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Am J Physiol Heart Circ Physiol 289: H114 –H130, 2005;
doi:10.1152/ajpheart.01045.2004.
Dynamic myocardial contractile parameters from
left ventricular pressure-volume measurements
K. B. Campbell,1 Y. Wu,1 A. M. Simpson,1 R. D. Kirkpatrick,1
S. G. Shroff,2 H. L. Granzier,1 and B. K. Slinker1
1
Department of Veterinary and Comparative Anatomy, Pharmacology, and Physiology,
College of Veterinary Medicine, Washington State University, Pullman, Washington;
and 2Department of Bioengineering, University of Pittsburgh, Pennsylvania
Submitted 10 October 2004; accepted in final form 8 December 2004
LINKING MYOCARDIAL CONTRACTILE parameters with measurements taken in intact heart continues as a longstanding challenge in cardiovascular research. A widely applied and venerable approach has been to associate the isometric force-length
relationship of isolated muscle with the various measures of the
Frank-Starling mechanism in beating heart. Although this is a
valid association, it is primarily an intuitive or qualitative
notion that works well in reasoned explanations but does not
offer a basis upon which to build a comprehensive system for
quantitative connections. In the 1960s and 1970s, an intense
effort was made to link muscle contractile behavior with whole
heart behavior by relating the maximal velocity of unloaded
cardiac muscle contraction with the time rate of change of
pressure (dP/dt) during isovolumic contraction. This linkage
was based on the Hill model of cardiac muscle and on the
existence of a unique value for that model’s series elastic
element (55, 56). Whereas dP/dt continues as a valuable index
for assessing the global contractile status of heart, the association between it and maximal shortening velocity could not be
substantiated when it was shown that the Hill model was not a
good representation of cardiac muscle and that the apparent
series elastic element in this model was not uniquely valued
(38, 40, 41). In the 1970s and 1980s, the apparent linear
relationship between isochronal left ventricular (LV) pressure
and volume led to the time-varying elastance [E(t)] concept for
representing global LV mechanodynamic properties (47, 58,
61). Experimental evidence for the validity of E(t) led to efforts
to link this global LV mechanical property to underlying
contractile properties of muscle (47, 48, 59). One successful
outcome of these efforts was the prediction and then the
repeated confirmation of a strong empirical association between pressure-volume area and myocardial O2 consumption
(47). However, it now appears that although E(t) is a useful
descriptor of simultaneous LV pressure and volume events, it
does not represent actual LV physical properties (6, 10, 14, 49,
52). Thus further attempts to link E(t) to contractile features of
muscle are not likely to yield satisfactory results.
A long-term approach for linking heart and muscle has
been to describe similarities in muscle and whole heart
behaviors; similarities between the isometric force-length
relationship of frog skeletal muscle and the isovolumic
pressure-volume relationship of frog heart being the classical muscle-heart analogy (23). Many other similarity associations have been made of a broad scope of behaviors
ranging from similarities in the end-shortening muscle
force-length and end-systolic LV pressure-volume relationships (1, 20, 27, 48) to similarities in step and frequency
responses of constantly activated heart and muscle (8, 11,
14, 15). Just as common has been the use of simple geometric transformations to derive muscle contraction relationships from LV measurements (5, 7, 44) or to reconstruct
LV behaviors from muscle measurements (22, 39). Despite
these many efforts, an unambiguous linkage with quantitatively verified associations has never been achieved.
Address for reprint requests and other correspondence: K. Campbell, Dept. of
Veterinary and Comparative Anatomy, Pharmacology, and Physiology, Washington State Univ., Pullman, WA 99164-6520 (E-mail: [email protected]).
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. Section 1734 solely to indicate this fact.
heart function; muscle; mathematical model; cardiac fiber; force
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Campbell, K. B., Y. Wu, A. M. Simpson, R. D. Kirkpatrick,
S. G. Shroff, H. L. Granzier, and B. K. Slinker. Dynamic
myocardial contractile parameters from left ventricular pressurevolume measurements. Am J Physiol Heart Circ Physiol 289:
H114 –H130, 2005; doi:10.1152/ajpheart.01045.2004.—A new dynamic model of left ventricular (LV) pressure-volume relationships
in beating heart was developed by mathematically linking chamber
pressure-volume dynamics with cardiac muscle force-length dynamics. The dynamic LV model accounted for ⬎80% of the
measured variation in pressure caused by small-amplitude volume
perturbation in an otherwise isovolumically beating, isolated rat
heart. The dynamic LV model produced good fits to pressure
responses to volume perturbations, but there existed some systematic features in the residual errors of the fits. The issue was whether
these residual errors would be damaging to an application where
the dynamic LV model was used with LV pressure and volume
measurements to estimate myocardial contractile parameters. Good
agreement among myocardial parameters responsible for response
magnitude was found between those derived by geometric transformations of parameters of the dynamic LV model estimated in
beating heart and those found by direct measurement in constantly
activated, isolated muscle fibers. Good agreement was also found
among myocardial kinetic parameters estimated in each of the two
preparations. Thus the small systematic residual errors from fitting
the LV model to the dynamic pressure-volume measurements do
not interfere with use of the dynamic LV model to estimate
contractile parameters of myocardium. Dynamic contractile behavior of cardiac muscle can now be obtained from a beating heart by
judicious application of the dynamic LV model to information-rich
pressure and volume signals. This provides for the first time a
bridge between the dynamics of cardiac muscle function and the
dynamics of heart function and allows a beating heart to be used in
studies where the relevance of myofilament contractile behavior to
cardiovascular system function may be investigated.
CONTRACTILE PARAMETERS FROM LV PRESSURE AND VOLUME
Glossary
Ai
b
c
D
E(t)
E(t)-R
E{}
Magnitude scalar for ith sine wave component of
⌬V(t) signal
Rate constant of recruitment
Rate constant of distortion
Time derivative operator (⫽d/dt)
Time-varying left ventricular (LV) elastance
Elastance-resistance LV model
Dynamic elastance operator
AJP-Heart Circ Physiol • VOL
E0
E⬁
EP
e(i)
F(t)
f0
fmin
H{}
I{}
I0
I⬁
L(t)
LBL
ni
P(t)
P៮
PFf
PI(t)
Piso(t)
Pp(t)
PR(t)
r
R{}
R0
V(t)
VBL
Vw
x(t)
y(t)
Q10
⌬F(t)
⌬L(t)
␣, ␤
␹0
␳0
ε{}
ε0
ε⬁
ε( j␻)
␾(t)
␩(t)
␯(t)
␺(t)
␨(t)
Zero-frequency LV elastance
Infinite-frequency LV elastance
Passive elastance
Residual errors
Midwall fiber force
Heart pacing frequency
Frequency of minimal stiffness
Dynamic operator that maps LV inputs to LV
pressure
Dynamic interactance operator
Zero-frequency LV interactance
Infinite-frequency LV interactance
Length of midwall circumference
Baseline midwall circumference
Frequency multiplier of f0 for ith sine wave component of ⌬V(t) signal
LV chamber pressure
Average pressure during isovolumic beat
Force-to-pressure transforming factor
LV pressure component due to interactance
Isovolumic pressure
Passive pressure
LV pressure component due to remainder terms in
Taylor series
Rate constant of R{}
Dynamic operator that maps y(t) into PR(t)
Magnitude of R{}
LV chamber volume
Baseline LV chamber volume
LV wall volume
Cardiac muscle distortion variable
Inputs responsible for PR(t)
Ratio of reaction rate at one temperature and
reaction rate at a temperature 10°C lower
Changes in cardiac muscle force
Changes in cardiac muscle length
Viscoelastic parameters of passive left ventricle
Scalar parameter for elastance
Scalar parameter for resistance
Dynamic stiffness operator
Zero-frequency cardiac muscle stiffness
Infinite-frequency cardiac muscle stiffness
Frequency-dependent muscle fiber stiffness
LV interactance recruitment variable
LV volume recruitment variable
Cardiac muscle recruitment variable
LV volume distortion variable
LV interactance distortion variable
Building a Quantitative Link Between Cardiac Muscle
Dynamics and LV Dynamics
Cardiac muscle dynamics: constant activation. The kernel
for LV pressure-volume dynamics resides in the force-length
dynamics of contracting cardiac muscle. To investigate these
dynamics, we dissected bundles of fibers from the papillary
muscle of rat hearts, removed the cell membranes with detergent to control activation at constant levels, and mounted these
skinned fibers in an apparatus that allowed servo control of
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Simultaneous with these experimentally based attempts were
several modeling efforts in which elemental muscle contractile
behavior was integrated mathematically with wall material
properties, wall architecture and geometry, and chamber geometry in attempts to synthesize global organ function (25, 31,
33, 34, 36, 65). These efforts continue today with promise of
eventual success (19, 26, 37), but because of the massive
complexity of the problem, they are presently without practical
results that may easily be implemented either experimentally or
clinically.
A major problem in linking muscle contraction with LV
mechanical behavior has been the reliance on inappropriate
characterizations at the muscle level for making this link. For
instance, the two most commonly used descriptors of muscle
contraction, length-tension and force-velocity, are actually special cases with respect to contraction time and load, i.e., peak
force during isometric contraction in the case of length-tension
and initial shortening velocity against isotonic load in the case
of force-velocity. These descriptors are not necessarily applicable to the dynamic history throughout a contraction event.
An alternative to length-tension and force-velocity descriptors of contraction is the dynamic stiffness of constantly
activated muscle. Dynamic stiffness focuses on frequencydependent force-length relations during small length changes
and is profoundly sensitive to myofilament kinetic processes
(3, 4, 30, 35, 46, 50, 51, 57, 62, 63, 66). Importantly, the
frequency-domain expression of dynamic stiffness may be
easily converted into an equivalent time-domain expression
that allows prediction of the transient time course of muscle
force in response to muscle length perturbations. Using the
notion that myocardial dynamics are governed by both the
dynamics of cross-bridge recruitment and the separate dynamics of cross-bridge distortion (12, 42), we recently constructed
a simple differential equation representation of dynamic stiffness that accurately reproduces both transient and steady-state
length-induced myocardial dynamic behaviors between 0.1 and
40 Hz (9). Interestingly, this model of muscle has the same
mathematical form and dynamic time constants as an earlier
LV model we developed from purely phenomenological evidence to describe the dynamic pressure-volume relationship in
constantly activated heart (15). The implication of this model
equivalence is that contractile force-length dynamics of myocardium are expressed in unaltered form in pressure-volume
dynamics of the LV chamber. Thus the challenge became one
of extending this analogy to beating heart. In this study, we
show how to make this extension, allowing myocardial contractile parameters to be estimated from pressure-volume measurements taken in beating heart. This forges the long-sought
link between myocardial contractile dynamics and whole heart
pressure-volume behavior.
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CONTRACTILE PARAMETERS FROM LV PRESSURE AND VOLUME
⌬F共t兲 ⫽ ε0␯共t兲 ⫹ ε⬁x共t兲
recruitment
force response ⫽ recruitment
distortion
⫹ distortion
␯˙ 共t兲 ⫽ ⫺b关␯共t兲 ⫺ ⌬L共t兲兴
recruitment dynamics 共slow兲
(2)
⫺cx共t兲
dissipation of distortion by replacement of distorted cross-bridge
⫹
⌬L̇共t兲
distortion dynamics 共fast兲
冋
(1)
net recruitment drive
ẋ共t兲 ⫽
distortion variable. Both ␯(t) and x(t) possess units of length.
Parameters ε0 and b represent the magnitude and rate constant
of the slow recruitment response; parameters ε⬁ and c represent
the magnitude and rate constant of the fast distortion response.
Both ε0 and ε⬁ possess units of stiffness.
When this model was fit to force responses in 118 records
obtained from 19 fibers collected from 4 species with widely
varying myofilament protein compositions, ⬎98% of all the
measured variation in the force response was explained (Fig.
1). Furthermore, the remaining unexplained variation was not
correlated with the ⌬L(t) input and thus could not be accounted
for by any improvement in the model. Thus Eqs. 1–3 were
accepted as suitable representation of force-length dynamics in
constantly activated cardiac muscle.
To be consistent with mathematical developments, in the
APPENDIX we consider the alternative expression for Eqs. 1–3 by
writing the relation between ⌬F(t) and ⌬L(t) in input-output
terms. For this, we use a dynamic operator that carries the
physical units of stiffness. Employing the symbol D to represent differentiation with respect to time, D ⫽ (d/dt){}, Eqs. 1–3
may be written in the alternative but equivalent form
(3)
⌬F共t兲 ⫽ ε0
(4)
In Eq. 4, all terms within the square brackets constitute a
dynamic operator that operates on the input, ⌬L(t), to produce
an output, ⌬F(t). Let ε represent the operator in square brackets. Then the dynamic force-length relation of cardiac muscle
may be succinctly written as
⌬F共t兲 ⫽ ε兵⌬L共t兲其
driving force for distortion
In these equations, a dot over a variable represents the first time
derivative; ⌬L(t) is the measured change in length imposed
upon the muscle; ⌬F(t) is the model-predicted force in response to ⌬L(t); ␯(t) is the recruitment variable; and x(t) is the
册
b
D
兵⌬L共t兲其
⫹ ε⬁
D⫹b
D⫹c
(5)
The dynamic operator ε{} has physical units of stiffness.
Conversion of ε{} to the frequency domain changes D to the
complex frequency variable j␻ and allows construction of the
stiffness frequency spectrum.
Fig. 1. Force response (⌬F) from cardiac muscle fiber (top) during frequency-sweep length-change (⌬L) protocol (bottom). Measured force response (top; gray
tracing) and model prediction from curve fitting Eqs. 1–3 to the measured force signal (black lines overlaying gray tracing) are compared. Expanded time scale
at end of time period displays the last 0.2 s of the record during high-frequency (40 Hz) length change. Note the good model reproduction of amplitude and phase
throughout the record from lowest to highest frequencies. Time point during frequency-sweep protocol of minimal-amplitude force response is shown (arrow).
[Figure constructed from previously published data (8).]
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14fiber length (9). We then measured force development and
response to length change during constant Ca2⫹ activation.
Fiber length was changed over 45 s according to a command
signal that specified small-amplitude, sinusoidal length variations in a frequency sweep from 0.1 to 40 Hz (Fig. 1).
Measured force-response signals to these imposed length
changes were fit with a model developed from two fundamental considerations, including 1) a kinetic scheme for molecular
contractile processes within the myofibers, and 2) the notion
that a force generated by contractile units is equal to the
number of parallel force generators in the force-generating
state times the average elastic force per force generator (3, 12,
42). Change in the number of force generators was designated
“recruitment,” and change in the elastic force per generator
was designated “distortion.” Development based on the underlying molecular kinetic scheme yielded distinct dynamical
equations for recruitment and distortion. We then reduced the
recruitment-distortion model to incorporate one dynamic mode
(one amplitude and rate constant) for recruitment and one
dynamic mode (one amplitude and rate constant) for distortion.
Expressed in differential equation format, the reduced model is
H117
CONTRACTILE PARAMETERS FROM LV PRESSURE AND VOLUME
Dynamic Muscle Stiffness (Constant Activation) is Converted
to Dynamic LV Elastance (Constant Activation) by
Geometric Transformation
再 冋
册冎
and a circumferential stress (force per cross-sectional area) to
chamber-pressure transformation is given by
2⁄3
Vw
⫹1
P共t兲 ⫽
⫺ 1 F共t兲
(7)
V共t兲
再冋
册 冎
Although spherical geometry was used in the wall/chamber
geometric transforming factors, small volume variation results
in these factors being essentially constant regardless of the
geometry. Thus small volume variations, because they do not
cause appreciable variation in the transforming factors, allow a
linear transformation of the model for force-length relations of
the myofiber into an analogous model for pressure-volume
relations in the left ventricle. For example, consider that
Fig. 3. Logic for transforming LV chamber volume [V(t)] into chamber
pressure [P(t)] through the dynamic force-length [F(t)-L(t)] characteristics of
cardiac muscle.
measurements are made of LV pressure and volume. Then, for
small ⌬V around a baseline volume (VBL), the baseline length
(LBL) of a representative midwall circumference will be
1
Vw ⁄3
2
(8)
L BL ⫽ 6␲ VBL ⫹ 2
再 冋
册冎
Furthermore, the force per unit cross-sectional area at the
midwall is given by
F⫽
1
P
具PFf典
where the pressure-force transforming factor 具PFf典 is (from
Eq. 7)
2⁄3
Vw
具PF f典 ⫽
⫹1
⫺1
(9)
VBL
再冋
册 冎
With the use of these transformations, it can be shown that the
relationship between a generic midwall muscle stiffness, ε ⫽
dF/dL, and a generic LV chamber elastance, E ⫽ dP/dV, is
ε⫽
2
L BL
E
2␲2具PFf典
(10)
Thus the pressure-volume relationship of the left ventricle,
given by E, may be converted to the force-length relationship
of muscle, given by ε, and vice versa.
Constant activation of muscle fibers to produce constant
active force in heart occurs when Ba2⫹ is used as the activating
agent (8). Because small volume perturbations allow linear
transformation between the force-length relationship and the
pressure-volume relationship according to Eq. 10, we can take
advantage of the mathematical structure of ε in Eq. 5 and by
analogy write for the analogous property of constantly activated whole heart, i.e., the dynamic elastance, E
冋
E兵其 ⫽ E 0
b
D
⫹ E⬁
D⫹b
D⫹c
册再冎
(11)
where E0 is the static (zero-frequency) elastance, E⬁ is the
instantaneous (infinite-frequency) elastance, and b and c retain
the definitions originally given in Eqs. 2 and 3. Thus the
geometric transformation changes the values of the multipliers
ε0 to E0 and ε⬁ to E⬁ but does not affect the values of the
dynamic constants b and c.
With the definition for E{} given in Eq. 11, the pressure due
to elastance (PE) and in response to a ⌬V(t) perturbation in a
constantly activated heart may be written as
Fig. 2. Muscle fibers are arranged circumferentially at the left ventricular (LV)
midwall. Assuming a thick-walled sphere with homogenous material and
physiological properties, a simple geometric transformation allows chamber
volume [V(t)] to be converted to midwall length [L(t)], and force per unit area
at the midwall circumference [F(t)] to be converted to chamber pressure [P(t)].
AJP-Heart Circ Physiol • VOL
P E共t兲 ⫽ E兵⌬V共t兲其
(12)
Alternatively, it may be written in differential equation form
analogous to Eq. 1 as
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Cardiac muscle fibers constitute the majority of the material
of the LV wall and are arranged in a complicated pattern of
spiraling sheets. However, in the midwall, fibers are circumferentially arranged and serially connected along the midwall
circumference (Fig. 2). The centrality of these circumferential
midwall fibers in the overall arrangement allows them to be
taken as representative of all muscle fibers in heart providing
that the distribution of material and physiological properties
satisfies the condition of essential homogeneity as documented
by uniform sarcomere-shortening patterns throughout myocardium (2, 21, 24, 43). Thus the force and length of an average
circumferentially oriented fiber are representative of the forcelength behavior of all the muscle fibers in the wall. By
assuming spherical geometry, the transformation of force and
length of the representative muscle fiber into pressure and
volume of the LV chamber may be carried out by steps as
indicated in Fig. 3.
Reading Fig. 3 from left to right, there are two geometric
transformation steps. A volume-to-midwall circumferential
length transformation is given by
1
Vw ⁄3
(6)
L共t兲 ⫽ 6␲ 2 V共t兲 ⫹ 2
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CONTRACTILE PARAMETERS FROM LV PRESSURE AND VOLUME
P E共t兲 ⫽ E0␩共t兲 ⫹ E⬁␺共t兲
(13)
where ␩(t) is the volume recruitment variable, analogous to the
length recruitment variable ␯(t) in Eq. 2, and ␺(t) is the volume
distortion variable, analogous to the lineal distortion variable
x(t) in Eq. 2. The corresponding differential equations, which
are analogous to Eqs. 2 and 3, are
␩˙ 共t兲 ⫽ ⫺b关␩共t兲 ⫺ ⌬V共t兲兴
(14)
␺˙ 共t兲 ⫽ ⫺c␺共t兲 ⫹ ⌬V̇共t兲
(15)
Dynamic LV Elastance (Constant Activation) is Converted to
Dynamic LV Interactance (Variable Activation) in
Beating Heart
Now the challenge is to apply the above framework for
myocardial-based dynamics of constantly activated left ventricle to beating heart, where activation is not constant. We
summarize here the results of a more complete mathematical
analysis of LV pressure-volume relationships during small
volume perturbations of the otherwise isovolumic beating heart
given in the APPENDIX. In this analysis, the pressure [P(t)]
generated during a beat undergoing volume perturbation
[⌬V(t)] depends on both ⌬V(t) and the isovolumic pressure
[Piso(t)] that would have developed had no ⌬V(t) been administered.
In general, the dependence of P(t) on ⌬V(t) and Piso(t)
involves history (memory) of the system. To include history
effects, it is useful to write the P(t) dependence in terms of a
dynamic operator, H{}, i.e.
P共t兲 ⫽ H兵Piso共t兲, ⌬V共t兲其
(16)
Equation 16 says that P(t) is the result of the mathematical
operation H{} on two input functions, ⌬V(t) and Piso(t). The
dynamic operator H{} has a mathematical character similar to
E{} in that it contains the time-derivative operator D as one of
its primitive components; other primitive operators include
scalars, summers, and multipliers.
A sequence of mathematical steps including a Taylor series
expansion of Eq. 16 enables us to write an expression for the
pressure response [⌬P(t)] to a small-amplitude volume perturbation in terms of the following: 1) an elastance pressure
[PE(t)] due to the action of ⌬V(t) alone as if ⌬V(t) were applied
to the left ventricle, generating a constant pressure equal to the
mean of Piso(t); 2) an interactance pressure [PI(t)] due to the
interaction of ⌬V(t) and the variation of Piso(t) around its mean
value[⌬Piso(t)]; and 3) a residual pressure [PR(t)] due to the
sum of all the residual higher-order terms in the Taylor series.
Thus
⌬P共t兲 ⫽ PE共t兲 ⫹ PI共t兲 ⫹ PR共t兲
(17)
In this, the elastance pressure PE(t) is given by Eqs. 12–15 as
if the heart is constantly activated to generate a pressure equal
to the mean of Piso(t). The interactance pressure PI(t), which is
AJP-Heart Circ Physiol • VOL
P I共t兲 ⫽ I兵⌬V共t兲⌬Piso共t兲其
(18)
where the operator I{} operates dynamically on the product
⌬V(t)⌬Piso(t), which may be treated as a time signal defined as
u(t) ⫽ ⌬V(t)⌬Piso(t). Furthermore, it is shown in the APPENDIX
that I relates to E through differentiation with respect to Piso(t)
as follows:
I⫽
⳵
关E兵其兴
⳵Piso
(19)
We make use of the specific formulation for E{} in Eq. 11 to
derive a specific formulation for I{} from Eq. 19. We assume
that of the E parameters, only the scaling coefficients E0 and E⬁
change with Piso, whereas the recruitment rate constant b and
the distortion rate constant c do not. With this assumption, I
may be represented as
I⫽
⳵E⬁ D
⳵E0 b
⫹
⳵Piso D ⫹ b ⳵Piso D ⫹ c
(20)
where ⳵E0/⳵Piso is the slope of E0 dependence on Piso (let I0 ⫽
⳵E0/⳵Piso), and ⳵E⬁/⳵Piso is the slope of i⬁ dependence on Piso
(let I⬁ ⫽ ⳵E⬁/⳵Piso). Substitution yields
I ⫽ I0
b
D
⫹ I⬁
D⫹b
D⫹c
(21)
In the manner of the elastance example, we can now write
the dynamic operations of Eq. 18 in differential equation
format
P I共t兲 ⫽ I0␾共t兲 ⫹ I⬁␨共t兲
(22)
where ␾(t) is a time function with units of work (mmHg䡠ml)
given by the first-order differential equation
␾˙ 共t兲 ⫽ ⫺b关␾共t兲 ⫺ u共t兲兴
(23)
and ␨(t) is a time function also with units of work (mmHg䡠ml)
given by
␨˙ 共t兲 ⫽ ⫺c␨共t兲 ⫹ u̇共t兲
(24)
The rate constants b and c are the same recruitment and
distortion rate constants as in the elastance equations, Eqs. 14
and 15. The physical units of I0 and I⬁ are reciprocal volume
(ml⫺1).
In the APPENDIX, it is shown that for the mean of the
isovolumic pressure over the beat period (P៮ ) that
E 0 ⫽ I 0P៮
(25)
and
E ⬁ ⫽ I ⬁P៮
(26)
The remainder term [PR(t)], because it consists of lumping
several higher-order terms in the Taylor series together, has
few guidelines for representing its operator and the signal upon
which it operates. Because it is a remainder of higher-order
terms, it is assumed that PR(t) is small relative to PI(t), and its
exact representation is less important to the problem than a
correct representation of PI(t). Here we take an ad hoc ap289 • JULY 2005 •
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In fact, we have previously acquired experimental evidence
that Eqs. 13–15 satisfactorily describe the steady-state pressure
response to sinusoidal volume perturbations in constantly activated heart, and these are analogous to the equations that
describe the steady-state force response to sinusoidal length
perturbations of constantly activated papillary muscle (15).
of prime importance in this problem, may be couched in a form
similar to our earlier expression, Eq. 12, for PE(t)
CONTRACTILE PARAMETERS FROM LV PRESSURE AND VOLUME
proach and express the pressure due to these remainder terms,
PR(t), in a dynamic form, i.e.
P R共t兲 ⫽ R兵y共t兲其
(27)
y共t兲 ⫽ 关E ⬁ ␺共t兲 ⫹ I ⬁␨共t兲兴2
With this, R{} was represented as
冋
R兵y共t兲其 ⫽ R 0
册
r
兵y共t兲其
D⫹r
(28)
(29)
where R0 is a magnitude parameter and r is a rate constant. The
corresponding differential equation is
Ṗ R共t兲 ⫽ ⫺r关PR共t兲 ⫺ R0y共t兲兴
(30)
Summary of Quantitative Muscle-Left Ventricle Linkage
A dynamic kernel for LV dynamics was derived from
dynamic force-length relations in constantly activated cardiac
muscle fibers. Geometric transformation was used to convert
dynamic force-length relationship of a representative midwall
myofiber into dynamic pressure-volume relationship in the left
ventricle of constantly activated heart. Applying the calculus of
variation and a Taylor series expansion to the dependence of
LV pressure on both the volume perturbation and the isovolumic pressure during beating (see APPENDIX), the pressure response to a volume perturbation was found to consist of three
components as follows: a component due to the dynamic
elastance referenced to the mean isovolumic pressure, a component due to the interactance between the volume variation
and the pressure variation around its mean, and a component
due to a remainder term consisting of higher-order terms in the
Taylor series.
Summarizing the model equations
⌬P共t兲 ⫽ PE共t兲 ⫹ PI共t兲 ⫹ PR共t兲
(a)
P E共t兲 ⫽ E0␩共t兲 ⫹ E⬁␺共t兲
(b)
␩˙ 共t兲 ⫽ ⫺b关␩共t兲 ⫺ ⌬V共t兲兴
␺˙ 共t兲 ⫽ ⫺c␺共t兲 ⫹ ⌬V̇共t兲
(b1)
(b2)
P I共t兲 ⫽ I0␾共t兲 ⫹ I⬁␨共t兲
␾˙ 共t兲 ⫽ ⫺b关␾共t兲 ⫺ u共t兲兴; u共t兲 ⫽ ⌬V共t兲⌬Piso共t兲
␨˙ 共t兲 ⫽ ⫺c␨共t兲 ⫹ u̇共t兲
Ṗ R共t兲 ⫽ ⫺r关PR共t兲 ⫺ R0y共t兲兴; y共t兲 ⫽ 关E⬁␺共t兲 ⫹ I⬁␨共t兲兴2
(c)
(c1)
(c2)
(d)
We refer to Eqs. a–d as the dynamic LV model. The dynamic
LV model equations contain five state variables [␩(t), ␺(t),
␸(t), ␨(t), and PR(t)] and eight parameters including five magAJP-Heart Circ Physiol • VOL
nitude-scaling parameters (E0, E⬁, I0, I⬁, and R0) and three rate
constants (b, c, and r). Equations 25 and 26 demonstrate that
E0 and E⬁ are not independent of I0 and I⬁. Thus there are only
six free parameters in the model equation set. (Note that as a
consequence of the interdependence of E0 and I0 and E⬁ and I⬁,
a simpler rendition of Eqs. a–d can be written; see APPENDIX.
However, this simpler rendition obscures the origins of the
components of the pressure response, which are important for
understanding the model and how it arises.)
METHODS
Experimental Protocol
Hearts for these studies were obtained from rats according to a
protocol approved by the Washington State University Institutional
Animal Care and Use Committee. All animals in this study received
humane care in compliance with the animal use principles of the
American Physiological Society and the Principles of Laboratory
Animal Care formulated by the National Society of Medical Research
and the National Institutes of Health’s Guidelines for the Care and
Use of Laboratory Animals (NIH Publication No. 85-23, Revised
1985).
Hearts were isolated from young adult male rats (2– 6 mo of age;
300 –500 g body wt) after administration of anesthesia [that contained
(in mg/kg im) 50 ketamine, 5 xylazine, and 1 acepromazine]. Upon
excision of the heart, the aorta was quickly cannulated and the heart
was immediately perfused through the aortic cannula with KrebsHenseleit solution (Ca2⫹ concentration, 1.25 mM) that contained high
levels of dissolved O2 (PO2 ⬎ 600 mmHg) at a constant perfusion
pressure of 100 mmHg. Hearts were mounted onto an experimental
system that consisted of a constant-pressure perfusion subsystem, an
environmental control subsystem, a pacing-control subsystem, and a
volume servo subsystem. A latex balloon attached to the obturator of
the volume servo subsystem and sized so as to not develop measurable
pressure when inflated to 500 ␮l was inserted into the LV chamber
through the mitral orifice. The mitral annulus was secured to the
obturator. Mounted and perfused hearts were then submerged in
perfusate in a temperature-controlled environmental chamber that
kept the epicardial surface wet and allowed for field stimulation.
Electrical pacing was initiated using a computer-controlled stimulator
and field electrodes placed in the bath of the environmental chamber
on either side of the heart. The balloon was inflated to a reference
volume [VBL], which was defined as the volume necessary to achieve
an end-diastolic pressure of ⬃5 mmHg. LV pressure [P(t)] was
measured with a 3-Fr catheter-tip Millar pressure transducer that was
passed through a port in the volume servo system through the lumen
of the obturator and positioned in the balloon within the LV chamber.
Experiments were conducted at both 37 and 25°C by adjusting the
temperature of the perfusate and the water circulating through the
jackets of the experimental subsystems.
The LV balloon was connected to a computer-controlled volume
servo system with a displacement piston in a servo chamber. Movement of the piston displaced volume out of or into the LV balloon
according to the servo command. This allowed dynamic control and
perturbation of LV volume with simultaneous measurement of LV
pressure.
The pace period was set at 500 ms in experiments conducted at
37°C and 1,500 ms in experiments conducted at 25°C. Once stable
isovolumic beating was achieved at VBL, a single-beat Frank-Starling
protocol (16) was administered to evaluate both systolic and diastolic
LV functions. Criteria for acceptable preparations included the functional indices of developed pressure ⱖ 100 mmHg and passive
stiffness ⱕ 0.7 mmHg/␮l at VBL. After we established that functional
criteria were met, a volume-perturbation protocol was administered.
The volume-perturbation protocol was designed to provide pressure-response data from which model parameters could be estimated.
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where it is understood that R{}, like E{} and I{}, is a dynamic
operator, and y(t) is an appropriate signal to be defined.
We imposed the constraints that R{} be of low dynamic
order and that it add no more than two unknown parameters to
the overall model. Furthermore, likening PR(t) to a deactivation
effect (19, 28, 29), we assumed that deactivation comes about
as a result of distortion. Distortion is expressed in PE(t) and
PI(t) by the terms E⬁␺(t) and I⬁␨(t). We caused this deactivation to be independent of direction by assigning the forcing
function y(t) to the square of the sum of these distortion-related
terms
H119
H120
CONTRACTILE PARAMETERS FROM LV PRESSURE AND VOLUME
Because dynamic behaviors associated with recruitment occur at low
frequencies, whereas dynamic behaviors associated with distortion
occur at higher frequencies (see Fig. 1), the perturbation protocol for
parameter estimation necessarily needed to generate information at
both low and high frequencies. To satisfy this requirement, six ⌬V(t)
signals were constructed, each of which consisted of one of two
frequency compositions and one of three mean values. Each of the two
frequency composites consisted of five summed sinusoids (Table 1).
Separate records were taken of the response to each applied signal
5
៮j⫹
⌬V共t兲 ⫽ ⌬V
兺 A sin共n 2␲f 兲
i
i
(31)
0
i⫽1
Removing Passive Pressure Response From Measured Data
er ⫽
ē 2
s p2
where, for npts data points in the sample,
Pacing periods were chosen to be long enough to allow a welldefined diastolic period where pressure existed at the diastolic (passive) level without increasing or decreasing. During these diastolic
periods, the pressure response to the volume perturbations was considered to be entirely due to passive LV properties. The passive left
ventricle was modeled as a generalized viscoelastic body according to
Ṗ P共t兲 ⫽ ⫺␣Pp共t兲 ⫹ EP␣⌬V共t兲 ⫹ ␤⌬V̇共t兲
The dynamic LV model was fit to the active part of the measured
pressure response by solving the model equations (Eqs. a–d) for ⌬P(t)
using measured ⌬V(t) and ⌬Piso(t) as the forcing functions according
to their roles in each model component. The differential equations
were solved for the respective state variables by numerical integration
(fourth-order Runge-Kutta) using an integration step size equal to the
sampling interval of 0.001 s.
Model-generated ⌬P(t) was fit to measured ⌬P(t) by adjusting the
six free model parameters (E0, b, E⬁, c, R0, and r) using a modified
Levenberg-Marquardt algorithm (MINPACK; Argonne National Laboratory) to minimize the sum of square residual errors. Outcomes
from this fitting procedure included the following: 1) the set of
parameters (E0, b, E⬁, c, R0, and r) that provides the best fit of the
model to the measured pressure response; 2) the standard errors of the
estimates for each of these parameters; 3) the model generation of best
fit, ⌬P̂(t); and 4) the time series of residual errors, e(i) ⫽ ⌬P̂(i) ⫺
⌬P(i), where the i index indicates sampled values of ⌬P̂(t) and ⌬P(t).
Model evaluation consisted of indices of descriptive validity, i.e.,
measures of how well the model fit the data. Two measures of
descriptive validity were used, including 1) the linear regression of
⌬P̂(t) on ⌬P(t), ⌬P̂(t) ⫽ b⌬P(t) ⫹ p0, and an evaluation of the
regression parameters for their ideal values of b ⫽ 1 and p0 ⫽ 0; and
2) a relative error, er, defined as
ē 2 ⫽
npts
兺 关e共i兲兴
2
i⫽1
is the mean square error of residuals and
(32)
where Pp(t) is passive pressure, EP is the DC passive elastance, and ␣
and ␤ are viscoelastic parameters.
This passive-pressure model was fit (by nonlinear least squares; see
below) to just that portion of the pressure response during the
identified diastolic periods [Piso(t) ⬍ 10 mmHg; two diastolic periods
in each record of two beats] and the passive parameters (EP, ␣, and ␤)
were thus estimated. The passive LV properties were taken to be in
parallel with the active LV properties. With the use of Eq. 32 and the
estimated passive parameters, that part of the total response due to
passive properties was predicted throughout the data record, and the
predicted passive pressure was then subtracted from the total ⌬P(t)
response to leave just the active response. The active response was the
signal to which the following data-fitting and parameter-estimation
techniques were applied.
1
npts
1
s ⫽
npts ⫺ 1
2
p
npts
兺 关⌬P共i兲 ⫺ ⌬P៮ 兴
2
i⫽1
is the total variance in ⌬P(t). Note that because the residual errors in
this model-fitting exercise did not prove to be randomly distributed,
the variance in the data that was not explained by the model was not
simply 1 ⫺ R2. Thus er was valuable as an independent measure of
goodness of fit.
At the end of the experiment, all atrial and great vessel tissue was
trimmed from the heart, and the heart was blotted dry. The right
ventricular free wall was trimmed from the heart, and the septum plus
the LV free wall was weighed. The dissected right ventricular free
wall was added to the septum plus the LV free wall, and all ventricular
tissue was weighed.
RESULTS AND DISCUSSION
Table 1. Frequency composition of ⌬V(t)
Volume-Perturbation Protocol Resulted in PressureResponse Signals With Rich Dynamic Content
ni
Ai
Frequency Composition 1
Frequency Composition 2
0.02VBL
⫺0.02VBL
0.01VBL
⫺0.01VBL
0.005VBL
0.5
2
4
8
32
0.5
1
6
16
32
Each of the two frequency composites consisted of five summed sinusoids.
⌬V(t), volume-perturbation signal; Ai, magnitude scalar for ith sine wave
component of ⌬V(t); ni, frequency multiplier of heart pacing frequency for ith
sine wave component of ⌬V(t); VBL, baseline left ventricular chamber volume.
AJP-Heart Circ Physiol • VOL
The volume-perturbation protocol was successful in generating pressure responses with dynamically rich information
content. An example of typical pressure responses obtained
from the volume-perturbation protocol (at 25°C; pacing period,
1.5 s) is given in Fig. 4. For clarity, this example shows only
a subset of two of the full ensemble of six records to which the
៮ j ⫽ 0, frequency compositions 1 and 2). Note
model was fit (⌬V
that at this pacing rate, there is ⬃0.5 s of diastole during which
passive LV properties could be estimated using Eq. 32. The
⌬P(t) signal shown in Fig. 4 (bottom) consisted of just the
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៮ j ⫽ ⫺0.02VBL, 0, or 0.02VBL, and f0 ⫽ 1/(pace period). The
where ⌬V
magnitude scalar (Ai) and frequency multiplier of f0 (ni) for the ith sine
wave component of the two different composite frequency signals
were as indicated in Table 1.
Because the frequency of the slowest component in each frequency
composite was 0.5 f0, the volume perturbation in both signals with
differing frequency compositions lasted over 2 beats. In addition to
recording the volume-perturbed beats, it was necessary to record the
Piso(t) value in the beat immediately preceding the volume-perturbed
beats. This was taken to be the Piso(t) value that would have been
generated by the two perturbed beats if ⌬V(t) had not been applied.
៮ j values, an ensemble
With two frequency compositions and three ⌬V
of six pressure-response records was generated.
Data Fitting, Parameter Estimation, and Model Evaluation
CONTRACTILE PARAMETERS FROM LV PRESSURE AND VOLUME
H121
Fig. 4. Two 3-s records of composite frequency volume-perturbation protocol. Pacing
frequency (f0) equaled 0.67 Hz. Volume perturbation consisted of a different combination
of five frequencies between 0.5 and 32 f0 in
each record (left vs. right). Top: pressure
[P(t)] was recorded during isovolumic beating (thin lines) and volume perturbation
(thick lines). Middle: volume-perturbation
signals. Bottom: pressure response [⌬P(t)] to
volume perturbation was calculated as the
difference between pressure during volume
perturbation and isovolumic pressure; different shapes of ⌬P(t) (left vs. right) are due to
different frequency contents of ⌬V(t).
Dynamic LV Model Accounts for Most Features and
Majority of Variance in Pressure Response to
Volume Perturbation
The dynamic LV model, when fit to the data, reproduced all
identifiable qualitative features in the response records. Qualitative comparison of a measured response during a single beat
with a model-generated response for that beat demonstrated
feature-by-feature reproduction (Fig. 5) including 1) largeamplitude responses during systole and virtually no response
during diastole, 2) variation in response amplitude over the
time course of the systolic interval, and 3) characteristic
differences in shapes of the responses to volume perturbations
with differing frequency content. When comparison is made
between the first and second beats in the record, the trajectory
of response during beat 1 (during the positive half of the
slowest volume sine wave in the perturbation signal) was
always above that of response during beat 2 (during the
negative half of the slowest volume sine wave in the perturbation signal) in both the measured and model-generated signals. When comparison was made of the responses in records
៮ j, the trajectory of response at ⌬V
៮ j ⫽ 0.02VBL
at different ⌬V
៮
was always above that at ⌬Vj ⫽ 0, which was above the
៮ j ⫽ ⫺0.02VBL in both measured and modeltrajectory at ⌬V
generated records.
The fit of the dynamic LV model to the full ensemble of six
៮ j times two frequency
pressure-response records (three ⌬V
compositions) generated the goodness-of-fit measures in Table
2. An example of a fit to an ensemble of six records (animal 29;
Fig. 5. Comparison of measured and model-generated pressure responses. Systolic and diastolic periods of the beat are
identified; diastole includes late relaxation. In each graph,
the first (solid line) and second (dashed line) beats of the
two beats in each response record are shown to demonstrate
the differences in response trajectories of these two beats.
Differences in responses to volume perturbations consisting
of frequency compositions 1 and 2 are also reproduced by
the model. See text for additional explanation.
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active part of the response (i.e., the passive response has been
removed). Note the difference in ⌬P(t) trajectory on the first
and second beats in each of the two records, as these correspond to the positive- and negative-going halves of the slowest
frequency sinusoid (f0, 0.5) in the volume-perturbation signal
(Table 1). Also, note the different shape and form of ⌬P(t)
when ⌬V(t) consisted of frequency composition 1 (Fig. 4, left)
compared with when it consisted of frequency composition 2
(Fig. 4, right). With the combination of the trajectories and
shapes represented in the full set of six records in the volumeperturbation protocol, sufficient information was present in the
combined records to estimate model parameters that selectively
influenced low- and high-frequency behaviors.
H122
CONTRACTILE PARAMETERS FROM LV PRESSURE AND VOLUME
Table 2. Goodness-of-fit statistics
Animal No.
er
b
0.794
0.856
0.819
0.782
0.846
0.826
0.845
0.850
0.878
0.833
0.832
0.029
0.782
0.819
0.789
0.772
0.834
0.791
0.822
0.835
0.865
0.800
0.800
0.023
0.881
0.651
0.847
0.811
0.861
0.769
0.733
0.837
0.806
0.797
0.800
0.068
0.875
0.571
0.844
0.775
0.869
0.752
0.754
0.841
0.785
0.745
0.781
0.089
37°C
17
24
27
28
29
31
43
44
49
50
Mean
SD
0.205
0.147
0.182
0.218
0.153
0.177
0.157
0.150
0.123
0.170
0.168
0.028
33
34
35
36
37
38
45
46
47
48
Mean
SD
0.119
0.387
0.152
0.194
0.139
0.233
0.269
0.163
0.196
0.217
0.207
0.078
25°C
er, Square root of mean sum of square of residual errors; R2, from linear
regression of model-generated vs. measured pressure response; b, slope of
regression line.
37°C; Table 2) is given by the overlay of model-generated
responses on measured responses in Fig. 6. For much of the
systolic period, it was difficult to discern the difference between measured and model-generated signals. This means that
the trajectory of the low-frequency component of the measured
⌬P共t兲 ⫽ E共t兲⌬V共t兲 ⫹ R共t兲⌬V̇共t兲
Because time variation in E(t) and i(t) follows that of Piso(t)
(53, 54), Eq. 36 was written in terms of Piso(t) and two
Fig. 6. Fit of dynamic LV model (solid lines) to
measured pressure responses (broken lines) over
two beats. One set of parameters was used in the
model generation of all six responses. Responses
to ⌬V perturbations of one composition of five
៮ ; rows) valfrequencies around three mean (⌬V
ues (left) and responses to another composition
of five frequencies around the same three mean
values (right) are shown. VBL, baseline LV
chamber volume. See text for details.
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R2
response was well reproduced, as was the timing of the peaks
and valleys in the high-frequency aspects of the response.
Importantly, the reproduction of all six pressure-response
records in Fig. 6 was with a single set of parameters that were
estimated from fitting to all six records simultaneously.
Quantitative statistics from the fitting procedure are summarized in Table 2. The mean R2 from fits to records at 37°C was
0.83; the mean R2 from fits to records at 25°C was 0.80. The
mean er (mean sum of squared residual errors relative to signal
variance) was 0.17 from fits of records at 37°C and was 0.21
from fits of records at 25°C. The difference between mean
goodness-of-fit parameters at 37 and 25°C was largely due to
a relatively poor fit (R2 ⫽ 0.65; er ⫽ 0.39) in 1 of the 10 hearts
(animal 34) in the 25°C dataset. Although this heart appeared
to be an outlier, we had no objective reason for excluding it.
Other than this one heart, fits to pressure responses at both
temperatures were equally good. Consistent with the close
overlay of measured and model-generated pressure responses
in Fig. 6, the goodness-of-fit statistics indicated that there was
only a small amount of residual variance in the data that was
not accounted for by the model.
To appreciate the capability of the dynamic LV model to
closely reproduce a set of dynamically complex pressureresponse records, i.e., records with representation of both low
and high frequencies (Fig. 6), comparison needs to be made to
the best fit to this same set of data that can be obtained using
an alternative model. For this alternative, we choose the
broadly applied time-varying elastance-resistance [E(t)-R] LV
model. In accord with previous work (13, 29, 34, 53, 54, 60,
64), we couched the E(t)-R formulation as
CONTRACTILE PARAMETERS FROM LV PRESSURE AND VOLUME
parameters including a scalar for elastance (␹0) and another for
resistance (␳0)
⌬P共t兲 ⫽ Piso共t兲关␹0⌬V共t兲 ⫹ ␳0⌬V̇共t兲兴
(37)
phase of the cardiac cycle. These errors briefly became predominantly negatively valued at the time of minimum dPiso/dt
and then briefly became predominantly positively valued during the late phases of relaxation (data not shown). The issue
raised by these systematic residual errors is whether their
existence becomes damaging to the intended application of the
model, i.e., the use of the model to allow estimates of cardiac
muscle-contraction parameters from pressure and volume measurements taken in whole heart. To address this issue, we
compared muscle-contraction parameters estimated from this
study, using measurements taken in beating heart, with corresponding contraction parameters previously obtained in a study
that used constantly activated, isolated cardiac muscle fibers
(9). First, however, model parameters need to be organized to
facilitate the comparison.
Dynamic Parameters May Be Separated Into TemperatureInsensitive (Magnitude Scaling) and Temperature-Sensitive
(Kinetic Rate Constant) Groups
The estimates of parameters for all hearts and the averages
of these estimates are given in Table 3. Because the most
relevant of the six estimated model parameters (E0, b, E⬁, c, R0,
and r) were the four derived from dynamic behavior of muscle
(E0, b, E⬁, c), we focused on these four. These four parameters
may be partitioned two ways, as follows: 1) those associated
with the relatively slow dynamics of recruitment of forcegenerating units (i0 and b) and those associated with the
relatively fast dynamics of distortion of force-generating units
(E⬁ and c), or 2) parameters that scale the magnitude of the
recruitment and distortion components of the response (E0 and
E⬁) and parameters that govern the speed (kinetics) of the two
components of the response (b and c). Magnitude-scaling parameters (E0 and E⬁) represent different aspects of the response than
Fig. 7. Fit of elastance-resistance [E(t)-R]
model (solid lines) to the same pressure responses (broken lines) shown in Fig. 6. Inability
of the E(t)-R model to reproduce the measured
response demonstrates the challenge of representing LV properties responsible for a broad
range of dynamic behaviors. One set of parameters was used in the model generation of all six
responses.
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The same optimization techniques used to fit the dynamic LV
model to the pressure-response records were used to fit with the
E(t)-R model and to estimate the two parameters ␹0 and ␳0. The
results from the fitting procedure (Fig. 7) exhibit a clear
separation between measured and E(t)-R-predicted responses
throughout systole. Not only did the low-frequency trajectories
of the model-generated and measured pressure responses differ, but also, the timing of the peaks and valleys of the higher
frequency components within the model-generated and measured responses did not coincide as they did with the dynamic
LV model. The average R2 from fitting the measured pressure
responses with the E(t)-R model was only 0.37 compared with
the R2 of ⬎0.80 for the dynamic LV model. Although the
E(t)-R model possessed only two parameters and the dynamic
LV model possessed six, we argue that the E(t)-R model could
not reproduce the measured response pattern with acceptable
fidelity not because of a lack of parameters, but because it did
not possess the dynamic mechanisms necessary for such reproduction. The point to be made in comparing Figs. 6 and 7
is not to denigrate the E(t)-R model, but rather, to demonstrate
the challenge of using a model to reproduce a dynamically rich
pressure-response signal from a beating heart, and to emphasize that we have largely met that challenge with our present
dynamic LV model.
Despite the overall good fit by the dynamic LV model, we
found that the slopes of the regression lines of model-generated
vs. measured pressure response were ⬍1 in every heart at both
temperatures (see Table 2). This implied that there was a
systematic character to the residual errors from the model fit.
Indeed, when plotted as a function of time, the residual errors
appeared to be nonrandomly distributed during the relaxation
H123
H124
CONTRACTILE PARAMETERS FROM LV PRESSURE AND VOLUME
Table 3. Estimated parameters of dynamic LV model
Animal No.
P៮ , mmHg
I0,* ␮l⫺1 ⫻ 10⫺3
E0, mmHg ⫻ ␮l⫺1
I⬁,† ␮l⫺1 ⫻ 10⫺2
b, s⫺1
E⬁, mmHg ⫻ ␮l⫺1
c, s⫺1
R0, mmHg ⫻ ␮l2
r, s⫺1
1.82
1.73
1.97
2.98
1.95
2.11
2.03
1.81
2.22
2.08
2.07
0.35
0.830
0.779
1.041
1.798
0.995
0.926
1.300
0.959
0.770
0.889
1.028
0.311
154.8
125.6
124.2
115.4
118.4
141.4
119.8
139.7
103.7
137.3
128.0
15.1
⫺1.45
⫺1.48
⫺1.54
⫺1.26
⫺1.42
⫺1.83
⫺1.28
⫺1.54
⫺1.13
⫺1.85
⫺1.48
0.231
53.7
29.3
19.3
16.1
15.7
29.5
24.6
30.7
36.1
28.0
28.3
11.1
2.35
1.78
2.11
1.72
1.96
2.07
3.68
3.48
2.30
2.03
2.35
0.68
1.42
1.68
1.38
1.01
1.18
1.56
1.97
1.77
0.92
0.92
1.38
0.37
43.8
70.8
43.7
68.9
46.5
53.9
47.9
57.5
60.0
70.2
56.3
10.7
⫺0.902
⫺0.925
⫺1.082
⫺1.159
⫺1.142
⫺0.704
⫺0.961
⫺0.947
⫺1.030
⫺0.979
⫺0.983
0.132
17.9
18.7
11.7
10.1
8.13
16.0
7.91
12.1
19.8
17.8
14.0
4.53
37°C
45.7
45.0
52.8
60.3
51.1
43.9
64.1
53.1
34.7
42.7
49.3
8.73
5.01
4.84
3.60
5.71
4.15
4.76
5.37
4.82
9.09
5.78
5.31
1.48
0.229
0.218
0.190
0.344
0.212
0.209
0.344
0.256
0.315
0.247
0.256
0.057
19.0
19.2
13.7
13.9
13.8
13.2
10.0
11.5
18.4
11.6
14.4
3.30
33
34
35
36
37
38
45
46
47
48
Mean
SD
60.4
94.4
65.5
58.4
60.0
75.3
53.6
50.9
40.0
45.5
60.4
15.6
9.04
1.99
6.03
4.38
5.20
3.73
8.77
8.68
5.99
4.75
5.86
2.35
0.546
0.188
0.395
0.256
0.312
0.281
0.470
0.442
0.240
0.216
0.335
0.121
7.06
5.35
7.71
4.68
5.84
5.98
6.97
6.19
5.42
4.99
6.01
0.98
25°C
P៮ , average pressure during isovolumic beat; I0, zero-frequency left ventricular (LV) interactance; E0, zero-frequency LV elastance; b, recruitment rate constant;
I⬁, infinite-frequency LV interactance; E⬁, infinite-frequency LV elastance; c, distortion rate constant; R0, magnitude of residual term; r, rate constant of residual
term. *Not estimated; calculated from E0 and P៮ according to Eq. 25. †Not estimated; calculated from E⬁ and P៮ according to Eq. 26.
those aspects represented by the kinetic parameters (b and c). We
asked how the magnitude-scaling parameters differ from the
kinetic parameters in their dependence on temperature.
In comparing the magnitude-scaling parameters at different temperatures, it was necessary to account for the influence of temperature on pressure development. Based on the
average pressure (P៮ ), over the course of an isovolumic beat,
hearts at 25°C generated more pressure (P៮ ⫽ 60.4 mmHg)
than those at 37°C (P៮ ⫽ 49.3 mmHg). To account for this
difference in the comparison, we plotted E0 and E⬁ vs. P៮ and
analyzed differences in the E0 vs. P៮ and E⬁ vs. P៮ relationships at the two temperatures using ANOVA. No demonstrable differences in each of these relationships at the two
temperatures were found; P ⫽ 0.21 for E0 and P ⫽ 0.12 for
E⬁. Therefore, we concluded that the magnitude-scaling
parameters E0 and E⬁ were temperature independent. Thus
estimates of these parameters obtained in the heart at 37 and
25°C in this study could be compared with estimates of
analogous parameters obtained from earlier muscle experiments conducted at other temperatures.
In contrast, the kinetic parameters b and c were strongly
temperature dependent. The average value of b at 37°C (14.4
s⫺1) was 2.4 times the average value at 25°C (6.01 s⫺1), and
the average value of c at 37°C (128.0 s⫺1) was 2.3 times the
average value at 25°C (56.3 s⫺1); P ⬍ 0.001 by t-test for both
b and c. Thus when comparing with equivalent kinetic parameters obtained from earlier cardiac muscle studies conducted at
other temperatures, the temperature dependence of these kinetic parameters had to be taken into account.
AJP-Heart Circ Physiol • VOL
Magnitude-Scaling LV Parameters Are Consistent With
Magnitude-Scaling Muscle Parameters
Because the magnitude-scaling parameters were not affected
by temperature, we lumped E⬁ values determined at 37 and
25°C together. E⬁ is referenced to the mean pressure level
during isovolumic beating as if the heart muscle were constantly activated to produce the level of myocardial force
commensurate with the mean pressure (P៮ ) during the beat.
Geometric transformation (Eq. 7) was used to convert P៮ to a
representative midwall force (F). Additional geometric transformation (Eq. 10) was used to transform the estimated chamber E⬁ to the corresponding midwall ε⬁. The resulting ε⬁ vs. F
relation from calculations in all 20 hearts is given by the data
points in Fig. 8. Theoretically and experimentally (9), there is
a linear relationship between steady-state active force produced
by a constantly activated muscle and the infinite frequency
stiffness (ε⬁) of the muscle fiber. The regression line describing this linear relationship for data previously collected in 19
cardiac muscle fibers at varying levels of activation is plotted
in Fig. 8 along with the 95% tolerance limits (9) for observations around the regression line. Of the 20 ε⬁ vs. F points
derived by geometric transformation of E⬁ and P៮ from the 20
hearts studied, 18 fall on or within the 95% tolerance limits for
the muscle fiber regression. That is, the observations of ε⬁ vs.
F obtained from estimates in beating heart using our dynamic
LV model appear to be members of the same population as
defined by the relationship between ε⬁ and F found from
experiments using constantly activated, isolated cardiac muscle
fibers. This remarkable result leads to the conclusion that
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17
24
27
28
29
31
43
44
49
50
Mean
SD
CONTRACTILE PARAMETERS FROM LV PRESSURE AND VOLUME
parameters estimated from fitting the dynamic LV model to
dynamic pressure-volume behaviors may be reliably used to
estimate stiffness vs. force relations in representative midwall
muscle fibers.
Kinetic LV Parameters Are Consistent With Kinetic
Muscle Parameters
The estimates of kinetic parameters b and c obtained in these
beating-heart studies, which do not change with geometric
transformation, are also in good agreement with findings from
isolated muscle studies. Representative values for b and c
obtained in rat skinned cardiac muscle fibers at 20°C were 4.2
and 31.9 s⫺1, respectively (data taken from results reported in
Ref. 9). For the rat hearts of these studies, average estimates of
b and c were 6.1 and 54.1 s⫺1, respectively, at 25°C and were
14.3 and 139.9 s⫺1, respectively, at 37°C. These values, obtained at different temperatures in beating heart and in constantly activated isolated muscle fibers, were considered together in an Arrhenius plot. The individual plots for b and c
(Fig. 9) demonstrate that estimates from beating heart and
constantly activated isolated muscle fall along one line for each
kinetic constant. The calculated Q10 values from these Arrhenius lines were 2.02 for b and 2.21 for c. These Q10 values are
characteristic of myofilament kinetic behaviors. Thus we conclude that isolated beating heart can be used with our dynamic
LV model to estimate parameters of the same underlying
myofilament kinetic processes as could be evaluated using
constantly activated isolated muscle.
Estimates of Frequency-Dependent Dynamic Muscle
Stiffness: Beating-Heart Measurements Are Consistent With
Constantly Activated Muscle Fiber Measurements
Frequency-dependent muscle fiber stiffness [ε(j␻)] is widely
used for studying cardiac myofilament function and changes in
myofilament function induced by myofilament phosphorylation, changes in myofilament protein composition, and actions
on the myofilament by various inotropic agents (3, 4, 30, 35,
46, 50, 51, 57, 62, 63, 66). There are several features of ε( j␻)
to which physiological significance may be attached including
AJP-Heart Circ Physiol • VOL
Fig. 9. Arrhenius plots of logarithm of kinetic constants for both cross-bridge
distortion (c) and recruitment (b) vs. 1/temperature (T). Points obtained from
muscle fiber at 20°C (*) and from isolated heart at 25 and 37°C (E) fall on a
common line.
the ratio of low-frequency to high-frequency asymptotes; the
kinetic constant of the low-frequency process; the kinetic
constant of the high-frequency process; and the dip frequency
(fmin) or frequency at which ε(j␻) becomes minimally valued.
Until now, ε(j␻) could only be obtained from isolated, constantly activated muscle fiber preparations. There has been no
way to obtain related information from measurements taken in
beating heart. However, with our demonstration in Figs. 8 and
9 that the dynamic behavior of beating heart, as expressed by
the dynamic LV pressure-volume relationship, may be transformed to apply to an equivalent midwall muscle fiber, it is
now possible to construct frequency-dependent muscle fiber
stiffness from data obtained in beating heart using the dynamic
LV model.
To show the equivalence of ε(j␻) estimated from beating
heart and constantly activated muscle, the magnitude frequency spectrum of ε(j␻) for an average half-activated (pCa
5.7) rat cardiac muscle fiber at 20°C obtained in an earlier
study is compared with the magnitude frequency spectra of
ε(j␻) for a representative midwall fiber at 25 and 37°C as
calculated from pressure-volume measurements obtained in
beating hearts in this study (Fig. 10; to facilitate comparison,
all frequency spectra in Fig. 10 have been normalized by ε⬁).
Note the similarity in shapes of the three frequency spectra
with respect to relative values of low- and high-frequency
asymptotes and the presence of the well-defined dips. Also,
note that increasing temperature shifts the spectrum progressively to the right as one would expect. The fmin for these three
spectra are 0.8 Hz at 20°C (obtained from constantly activated
muscle fiber), 1.4 Hz at 25°C (obtained from beating heart),
and 3.3 Hz at 37°C (obtained from beating heart).
Taken together, the foregoing analyses show that the myofilament-based model we derived for LV dynamics can be used
to derive accurate estimates of myofilament kinetic parameters
and behaviors based on pressure and volume measurements
taken in intact, beating heart. Such estimates could previously
be derived only from studies of isolated, constantly activated
cardiac muscle. This new ability now allows the direct testing
of many hypotheses in intact beating heart that could only be
done previously by indirect inference from studies of constantly activated isolated muscle.
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Fig. 8. Plots of infinite-frequency (instantaneous) muscle fiber stiffness (ε⬁)
vs. muscle fiber force (F). Regression from fit to data obtained in isolated
muscle (solid line) and 95% tolerance limits on the observations around the
regression line (dashed lines) are shown. Data points were derived by geometric transformation of model-derived LV infinite-frequency elastance and
measured mean pressure during a heart beat.
H125
H126
CONTRACTILE PARAMETERS FROM LV PRESSURE AND VOLUME
mogenous distribution of wall material and physiological properties.
The dynamic LV model accounted for ⬎80% of the measured variation in pressure caused by small-amplitude volume
perturbation in the otherwise isovolumically beating heart. The
widely used E(t)-R model, which does not possess dynamic
features, could account for only 37% of the pressure variation.
Thus the importance of dynamic features in the LV model was
underscored.
Although able to generate good fits to pressure responses to
volume perturbations, there existed some systematic features in
the residual errors of the fit with the dynamic LV model. The
issue arose as to whether these residual errors would be
damaging to an application of the model wherein myocardial
contractile parameters were estimated.
Good agreement was found between magnitude-scaling
myocardial parameters derived by geometric transformation of
parameters of the dynamic LV model estimated in beating
heart and those found by direct measurement in constantly
activated isolated muscle fibers. Good agreement was also
found between temperature-sensitive kinetic parameters estimated in the two preparations. Thus the small systematic
residual errors from fitting the LV model to dynamic pressurevolume measurements do not interfere with the use of the
model to estimate contractile parameters in myocardium.
Dynamic contractile behavior of cardiac muscle can now be
obtained from beating heart by judicious application of the
dynamic LV model to information-rich pressure and volume
signals. This provides, for the first time, a bridge between
cardiac muscle function and heart function and allows beating
heart to be used in studies where the relevance of myofilament
contractile behavior to cardiovascular system function may be
investigated.
APPENDIX A
Relationship Between Pressure and Volume in Beating Heart
is Given in Terms of Dynamic Operations
LV pressure [P(t)] depends at least on both the volume changes
[⌬V(t)] that occur during a contraction event and the pressure that
would have developed [Piso(t)] had no volume change occurred. In
general, P(t) dependence on ⌬V(t) and Piso(t) involves history (memory) of the system. To include history effects, we write the dependence in terms of the dynamic operator H{}, i.e.
P共t兲 ⫽ H兵Piso共t兲, ⌬V共t兲其
Fig. 10. Frequency spectra of myocardial stiffness magnitude [ε(j␻)]. One
spectrum (left) was obtained from force-length measurements in isolated,
constantly activated cardiac muscle at 20°C. Other two spectra were derived
from dynamic LV model applied to pressure and volume measurements taken
in isolated beating heart conducted at 25 (middle) and 30°C (right).
AJP-Heart Circ Physiol • VOL
(A1)
Equation A1 may be read as follows: P(t), as the output of a dynamic
system, is the result of the mathematical operation H{} on two inputs
to the system, ⌬V(t) and Piso(t).
The dynamic operator H{} is not an algebraic function, which we
represent as F( ), but it is similar to F( ) in that both H{} and F( ) map
system inputs to an output. However, H{} and F( ) are dissimilar in
that F( ) maps instantaneous values of the input to the output, whereas
H{} maps an entire input function into an output function.
For instance, consider the relationship between electrical current
[i(t)] and voltage [v(t)] in two systems, one consisting of a single
electrical resistor (R) and the other of a parallel combination of an R
and an electrical capacitor (C). In the case of the system of a single R,
the input-output relation between v(t) and i(t) may be written in terms
of the well-known functional relationship of Ohm’s law
␯共t兲 ⫽ Ri共t兲
(A2)
Here, instantaneous values of i(t) are mapped onto instantaneous
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For example, we have documented (9) that fmin represents
the frequency at which there is a transition in dominance of
mechanisms responsible for muscle fiber stiffness (i.e., the
⌬F/⌬L ratio); at lower frequencies, the dominant mechanism(s)
is the length-dependent process responsible for cross-bridge
recruitment, whereas at higher frequencies, the dominant
mechanism is that responsible for cross-bridge distortion. This
transition frequency, fmin, which identifies the frequency of
minimum muscle fiber stiffness, has been interpreted by others
to define the heart rate that leads to optimal cardiac function
(45, 57, 62, 63). Based on our finding that fmin is 3.3 Hz at 37°C,
our data suggest that the heart rate of optimal function for rats at
physiological temperature would be ⬃200 min⫺1. Because the
normal resting heart rate in rats of the kind used in this study is on
the order of 350 min⫺1, our finding of an optimal myofilament
heart rate of 200 min⫺1 means that myofilament mechanisms
alone are not dictating resting heart rates.
One can now hypothesize, for example, either that Ca2⫹handling mechanisms such as those responsible for forcefrequency and mechanical restitution also dictate the optimal
frequency, or that the resting heart rate is not optimal with
respect to any of these underlying cellular mechanisms. By
directly linking cardiac muscle function with whole heart
function, our myofilament-based LV dynamic model provides
the tool needed to formulate and test hypotheses about matching of heart rate to underlying myofilament kinetic mechanism
in intact beating heart. This and other related questions about
matching and tuning of myofilament mechanisms to cardiovascular system function could not previously be addressed in a
manner that lent themselves to experimental testing at the level
of intact heart. With the results of this study, however, it is now
possible to formulate hypotheses and perform, in intact beating
heart rather than in constantly activated (nonbeating) isolated
muscle, the direct experimental tests of hypotheses that require
integrating dynamic muscle function with whole heart function.
In conclusion, to summarize this study, a new dynamic
model of LV pressure-volume relationships in beating heart
was developed by linking chamber pressure-volume dynamics
with cardiac muscle force-length dynamics and using the
assumptions of simple spherical geometry and essential ho-
H127
CONTRACTILE PARAMETERS FROM LV PRESSURE AND VOLUME
values of v(t) by the function R( ). Time variation in the output, v(t),
depends only on time variation in the input, i(t), and nothing about the
system contributes to this time-variation. However, in the case of the
parallel R-C circuit, the input-output relation between v(t) and i(t)
must be written with differential equations either in the familiar form
冉 冊
d␯共t兲
1
1
␯共t兲 ⫽ i共t兲
⫹
dt
RC
C
(A3)
or in an alternative form where we let D ⫽ d/dt and rewrite the
relationship of Eq. A3 as
冤 冥
D⫹
1
RC
⫽ H兵P៮ , 0其 ⫹ terms dependent on ⌬V alone
⫹ terms dependent on ⌬Piso alone
⫹ terms dependent on the interaction
of ⌬V and ⌬Piso
where
Terms dependent on ⌬V alone ⫽
⳵H兵Piso, ⌬V其
⌬V
⳵V
(A5a)
⫹ higher-order terms in ⌬V
Terms dependent on ⌬Piso alone ⫽
兵i共t兲其
(A5)
⳵H兵Piso, ⌬V其
⌬Piso
⳵Piso
(A5b)
⫹ higher-order terms in ⌬Piso
(A4)
Terms dependent on interaction of ⌬V and ⌬Piso
⫽2
The term in square brackets on the right side of Eq. A4 behaves as a
dynamic operator in the sense in which we used it above, i.e., it maps
the complete function i(t) onto v(t). In general, at any instant, the
output of the dynamic system represented by the dynamic operator
H{} depends not only on the present value of the input but on the
complete time history of the system.
History effects enter the general relationship of Eq. A1 and the
specific relationship of Eq. A4 because the relevant dynamic operator
in each equation contains the time-derivative operator
(A5c)
⳵2H兵Piso, ⌬V其
⌬Piso⌬V ⫹ higher-order terms in ⌬Piso⌬V
⳵Piso⳵V
Note that all partial derivatives in Eqs. A5a–A5c are evaluated at the
reference values (P៮ , 0). Time variation is implicit in all terms on the
right-hand side.
Consider now the isovolumic condition where ⌬V ⫽ 0. Here, all
⌬V terms in Eq. A5 including the interaction terms drop out, and
pressure variation takes place as a result of ⌬Piso(t) alone. Under these
isovolumic conditions
⳵H兵Piso, ⌬V其
⳵2H兵Piso, ⌬V其 2
⌬Piso ⫹
⌬Piso
2
⳵Piso
⳵Piso
(A6)
⫹ all higher-order terms involving only ⌬Piso variation
P共t兲 ⫽ Piso共t兲 ⫽ H兵P0, 0其 ⫹
d
D ⫽ 兵其
dt
as one of its primitive operator components. Other primitive operator
components include scalars, summers, and multipliers. We constrain
this analysis to situations in which the presence of primitive operators
in H{}, including D, allows Eq. 1A to be manipulated as if it were an
algebraic equation.
Thus all the ⌬Piso terms in Eq. A5 may be collected into Piso(t) and the
equation rewritten as
P共t兲 ⫽ Piso共t兲 ⫹ terms dependent on ⌬V alone
⳵H兵Piso, ⌬V其
⳵V
Taylor Series Expansion Allows Dividing of Response Into
Components and Then Grouping of Response Components
Into Three Categories
We proceed using an incremental analysis by writing the input
variables as time-dependent variations around mean values. Because
⌬V(t) represents change in volume only and does not include the
baseline volume (VBL), it is already in an incremental form. VBL
enters the relationship through the Frank-Starling mechanism wherein
Piso(t) depends on VBL. Thus the effect of VBL on P(t) is implicit in
the functional dependence of P(t) on Piso(t). By separating the LV
volume into VBL and ⌬V(t) as we have done, Piso(t) and ⌬V(t) in Eq.
A1 are truly independent variables, which facilitates further mathematical decomposition.
An incremental form of Piso(t) is obtained by taking it to be a
beat-period mean value (P៮ ) plus variation [⌬Piso(t)] around this mean, i.e.
P iso共t兲 ⫽ P៮ ⫹ ⌬Piso共t兲
Similarly, the incremental form of ⌬V(t) is written as variation around
zero mean
⌬V共t兲 ⫽ 0 ⫹ ⌬V共t兲
because the reference level of ⌬V(t) is zero.
With freedom to treat H{} algebraically and a representation of the
two input variables, Piso(t) and ⌬V(t), in incremental form (i.e.,
variation around reference levels), it becomes possible to perform a
Taylor series expansion of Eq. A1 around the reference level (P៮ , 0)
AJP-Heart Circ Physiol • VOL
(A7)
⌬V ⫹ higher-order terms
⫹ terms dependent on interaction of ⌬V and ⌬Piso
2
⳵2H兵Piso, ⌬V其
⳵Piso⳵V
⌬Piso⌬V ⫹ higher-order terms
We group the terms in Eq. A7 into the following categories:
P共t兲 ⫺ Piso共t兲 ⫽
⳵H兵Piso, ⌬V其
⌬V
⳵V
⫹2
linear volume effects at average pressure
⫹
⳵2H兵Piso, ⌬V其
⌬Piso⌬V
⳵Piso⳵V
time-varying effects due to Piso共t兲
再
冎
all remaining
higher-order terms
(A8)
nonlinear effects
where P(t) minus Piso(t) ⫽ ⌬P(t) represents the pressure response to a
⌬V(t) administered during a heartbeat
⳵H兵Piso, ⌬V其
⌬V
⳵V
represents the component of the pressure response PE(t) due to linear
⌬V effects as they would occur if pressure were constant at P៮
⳵ 2 H兵Piso, ⌬V其
⌬Piso⌬V
⳵Piso⳵V
represents the component of the pressure response PI⬘(t) due to the
first interaction between ⌬V and ⌬Piso, which embraces the timevarying features of the left ventricle, and
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␯共t兲 ⫽
1
C
P共t兲 ⫽ H兵P៮ ⫹ ⌬Piso, 0 ⫹ ⌬V其
H128
CONTRACTILE PARAMETERS FROM LV PRESSURE AND VOLUME
再
higher-order
terms
冎
I⫽
represents the component of the pressure response PR(t) due to all
remaining terms in the Taylor series including nonlinear features of
the left ventricle.
APPENDIX B
Partial Derivative Terms in Taylor Series Act as
Dynamic Operators
to stress that the pressure from its effect arises from the interaction
between isovolumic pressure and volume change.
The interactance operates on a time signal that is the product of
⌬Piso(t) and ⌬V(t) and carries units of energy. The component of the
pressure response due to the interaction between ⌬Piso(t) and ⌬V(t)
may be succinctly written as
Pressure due to interactance ⫽ PI共t兲 ⫽ I兵⌬Piso共t兲⌬V共t兲其
It is important to recognize that the partial derivatives of H{Piso,
⌬V} in Eq. A8 are, like the parent expression, dynamic operators that
operate on the variables ⌬V(t) and/or ⌬Piso(t)⌬V(t). Thus
Because ⌬Piso(t)⌬V(t) carries units of energy, I{} carries units of
reciprocal volume.
Because I relates to E through differentiation with respect to Piso(t),
we use this to derive a specific formulation for I{}. We assume that of
the E parameters, only the scaling coefficients E0 and E⬁ change with
Piso, whereas the recruitment rate constant b and the distortion rate
constant c do not. With this assumption, I may be written as
operates on ⌬V in a way that includes dynamic effects. A specific
mathematical form of this operation is derived in the text as
(B1)
For now, we are concerned with the physical meaning of the
operation. Note that
expresses the change in pressure ⳵H{Piso,⌬V} with a change in
volume ⳵V (and carries the units of mmHg/ml). With these units and
the fact that this operator includes dynamic effects, this partial
derivative is properly designated as a dynamic elastance. To simplify
notation, this may be written as
⳵E0 b
⳵E⬁ D
⫹
⳵Piso D ⫹ b ⳵Piso D ⫹ c
where E{⌬V} is the symbolic representation for the elastance operation. Thus the component of the pressure response due to linear
dynamic elastance may be given in succinct notation as
Pressure due to elastance ⫽ E兵⌬V共t兲其
(B2)
We now look at the interaction term, which we judge to be of prime
importance in this problem. Because ⌬Piso(t) and ⌬V(t) are independent of one another, the sequence of derivative operations in the
interaction term may be performed in any order. Consider the differentiation sequence
册
⳵ 2 H兵Piso, ⌬V其
⳵
⳵ ⳵H兵Piso, ⌬V其
⫽
关E兴
⫽
⳵Piso⳵V
⳵Piso
⳵V
⳵Piso
b
D
⫹ I⬁
D⫹b
D⫹c
In as much as [E] is a dynamic operator
⳵
关E兴
⳵P iso
P I共t兲 ⫽ I0␾共t兲 ⫹ I⬁␨共t兲
⳵
关E兴
⳵P iso
(B7)
where ␾(t) is a time function with units of work (mmHg 䡠 ml) given by
the first-order differential equation
(B8)
and ␨(t) is a time function also with units of work (mmHg 䡠 ml) given
by
␨˙ 共t兲 ⫽ ⫺c␨共t兲 ⫹ u̇共t兲
(B9)
The physical units of I0 and I⬁ are reciprocal volume (ml⫺1).
The remainder term, because it consists of lumping several terms in
the Taylor series together, has few guidelines for representing its
operator and the signal upon which it operates. Here, we take an ad
hoc approach and express the pressure due to these remainder terms,
PR(t), in a form analogous to that used for expressing pressure due to
elastance and interactance, i.e.
(B10)
where it is understood that i{}, like E{} and I{}, is a dynamic
operator, and y(t) is an appropriate signal to be defined later.
With this notation, we may now couch the pressure response to a
volume perturbation ⌬P(t) in terms of its components as
⌬P共t兲 ⫽ E兵⌬V共t兲其 ⫹ I兵⌬Piso共t兲⌬V共t兲其 ⫹ R兵y共t兲其
is also a dynamic operator where the coefficients in [E], as they are
given in Eq. B1, become modified in
(B6)
In the manner of the elastance example, we can now write the
dynamic operations of Eq. B6 in differential equation format
P R共t兲 ⫽ R兵y共t兲其
(B3)
(B5)
where ⳵E0/⳵Piso is the slope of E0 dependence on Piso (let I0 ⫽
⳵E0/⳵Piso) and ⳵E⬁/⳵Piso is the slope of E⬁ dependence on Piso (let
I⬁ ⫽ ⳵E⬁/⳵Piso). Substitution yields
␾˙ 共t兲 ⫽ ⫺b关␾共t兲 ⫺ u共t兲兴
⳵H兵Piso, ⌬V其
兵⌬V其 ⫽ E兵⌬V其
⳵V
冋
I⫽
I ⫽ I0
⳵H兵Piso, ⌬V其
⳵V
(B4)
(B11)
A simplified expression for ⌬P(t) may be derived from the relations
between the elastance and interactance components. Note that
⌬P iso共t兲 ⫽ Piso共t兲 ⫺ P៮
(B12)
⌬V共t兲⌬Piso共t兲 ⫽ ⌬V共t兲Piso共t兲 ⫺ ⌬V共t兲P៮
(B13)
thus
according to their variation with respect to Piso. We designate this new
dynamic operator the dynamic interactance
AJP-Heart Circ Physiol • VOL
and
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⳵H兵Piso, ⌬V其
⳵V
⳵H兵Piso, ⌬V其
b
D
⫽ E0
⫹ E⬁
⳵V
D⫹b
D⫹c
⳵
关E兴
⳵Piso
CONTRACTILE PARAMETERS FROM LV PRESSURE AND VOLUME
I兵⌬V共t兲⌬Piso共t兲其 ⫽ I兵⌬V共t兲Piso共t兲其 ⫺ I兵⌬V共t兲P៮ 其
(B14)
Furthermore,
I兵⌬V共t兲P៮ 其 ⫽ IP៮ 兵⌬V共t兲其
(B15)
Substituting Eq. B15 into Eq. B14
⌬P共t兲 ⫽ E兵⌬V共t兲其 ⫹ I兵⌬V共t兲Piso共t兲其 ⫺ IP៮ 兵⌬V共t兲其 ⫹ R兵y共t兲其 (B16)
Because
I⫽
then
⳵E⬁ D
⳵E0 b
⫹
⳵Piso D ⫹ b ⳵Piso D ⫹ c
冋 册
It can be shown that
冋 册
(B18)
冋 册
(B19)
⳵E 0 ៮
P ⫽ E0
⳵P iso
and
(B17)
⳵E ⬁ ៮
P ⫽ E⬁
⳵P iso
thus
E兵⌬V共t兲其 ⫽ IP៮ 兵⌬V共t兲其
(B20)
Substituting Eq. B20 into Eq. B16 gives
⌬P共t兲 ⫽ I兵⌬V共t兲Piso共t兲其 ⫹ R兵y共t兲其
(B21)
which is an alternative equation for ⌬P(t) to that used in the text.
ACKNOWLEDGMENTS
The authors acknowledge the help of Robert Hutchinson in the machining
and construction of the experimental device used in these studies.
GRANTS
This work was supported in part by a generous grant from the Washington
State Fraternal Order of Eagles, Pasco Erie (to K. B. Campbell); by National
Heart, Lung, and Blood Institute Grant RO1 HL-61487/62881 (to H. Granzier);
and by American Heart Association Grant 0435241N (to Y. Wu).
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