Download Modeling ventricular contraction with heart rate changes

Journal of Theoretical Biology 222 (2003) 337–346
Modeling ventricular contraction with heart rate changes
J.T. Ottesen*, M. Danielsen
Department of Mathematics and Physics, Roskilde University, Postbox 260, Roskilde DK-4000, Denmark
Received 12 February 2002; received in revised form 29 November 2002; accepted 6 December 2002
Abstract
Recently, a mathematical model of the pumping heart has been proposed describing the heart as a pressure source depending on
time, volume and flow. The underlying concept is based on a new two-step paradigm that allows separation between isovolumic
(non-ejecting) and ejecting heart properties. The first step describes the ventricular pressure in the isovolumic ventricle. In the
following step, the isovolumic description is extended with the ejection effect in order to embrace the pumping heart during actual
blood ejection. The description of the isovolumic heart properties plays a crucial role in this paradigm. However, only a single
isovolumic model has previously been used restricting the heart rate to 1 Hz: In this paper, a family of models describing the
isovolumic contracting ventricle are critically examined. A characterization of what constitutes an optimal model is given and used
as a criteria for choosing the optimal model in this family. Moreover, and this is indeed a point, the proposed model in this study is
valid for arbitrary heart rates and based on experimental data. The model exhibits all major features of the ejecting heart, including
how ventricular pressure and flow vary in time for various heart rates and how stroke volume and cardiac output vary with heart
rate. The modeling strategy presented embraces the same steps and demarcations as those suitable for clinical examination whereby
new experiments are suggested.
r 2003 Elsevier Science Ltd. All rights reserved.
Keywords: Mathematical modeling; Ventricular contraction; Variable heart rate; Ejection effect
1. Introduction
In 1994, Mulier described the isovolumic ventricle as a
pressure source depending on time and volume (Mulier,
1994). This mathematical model was based on the Frank
mechanism from 1895 stating that the isovolumic
ventricular pressure increases with volume (Frank,
1959) The model also embraces ejecting heart properties
when it is extended with a description of the ejection
effect (Danielsen, 1998). In this formulation, the
ventricle is considered as a pressure source depending
on time, volume and outflow. The model is based on a
two-step paradigm in which the first step describes
ventricular pressure in the isovolumic (non-ejecting)
ventricle as a pressure source depending on time and
ventricular volume (Danielsen et al., in preparation).
The second step involves actual blood ejection and
inclusion of the ejection effect. When an isovolumic
model predicts ventricular pressure during ejection, an
ejection effect is identified that consists of positive and
negative effects of ventricular blood ejection (Danielsen,
*Corresponding author. Tel.: +45-46-74-22-98; fax: +45-46-74-30-20.
E-mail address: [email protected] (J.T. Ottesen).
1998). The computed results embrace all major features
of the ejecting ventricle and agree with various experimental results (Danielsen and Ottesen, 2001; Danielsen
et al., in preparation). Also, the definition of ventricular
elastance, based on the classical approach @pv =@Vv ; is
valid during the entire heart beat (Palladino et al., 1998;
Danielsen et al., in progress). Since the two-step
paradigm allows separation between isovolumic and
ejecting heart properties these basic heart properties can
be investigated individually.
Isovolumic models play a crucial role in this two-step
paradigm. However, analysis of alternative analytical
formulations have only received very limited attention.
Thus, our study presents a warranted investigation of
various experimentally based descriptions. In particular,
a simple isovolumic model is presented that contains
fewer physiological parameters (the number is reduced
from 6 to 4) and exhibits the same close agreement with
actual measured ventricular pressure in dog hearts as
the model by Mulier (1994). Our model is based on
experimentally obtained data and embraces arbitrary
heart rate in contrast to earlier models. By introducing
arbitrary heart rate, the model paradigm may be useful
in a broader range of applications. This includes
0022-5193/03/$ - see front matter r 2003 Elsevier Science Ltd. All rights reserved.
doi:10.1016/S0022-5193(03)00040-7
338
J.T. Ottesen, M. Danielsen / Journal of Theoretical Biology 222 (2003) 337–346
treatment of pathological situations like hypertension,
enlarged hearts and other cardiovascular deceases.
Section 2 describes the model of the pumping
ventricle. First, the structure of the isovolumic model
is developed based on experimental findings. Subsequently, the actual analytical expression describing the
ventricular contraction is developed and investigated.
The optimal description is the simplest possible expression containing the fewest parameters each having
physiological interpretation and that generates computed results in close agreement with experimental
findings. Furthermore, the description should be valid
for arbitrary heart rates. The extension allowing
arbitrary heart rates is presented in Section 3. Finally,
the model is coupled to a three-element windkessel
model of the systemic part of the cardiovascular system
in Section 4. It is shown that the model exhibits all
major features of the ejecting heart, including how
ventricular pressure and flow depend on heart rate and
how stroke volume and cardiac output vary with heart
rate. Section 5 gives a discussion of the model and the
modeling process. It is revealed that the approach taken
gives an optimal description which exhibits excellent
agreement with known experimental findings including
isovolumic contraction in dog hearts with arbitrary
heart rates. However, a complete ventricular description
is only possible with the ejection effect. The strategy
presented here is suitable for experimental examination,
since it embraces the same steps and demarcations as
those frequently adopted in experiments. In addition to
the types of experiments used when developing and
examining the isovolumic ventricular model, the ejection
effect requires further experimental investigations.
2. Isovolumic ventricular contraction
Otto Frank demonstrated in 1895 that ventricular
pressure in the isovolumic (non-ejecting) contracting
frog heart increased with ventricular volume Vv to an
upper physiological limit. The pressure varies between
an isovolumic minimum pd ðVv Þ and an isovolumic
maximum ps ðVv Þ which may be expressed as
pv ðt; Vv Þ ¼ pd ðVv Þ þ ½ps ðVv Þ pd ðVv Þ f ðtÞ;
ð1Þ
where t represents time and f ðtÞ is a continuous
function, termed the activation function. This function
f generates a bell-shaped curve that varies between 0
and 1: Mulier adopted this approach and developed his
mathematical description from experiments on dog
hearts with a fixed heart frequency equal to 1 Hz
(Mulier, 1994). He showed that
pd ðVv Þ ¼ aðVv bÞ2 ;
ð2Þ
where the parameter a relates to ventricular elastance
during relaxation and b represents ventricular volume
for zero diastolic pressure. Measurements of peak
developed pressure revealed that
ps ðVv Þ pd ðVv Þ ¼ cVv d;
ð3Þ
where the parameters c and d relate to the volumedependent and volume-independent components of
developed pressure, respectively. Of course, Eqs. (2)
and (3) hold solely in the limited physiological range
considered here. Using Eqs. (1)–(3), the isovolumic
ventricular pressure can be described by
pv ðt; Vv Þ ¼ aðVv bÞ2 þ ðcVv dÞf ðtÞ:
ð4Þ
The activation function f is selected such that the
mathematical description of the ventricle agrees with
actual measured isovolumic ventricular pressure for
different volumes. In the following, a number of
activation functions are discussed. For each activation
function, computed ventricular pressure is compared
with the actual measured counterpart in dog hearts
adopted from Mulier (1994). The method used is the socalled Lavenberg–Marquardt algorithm which has
become a standard for dealing with least-square method
for non-linear models. The Lavenberg–Marquardt
algorithm is an iterative method. The step is in the
direction of the gradient and in calculating the step size
it uses the diagonal elements in the Hessian (Press et al.,
1989).
(a) Muliers approach: In 1994, Mulier suggested the
activation function to be described by
(
a
ð1 eðt=tc Þ Þ;
0ptptb ;
gðtÞ ¼
ð5Þ
ðt=tc Þa ððttb Þ=tr Þa
Þe
; tb ototh ;
ð1 e
such that f ðtÞ ¼ gðtÞ=gðtp Þ; where tp represents time for
peak ventricular pressure, t is time and tc ; tr ; a are
ventricular parameters (Mulier, 1994). The parameters
tc and tr represent contraction (increase in pressure) and
relaxation (pressure decreases), respectively. The parameter a describes the overall rate of onset of these
processes and tb denotes the time when the relaxation
process starts. The heart period is represented by th :
Fig. 1 compares measured isovolumic ventricular pressure with the model computed pressure using Eqs. (4)
and (5). Table 1 depicts the parameter values and the
correlation coefficient R2 between computed and
measured results. Notice that the onset of contraction
and heart rate are additional parameters. For the
isovolumetric contraction, the onset of contraction is a
fitting parameter too, but the heart rate is fixed. Later on
when allowing varying heart rates it becomes a fitting
parameter too. On the other hand, the parameter tb is
calculated from the others since f 0 ðtp Þ ¼ 0 implies
(
a=ða1Þ ðtp =tc Þa 1=ða1Þ )
tr
e
tb ¼ tp 1 :
ð6Þ
a
tc
1 eðtp =tc Þ
J.T. Ottesen, M. Danielsen / Journal of Theoretical Biology 222 (2003) 337–346
339
160
line 1
line 2
140
120
100
80
60
40
20
0
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Fig. 1. Measured isovolumic ventricular pressure for seven different volumes in dog hearts ðÞ taken from Mulier (1994). Superimposed is the best fit
using Eq. (4) and the Mulier activation function (5) (dashed). The nearly horizontal lines are included solely to indicate that all seven curves are fitted
using the same parameters.
Table 1
Parameter values for the Mulier activation function (5) which
generates the best fit to measured isovolumic ventricular pressures in
dog hearts for seven different ventricular volumes
Parameter
All seven curves
tc
tr
a
tp
tb
R2
0:13561 s
0:20441 s
2.68440
0:23705 s
0:05366 s
0.99753
R2 is the correlation coefficient for all seven curves in Fig. 1.
Experimental results are adopted from Mulier (1994).
and relaxation, respectively. Later on when allowing
varying heart rate, b becomes in one-to-one correspondence with the heart rate. The parameters n and m; to be
discussed in detail later on, characterize the contraction
and relaxation phases of the ventricle. The number of
parameters in this description is reduced to four. The
polynomial function has compact support and is a very
simple algebraic expression.
(c) Gamma distribution: The activation function may
also be described by a gamma distribution such that
ðt td Þ
m1
exp m 1 gðtÞ ¼ ðt td Þ
;
ð8Þ
a
Thus, the number of parameters is equal to six including
the onset of contraction.
(b) Polynomial expression: A polynomial of degree
(n; m) provides a simple expression for the activation
function f ðtÞ ¼ gðtÞ=gðtp Þ with
(
ðt aÞn ðb tÞm ; aptpbðHÞ;
gðtÞ ¼
ð7Þ
0;
bðHÞototh ;
where t represents time and td ; m and a are ventricular
parameters. In this case, it is difficult to give sensible
interpretations of the parameters. Moreover, as in the
Mulier approach, the gamma distribution lacks compact
support.
(d) Combined exponentials: Two exponential functions
may provide a description of the isovolumic ventricular
pressure by
xðt td Þ
gðtÞ ¼
ð9Þ
expð1 þ e ðt td ÞÞa þ expðt td 1Þ
where a; b; n and m are ventricular parameters, and th
and H are the heart period and frequency, respectively.
The parameter tp fulfills g0 ðtp Þ ¼ 0: Thus, tp ¼ ðbn þ
amÞ=ðn þ mÞ and gðtp Þ ¼ nn mm ½ðb aÞ=ðn þ mÞnþm :
Fig. 2 displays the best fit between computed isovolumic
ventricular pressure and the corresponding measured
ventricular pressure, using the ventricular pressure
model (4) and function gðtÞ in Eq. (7). Table 2 displays
the parameter values and the correlation coefficient R2 :
The parameters a and b denote the onset of contraction
in which the ventricular properties are embedded in the
parameters x; e; a and td : Like the gamma distribution
(8) and Mulier’s activation function (5), the function (9)
does not offer compact support nor is it build solely of
algebraic functions.
(e) Product of Hill functions: By using two Hill
functions, the activation function can be described by
f ðtÞ ¼ gðtÞ=gðtp Þ with
ðt td Þa
Bb
gðtÞ ¼
;
ð10Þ
Aa þ ðt td Þa Bb þ ðt td Þb
J.T. Ottesen, M. Danielsen / Journal of Theoretical Biology 222 (2003) 337–346
340
160
line 1
line 2
140
120
100
80
60
40
20
0
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Fig. 2. Measured isovolumic ventricular pressure for seven different volumes in dog hearts ðÞ taken from Mulier (1994). Superimposed is the best fit
using Eq. (4) and the polynomial activation function (7) (dashed). The nearly horizontal lines are included solely to indicate that all seven curves are
fitted by use of the same parameters.
Table 2
Parameter values for the polynomial activation function (7) which
generates the best fit to measured isovolumic ventricular pressure in
dog hearts for seven different ventricular volumes
Parameter
All seven curves
a
b
n
m
R2
0:07642 s
0:68124 s
2.05023
3.68662
0.99709
2
R is the correlation coefficient for all seven curves. Experimental
results are adopted from Mulier (1994).
Table 3
Comparison of correlation coefficients among the activation functions
(5)–(10)
Function
R2
Max. res. (mmHg)
(a) Mulier’s
(b) Gamma
(c) Exponential
(d) Hill functions
(e) nm-polynomial
0.99753
0.99310
0.99753
0.99522
0.99709
5
8
5
7
6
R2 is the correlation coefficient for all seven curves. The maximum
residuals are given by Max. res.
where
where a; b; A; B and td are ventricular parameters. The
function (10) is build of algebraic functions and involves
fractions of power functions. Consequently, it is more
complicated than a polynomial function. In addition,
Eq. (10) does not have compact support.
All ventricular models (5)–(10) exhibit a close agreement between computed and measured ventricular
pressure curves as shown in Figs. 1 and 2. Also, the
correlation coefficients and maximum of the residuals
are almost equal as displayed in Table 3. In addition, all
the parameters of the ventricular models, apart from the
gamma distribution, are given physiological interpretations. The polynomial model (7) is the simplest possible
choice among the candidates. With only four parameters
it has the lowest number of parameters. Consequently,
the polynomial description of the activation function is
used in the following to describe the isovolumic
ventricular pressure:
pv ðt; Vv Þ ¼ aðVv bÞ2 þ ðcVv dÞf ðtÞ;
ð11Þ
8
>
<
ðt aÞn ðb tÞm
mþn ;
f ðtÞ ¼ nn mm ½ðb aÞ=ðm þ nÞ
>
: 0;
aptpbðHÞ;
ð12Þ
bðHÞototh ;
where nn mm ½ðb aÞ=ðm þ nÞmþn is a normalizing factor,
such that maxff ðtÞg ¼ 1: The parameter a represents
time for onset of contraction and b time for end of the
active force generation. The fact that the nm-polynomial
function f has compact support is very important. This
is the reason why we are able to allow arbitrary heart
rate, as we will show in the next section. The other
models lack compact support, explaining why attempts
to extend such models have failed. The parameters n and
m characterize contraction and relaxation, respectively.
They are not completely uncoupled, however, similar
problems appear in the other models. A higher value of
n results in lower slope on the ascending and an
increased slope on the descending part of f as shown
in Fig. 3. When the parameter m rises, the slope on the
J.T. Ottesen, M. Danielsen / Journal of Theoretical Biology 222 (2003) 337–346
341
200
pv [mmHg]
150
100
50
0
0
Fig. 5. Measured isovolumic ventricular pressure in dog hearts for
different heart rates (Regen et al., 1993).
0.2
0.4
0.6
0.8
1
t [s]
220
200
line 1
line 2
210
Peak time (msec)
Fig. 3. Isovolumic ventricular pressure computed using the isovolumic
model (11) when m ¼ 2:2 and n ¼ 1 (dotted), n ¼ 2 (solid) to n ¼ 3
(dashed). The peak pressure moves to the right as n increases for
m fixed.
200
190
180
170
160
pv [mmHg]
150
150
40
60
80
100
120
140
160
180
200
Heart rate H (b/min)
Fig. 6. Experimental data extracted from Fig. 5 showing time for peak
pressure tp as a decreasing sigmoidal function in heart rate H ðÞ
(Regen et al., 1993). Superimposed is the best fit using the Hill function
(13) (dashed). The peak times and pressures corresponding to the two
curves in Fig. 5 with heart rates larger than 150 beats=min are
excluded, since these data can only be uncertainly estimated based
on the figure.
100
50
0
0
0.2
0.4
0.6
0.8
1
t [s]
Fig. 4. Computed isovolumic pressure also using Eq. (11) when n ¼ 2
and m ¼ 1 (dotted), m ¼ 2:2 (solid) and m ¼ 3 (dashed). The peak
pressure moves to the left as m increases for n fixed.
ascending part increases while the slope on the
descending part decreases as demonstrated in Fig. 4.
3. Isovolumic model and change in heart rate
In an experiment on dogs in 1993, Regen et al. found
that peak isovolumic pressure drops and the pressure
curve becomes narrower when heart rate increases as
shown in Fig. 5 (Regen et al., 1993). They also showed
that the measured isovolumic ventricular pressure
curves, shown in Fig. 5, are almost identical when
normalized with respect to time and peak isovolumic
pressure. Consequently, the isovolumic ventricular
model (11) can be modified to include changes in heart
rate by scaling time and peak values of the activation
function f : This is possible since the compact support
causes the ventricular end of contraction, b; to appear
explicitly in Eq. (12). We will introduce these modifications by extracting two sigmoidal scaling relations. The
experimental data displayed in Fig. 5 reveal a clear
decreasing sigmoidal relation between time for peak
pressure tp and heart rate H as shown in the Fig. 6.
Fig. 6 also displays a close agreement between data and
the sigmoidal curve given by the Hill function:
yn
tp ðHÞ ¼ tp; min þ
ð13Þ
ðtp; max tp; min Þ;
H n þ yn
where y represents the median and n the steepness of the
relation, tp; min and tp; max denote the minimum and
maximum values, respectively. Recognizing that time
for peak pressure tp is related to the parameter b in the
isovolumic pressure model (11) as
tp ðHÞ ¼ a þ
n
ðb aÞ;
nþm
ð14Þ
J.T. Ottesen, M. Danielsen / Journal of Theoretical Biology 222 (2003) 337–346
342
220
250
line 1
line 2
200
200
190
piso [mmHg]
Peak pressure (msec)
210
180
150
100
170
160
40
50
60
80
100
120
140
160
180
200
Heart rate H (b/min)
Fig. 7. Experimental data extracted from Fig. 5 showing peak pressure
pp as an increasing sigmoidal function in heart rate H ðÞ (Regen et al.,
1993). Superimposed is the best fit using the Hill function (16)
(dashed). The peak times and pressures corresponding to the two
curves in Fig. 5 with heart rates larger than 150 beats=min are
excluded, since these data can only be uncertainly estimated based
on the figure.
the change in tp with heart rate can be introduced into
the isovolumic pressure model by modifying the parameter b such that it becomes a function of heart rate H:
nþm
am
tp ðHÞ ð15Þ
bðHÞ ¼
n
n
with tp ðHÞ as in Eq. (13).
The experimental results in Fig. 5 show that peak
ventricular pressure pp and heart rate H are related by
an increasing sigmoidal curve as illustrated in Fig. 7. It
can be described by the Hill function
HZ
pp ðHÞ ¼ pp; min þ
ð16Þ
ðpp; max pp; min Þ;
H Z þ fZ
where f is the median, Z represents the steepness of the
relation and pp; min and pp; max represent the minimum
and maximum values, respectively. As shown in Fig. 7,
the sigmoidal relation (16) is in close agreement with
actual measured results.
By combining the isovolumic model (11), the relation
between b and heart rate (15) and the function relating
peak pressure and heart rate (16), the isovolumic
pressure as a function of time, volume and heart rate
is given by
pv ðVv ; t; HÞ ¼ aðVv bÞ2 þ ðcVv dÞf ðtÞ;
where
f ðt; HÞ ¼
8
>
< pp ðHÞ
>
: 0;
ðt aÞn ðbðHÞ tÞm
;
aÞ=ðm þ nÞmþn
nn mm ½ðbðHÞ
ð17Þ
0
0
0.1
0.2
0.3
0.4
0.5
0.6
t [s]
Fig. 8. Isovolumic ventricular pressure when heart rate H is 1 Hz
(solid). Peak pressure increases and pressure curves become more
narrow when heart rate is augmented (H ¼ 1:1 Hz; dashed, H ¼ 1:25;
dotted and H ¼ 1:3; dash-dot). Pressure drops and pressure curves
become broader as heart rate decreases (H ¼ 0:9 Hz; dashed, H ¼
0:75; dotted and H ¼ 0:6; dash-dot).
Table 4
Control parameter values for the isovolumic model (17)
Parameter
a
b
c
d
n
m
a
Value
0:007 mmHg=ml
5 ml
1:6 mmHg=ml
1 mmHg
2
2.2
0s
2
Parameter
Value
tp; min
tp; max
f
n
pp; min
pp; max
y
Z
0:1859 s
0:2799 s
1
9.9
0.842
1.158
1
17.5
pressure increases and the pressure curves become more
narrow when heart rate is raised and vice versa in
agreement with experiments. Control parameter values
for the isovolumic model (17) can be found in Table 4.
The total number of parameters becomes 15. Notice that
one may choose to use linear functions to describe tp ðHÞ
and pp ðHÞ instead of sigmoidal Hill functions in a
limited range. As a consequence, the total number of
parameters is reduced by 4. In any case, the structure of
the model is very simple, each part may be decoupled
and investigated individually and the coupling between
the parts is obvious and simple.
aptpbðHÞ;
bðHÞototh
ð18Þ
and the function bðHÞ is given by Eq. (15) and pp by
Eq. (16).
Fig. 8 displays isovolumic pressure curves during
various heart rates calculated from the model (17). Peak
4. Extension of the model allowing ejection
The isovolumic model (17) exhibits the major features
of an ejecting human heart when it is coupled to a
description of the arterial tree as the three-element
windkessel model and filled from a constant pressure
reservoir pr ; as shown in Fig. 9. Thus, during ejection
J.T. Ottesen, M. Danielsen / Journal of Theoretical Biology 222 (2003) 337–346
pa
Rin
700
Qs
Qv
Q in
ps
600
R0
pv
pr
343
500
Rs
Cs
Qv [ml/s]
Qc
400
300
200
Fig. 9. The isovolumic pressure model (17) coupled to a three-element
modified windkessel arterial load and a venous pressure reservoir pr :
The gray region represents both the heart and the venous pressure
reservoir, the white region contains the three-element modified
windkessel arterial load. R0 represents the characteristic aortic
impedance, Rs the total peripheral resistance, Cs the total arterial
compliance, Qin the flow into the ventricle, Qv the flow out of the
ventricle, Qs the flow through the peripheral system and Qc the flow
stored in the elastic arteries. The mitral (left) and the aortic (right)
valves are indicated as diodes.
100
0
0
0.1
0.2
0.3
0.4
t [s]
0.5
0.6
0.7
0.8
Fig. 11. Computed ventricular outflow Qv during blood ejection when
H ¼ 0:8 Hz; dotted, H ¼ 1 Hz; solid and H ¼ 1:2 Hz; dashed.
80
140
75
120
70
SV [ml]
80
65
60
60
v
p &p
ao
[mmHg]
100
55
40
50
20
0
0
45
0.8
0.1
0.2
0.3
0.4
t [s]
0.5
0.6
0.7
0.8
Fig. 10. Computed ventricular pressure pv and root aortic pressure pao
during blood ejection when H ¼ 0:8 Hz; dotted, H ¼ 1 Hz; solid and
H ¼ 1:2 Hz; dashed.
ventricular pressure pv and root aortic pressure pa are
guided by
Rs þ R0
1
p’s ¼ ps þ
pv ðt; Vv Þ;
ð19Þ
R0 Cs
R0 Rs Cs
1
1
V’v ¼
ps pv ðt; Vv Þ;
R0
R0
ð20Þ
pa ¼ ps R0 V’ v ;
ð21Þ
where R0 is the characteristic aortic impedance, Rs is the
total peripheral resistance and Cs is the total arterial
compliance, see Danielsen (1998) for further details.
Figs. 10 and 11 show computed ventricular pressure and
outflow during ejection for a normal (H ¼ 1 Hz; solid),
1
1.2
1.4
1.6
1.8
2
2.2
2.4
H [Hz]
Fig. 12. Computed stroke volume SV when heart rate is raised from
0:8 to 2:5 Hz in steps of 0:1 Hz:
a lower (H ¼ 0:8 Hz; dotted) and higher heart rate
(H ¼ 1:2 Hz, dashed). Peak ventricular pressure and
outflow rise and the curves become more narrowed as
heart rate increases and vice versa as in experiments.
Stroke volume SV (the amount of blood volume ejected
during one heart rate) is almost unchanged while cardiac
output CO (CO ¼ SV H) increases with heart rate.
Figs. 12 and 13 illustrate the result of heart pacing in
which the heart rate is increased significantly. Stroke
volume drops while cardiac output continues to
increase. This development agrees with experiments
except that cardiac output is supposed to drop for high
heart rates (Melbin et al., 1982). This may be related to
the fact that the ventricle is not coupled to a closed
J.T. Ottesen, M. Danielsen / Journal of Theoretical Biology 222 (2003) 337–346
344
human circulation and that other effects such as the
various control mechanisms are lacking in this cardiovascular model or it may be related to the fact that
Eqs. (2) and (3) are only valid in a limited range.
motivation behind this model is guided by a new
two-step paradigm for ventricular modeling in which
isovolumic models play a crucial role. The first step
focuses on developing a mathematical model of the
isovolumic ventricular pressure. In the second step, an
analytical model of the ejection effect is added in order
to allow the model to embrace actual blood ejection.
Previously only the isovolumic model by Mulier has
been used which restricts the heart rate to 1 Hz:
Consequently, we examined a number of different
isovolumic models and found that the ventricular model
in Eq. (11) gives the simplest mathematical description
and contains the lowest number of parameters all given
physiological interpretation. In addition, this choice
allows arbitrary heart rate. By using experimental data
showing isovolumic ventricular pressure for a wide
range of different heart rates we extended the isovolumic
model (11) to include varying heart rates. When this
final model (17) is allowed to eject into a description of
the vasculature given by the three-element windkessel
model, it exhibits all the major features of the ejecting
ventricle, including changes in pressure and flow curves
during various heart rates. It also showed how stroke
volume and cardiac output varies with heart rate in
a limited range. Hereby an optimal description is
developed which exhibits an excellent agreement with
experiments during both isovolumic contractions and
ejecting heart beats for arbitrary heart rates.
Closer scrutiny of the computed ventricular pressure
and outflow curves reveals however a number of minor
discrepancies. Ventricular outflow curve is concave on
the descending part though it appears often convex in
5. Summary, discussion and outlook
We established a new mathematical model (17)
describing the isovolumic ventricular pressure as a
pressure source depending on both ventricular volume
and time in close agreement with experiments. The
130
120
110
CO [ml/s]
100
90
80
70
60
50
1
1.5
2
2.5
H [Hz]
Fig. 13. Computed cardiac output CO when heart rate is raised from
0:8 to 2:5 Hz in steps of 0:1 Hz:
150
140
1
2
130
120
Pressure[mmHg]
110
100
3
90
4
80
3
70
2
60
50
40
30
20
10
0
100
200
300
400
500
Time [ms]
Fig. 14. Positive and negative effects of ventricular ejection. Curve 1 is measured isovolumic pressure at fixed-end diastolic volume and curve 2 is the
computed pressure during ejection obtained from Mulier (1994) using measured volume Vv : Curve 3 shows measured ventricular pressure and curve 4
displays root aortic pressure for an ejecting beat. Comparing curves 2 and 3, measured ventricular pressure is lower in early systole (deactivation) and
higher later (hyperactivation) when compared to computed pressures. Adapted from Mulier (1994), Danielsen et al. (in preparation).
J.T. Ottesen, M. Danielsen / Journal of Theoretical Biology 222 (2003) 337–346
nature and the ejection period (systole) is too narrow.
Also, the ventricular pressure is concave at the top
descending part. These results are not related to the
specific choice of activation function (a)–(e), but caused
by the lack of the ejection effect identified when a model
based on isovolumic heart properties alone predicts
ventricular pressure during actual blood ejection. Model
predicted pressure is lower than measured pressure
during early ejection, denoted deactivation, and higher
later, termed hyperactivation. This is also observed
experimentally by Mulier as for instance shown in
Fig. 14. Deactivation may be related to muscle shortening during early ejection forcing crossbridge bonds
to detach and thus pressure to diminish. Formation of
new crossbridge bonds in the muscle fibers may explain
hyperactivation. The strength of hyperactivation is
subsequently guided by the available biochemical
energy. Also, De Tombe and Little (1994) concluded
from muscle experiments that this behavioral pattern
results from myocardial properties. It was previously
shown that arterial wave reflections and inertial effects
cannot fully explain the ejection effect (Danielsen, 1998;
Danielsen et al., 2000a).
The ejection effect has been introduced by a
modification of the activation function f such that it
becomes a function F of time t and ventricular outflow
Qv (Danielsen and Ottesen, 2001; Danielsen et al.,
2000b, in preparation):
F ðt; Qv Þ ¼ f ðtÞ k1 Qv ðtÞ þ k2 Q2v ðt tÞ;
t ¼ kt:
345
pathological situations, developments in the heart’s
contractile state may be observed using this paradigm.
Individuals with enlarged hearts may undergo partial
left ventriculectomy during which their chamber size is
reduced. This is believed to improve the heart function
(Rabbany et al., 2002). Improvements in the contractile
state may be observed by determining the parameter c
before and after partial left ventriculectomy. Similar
methods may be used during other pathological cases.
In all these situations, the contractile properties
embedded in the ejection effect have to be considered
carefully.
The strategy used to establish the isovolumic model
can be directly adopted in experiments, since the model
embraces the same steps and demarcations as frequently
adopted in experiments. With the new isovolumic
ventricular model in place the ejection effect can be
examined for all heart rates both experimentally and
theoretically. The two-step paradigm can also be applied
to various cardiovascular models and may be used to
gain new insights into cardiovascular diseases.
Acknowledgements
This work was supported by a grant from the Danish
Heart Foundation (99-1-2-14-22675) and by Trinity
College, Hartford, CT, USA.
ð22Þ
The two positive parameters k1 and k2 represent the
strength of deactivation and hyperactivation, respectively, while t is a time-varying time delay. The second
term on the right-hand side of Eq. (22) describes
deactivation and the term k2 Q2v ðt tÞ represents hyperactivation which becomes active t later in time than
k1 Qv : Thus, the time delay t ¼ kt allows for almost
immediate cycling of crossbridges in early systole and a
slower formation of bonds in late systole. The parameter
k (0oko1) relates to the change in the rate of
formation of new bonds with time.
The introduction of arbitrary heart rate will expand
the application potential for the modeling concept to
broader range of applications including many pathological situations. This requires determination of parameter values and knowledge about the physiological
significance of the parameters. The parameter values of
the model may vary among individuals and have to
be determined individually in each case. The degree of
variations among individuals requires further studies.
The physiological significance of each parameter is
carefully explained in this study. The significance of the
parameter c in the isovolumic model (4) is e.g. closely
linked to the contractile properties of the heart such that
a higher value relates to a better contractile state. In
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