## Abstract

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.

- heart function
- muscle
- mathematical model
- cardiac fiber
- force

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/d*t*) 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/d*t* 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 O_{2} 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.

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 frequency-dependent 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.

## Glossary

*A*_{i}- Magnitude scalar for
*i*th sine wave component of ΔV(*t*) signal *b*- Rate constant of recruitment
*c*- Rate constant of distortion
- D
- Time derivative operator (=d/d
*t)* *E*(*t*)- Time-varying left ventricular (LV) elastance
*E*(*t*)-*R*- Elastance-resistance LV model
*E*{}- Dynamic elastance operator
*E*_{0}- Zero-frequency LV elastance
*E*_{∞}- Infinite-frequency LV elastance
*E*_{P}- Passive elastance
*e(i)*- Residual errors
- F(
*t*) - Midwall fiber force
*f*_{0}- Heart pacing frequency
*f*_{min}- Frequency of minimal stiffness
- H{}
- Dynamic operator that maps LV inputs to LV pressure
- I{}
- Dynamic interactance operator
- I
_{0} - Zero-frequency LV interactance
- I
_{∞} - Infinite-frequency LV interactance
*L(t)*- Length of midwall circumference
*L*_{BL}- Baseline midwall circumference
*n*_{i}- Frequency multiplier of
*f*_{0}for*i*th sine wave component of ΔV(*t*) signal - P(
*t*) - LV chamber pressure
- P̄
- Average pressure during isovolumic beat
- PF
_{f} - Force-to-pressure transforming factor
- P
_{I}(*t*) - LV pressure component due to interactance
- P
_{iso}(*t*) - Isovolumic pressure
- P
_{p}(*t*) - Passive pressure
- P
_{R}(*t*) - LV pressure component due to remainder terms in Taylor series
*r*- Rate constant of R{}
- R{}
- Dynamic operator that maps
*y(t)*into P_{R}(*t*) - R
_{0} - Magnitude of R{}
- V(
*t*) - LV chamber volume
- V
_{BL} - Baseline LV chamber volume
- V
_{w} - LV wall volume
*x(t)*- Cardiac muscle distortion variable
*y(t)*- Inputs responsible for P
_{R}(*t*) - Q
_{10} - Ratio of reaction rate at one temperature and reaction rate at a temperature 10°C lower
- ΔF(
*t*) - Changes in cardiac muscle force
- Δ
*L*(*t*) - Changes in cardiac muscle length
*α, β*- Viscoelastic parameters of passive left ventricle
- χ
_{0} - Scalar parameter for elastance
- ρ
_{0} - Scalar parameter for resistance
- ε{}
- Dynamic stiffness operator
- ε
_{0} - Zero-frequency cardiac muscle stiffness
- ε
_{∞} - Infinite-frequency cardiac muscle stiffness
- ε(
*j*ω) - Frequency-dependent muscle fiber stiffness
- φ(
*t*) - LV interactance recruitment variable
- η(
*t*) - LV volume recruitment variable
- ν(
*t*) - Cardiac muscle recruitment variable
- ψ(
*t*) - LV volume distortion variable
- ζ(
*t*) - 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 of14fiber length (9). We then measured force development and response to length change during constant Ca^{2+} 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 (1) (2) (3) In these equations, a dot over a variable represents the first time derivative; Δ*L(t)* is the measured change in length imposed upon the muscle; ΔF(*t*) is the model-predicted force in response to Δ*L(t)*; ν*(t)* is the recruitment variable; and *x(t*) is the distortion variable. Both ν*(t)* and *x(t)* possess units of length. Parameters ε_{0} 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 ε_{0} 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 Δ*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 ΔF*(t)* and Δ*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/d*t*){}, *Eqs. 1*–*3* may be written in the alternative but equivalent form (4) In *Eq. 4*, all terms within the square brackets constitute a dynamic operator that operates on the input, Δ*L(t)*, to produce an output, ΔF*(t)*. Let ε represent the operator in square brackets. Then the dynamic force-length relation of cardiac muscle may be succinctly written as (5) The dynamic operator ε{} has physical units of stiffness. Conversion of ε{} to the frequency domain changes D to the complex frequency variable *jω* and allows construction of the stiffness frequency spectrum.

### Dynamic Muscle Stiffness (Constant Activation) is Converted to Dynamic LV Elastance (Constant Activation) by Geometric Transformation

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 force-length 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 (6) and a circumferential stress (force per cross-sectional area) to chamber-pressure transformation is given by (7)

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 measurements are made of LV pressure and volume. Then, for small ΔV around a baseline volume (V_{BL}), the baseline length (*L*_{BL}) of a representative midwall circumference will be (8) Furthermore, the force per unit cross-sectional area at the midwall is given by where the pressure-force transforming factor 〈PF_{f}〉 is (from *Eq. 7*) (9) With the use of these transformations, it can be shown that the relationship between a generic midwall muscle stiffness, ε = dF/d*L*, and a generic LV chamber elastance, *E* = dP/dV, is (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 Ba^{2+} 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* (11) where *E*_{0} 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 ε_{0} to *E*_{0} 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 (P_{E}) and in response to a ΔV(*t*) perturbation in a constantly activated heart may be written as (12) Alternatively, it may be written in differential equation form analogous to *Eq. 1* as (13) where η(*t*) is the volume recruitment variable, analogous to the length recruitment variable ν(*t*) in *Eq. 2*, and ψ(*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 (14) (15) 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).

### 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 [ΔV(*t*)] depends on both ΔV(*t*) and the isovolumic pressure [P_{iso}(*t*)] that would have developed had no ΔV(*t*) been administered.

In general, the dependence of P(*t*) on ΔV(*t*) and P_{iso}(*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. (16) *Equation 16* says that P(*t*) is the result of the mathematical operation H{} on two input functions, ΔV(*t*) and P_{iso}(*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 [ΔP(*t*)] to a small-amplitude volume perturbation in terms of the following: *1*) an elastance pressure [P_{E}(*t*)] due to the action of ΔV(*t*) alone as if ΔV(*t*) were applied to the left ventricle, generating a constant pressure equal to the mean of P_{iso}(*t*); *2*) an interactance pressure [P_{I}(*t*)] due to the interaction of ΔV(*t*) and the variation of P_{iso}(*t*) around its mean value[ΔP_{iso}(*t*)]; and *3*) a residual pressure [P_{R}(*t*)] due to the sum of all the residual higher-order terms in the Taylor series.

Thus (17)

In this, the elastance pressure P_{E}(*t*) is given by *Eqs. 12*–*15* as if the heart is constantly activated to generate a pressure equal to the mean of P_{iso}(*t*). The interactance pressure P_{I}(*t*), which is of prime importance in this problem, may be couched in a form similar to our earlier expression, *Eq. 12*, for P_{E}(*t*) (18) where the operator I{} operates dynamically on the product ΔV(*t*)ΔP_{iso}(*t*), which may be treated as a time signal defined as *u*(*t*) = ΔV(*t*)ΔP_{iso}(*t*). Furthermore, it is shown in the appendix that I relates to E through differentiation with respect to P_{iso}(*t*) as follows: (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 *E*_{0} and *E*_{∞} change with P_{iso}, whereas the recruitment rate constant *b* and the distortion rate constant *c* do not. With this assumption, I may be represented as (20) where ∂*E*_{0}/∂P_{iso} is the slope of *E*_{0} dependence on P_{iso} (let I_{0} = ∂*E*_{0}/∂P_{iso}), and ∂*E*_{∞}/∂P_{iso} is the slope of i_{∞} dependence on P_{iso} (let I_{∞} = ∂*E*_{∞}/∂P_{iso}). Substitution yields (21)

In the manner of the elastance example, we can now write the dynamic operations of *Eq. 18* in differential equation format (22) where φ(*t*) is a time function with units of work (mmHg·ml) given by the first-order differential equation (23) and ζ*(t)* is a time function also with units of work (mmHg·ml) given by (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 I_{0} 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 (P̄) that (25) and (26)

The remainder term [P_{R}(*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 P_{R}(*t*) is small relative to P_{I}(*t*), and its exact representation is less important to the problem than a correct representation of P_{I}(*t*). Here we take an ad hoc approach and express the pressure due to these remainder terms, P_{R}(*t*), in a dynamic form, i.e. (27) 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 P_{R}(*t*) to a deactivation effect (19, 28, 29), we assumed that deactivation comes about as a result of distortion. Distortion is expressed in P_{E}(*t*) and P_{I}(*t*) by the terms *E*_{∞}ψ(*t*) and I_{∞}ζ(*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 (28) With this, R{} was represented as (29) where R_{0} is a magnitude parameter and *r* is a rate constant. The corresponding differential equation is (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 (a) (b) (b1) (b2) (c) (c1) (c2) (d) We refer to *Eqs. a*–*d* as the dynamic LV model. The dynamic LV model equations contain five state variables [η(t), ψ(t), φ(t), ζ(t), and P_{R}(t)] and eight parameters including five magnitude-scaling parameters (*E*_{0}, *E*_{∞}, I_{0}, I_{∞}, and R_{0}) and three rate constants (*b*, *c*, and *r*). *Equations 25* and *26* demonstrate that *E*_{0} and *E*_{∞} are not independent of I_{0} and I_{∞}. Thus there are only six free parameters in the model equation set. (Note that as a consequence of the interdependence of *E*_{0} and I_{0} 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 Krebs-Henseleit solution (Ca^{2+} concentration, 1.25 mM) that contained high levels of dissolved O_{2} (Po_{2} > 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 μl 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 [V_{BL}], 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 V_{BL}, 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 ≥ 100 mmHg and passive stiffness ≤ 0.7 mmHg/μl at V_{BL}. 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. 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 ΔV*(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 (31) where ΔV̄_{j} = −0.02V_{BL}, 0, or 0.02V_{BL}, and *f*_{0} = 1/(pace period). The magnitude scalar (*A*_{i}) and frequency multiplier of *f*_{0} (*n*_{i}*)* for the *i*th 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 *f*_{0}, 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 P_{iso}(*t*) value in the beat immediately preceding the volume-perturbed beats. This was taken to be the P_{iso}(*t*) value that would have been generated by the two perturbed beats if ΔV(*t*) had not been applied. With two frequency compositions and three ΔV̄_{j} values, an ensemble of six pressure-response records was generated.

### Removing Passive Pressure Response From Measured Data

Pacing periods were chosen to be long enough to allow a well-defined 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 32 where P_{p}(*t*) is passive pressure, *E*_{P} is the DC passive elastance, and α and β 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 [P_{iso}(*t*) < 10 mmHg; two diastolic periods in each record of two beats] and the passive parameters (*E*_{P}, α, and β) 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 ΔP(*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.

### Data Fitting, Parameter Estimation, and Model Evaluation

The dynamic LV model was fit to the active part of the measured pressure response by solving the model equations (*Eqs. a*–*d*) for ΔP(*t*) using measured ΔV(*t*) and ΔP_{iso}(*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 ΔP(*t*) was fit to measured ΔP(*t*) by adjusting the six free model parameters (*E*_{0}, *b*, *E*_{∞}, *c*, R_{0}, 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 (*E*_{0}, *b*, *E*_{∞}, *c*, R_{0}, 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, ΔP̂(*t*); and *4*) the time series of residual errors, *e(i*) = ΔP̂(*i*) − ΔP(*i*), where the *i* index indicates sampled values of ΔP̂(*t*) and ΔP(*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 ΔP̂(*t*) on ΔP(*t*), ΔP̂(*t*) = *b*ΔP(*t*) + *p*_{0}, and an evaluation of the regression parameters for their ideal values of *b* = 1 and *p*_{0} = 0; and *2*) a relative error, *e*_{r}, defined as where, for *npts* data points in the sample, is the mean square error of residuals and is the total variance in Δ*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 − *R*^{2}. Thus *e*_{r} 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

### Volume-Perturbation Protocol Resulted in Pressure-Response Signals With Rich Dynamic Content

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 model was fit (ΔV̄_{j} = 0, *frequency compositions 1 and 2*). Note that at this pacing rate, there is ∼0.5 s of diastole during which passive LV properties could be estimated using *Eq. 32*. The ΔP(*t*) signal shown in Fig. 4 (*bottom*) consisted of just the active part of the response (i.e., the passive response has been removed). Note the difference in ΔP(*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 (*f*_{0}, 0.5) in the volume-perturbation signal (Table 1). Also, note the different shape and form of ΔP(*t*) when ΔV(*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 volume-perturbation protocol, sufficient information was present in the combined records to estimate model parameters that selectively influenced low- and high-frequency behaviors.

### 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*) large-amplitude 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 at different ΔV̄_{j}, the trajectory of response at ΔV̄_{j} = 0.02V_{BL} was always above that at ΔV̄_{j} = 0, which was above the trajectory at ΔV̄_{j} = −0.02V_{BL} in both measured and model-generated records.

The fit of the dynamic LV model to the full ensemble of six pressure-response records (three ΔV̄_{j} times two frequency compositions) generated the goodness-of-fit measures in Table 2. An example of a fit to an ensemble of six records (*animal 29*; 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 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 *R*^{2} from fits to records at 37°C was 0.83; the mean *R*^{2} from fits to records at 25°C was 0.80. The mean *e*_{r} (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 (*R*^{2} = 0.65; *e*_{r} = 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 pressure-response 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 (36) Because time variation in *E*(*t*) and i(*t*) follows that of P_{iso}(*t*) (53, 54), *Eq. 36* was written in terms of P_{iso}(*t*) and two parameters including a scalar for elastance (χ_{0}) and another for resistance (ρ_{0}) (37) 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 χ_{0} and ρ_{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 *R*^{2} from fitting the measured pressure responses with the *E*(*t*)-*R* model was only 0.37 compared with the *R*^{2} 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 phase of the cardiac cycle. These errors briefly became predominantly negatively valued at the time of minimum dP_{iso}/d*t* 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 Temperature-Insensitive (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 (*E*_{0}, *b*, *E*_{∞}, *c*, *R*_{0}, and *r*) were the four derived from dynamic behavior of muscle (*E*_{0}, *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 force-generating units (*i*_{0} 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 (*E*_{0} and *E*_{∞}) and parameters that govern the speed (kinetics) of the two components of the response (*b* and *c*). Magnitude-scaling parameters (*E*_{0} and *E*_{∞}) represent different aspects of the response than 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 (P̄), over the course of an isovolumic beat, hearts at 25°C generated more pressure (P̄ = 60.4 mmHg) than those at 37°C (P̄ = 49.3 mmHg). To account for this difference in the comparison, we plotted *E*_{0} and *E*_{∞} vs. P̄ and analyzed differences in the *E*_{0} vs. P̄ and *E*_{∞} vs. P̄ 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 *E*_{0} and *P* = 0.12 for *E*_{∞}. Therefore, we concluded that the magnitude-scaling parameters *E*_{0} 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.

### 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 (P̄) during the beat. Geometric transformation (*Eq. 7*) was used to convert P̄ 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 P̄ 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 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 Q_{10} values from these Arrhenius lines were 2.02 for *b* and 2.21 for *c*. These Q_{10} 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*ω)] 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*ω) to which physiological significance may be attached including 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 (*f*_{min}) or frequency at which ε(*j*ω) becomes minimally valued. Until now, ε(*j*ω) 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*ω) estimated from beating heart and constantly activated muscle, the magnitude frequency spectrum of ε(*j*ω) 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*ω) 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 *f*_{min} 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.

For example, we have documented (9) that *f*_{min} represents the frequency at which there is a transition in dominance of mechanisms responsible for muscle fiber stiffness (i.e., the ΔF/Δ*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, *f*_{min}, 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 *f*_{min} 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 Ca^{2+}-handling mechanisms such as those responsible for force-frequency 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 homogenous 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 pressure-volume 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 [ΔV(*t*)] that occur during a contraction event and the pressure that would have developed [P_{iso}(*t*)] had no volume change occurred. In general, P(*t*) dependence on ΔV(*t*) and P_{iso}(*t*) involves history (memory) of the system. To include history effects, we write the dependence in terms of the dynamic operator H{}, i.e. (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, ΔV(*t*) and P_{iso}(*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 (A2) Here, instantaneous values of i(t) are mapped onto instantaneous 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 (A3) or in an alternative form where we let D = d/d*t* and rewrite the relationship of *Eq. A3* as (A4) 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 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.

### 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 ΔV(*t*) represents change in volume only and does not include the baseline volume (V_{BL}), it is already in an incremental form. V_{BL} enters the relationship through the Frank-Starling mechanism wherein P_{iso}(*t*) depends on V_{BL}. Thus the effect of V_{BL} on P(*t*) is implicit in the functional dependence of P(*t*) on P_{iso}(*t*). By separating the LV volume into V_{BL} and ΔV(*t*) as we have done, P_{iso}(*t*) and ΔV(*t*) in *Eq. A1* are truly independent variables, which facilitates further mathematical decomposition.

An incremental form of P_{iso}(*t*) is obtained by taking it to be a beat-period mean value (P̄) plus variation [ΔP_{iso}(*t*)] around this mean, i.e. Similarly, the incremental form of ΔV(*t*) is written as variation around zero mean because the reference level of ΔV(*t*) is zero.

With freedom to treat H{} algebraically and a representation of the two input variables, P_{iso}(*t*) and ΔV(*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 (P̄, 0) (A5) where (A5a) (A5b) (A5c) Note that all partial derivatives in *Eqs. A5a*–*A5c* are evaluated at the reference values (P̄, 0). Time variation is implicit in all terms on the right-hand side.

Consider now the isovolumic condition where ΔV = 0. Here, all ΔV terms in *Eq. A5* including the interaction terms drop out, and pressure variation takes place as a result of ΔP_{iso}(*t*) alone. Under these isovolumic conditions (A6) Thus all the ΔP_{iso} terms in *Eq. A5* may be collected into P_{iso}(*t*) and the equation rewritten as (A7) We group the terms in *Eq. A7* into the following categories: (A8) where P(*t*) minus P_{iso}(*t*) = ΔP(*t*) represents the pressure response to a ΔV(*t*) administered during a heartbeat represents the component of the pressure response P_{E}(*t*) due to linear ΔV effects as they would occur if pressure were constant at P̄ represents the component of the pressure response P_{I}′(*t*) due to the first interaction between ΔV and ΔP_{iso}, which embraces the time-varying features of the left ventricle, and represents the component of the pressure response P_{R}(*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

It is important to recognize that the partial derivatives of H{P_{iso}, ΔV} in *Eq. A8* are, like the parent expression, dynamic operators that operate on the variables ΔV(*t*) and/or ΔP_{iso}(*t*)ΔV(*t*). Thus operates on ΔV 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{P_{iso},ΔV} 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 where *E*{ΔV} 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 (B2)

We now look at the interaction term, which we judge to be of prime importance in this problem. Because ΔP_{iso}(*t*) and ΔV(*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 (B3) In as much as [*E*] is a dynamic operator is also a dynamic operator where the coefficients in [*E*], as they are given in *Eq. B1*, become modified in according to their variation with respect to P_{iso}. We designate this new dynamic operator the *dynamic interactance* 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 ΔP_{iso}(*t*) and ΔV(*t*) and carries units of energy. The component of the pressure response due to the interaction between ΔP_{iso}(*t*) and ΔV(*t*) may be succinctly written as (B4) Because ΔP_{iso}(*t*)ΔV(*t*) carries units of energy, I{} carries units of reciprocal volume.

Because I relates to *E* through differentiation with respect to P_{iso}(*t*), we use this to derive a specific formulation for I{}. We assume that of the *E* parameters, only the scaling coefficients *E*_{0} and *E*_{∞} change with P_{iso}, whereas the recruitment rate constant *b* and the distortion rate constant *c* do not. With this assumption, I may be written as (B5) where ∂*E*_{0}/∂P_{iso} is the slope of *E*_{0} dependence on P_{iso} (let I_{0} = ∂*E*_{0}/∂P_{iso}) and ∂*E*_{∞}/∂P_{iso} is the slope of *E*_{∞} dependence on P_{iso} (let I_{∞} = ∂*E*_{∞}/∂P_{iso}). Substitution yields (B6)

In the manner of the elastance example, we can now write the dynamic operations of *Eq. B6* in differential equation format (B7) where φ(*t*) is a time function with units of work (mmHg·ml) given by the first-order differential equation (B8) and ζ(*t*) is a time function also with units of work (mmHg·ml) given by (B9) The physical units of I_{0} 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, P_{R}(*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 ΔP(*t*) in terms of its components as (B11)

A simplified expression for ΔP(*t*) may be derived from the relations between the elastance and interactance components. Note that (B12) thus (B13) and (B14) Furthermore, (B15) Substituting *Eq. B15* into *Eq. B14* (B16) Because then (B17) It can be shown that (B18) and (B19) thus (B20) Substituting *Eq. B20* into *Eq. B16* gives (B21) which is an alternative equation for ΔP(*t*) to that used in the text.

## 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).

## Acknowledgments

The authors acknowledge the help of Robert Hutchinson in the machining and construction of the experimental device used in these studies.

## Footnotes

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- Copyright © 2005 by the American Physiological Society