Vol. 276, Issue 2, H582-H594, February 1999
Disparate effects of three types of extracellular acidosis on
left ventricular function
David S.
Berger,
Susan K.
Fellner,
Kimberly A.
Robinson,
Katherine
Vlasica,
Ivan E.
Godoy, and
Sanjeev G.
Shroff
Cardiology and Nephrology Sections, Department of Medicine,
University of Chicago, Chicago, Illinois 60637
 |
ABSTRACT |
Effects of
acidosis on muscle contractile function have been studied extensively.
However, the relative effects of different types of extracellular
acidosis on left ventricular (LV) contractile function, especially the
temporal features of contraction, have not been investigated in a
single model. We constituted perfusion buffers of identical ionic
composition, including Ca2+
concentration
([Ca2+]), to mimic
physiological control condition (pH 7.40) and three types of acidosis
with pH of 7.03: inorganic (IA), respiratory (RA), and lactic (LA).
Isolated rabbit hearts (n = 9) were
perfused with acidotic buffers chosen at random, each preceded by the
control buffer. Under steady-state conditions, instantaneous LV
pressure (Pv) and volume
(Vv) were recorded for a range
of Vv. The results were as
follows. 1) LV passive
(end-diastolic) elastance increased with IA and RA. However, this
increase may not be a direct effect of acidosis; it can be explained on
the basis of myocardial turgor. 2)
Although LV inotropic state (peak active
Pv and elastance) was depressed by
all three acidotic buffers, the magnitude of inotropic depression was
significantly less for LA. 3)
Temporal features of Pv were
altered differently. Whereas IA and RA reduced time to peak
Pv
(tmax) and
hastened isovolumic relaxation at a common level of LV wall stress, LA
significantly increased
tmax and retarded
relaxation. These results and a model-based interpretation suggest that
cooperative feedback (i.e., force-activation interaction) plays an
important role in acidosis-induced changes in LV contractile function.
Furthermore, it is proposed that LA-induced responses comprise two
components, one due to intracellular acidosis and the other due to
pH-independent effects of lactate ions.
pH; left ventricular inotropic state; left ventricular relaxation; cooperative feedback; activation-cross bridge dynamics; lactate ion
| |
INTRODUCTION |
MOST BIOCHEMICAL PROCESSES are sensitive to the pH of
the environment in which the processes occur. Such dependence on pH makes alteration of H+
concentration ([H+]) a
useful tool for examining muscle physiology. Experimental models of
acidosis usually employ inorganic acidosis (IA; decreasing the amount
of bicarbonate buffer in the perfusate or superfusate) or respiratory
acidosis (RA; increasing the percentage of
CO2 in the perfusion gas mixture).
Clinically, RA can occur when blood pCO2 rises in
the setting of pulmonary dysfunction, whereas bicarbonate wasting from
the kidneys or small intestine can cause IA. A form of metabolic
organic acidosis, lactic acidosis (LA), occurs as a result of anaerobic
metabolism, often because of cardiovascular dysfunction.
Various forms of acidosis have been studied extensively in skeletal and
cardiac muscle preparations, all sharing a common observation: acidosis
reduces inotropic state as measured by the ability of the muscle to
generate active force (17, 34, 45). Reduced myofilamental
Ca2+ sensitivity (rightward shift
of steady-state force-pCa relationship) is thought to be the primary
cause of negative inotropy; however, changes in cross-bridge dynamics
(reduced cycling rate and force per unit attached cross bridge) may
also play a role. Most of these studies have used RA or IA and have
focused on the magnitude of active force generation. Temporal features
of force generation (e.g., rates of rise and relaxation) can provide
additional insights into muscle contractile function. These aspects,
when studied, have been shown to be influenced variably by pH (34, 49,
51). However, recent studies have demonstrated that the proper analysis of certain temporal features (e.g., those describing relaxation) should
incorporate changes in the magnitude of active pressure (force) (4, 23,
50).
In the present study, we examined the effects of all three types of
acidosis on left ventricular contractile function in isolated rabbit
hearts. Identical levels of extracellular acidosis were produced, with
perfusion buffers varying only in bicarbonate, CO2, or lactate concentration;
Na+,
K+,
Mg2+,
Pi, and
Ca2+ concentrations
([Na+],
[K+],
[Mg2+],
[Pi], and
[Ca2+], respectively)
were identical. Our results and model-based interpretations suggest
that cooperative feedback (i.e., force-activation interaction) plays an
important role in acidosis-induced changes in left ventricular contractile function. In addition, we found that effects of LA were
markedly different from those of RA or IA, especially regarding the
temporal aspects of contraction. We hypothesize that these disparities
are caused by pH-independent effects of lactate ions.
| |
METHODS |
All protocols were reviewed and approved by The University of Chicago
Institutional Animal Care and Use Committee and conform with the
Guide for the Care and Use of Laboratory
Animals published by the National Institutes of Health
[DHHS Publication No. (NIH) 85-23, Revised 1985, Office of
Science and Health Reports, Bethesda, MD 20892].
Experimental Preparation and Isolated Heart Setup
Experiments were performed on hearts isolated from adult male rabbits
(New Zealand White) weighing 2.0-3.0 kg. Rabbits were preanesthetized with 5.0 mg/kg xylazine (Ben Venue Laboratory, Bedford,
OH) and 0.01 mg/kg glycopyrrolate (Robinul-V; Elkins-Sinn, Cherry Hill,
NJ). After 10 min, rabbits were anesthetized with 30-50 mg/kg
ketamine (Kedalar; Parke-Davis, Morris Plains, NJ) and 1.0 mg/kg
acepromazine (Fermenta Animal Health, Kansas City, MO). Tracheotomy was
performed after anesthesia, and rabbits were artificially ventilated
(Harvard Ventilator, model 683; Harvard Apparatus, South Natick, MA)
with a 95% O2-5%
CO2 mix at a respiratory rate of
43 breaths/min and a tidal volume of 25-30 ml. After median sternotomy and ligation of great vessels, a metal cannula connected to
the perfusion system was inserted into the brachiocephalic artery and
immediately flushed with heparinized saline (3.0 ml, 1,000 U/ml).
Retrograde perfusion of the coronary arteries was then begun at a
constant perfusion pressure of 80 mmHg. The heart was perfused at
37°C with oxygenated modified Krebs-Henseleit solution (see
Perfusate Formulary), which was not
recirculated. Connective tissue was cut away and the heart removed from
the chest while being constantly perfused. Thus at no time was coronary circulation interrupted. Nine animals were studied.
A thin latex balloon, secured at the end of a piston-cylinder device
attached to a linear motor, was positioned in the left ventricle via
the mitral orifice. A thread tied to the end of the balloon was passed
through the apex of the left ventricle to secure the balloon in the
chamber. A purse string was tied around the mitral orifice to secure
the heart to the piston-cylinder device. The piston position was sensed
by a linear voltage-displacement transformer. All hearts were paced
using unipolar electrodes attached to the apex of the left ventricle.
More comprehensive details of the isolated heart preparation have been
described previously (3, 44).
Perfusate Formulary
The control perfusate (CT) was made by adding the ingredients listed in
Table 1 to double-distilled water at a
temperature of 30°C. When the perfusate was warmed to 37°C and
bubbled through the appropriate gas mixture (Table 1), free ionic
concentrations of H+,
Ca2+, and other electrolytes
reached their desired control values. The three acidotic perfusates
were modified from the control solution as follows (Table 1).
Respiratory acidosis.
RA in this perfusate was achieved by increasing the
%CO2-to-%O2
ratio from its control value.
Inorganic acidosis.
IA perfusate was formulated by reducing the concentration of
bicarbonate (NaHCO3) in solution.
Lactic acidosis.
LA perfusate was created by adding racemic lactic acid to the solution.
The dissociation of this compound results in increased [H+].
All three perfusion buffers were designed to achieve identical
[Na+],
[K+],
[Mg2+],
[Pi], and
[Ca2+]. Only sodium
bicarbonate- and sodium chloride-containing compounds were varied to
achieve this (Table 1). It should be noted that the final pH and free
ionic concentrations are very sensitive to temperature and actual gas
mixture. For this reason, high-tolerance gas mixtures were used
(±5% instead of the typical ±10%) and pH, electrolytes, and
free Ca2+ were monitored
throughout the experiment. In addition, perfusion buffers were prepared
in 4-liter batches so that an entire experiment could be performed
using the same buffers.
Experimental Protocol and Data Collection
In this study, the single-beat Frank-Starling (SBFS) protocol (10) was
used from which active and passive functional states of the heart were
evaluated for each experimental condition (control and acidosis). In
the SBFS protocol, the heart was allowed to beat isovolumically at a
constant reference volume (Vref)
until it reached steady state, at which time the left ventricular
chamber volume (Vv) was changed
over a short period of time in late diastole. After several cardiac
cycles occurred at this perturbed volume, Vv was changed back to
Vref, again in late diastole. The
left ventricular pressure (Pv)
and Vv from two cardiac cycles
were sampled: one steady state at
Vref and the other as the first
beat after the volume change (the beat that will be analyzed). A full SBFS protocol consisted of 10-12 equispaced volume changes
centered around Vref. If an
arrhythmia occurred during the data collection, for example, a
mechanically induced premature contraction, that Pv-Vv
pair was omitted from the analysis. This was not common with the
control perfusate but occurred more frequently with acidosis, not a
surprising outcome given that acidosis is known to be arrhythmogenic.
The following data-collection protocol was used for all hearts. After
the heart was isolated and instrumented, an SBFS protocol was performed
and analyzed (see Data Analysis) to
identify Vmax, the volume that
resulted in maximum active Pv. The
volume was adjusted to be 80% of
Vmax and served as
Vref for the rest of the
experiment. The heart was then subjected to six SBFS protocols, with
the perfusate being changed after each one. The order of the type of
acidosis was chosen at random with each acidotic perfusate used once
and preceded by control perfusate, for example, CT-IA-CT-LA-CT-RA. Ample time, typically 10-12 min, was provided after a perfusate switch so that the heart reached steady state as identified by the
stability of the Pv time course.
Thus, after surgery, each experiment lasted ~75 min. Also, pH of the
perfusate was monitored continuously (pH Meter 59003 20; Cole-Parmer,
Vernon Hills, IL). A perfusate sample (14 ml) was taken just before
each SBSF protocol, from which relevant ionic concentrations were
measured (NOVA 2 Ionized Calcium Analyzer; NOVA Biomedical, Waltham,
MA, and Clinical System Synchron CX5CE; Beckman, Schaumburg, IL).
Data Analysis
All analyses were performed using left ventricular pressure and volume
data obtained from the first perturbed beat of the SBFS protocol. The
effect of acidosis on the left ventricular end-diastolic (passive)
pressure-volume relationship was evaluated quantitatively by comparing
end-diastolic Pv
(Ped) and passive elastance
(Ep) at
Vref.
Ep was determined
by first fitting the Ped-Vv
data to the following exponential function (Fig.
1A)
![P<SUB>ed</SUB> = &agr;<SUB>1</SUB>[<IT>e</IT><SUP>&agr;<SUB>2</SUB>(V<SUB>v</SUB>−V<SUB>o</SUB>)</SUP> − 1]](/content/vol276/issue2/fulltext/H582/img001.gif)
|
(1)
|
where
1,
2, and
Vo are parameters to be
estimated. From Eq. 1,
Ep was defined as
|
(2)
|
Left ventricular active pressure
(Pact) was determined for all
Vv as the difference between
measured Pv and
Ped, and its peak value was
denoted Pmax. Left ventricular
Pmax-Vv
relationships were examined quantitatively by comparing
Pmax and active elastance (Ea) at
Vref. The
Pmax-Vv
relationship (Fig. 1A)
was fit to a second-degree polynomial
(Pmax = a0 + a1Vv + a2V2v, where a0,
a1, and
a2 are
parameters) from which
Ea was defined as
|
(3)
|
The left ventricular isovolumic relaxation process was
quantified by relating relaxation time
(Tr) to maximum
active stress (
max) for the
full range of Vv. Left ventricular
active wall stress (act) was
estimated using a thick-walled spherical model
|
(4)
|
where
Mv and 
are
left ventricular muscle mass and density ( = 1.05 g/ml),
respectively (18).
Tr was determined
as
|
(5)
|
where
t75 and
t25 are the times
at which stress falls to 75% and 25% of
max, respectively (Fig.
1B) (50). The relationship between
Tr and
max defines the relaxation
state of the left ventricle, and any increase or decrease in
Tr at a given
max would indicate slower or
faster relaxation, respectively (4, 50).
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Fig. 1.
A: left ventricular (LV) active ( )
and passive ( ) pressure-volume data derived from single-beat
Frank-Starling (SBFS) protocol. Passive (end-diastolic) and active
pressure-volume relationships were fit to an exponential function
(Eq. 1) and a second-degree
polynomial, respectively (solid lines). Passive and active elastance
were calculated from these fits according to Eqs.
2 and 3.
B: quantitation of temporal features
of LV active stress waveform. Except for an amplitude scale factor,
stress and pressure waveforms are identical for an isovolumic
contraction. Maximum active stress value
(max) was attained at
t = tmax. Relaxation
time (Tr) was
calculated as difference between times at which active stress
(act) falls to 75 and 25% of
max. Rise time (Trise) was
defined similarly as time difference between rise to 75% and rise to
25% of max.
|
|
Additional temporal features of the left ventricular pressure waveform
were determined. These were the time at which developed pressure
(stress) reached its maximum value
(tmax) and the
rise time
(Trise).
Trise was defined
similarly to Tr
except that the rising portion of the
Pv time course is used (Fig.
1B).
Statistical Analysis
One-way analysis of variance (ANOVA) was performed to compare
compositions of the different perfusate solutions. If data did not
satisfy the normality condition, Kruskal-Wallis one-way ANOVA on ranks
was performed (noted in Table 2). If
differences among the groups were statistically significant
(P < 0.05), all pairwise multiple comparisons were made using either the Student-Newman-Keuls or
Dunn's (nonnormal data) method.
To determine whether acidosis affects physiological function, values of
physiological variables for control and acidosis, as well as their
percent changes from control, were analyzed using one-way
repeated-measures ANOVA (RM-ANOVA). If the normality condition was not
satisfied, Friedman RM-ANOVA on ranks was performed. If differences
among the groups were statistically significant
(P < 0.05), all pairwise multiple
comparisons were made using either the Student-Newman-Keuls or Dunn's
(nonnormal data) method.
To further investigate the
Tr-max
relationship over the entire
max range, repeated-measures
analysis of covariance (RM-ANCOVA) was performed. RM-ANCOVA was
implemented using multiple linear regression with dummy variables (19).
With this implementation, effects of individual experimental conditions
on the
Tr-max
relationship (slope and intercept) could be identified readily. Effects
coding was used to construct the dummy variables to identify
intercondition (Cj) and
interrabbit (Ri) differences
|
(6)
|
where
j = 1, ..., nc 
1, with nc as the
number of experimental conditions
(nc = 6; three
acidotic and three control) and i = 1, ..., nr 1, with nr as the
number of rabbits
(nr = 9). The
experimental conditions were coded for
j as 1, control IA; 2, IA; 3, control
RA; 4, RA; 5, control LA; and 6, LA. The multiple regression model
reads
|
(7)
|
The
overall
Tr-max
relationship (quadratic function) is quantified by coefficients
b0,
b1, and
b2, which
correspond to the mean values for the entire data set (i.e., all
conditions, all animals). Coefficients
c0,j,
c1,j,
and
c2,j quantify the deviations of the jth
experimental condition from the overall
Tr-max
relationship. Similarly, coefficients
r0,i, r1,i,
and
r2,i
represent deviations of the ith rabbit
from the overall
Tr-max
relationship. Thus the effect of a particular type of acidosis can be
identified readily by examining the statistical significance of
coefficients
c0,j, c1,j,
and
c2,j.
This analysis was carried out using the stepwise regression algorithm
(Minitab, version 10.2).
| |
RESULTS |
Table 2 contains values (means ± SD) for pH and
[Ca2+],
[Na+],
[K+], and
[Pi] measured for
control and each acidotic perfusate. The control values are those at
the start of the experiment just before the first SBFS. As shown, pH
values for the three acidotic perfusates were the same and
significantly lower than control value. Equally important,
concentrations of relevant electrolytes, particularly
Ca2+, were the same in each
acidotic perfusate. Although
[Na+] or
[K+] in certain
acidotic perfusates were different statistically from control values
(Table 2), these differences were very small and, therefore, not likely
to be functionally important.
Figure 2 contains raw
Pv and
Vv data (perturbed beat from the
SBFS protocol) from all six conditions from one representative experiment. These data were used to analyze left ventricular passive and active pressure-volume relationships and relaxation behavior.
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Fig. 2.
Examples of LV pressure and volume data from SBFS protocol for
inorganic (IA; A), respiratory (RA;
B), and lactic acidosis (LA;
C). Each acidosis was preceded by
its control condition.
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|
Left Ventricular End-Diastolic (Passive) Pressure-Volume
Relationships
Passive pressure-volume relationships from a single experiment (same as
in Fig. 2) are shown in Fig.
3A. Only
IA and RA showed a small but consistent increase in
Ped for all
Ved. When data from all hearts are
considered (Table 3), the control values of
Ped and
Ep at
Vv = Vref were not different. Relative
to their control conditions, only IA and RA increased
Ped (at
Vref) and Ep (at
Vref) significantly (Fig. 3,
B and
C); the absolute values of these
quantities are given in Table 3.
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Fig. 3.
A: effects of IA, RA, and LA on LV
passive pressure (Ped)-volume
(Vv) relationships in 1 heart.
Raw data for these plots are from Fig. 2.
B: percent change in
Ped at
Vv = reference volume
(Vref) for all hearts.
C: percent change in passive elastance
(Ep) at
Vv = Vref for all hearts. All
significant acidosis-induced changes in
Ped,
Ep, and
Ped-Vv
relationships could be explained by coronary turgor (see
DISCUSSION). Data are means ± SE. * P < 0.05 vs. control.
 P < 0.05 for RA or IA
vs. LA.
|
|
Left Ventricular Peak Isovolumic (Active) Pressure-Volume
Relationships
The active pressure-volume curve was shifted down compared with control
in all types of acidosis (Fig.
4A,
derived from the data in Fig. 2). Reduction of
Pmax for a given
Ved was greater for IA and RA
compared with LA. Once again, analysis of data from all animals
revealed that control values of
Pmax and
Ea at
Vv = Vref were not different (Table 3).
This group analysis also showed that both
Pmax and
Ea at
Vv = Vref were reduced significantly by
IA and RA, whereas only Pmax at
Vref was diminished significantly by LA. Furthermore, whereas the negative inotropic effects of IA and RA
were not different from each other, both had a significantly larger
negative inotropic effect than LA. The same statistical results were
obtained from the analysis of percent changes from control (Fig. 4,
B and
C).
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Fig. 4.
A: effects of IA, RA, and LA on LV
maximum active pressure
(Pmax)-volume
(Vv) relationships in 1 heart,
showing reduced Pmax at all
Vv.
B: percent change in
Pmax at
Vv = Vref for all hearts.
C: percent change in active elastance
(Ea) at
Vv = Vref for all hearts. Data are
means ± SE. * P < 0.05 vs.
control. P < 0.05 for RA
or IA vs. LA.
|
|
Left Ventricular Relaxation
Figure 5A
shows the
Tr-max
relationships from the same experimental data as in Fig. 2. Again,
effects of IA and RA were similar to each other; the
Tr-max
points were shifted downward such that relaxation was hastened at all
max. In contrast, the
Tr-max points were shifted upward in LA, indicating that LA slowed left ventricular relaxation.
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Fig. 5.
Effects of IA, RA, and LA on
Tr-max
relationships in 1 heart. Both IA and RA hastened relaxation, whereas
LA slowed it.
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|
Analysis of
Tr-max
relationships revealed that the model given by Eq. 7 fit the entire data set well (all
max points, all conditions, all
rabbits; n = 429;
R2 = 0.95).
Except for the intercept for RA control, none of the control condition
coefficients was significantly different from the overall mean values.
The difference in the intercept for RA control was very small (+1 ms).
In contrast, all intercepts for the acidotic conditions were
significantly different from the overall mean values
(P < 0.05), as indicated
by the significant coefficients
c0,2,
c0,4, and
c0,6. The
stress-associated coefficients for the acidotic groups were not
different from the mean values. Figure 6
shows the regression results for all six conditions plotted together.
Note that the three control conditions are together, whereas LA lies
above these values (slower relaxation) and both RA and IA lie below
them (faster relaxation). From this analysis, it is clear that the
entire
Tr-max
relationships are similarly affected by IA and RA and that LA affects
this relationship differently over the entire range of
max. Furthermore, from
continuity, we can conclude that the effects of LA on the
Tr-max
relationship are significantly different from those of IA or RA.
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Fig. 6.
Results of multiple linear regression analysis in which intercondition
differences (i.e., controls vs. different acidosis) in
Tr-max
relationships were identified. Relationships for all 3 control
conditions were statistically not different from each other. With
respect to control condition, IA and RA shifted this relationship
downward and LA shifted it upward. Thus effects of LA on
Tr-max
relationship were significantly different from those of IA or RA.
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|
We want to provide a quantitative feel for
max-dependent and -independent
components responsible for the acidosis-induced net changes in
Tr at
Vref. Because the effects of IA
and RA on max and
Tr were similar
(Table 3), data from these two conditions were combined. With respect
to the control condition, acidosis (IA and RA combined) reduced
max and
Tr at
Vref by 24% (from 52 to 40 mmHg)
and 11% (from 83 to 74 ms), respectively (measured data, Table 3).
According to the data presented in Fig. 6, a portion (5.6%) of this
fall in Tr was
due to the reduction in max;
the rest (5.4%) was due to the IA/RA-induced hastening of relaxation.
In contrast, LA reduced max at
Vref by 10%, whereas Tr increased by
5% (measured data, Table 3). Thus the expected fall in
Tr due to
reductions in both max and pH
with LA is more than offset by some factor that slows relaxation.
Additional Temporal Features of Left Ventricular Pressure Waveform
Normalized (i.e., magnitude ranging from 0 to 1)
Pact time-course curves for each
type of acidosis and its control (at
Vv = Vref) are shown in Fig.
7A. These
plots show the effects of acidosis on important temporal landmarks.
Specifically, RA and IA caused maximum pressure to occur sooner. Once
again, LA had different effects; pressure rise was slower, and the
occurrence of peak pressure was delayed. As shown in Table 3 (absolute
values) and Fig. 7, B and
C (percent changes), these
observations are borne out quantitatively for all hearts.
Trise and
tmax were not different among the three controls and were increased significantly by
LA. In contrast, IA and RA caused a small but significant reduction in
tmax.
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Fig. 7.
Acidosis-induced changes in temporal features of LV pressure waveform.
A: LV normalized active pressure
(i.e., peak value of 1) at Vv = Vref in 1 heart. Both IA and RA
hastened pressure rise and fall, whereas LA retarded rise and fall.
B and
C: percent change in
Trise and
tmax,
respectively, at Vv = Vref for all hearts. Data are
means ± SE. * P < 0.05 vs.
control. P < 0.05 for RA
or IA vs. LA.
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| |
DISCUSSION |
IA, RA, and LA were studied together in the whole heart while
concentrations of Ca2+ and other
relevant electrolytes in the perfusates were tightly controlled (Table
2). Two primary observations will be discussed in detail. The first is
that the effects of RA and IA on all aspects of left ventricular
function were broadly similar. We will discuss these findings in light
of previous studies that, in general, showed similar results. The
second observation, unique to our study, is that LA has distinctly
different, and in some cases opposite, effects on LV function compared
with those of IA and LA. Possible mechanisms underlying these
observations will be presented.
Acidosis: Effect on Passive Pressure-Volume Relationships
Both IA and RA rotated left ventricular end-diastolic (passive)
pressure-volume relationships leftward (Table 3 and Fig. 3),
indicating an increase in passive chamber stiffness. Because coronary
perfusion pressure (Pcor) was
constant, the decrease in Pv with
acidosis has a potential to increase myocardial volume (intravascular,
extravascular, or both), which can augment both systolic and diastolic
Pv at a fixed
Vv (22). In a previous study, we
showed that, at a given Vv,
Pmax reductions of 35-40%, induced with hypocalcemia
([Ca2+] = 0.63 mM),
lead to 18-25% increases in
Ped and 25-30% increases in
Ep (43). Similar
increases in Ped and
Ep were found
when Pcor was increased (4). In
the present study, RA caused a 23% reduction in
Pmax coincident with a 40%
increase in Ped and a 28%
increase in Ep.
Similarly, IA caused Pmax to fall
26% while Ped and
Ep increased 25 and 28%, respectively. Thus changing the pressure gradient between
Pcor and
Pv in this study was accompanied by changes in diastolic properties similar to those seen previously (43).1
Furthermore, acidosis does not appear to alter passive force at a fixed
muscle length in isolated cardiac muscle studies [e.g., see Fig.
1 in Ricciardi et al. (39) and Fig. 1 in Orchard and Kentish
(34)], indicating no direct effects of acidosis on passive muscle
properties. These considerations lead us to suggest that changes in
left ventricular passive behavior in the present study are secondary to
myocardial turgor; it is not necessary to invoke any direct effects of
pH. That LA showed little changes in
Ped or
Ep is probably
due to the significantly smaller decrease in Pmax.
Respiratory and Inorganic Acidosis: Effect on Active Pressure
Development and Relaxation
All data were collected during steady-state conditions, at least 12 min
after a change of perfusate. This is important in the light of
observations made by Orchard (33) in an isolated cardiac muscle
preparation. A rather long transient response was observed in both
intracellular free Ca2+ signal and
developed tension after exposure to respiratory acidosis. In the
eventual steady state, the free
Ca2+ signal was slightly elevated
and prolonged compared with that of the control, and tension was
reduced. Similar behavior was observed with inorganic and lactic
acidosis as well. If this prolongation was due to impaired
Ca2+ uptake by the sarcoplasmic
reticulum (SR), the relaxation should have slowed, not hastened
as observed here. Furthermore, caffeine or ryanodine pretreatment
greatly affected the transient responses (both free
Ca2+ and mechanical), yet they
resulted in similar steady-state tension reductions with acidosis.
Therefore, it is unlikely that the steady-state responses in
ventricular function observed in the present study are due to
SR-induced changes in activator
Ca2+.
The reduced active pressure seen with IA and RA is consistent with a
general negative inotropy observed in all previous studies of acidosis
in skeletal muscle (12-14, 17, 30), isolated cardiac muscle (17,
25, 39, 45), and the whole heart (15, 24, 42, 47, 48). Furthermore,
this reduced inotropy is consistent with the well-known
acidosis-induced reduction of maximally activated force
(Fmax) and a rightward shift of
the force-pCa curve [i.e., decreased pCa at half-maximal
activation (pCa50)]. To
facilitate understanding of acidosis-induced negative inotropy, it is
appropriate to consider the following four aspects of contractile and
activation processes: 1) reduced
force per attached cross bridge; 2)
altered kinetic properties of cross-bridge cycling such that
cross-bridge detachment is favored relative to its attachment;
3) reduced
Ca2+-induced myofilamental
activation; and 4) reduced
cooperative feedback. Although all of these factors may contribute to
acidosis-induced changes, we propose that reduced cooperative feedback,
induced by acidosis, can reconcile observed changes in the temporal
features of the pressure waveform and also be consistent with other
known effects of acidosis. This proposal is based in part on a simple model of activation/cross-bridge dynamics described in the
APPENDIX (Fig.
8). Briefly, the cross-bridge cycle
consists of three states, two attached (force producing and non-force
producing) and one unattached. The
Ca2+-induced myofilamental
activation is governed by the effective rate constants
Kon and
Koff. This
reversible reaction includes both
Ca2+ binding to troponin C and
switching of thin filament regulatory units. Cooperative feedback is
represented by modulating
Koff (parameter
in Fig. 8);
Koff decreases in
the presence of attached, force-producing cross bridges, thus
facilitating myofilamental activation. Other forms of cooperative
feedback, such as nearest-neighbor thin filament and/or
cross-bridge interaction and active force cross-bridge kinetic rate
constant interaction, are known to exist (46); our nonspecific and
simple formulation probably does not represent these processes well. In
the following discussion, force and pressure are used interchangeably.
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Fig. 8.
Cross bridge-based model of activation and contraction. Cross bridges
exist in 4 states: noncycling
(Xnc), in which
Ca2+ is not bound to thin filament
regulatory site; cycling unattached
(Xco), in which
Ca2+ is bound but myosin head is
unattached; and 2 cycling attached states
(Xcn and
Xcf), which are
non-force producing and force producing, respectively. Rate constants
Kon and
Koff govern
transition to and from noncycling and cycling states, respectively.
Transition among cycling states is governed by rate constants
f, d,
h, and
g. Finally, cooperative feedback is
provided by modulating
Koff as a
function of force (i.e.,
Xcf) and a
cooperativity parameter ().
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Reduced force per attached cross bridge can occur either by a
diminution of force per attached, force-producing cross bridge (17) or
by a redistribution of attached cross bridges that favors the
non-force-producing population (41). Because the first possibility simply scales all force values, it cannot alter any temporal features of the pulse. The redistribution of attached cross bridges can only be
achieved by changing cross-bridge kinetic properties, specifically
reduced h, increased
g, or both (see
APPENDIX, Eq. A7).
The model-based analysis is now used to evaluate the remaining
candidates (i.e., cross-bridge kinetic- and activation-related parameters). For this purpose, we attempt to reproduce the following experimental observations with acidosis (respiratory or inorganic). In
terms of the left ventricular pressure pulse (Table 3 and RESULTS), acidosis reduces
Pmax,
tmax, and
Tr at a fixed
volume (i.e., Vref) by 24, 7, and 11%, respectively. It is expected that the extracellular pH fall
of 0.37 pH units (Table 2) would result in an intracellular pH
reduction of ~0.2 pH units (26, 42). Muscle data from literature (32,
41) indicate that this drop in intracellular pH increases relative
stiffness (Srel) by ~10% and
affects the force pCa relationship such that
Fmax (26, 45) and
pCa50 (26) are reduced by ~15%
and ~0.15 pCa units, respectively. Effects of changes in individual
model parameters are presented in Table 4;
in each case, the model parameter is adjusted to yield a 24% reduction
in peak twitch force (Fpeak). In
the absence of cooperative feedback ( = 0), none of the kinetic
parameter changes yield the desired effect on
pCa50 or twitch timing parameters (tmax and
Tr). Thus the
presence of cooperative feedback appears to be necessary to reproduce
the experimental observations.
In the presence of cooperative feedback, increasing
g alone could reproduce the
acidosis-induced effects reasonably well (Table 4). However, the
experimental evidence in support of such an increase in
g is inconclusive. Most experimental
results, based on unloaded velocity of shortening or relaxation of
tetanic contractions, suggest that g
is either decreased or unchanged with acidosis (35, 41, 51). The only
evidence for increased g is based on
the observation that acidosis reduces both isometric force and ATPase
activity, with the reduction in ATPase activity less than that in
isometric force (13, 38). According to the Brenner scheme (7), this
would suggest an increase in g with
acidosis (38).
Evidence for acidosis-induced decrease in rate of force-producing
cross-bridge formation exists (31, 41). This could be achieved by a
decrease in f,
h, or both. However, because acidosis increases relative stiffness in both skeletal and cardiac muscle (29,
32, 41), a decrease in h is most
likely. Our model-based analysis indicated that a decrease in
h could reproduce acidosis-induced changes reasonably well, except perhaps for the too little change in
tmax (Table
4).
Finally, a reduction in
Ca2+-induced myofilamental
activation with acidosis has been observed in a variety of preparations
(5, 16, 26). A reduced
Kon-to-Koff
ratio would be necessary to yield observed changes in force pCa
relationship and inotropic state. Because
Kon is generally
thought to be diffusion limited, it is unlikely that
Kon is affected
by acidosis. In addition, a reduced
Kon would
increase tmax
(model-based analysis), which is contrary to our observations. An
increase in Koff,
implemented via a decrease in cooperative feedback (decreased ),
reproduces all of the acidosis-induced changes except for
Fmax and
Srel (Table 4). Thus it is
reasonable to propose that acidosis affects both h and ; a concomitant decrease in
h and reproduces all of the observed changes well (Table 4 and Fig. 9).
In summary, the modeling efforts, together with the experimental
observations, lead us to conclude that
1) the presence of cooperative
feedback is necessary, and 2)
inhibition of the transition to the force-producing state and reduced
cooperative feedback are likely mechanisms for the acidosis-induced
changes.
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Fig. 9.
Results of model simulations. A: force
(F; normalized so that maximum F for "control" = 1) for 2 conditions, "control" and RA or IA simulated by a concomitant
reduction in cooperative feedback [decrease in ( )
by 30%] and transition from non-force-producing to
force-producing state [decrease in
h
(h) by 16%].
B: normalized F (i.e., each F curve
has a maximum of 1) for same conditions as in
A. C:
F-pCa relationships (normalized so that maximally activated F for
"control" = 1) for same conditions as in
A. D:
normalized F (i.e., maximally activated F for each condition = 1)-pCa
relationships for same conditions as in
A. Control values of model parameters
(see Table 4 and APPENDIX) yielded a
twitch contraction morphology that mimicked temporal features of
measured pressure waveform (compare with Fig. 7 data). Combination of
and h yielded
results consistent with RA and IA: a reduction in peak developed F and
hastening of its rise and fall while causing a rightward and downward
shift in F-pCa relationship.
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Lactic Acidosis: Disparate Results
Effects on ventricular function due to LA are very different from those
due to IA and RA (Table 3, Figs. 3-7). The first observation is
that the negative inotropic effect (reduced
Pmax and
Ea) is mitigated in LA compared with IA and RA. Because intracellular pH is a
more important determinant of muscle function than extracellular pH
(37), one possible explanation for the differences in inotropic responses is that intracellular pH is not reduced as much in LA as in
IA and RA. Although we did not measure intracellular pH, such
differential changes in intracellular pH have been reported (42, 47,
53). Because the negative inotropic effect of reduced intracellular pH
follows a dose response (34), the above explanation would be warranted,
especially if one focuses on developed pressure data alone. However,
the effects of LA on
Trise,
tmax, and the Tr-max
relationship require further explanation. Less intracellular pH
reduction would, at most, move
Trise,
tmax, and
Tr toward their control values, not beyond. Directionally opposite changes in these
variables suggest that some factor in addition to an increase in
[H+] is involved with
the disparate LA response.
One obvious candidate for this additional factor is the lactate ion.
The cell membrane is permeable to lactate ions; thus addition of
lactate to the perfusion buffer will increase intracellular lactate
([lactate]i). It has
been shown that increased
[lactate]i, even
without any change in pH, reduces active force development in skeletal
muscle (2, 21). However, its effects on twitch characteristics of
cardiac muscle under normoxic conditions are not known. Xu et al. (52)
have shown that SR Ca2+ transport
is critically dependent on the ATP generated endogenously by
SR-associated glycolytic enzymes. Specifically, glycolytic ATP
supported a 15-fold greater Ca2+
transport than the exogenously supplied ATP at the same concentration. Under normoxic conditions, increased
[lactate]i is known to
suppress the glycolytic pathway, which could reduce glycolytic ATP
generation and, consequently, impair SR
Ca2+ uptake. Another possibility
is based on the observation that there is a positive correlation
between ATP phosphorylation potential (i.e., ATP concentration divided
by the product of ADP and Pi concentrations
{[ATP]/([ADP][Pi])})
and SR Ca2+ uptake and release
(28). Increased
[lactate]i, via its
effects on the cytosolic redox state, can reduce ATP phosphorylation
potential, resulting in impaired SR
Ca2+ uptake. Thus it is likely
that
[lactate]i-induced
changes in SR Ca2+ handling,
specifically reduced SR uptake, is the mechanism underlying the slowing
of relaxation and delayed time to peak pressure (force).
As a preliminary assessment of the independent effects of lactate ion,
two additional experiments were performed. Hearts were perfused first
with the control perfusate and then with a perfusate containing sodium
lactate (racemic mixture) such that the concentration of lactate ions
was the same as in LA (~11 mM). Both perfusates had the same pH
(7.40). Although Pmax at
Vref was not affected by lactate
(control, 137 mmHg; lactate, 132 mmHg), pressure wave morphology was
altered significantly. As shown in Fig.
10, lactate alone increases
tmax and greatly
slows relaxation. Although a more thorough evaluation of the
independent effects of lactate is necessary, we propose the following
hypothesis to reconcile the disparities observed between LA and both RA
and IA. LA-induced responses can be thought of as the sum of two
components, one due to intracellular acidosis and the other due to a
direct effect of lactate ions. The change in intracellular acidosis
with LA is likely to be not as great as that with RA or IA.
Nonetheless, the acidosis component of LA response will reduce
inotropic state (lower Pmax and
Ea) and hasten
pressure rise and fall (lower
Trise, tmax, and
Tr) compared
with control. Therefore, the direct component of the LA response (Fig.
10) must retard rate processes to such an extent that the net effect is
significant slowing of pressure rise and relaxation with respect to the
control state. Because we used a racemic mixture (i.e., both
L- and
D-lactate ions were present), we
cannot comment at this time on the mechanism for the direct effects of
lactate ions; both metabolic and nonmetabolic causes are possible.
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Fig. 10.
A: normalized LV active pressure under
control conditions and during perfusion with a solution containing
sodium lactate in 1 heart. Addition of sodium lactate to control
perfusate resulted in the same free ionic concentration of lactate as
in LA, without fall in pH. B:
Tr-max
relationship for control and perfusion with sodium lactate in same
heart as in A. These data demonstrate
that lactate ion has a direct effect that significantly slows pressure
rise and fall. Similar results were obtained from a 2nd heart.
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APPENDIX |
To gain some insight into the mechanisms for acidosis-induced changes
in left ventricular contractile function, a model-based interpretation
was pursued. A simple cross-bridge cycling-based model of cardiac
muscle force generation was used for this purpose (Fig. 8).
Model Description
The salient features of the model are as follows.
1) All states in which
troponin-tropomyosin regulatory units are off
(Ca2+ not bound to low-affinity
site on thin filament) are given by a single noncycling state
(Xnc).
2) All cycling, unattached states are given by Xco. Kon and
Koff are the apparent rate constants governing
the transition between noncycling and cycling states. It should be
noted that this reversible process includes both Ca2+ binding to troponin C and the
switching on and off of thin filament regulatory units.
3) Cycling, attached cross bridges
exist in either non-force- or force-producing states, given by
Xcn and Xcf. Only the
force-producing state gives rise to developed force (F). Thus, at any
time (t), F 
Xcf. Rate
constants f and
d govern the cycling between
unattached and attached non-force-producing states, whereas the
unidirectional transition between non-force-producing and
force-producing attached bridges, the power stroke, is governed by
h. Finally, the detachment of
force-producing cross bridges is governed by the rate constant
g.
4) The total number of attached cross bridges in the force-producing conformation governs the cooperative feedback (6, 20). This phenomenon was implemented by
modulating the off rate constant according to the function Koff = Koff,0/(1 + X2cf), where
Koff,0 is the initial off rate
(i.e., zero attached cross bridges) and is a fixed parameter.
Investigators have previously implemented this feedback as a linear
function of Xcf
(8, 27). We chose the quadratic function instead because it yields a
parabolic relationship between the apparent
Ca2+ binding coefficient
(Kon/Koff)
and normalized force, which is consistent with experimental findings
(20). Thus increasing will promote the transition from noncycling
to cycling states.
XT, the total
concentration of available cross bridges
(XT = Xnc + Xco + Xcn + Xcf), is
proportional to the concentration of low-affinity
Ca2+ binding sites on troponin C
and is fixed in this simulation of isometric (isovolumic) contraction.
With [Ca]i
representing intracellular free
Ca2+ concentration, this model can
be described completely by the following system of three coupled
differential equations
![− <IT>K</IT><SUB>on</SUB>[Ca]<SUB>i</SUB><IT>X</IT><SUB>cf</SUB> + <IT>K</IT><SUB>on</SUB>[Ca]<SUB>i</SUB><IT>X</IT><SUB>T</SUB>](/content/vol276/issue2/fulltext/H582/img011.gif)
|
(A1)
|
|
(A2)
|
|
(A3)
|
Activation
for this system is provided through
[Ca]i, which can be
either constant or time varying, to simulate excitation- contraction.
Constant Activation: Force and Stiffness
For the case of steady-state constant activation,
[Ca]i and
Xcf are constant
and all derivatives equal zero. Solving the resultant algebraic system
for Xcf yields
steady-state force for a given [Ca]i,
F([Ca]i), which
reads
![F([Ca]<SUB>i</SUB>) = F<SUB>max</SUB> <FR><NU><AR><R><C>16<SUP>1/3</SUP>[Ca]<SUP>2</SUP><SUB>i</SUB>(<IT>K </IT><SUP>2</SUP><SUB>2</SUB> − 3&bgr;<IT>K</IT> <SUP>2</SUP><SUB>1</SUB>) </C></R><R><C> + 2[Ca]<SUB>i</SUB> <FENCE><IT>K</IT><SUB>2</SUB><IT>K</IT> <SUP>1/3</SUP><SUB>c</SUB> − 16<SUP>1/3</SUP>&bgr;<IT>K</IT><SUB>1</SUB><IT>K</IT><SUB>3</SUB></FENCE> + (2<IT>K</IT><SUB>c</SUB>)<SUP>2/3</SUP></C></R></AR></NU><DE>6[Ca]<SUB>i</SUB><IT>K</IT><SUB>2</SUB><IT>K </IT><SUP>1/3</SUP><SUB>c</SUB></DE></FR>](/content/vol276/issue2/fulltext/H582/img014.gif)
|
(A4)
|
where
K1,
K2,
K3,
Kr, and
Kc are constants
given by
Two
additional solutions for
F([Ca]i) are
discarded because one yields F < 0 for all
[Ca]i > 0 and one is
complex. Fmax in Eq. A4 is the maximum possible force
under constant activation (limit as
[Ca]i 
)
and is given by
|
(A5)
|
The
factor multiplying Fmax in
Eq. A4 is the normalized force-pCa
relationship (bounded by 0 and 1). Thus, whereas the shape and position
of the force-pCa curve depend on all aspects of the model (cross-bridge
kinetics, activation, and
XT),
Fmax is independent of activation.
Equations A4 and A5 were used to construct the
force-pCa plots in Fig. 9 and to determine the pCa that yields
half-maximal activation (pCa50).
Whereas the Xcf
state provides force, both
Xcf and
Xcn contribute to
stiffness (S), and they do so equally such that S (Xcf + Xcn). For any level of
constant [Ca]i and steady-state conditions, Xcf and
Xcn are related
(Eq. A3 with
dXcf /dt = 0) as
|
(A6)
|
Thus
steady-state relative stiffness of the model under constant activation
(Srel) is
|
(A7)
|
Therefore,
a change in Srel can only be
achieved by altering h or
g.
Simulation With Time-Varying
[Ca]i
To simulate contraction, the following time-varying
[Ca]i forcing function
was applied
![[Ca]<SUB>i</SUB>(<IT>t</IT>) = [Ca]<SUB>i,max</SUB> <FENCE><FR><NU><IT>t</IT></NU><DE><IT>t</IT><SUB>Ca,max</SUB></DE></FR></FENCE><SUP>&tgr;</SUP> <IT>e</IT><SUP>1 − <FENCE><FR><NU><IT>t</IT></NU><DE><IT>t</IT><SUB>Ca,max</SUB></DE></FR></FENCE><SUP>&tgr;</SUP></SUP>](/content/vol276/issue2/fulltext/H582/img024.gif)
|
(A8)
|
where
[Ca]i,max,
tCa,max, and 
are maximum
[Ca]i, time to [Ca]i,max, and the parameter
controlling the shape of the [Ca]i waveform,
respectively. For all simulations (both control and acidotic conditions),
[Ca]i,max = 1.2 µM,
tCa,max = 0.15 s, and = 2.0, which
yielded a [Ca]i pulse
with a physiologically realistic time course and amplitude (1, 36).
With [Ca]i thus
described, free Ca2+ diminishes to
zero well before the end of the cycle. With this assumption, the
following initial conditions emerge
|
(A9)
|
For
control conditions, values for model parameters were
XT = 70 µM (40), Kon = 40 s
1 · µM1
(40), Koff,0 = 135 s1, = 0.035 µM2,
g = 75 s1, h = 50 s1, f = 400 s1, and
d = 400 s1.
Koff,0 and values were chosen so that the experimentally observed pressure wave
morphology was reproduced for the control state. The value of
g was based on the short-time scale
left ventricular dynamics data (9, 44). The
h-to-g
ratio is ~
(11, 54); this defined the value of
h. The reversible transition between
Xco and
Xcn is considered
to be rapid, i.e., values of f and
d should be large compared with
g or
h. As a first approximation, we made
f and
d equal. The nonlinear system in
Eqs. A1-A3 and A9 was integrated numerically
(Runge-Kutta 
rule) over a 1.0-s interval using a 0.005-s
step size.
| |
ACKNOWLEDGEMENTS |
We thank Dr. Yasushi Nakagawa for assistance in measuring free
ionic concentrations and Dr. Robert T. Mallet for helpful discussions regarding the effects of lactate ions.
| |
FOOTNOTES |
This study was supported in part by National Heart, Lung, and Blood
Institute Grant R01-HL-36185 and American Heart Association Grant-in-Aid No. 96009940. I. E. Godoy was supported by a grant from
Pontificia Universidad Católica de Chile.
Present address of S. K. Fellner: Dept. of Physiology, CB 7545, Univ.
of North Carolina, Chapel Hill, NC 27599.
The costs of publication of this
article were defrayed in part by the
payment of page charges. The article
must therefore be hereby marked
"advertisement"
in accordance with 18 U.S.C. §1734 solely to indicate this fact.
1
The Vref
at which changes in Ped and
Ep were compared
in the hypocalcemia study was the volume at which control
Ped was equal to 5 mmHg.
Vref in the present study was
larger, typically resulting in control
Ped equal to 8-10 mmHg.
Absolute Ped values and percent changes in Ped and
Ep are augmented
at higher volumes.
Address for reprint requests: S. G. Shroff, Cardiology Section, Dept.
of Medicine, The Univ. of Chicago Medical Center, 5841 S. Maryland
Ave., MC 5084, Chicago, IL 60637.
Received 22 May 1998; accepted in final form 7 October 1998.
| |
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Am J Physiol Heart Circ Physiol 276(2):H582-H594
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Abstract
Introduction
![+ <LIM><OP>∑</OP><LL><IT>j</IT>=1</LL><UL><IT>n</IT><SUB>c</SUB>−1</UL></LIM> [<IT>c</IT><SUB>0,<IT>j</IT></SUB>C<SUB><IT>j</IT></SUB> + <IT>c</IT><SUB>1,<IT>j</IT></SUB>(C<SUB><IT>j</IT></SUB>&sfgr;<SUB>max</SUB>) + c<SUB>2,<IT>j</IT></SUB>(C<SUB><IT>j</IT></SUB>&sfgr; <SUP>2</SUP><SUB>max</SUB>)]](/content/vol276/issue2/fulltext/H582/img008.gif)
![+ <LIM><OP>∑</OP><LL><IT>i</IT>=1</LL><UL><IT>n</IT><SUB>r</SUB>−1</UL></LIM> [<IT>r</IT><SUB>0,<IT>i</IT></SUB>R<SUB><IT>i</IT></SUB> + <IT>r</IT><SUB>1,<IT>i</IT></SUB>(R<SUB><IT>i</IT></SUB>&sfgr;<SUB>max</SUB>) + r<SUB>2,<IT>i</IT></SUB>(R<SUB><IT>i</IT></SUB>&sfgr; <SUP>2</SUP><SUB>max</SUB>)]](/content/vol276/issue2/fulltext/H582/img009.gif)
![<FR><NU>d<IT>X</IT><SUB>co</SUB></NU><DE>d<IT>t</IT></DE></FR> = −(<IT>f</IT> + <IT>K</IT><SUB>off</SUB> + <IT>K</IT><SUB>on</SUB>[Ca]<SUB>i</SUB>)<IT>X</IT><SUB>co</SUB> + (<IT>d</IT> − <IT>K</IT><SUB>on</SUB>[Ca]<SUB>i</SUB>)<IT>X</IT><SUB>cn</SUB>](/content/vol276/issue2/fulltext/H582/img010.gif)
![<IT>K</IT><SUB>1</SUB> = <IT>K</IT><SUB>on</SUB>[(<IT>d</IT> + <IT>f</IT>)<IT>g</IT> + (<IT>f</IT> + <IT>g</IT>)<IT>h</IT>]](/content/vol276/issue2/fulltext/H582/img015.gif)
![<IT>K</IT><SUB>r</SUB> = 4[3&bgr;<IT>K</IT><SUB>1</SUB>(<IT>K</IT><SUB>3</SUB> + [Ca]<SUB>i</SUB><IT>K</IT><SUB>1</SUB>) − [Ca]<SUB>i</SUB><IT>K</IT> <SUP>2</SUP><SUB>2</SUB>]<SUP>3</SUP>](/content/vol276/issue2/fulltext/H582/img018.gif)
![+[Ca]<SUB>i</SUB>K<SUP>2</SUP><SUB>2</SUB>[9&bgr;<IT>K</IT><SUB>1</SUB><IT>K</IT><SUB>3</SUB> − 2[Ca]<SUB>i</SUB>(9&bgr;<IT>K </IT><SUP>2</SUP><SUB>1</SUB> + <IT>K </IT><SUP>2</SUP><SUB>2</SUB>)]<SUP>2</SUP>](/content/vol276/issue2/fulltext/H582/img019.gif)
![<IT>K</IT><SUB>c</SUB> = 2[Ca]<SUP>3</SUP><SUB>i</SUB>(<IT>K</IT> <SUP>3</SUP><SUB>2</SUB> + 9&bgr;<IT>K </IT><SUP>2</SUP><SUB>1</SUB><IT>K</IT><SUB>2</SUB>) − 9&bgr;[Ca]<SUP>2</SUP><SUB>i</SUB><IT>K</IT><SUB>1</SUB><IT>K</IT><SUB>2</SUB><IT>K</IT><SUB>3</SUB> + <RAD><RCD>[Ca]<SUP>3</SUP><SUB>i</SUB><IT>K</IT><SUB>r</SUB></RCD></RAD>](/content/vol276/issue2/fulltext/H582/img020.gif)
