Research, 28, pp. 234–244, 1971.
[45] Spann, J.F., Buccino, R.A., Sonnenblick, E.H. & Braunwald, E., Contractile state of cardiac muscle obtained from cats with experimentally produced ventricular hypertrophy
and heart failure. Circulation Research, 21, pp. 341–357, 1967.
[46] Jacob, R., Vogt, M. & Rupp, H., Pathophysiological mechanisms in cardiac insufficiency
induced by chronic pressure overload: an attempt to analyze specific factors in animal
experiment. Basic Research in Cardiology, 81(Suppl 1), pp. 203–216, 1986.
[47] Hamrell, B.B. & Hultgren, P.B., Sarcomere shortening in pressure overload hypertrophy.
Federation Proceedings, 45, pp. 2591–2596, 1986.
[48] Capasso, J.M., Strobeck, J.E. & Sonnenblick, E.H., Myocardial mechanical alterations
during gradual onset long-term hypertension in rats. American Journal of Physiology,
241, pp. H435–H441, 1981.
[49] Schaible, T.F., Ciambrone, G.J., Capasso, J.M. & Scheuer, J., Cardiac conditioning
ameliorates cardiac dysfunction associated with renal hypertension in rats. Journal of
Clinical Investigation, 73, pp. 1086–1094, 1984.
[50] Takahashi, M., Sasayama, S., Kawai, C. & Kotoura, H., Contractile performance of the hypertrophied ventricle in patients with systemic hypertension. Circulation, 62, pp. 116–126, 1980.
[51] Spann, J.F., Jr., Buccino, R.A., Sonneblick, E.H. & Braunwald, E., Contractile state of
cardiac muscle obtained from cats with experimentally produced ventricular hypertrophy
and heart failure. Circulation Research, 21, pp. 341–354, 1967.
[52] Gaasch, W.H., Zile, M.R., Hoshino, P.K., Apstein, C.S. & Blaustein, A.S., Stress-shortening
relations and myocardial blood flow in compensated and failing canine hearts with pressureoverload hypertrophy. Circulation, 79, pp. 872–883, 1989.
[53] Shimizui, G., Hirota, Y., Kita, Y., et al., Left ventricular midwall mechanics in systemic
arterial hypertension: myocardial function is depressed in pressure-overload hypertrophy.
Circulation, 83, pp. 1676–1684, 1991.
[54] Petrie, M.C., Caruana, L., Berry, C. & McMurray, J.J., “Diastolic heart failure” or heart
failure caused by subtle left ventricular systolic function? Heart, 87, pp. 29–31, 2002.
[55] Thomas, J.D. & Popovic, Z.B., Assessment of left ventricular function by cardiac ultrasound. Journal of the American College of Cardiology, 48, pp. 2012–2025, 2006.
[56] Fukuta, H. & Little, W.C., Contribution of systolic and diastolic abnormalities to heart
failure with a normal and a reduced ejection fraction. Progress in Cardiovascular Disease,
49, pp. 229–240, 2007.
[57] Aurigemma, G.P., Silver, K.H., Preist, M.A. & Gaasch, W.H., Geometric changes allow
normal ejection fraction despite depressed myocardial shortening in hypertensive left ventricular hypertrophy. Journal of the American College of Cardiology, pp. 195–202, 1995.
WIT Transactions on State of the Art in Science and Engineering, Vol 35, © 2008 WIT Press
www.witpress.com, ISSN 1755-8336 (on-line)
WITPress_RRPS_ch004.indd 85
4/29/2008 5:09:37 PM
86 R EPAIR AND R EDESIGN OF PHYSIOLOGICAL SYSTEMS
[58] Vinch, C.S., Aurigemma, G.P., Simon, H.U., Hill, J.C., Tighe, D.A. & Meyer, T.E., Analysis
of left ventricular systolic function using midwall mechanics in patients >60 Years of age
with hypertensive heart disease and heart failure. American Journal of Cardiology, 96,
pp. 1299–1303, 2005.
[59] Shimizu, G., Hirota, Y., Kita, Y., Kawamura, K., Saito, T. & Gaasch, W.H., Myocardial function is depressed in pressure-overload hypertrophy. Left ventricular midwall mechanics
in systemic arterial hypertension. Circulation, 83, pp. 1676–1684, 1991.
[60] Perlini, S., Muiesan, M.L., Cuspidi, C., Sampieri, L., Trimarco, B., Aurigemma, G.P.,
Agabiti-Rosei, E. & Mancia, G., Ventricular hypertrophy and normalization of
chamber Geometry midwall mechanics are improved after regression of hypertensive left
ventricular hypertrophy and normalization of chamber geometry. Circulation, 103,
pp. 678–683, 2001.
[61] Yip, G.W., Zhang, Y., Tan, P.Y.H., Wang, M., Ho, P.Y., Brodin, L.A. & Sanderson, J.E.,
Left ventricular long-axis changes in early diastole and systole: impact of systolic function on diastole. Clinical Science, 102, pp. 515–522, 2002.
[62] Bursi, F., Weston, S.A., Redfield, M.M., Jacobsen, S.J., Pakhomov, S., Nkomo, V.T.,
Meverden, R.A. & Roger, V.L., Systolic and diastolic heart failure in the community. The
Journal of the American Medical Association, 296, pp. 2209–2216, 2006.
[63] Burkhoff, D., Maurer, M.S. & Packer, M., Heart failure with a normal ejection fraction.
is it really a disorder of diastolic function? Circulation, 107, p. 658, 2003.
[64] Kawaguchi, M., Hay, I., Fetics, B. & Kass, D.A., Combined ventricular and arterial
stiffening in patients with heart failure and preserved ejection fraction: implications for
systolic and diastolic reserve limitations. Circulation, 107, pp. 714–720, 2003.
[65] MacIver, D.H. & Townsend, M., A novel mechanism of heart failure with normal ejection
fraction. Heart, 94, pp. 446–449, 2008.
[66] Folland, E.D., Parisi, A.F., Moynihan, P.F., Jones, D.R., Feldman, C.L. & Tow, D.E.,
Assessment of left ventricular ejection fraction and volumes by real-time, two-dimensional
echocardiography. A comparison of cineangiographic and radionuclide techniques. Circulation, 60, pp. 760–766, 1979.
[67] Lentner, C., (Ed.). Geigy Scientific Tables. Heart and Circulation, Vol. 5, 8th ed., Ciba-Geigy,
p. 72.
[68] Aurigemma, G.P., Silver, K.H., Priest, M.A. & Gaasch, W.H., Geometric changes allow
normal ejection fraction despite depressed myocardial shortening in hypertensive left
ventricular hypertrophy. Journal of the American College of Cardiology, 26, pp. 195–202,
1995.
[69] Willenheimer, R., Cline, C., Erhardt, L. & Israelsson, B., Left ventricular atrioventricular
plane displacement: an echocardiographic technique for rapid assessment of prognosis in
heart failure. Heart, 78, pp. 230–236, 1997.
[70] Sanderson, J.E., Wang, M. & Yu, C.M., Tissue Doppler imaging for predicting outcome
in patients with cardiovascular disease. Current Opinions in Cardiology, 19, pp. 458–463,
2004.
[71] Wang, M., Yip, G.W., Wang, A.Y., Zhang, Y., Ho, P.Y., Tse, M.K., Yu, C.M. & Sanderson, J.E.,
Tissue Doppler imaging provides incremental prognostic value in patients with systemic
hypertension and left ventricular hypertrophy. Journal of Hypertension, 23, pp. 183–191,
2005.
[72] Wang, M., Yip, G., Yu, C.M., Zhang, Q., Zhang, Y., Tse, D., Kong, S.L. & Sanderson,
J.E., Independent and incremental prognostic value of early mitral annulus velocity in
patients with impaired left ventricular systolic function. Journal of the American College
of Cardiology, 45, pp. 272–277, 2005.
WIT Transactions on State of the Art in Science and Engineering, Vol 35, © 2008 WIT Press
www.witpress.com, ISSN 1755-8336 (on-line)
WITPress_RRPS_ch004.indd 86
4/29/2008 5:09:37 PM