INTRODUCTION

TOP ABSTRACT INTRODUCTION MODEL FOR MUSCLE DYNAMICS RESULTS DISCUSSION REFERENCES

GUNST AND COLLEAGUES (3, 7, 14) have reported extensive data on the dynamic force (F)-length (L) behavior of maximally activated trachealis muscle. They imposed L oscillations with a range of amplitudes and frequencies and measured the F exerted by the muscle. These data show that the dynamic F-L curves of smooth muscle are quite different from the isometric F (F_iso)-L curve. The dynamic curves depend on amplitude and rate of change of muscle L.

During the last decade, mathematical models of constricted lungs have been proposed (2, 10). Although these models are intended to describe the mechanics of ventilated lungs, they use the F_iso-L curve to describe the behavior of smooth muscle. An empirical model of dynamic smooth muscle behavior would be useful in modeling dynamic airway and lung mechanics, and here we propose such a model. The model is not intended to describe the basic molecular processes that underlie the dynamic F-L curve. Instead, it is an entirely empirical model that is intended to be simple enough to be useful in modeling lung mechanics. The model consists of a set of three first-order, ordinary differential equations. The solutions to these equations match the main features of the data on dynamic F-L relations. This model for maximally activated smooth muscle is incorporated into a model for an airway in the lung, and, as an example of the use of the model, the recovery of airway constriction after a deep breath is analyzed.

	MODEL FOR MUSCLE DYNAMICS

TOP ABSTRACT INTRODUCTION MODEL FOR MUSCLE DYNAMICS RESULTS DISCUSSION REFERENCES

In formulating the model, we focused on the data of Shen et al. (14) shown in Fig. 1. In these experiments, Shen et al. oscillated the L of maximally activated trachealis muscle and measured the periodic F-L relation that was established after a number of cycles. The muscle was shortened at a constant rate and then returned to its original L, with a rate of lengthening equal to the rate of shortening. For the loops shown in Fig. 1A, the amplitudes of L oscillation are the same, but the rates of shortening and lengthening are different. The rates extend over a range of a factor of 80. For the loops shown in Fig. 1B, the rates of shortening are the same, but the amplitudes are different.

View larger version (19K): [in this window] [in a new window]   Fig. 1.   Periodic force (F)-length (L) curves for maximally activated trachealis muscle for different frequencies (A) and different amplitudes (B). In all cases, maximum muscle L is 0.8 optimal length (Lo). Muscle F is nondimensionalized by the isometric force (Fiso) at maximum muscle L [Fiso(0.8 Lo)]. These curves are redrawn from data reported by Shen et al. (14).

We assume that measurements made at different muscle L values would be similar to the data shown in Fig. 1. That is, we assume that F at L scales with the F_iso at that L, and that F/F_iso depends on L as a fraction of optimal length (L_o). Thus, in terms of the dimensionless variables, l = L/L_o and f = F/F_iso, f(l) is assumed to be a universal function.

The model for the relation between f and l for the shortening limbs was guided by the following features of the data. As shown in Fig. 1B, the shortening limbs for muscle contracting at the same rate from the same initial F and L coincide, independent of the previous stretch amplitude. Thus the F-L behavior during shortening appears to be determined by the current F, L, and velocity of shortening, independent of history. As shown in Fig. 1A, the shortening limb for the fastest rate of shortening appears to be an exponential. The curves for slower rates peel off the curve for the fastest rate as if F were relaxing toward f = 1. In fact, plots of the slope of the descending limb (df/dl) vs. f are straight lines with different slopes for different rates of shortening and a common intersection at f = 1. The following equation was, therefore, chosen to describe the F-L relation for shortening

(1)

where k₁ and a₁ are constants, and and are time derivatives of f and l, respectively. The first term on the right side of Eq. 1 represents a nonlinear spring. The second term describes a relaxation process. That is, if were zero, this term would cause f to approach the equilibrium value of 1 with a rate that is proportional to the difference between the value of f and its equilibrium value. The effect of the second term is more pronounced at slower rates of shortening.

The lengthening limbs shown in Fig. 1 are more complicated. Despite beginning at different F and L values, all return to about the same F at the peak muscle L (0.8 L_o). The F-L behavior is thus dependent on history, effectively remembering the peak L attained during the cycle. Plots of df/dl vs. f for these curves are straight lines, but the slopes and intercepts depend not only on the rate of lengthening, but also on the amplitude of the oscillation. To describe the lengthening limbs, we, therefore, chose an equation similar to Eq. 1 but with an additional variable that depends on the history of the maneuver, g. This equation contains additional constants: k₂, k₃, a₂, a₃, and c₁

(2)

The three terms on the right side of Eq. 2 describe, in order, a nonlinear spring, a relaxation toward g, and a slower relaxation toward the F_iso. The exponential in the last term limits f from rising far above the F_iso.

The variable g in Eq. 2 contains information about the history of the maneuver. This variable is governed by the following equation in which the a's and c₂ are constants.

(3)

where is the time derivative of g. The first term on the right side of Eq. 3 describes a relaxation of g toward f. The rate constant for this relaxation is relatively small, and this term drives g toward the average value of f for the cycle. The second term describes a more complicated effect of f on g: for f < 1, this term is positive, and for f > 1, it is negative and large. Thus, for cyclic oscillations, this term adjusts the value of g so that the peak value of f is slightly bigger than 1. In the isometric limit, this term ensures that g, and hence f, equals 1.

Eqs. 1-3 contain a total of 10 constants. By an extensive trial-and-error process, the following values of these constants were chosen: k₁ = 28, k₂ = 10, k₃ = 12, a₁ = 0.12 s¹, a₂ = 0.16 s¹, a₃ = 0.001 s¹, a₄ = 0.001 s¹, a₅ = 0.002 s¹, c₁ = 22, and c₂ = 20.

The solution to Eqs. 1-3 gives muscle F as a fraction of F_iso, and F_iso depends on muscle L. For muscle L < L_o, F_iso(l) is described by the following equation, where F_o is the F_iso at L_o

(4)

To determine F, Eqs. 1-3 are solved for f, and f is multiplied by F_iso, as given by Eq. 4.

	RESULTS

TOP ABSTRACT INTRODUCTION MODEL FOR MUSCLE DYNAMICS RESULTS DISCUSSION REFERENCES

The curves F(l), calculated from Eqs. 1-4 for the same cyclic L maneuvers as those in Fig. 1, are shown in Fig. 2.

View larger version (18K): [in this window] [in a new window]   Fig. 2.   Periodic F-L curves computed from the model for the same oscillation parameters as those for the data shown in Fig. 1. A: different frequencies; B: different amplitudes.

An additional comparison between data and model is shown in Fig. 3. The data are redrawn from the results reported by Fredberg et al. (1). Fredberg et al. measured cyclic F-L curves for maneuvers that differed from those used by Shen et al. (14). The L oscillations were sinusoidal rather than saw-tooth, and mean L rather than peak L was held constant for different amplitudes.

View larger version (14K): [in this window] [in a new window]   Fig. 3.   A: F vs. L/Lo for sinusoidal L oscillation, redrawn from data reported by Fredberg et al. (1). B: F as a fraction of maximum Fiso at Lo (Fo) vs. L/Lo computed from the model for the same L oscillation parameters.

Airway model. A model for the constricted airway embedded in lung parenchyma was described by Gunst et al. (6), and others have used similar models (10, 11). In these models, the airway is pictured as consisting of a circumferential band of muscle surrounding a layer of incompressible tissue, as shown in Fig. 4. The adventitia is ignored, and the radius of the muscle is assumed to be the same as the outer radius of the airway. The maximum outer radius of the unconstricted airway, and hence, the maximum radius of the muscle, is denoted by the constant r_o. The radius of the muscle in the constricted state is denoted r_m, and the radius of the airway lumen is denoted r_l. The r_m and the r_l, nondimensionalized by r_o, are denoted _m and _l, respectively. We assume that the fraction of the area of the unconstricted airway occupied by tissue is 0.16 and that this area remains constant when the airway contracts. Thus    = 0.16, and the lumen is closed when _m = 0.4. 

View larger version (20K): [in this window] [in a new window]   Fig. 4.   Model of airway embedded in the parenchyma. The radius of the band of smooth muscle at the outer edge of the airway wall is denoted rm, and the airway lumen radius is denoted rl. Lumen pressure (Plumen) acts on the inner surface of the airway, and alveolar pressure (Palv) and parenchymal tethering stress () contribute to peribronchial pressure.

(5)

(6)

(7)

(8)

(9)

Example. As an example of an application of the model described above, we analyze the response of a constricted airway to a deep breath. Before the deep breath, the airway is assumed to be closed. In the closed state, _l = 0, and the inner walls of the airway are in contact. The contact pressure provides a value of P_lumen that is required to hold _m at the value _m = 0.4. It is assumed that the closed state has been held for sufficient time for F to relax to F_iso(0.4). Thus, in the initial state, the airway is closed and the F in the smooth muscle is the F_iso for _m = 0.4.

View larger version (11K): [in this window] [in a new window]   Fig. 5.   A: assumed form of the curve of PA vs. time during a deep breath. B: computed curve of F nondimensionalized by maximum Fiso Fo vs. muscle radius. The Fiso-L curve is shown by the dashed line. C: muscle radius vs. time. The lumen of the airway is closed at nondimensionalized muscle radius (m) = 0.4.

	DISCUSSION

TOP ABSTRACT INTRODUCTION MODEL FOR MUSCLE DYNAMICS RESULTS DISCUSSION REFERENCES

We set out to generate an empirical model that describes the dynamic F-L behavior of airway smooth muscle and that could be used in models of airway dynamics. Our model, unlike other models of smooth muscle dynamics (8, 12, 16), is not based on concepts of molecular dynamics of smooth muscle contraction. Ours is a purely empirical model, and its form is simpler than the models based on molecular dynamics.

The model consists of three ordinary differential equations. The first describes the F-L curve for muscle shortening. It consists of a combination of a nonlinear spring and a linear relaxation toward the F_iso. This simple relation is remarkably effective at matching the data for muscle shortening for a wide range of rates of shortening.

The model for muscle lengthening is more complicated because it appears that the F developed during muscle lengthening is history dependent. An internal variable g that depends on muscle history appears in the equation for lengthening. The equation for lengthening includes a nonlinear spring with a spring constant that depends on both f and g and two relaxation terms, a relaxation of f toward g and a slower relaxation of f toward 1.

A separate equation governs the evolution of g. The first term in this equation describes a slow relaxation of g toward f. For cyclic L histories, this term drives g toward the average value of f. However, we found that the data could not be matched without including a second term in this equation. This term adjusts g slightly. For slow, large-amplitude oscillations, g is slightly lower than the average value of f, and, for faster, smaller amplitude oscillations, g is slightly bigger than the average value of f. We have no physical analog for this term, akin to a nonlinear spring or F relaxation; it simply provides an ad hoc method for adjusting the F-L loops so that peak F and F_iso coincide.

The model contains 10 constants. This is a large, but not unreasonable, number of constants for a system of three equations. No biochemical significance is ascribed to these constants, and our aim is not to obtain values of biochemical rates or affinities by fitting the model to a limited set of data. Our aim is to generate an empirical model that can be used in modeling the mechanics of constricted airways. Therefore, questions about the uncertainty of the parameter values are not pertinent.

The curves computed from the model and shown in Figs. 2 and 3 match the experimental data for oscillatory forcing with a range of frequencies and amplitudes reasonably well. We would like to point out two other qualitative comparisons between data on smooth muscle F-L relations and the model. First, energy dissipation during a cycle was measured by Shen et al. (14). They report that hysteresivity decreases with increasing frequency of oscillation. This implies that the primary dissipative mechanism is a relaxation mechanism rather than a viscous mechanism, and the dissipative mechanisms in the model are relaxation mechanisms. Second, the variable g in the model is an unobservable internal variable. However, its value would be displayed by certain L histories. If, during a cyclic oscillation, the shortening were stopped at the midpoint, Eq. 2 predicts that f would relax relatively rapidly toward g and then f and g would relax slowly toward 1. Conversely, if muscle lengthening were interrupted, Eq. 2 predicts that f would initially decrease as f relaxes toward g, and then the two would relax more slowly toward 1. Data for interrupted shortening reported by Gunst (3) show the first behavior, and recent data for the second maneuver (7) show the second behavior.

As an application of the model, the effect of a deep breath on airway constriction was analyzed. The L history of the muscle during this maneuver is different from the L histories for which direct experimental measurements have been made, and this example, therefore, illustrates the usefulness of the model. In the initial state, muscle F equals the F_iso. In the first phase of the maneuver, transmural pressure rises rapidly, the muscle is forced to lengthen, and F rises above F_iso. After the inspiration phase, PA remains constant, and F relaxes toward F_iso. As F decreases, _m increases slightly. During the expiration phase, F drops rapidly, and muscle L and _m decrease, but the decrease in _m is small because the dynamic F-L curve for rapid shortening is steep. Muscle behavior during these two phases could be estimated from the data on F-L relations. After the expiration, PA remains constant as the airway closes. The value of the model lies in the prediction of the time course of muscle shortening during this phase of the maneuver. Eq. 1 is applicable during muscle shortening, and this equation together with the airway equations are solved for the two unknowns, f and l, as functions of time. The rate of recovery of f is affected both by muscle dynamics and by the stiffness of the load against which it shortens, just as the rate of relaxation of a viscoelastic element in series with a spring depends on both the time constant of the viscoelastic element and the spring constant. The time history of muscle F during shortening is peculiar. Although f recovers toward its equilibrium value of 1, the muscle shortens, and F_iso(l) decreases. As a result, F remains nearly constant. However, the hoop stress and the interdependence F both increase as r decreases and the airway closes.

The predicted time course of airway closure can be compared with data in the literature that describe the rise of lung impedance after a deep breath. Shinozuka et al. (15) measured the impedance of constricted dog lungs after a deep breath, and Pellegrino et al. (13) measured the recovery of impedance after a deep breath in bronchoconstricted human subjects. The time course of recovery of impedance was 20-30 s in both of these experiments. The time of recovery of airway closure obtained from the model and shown in Fig. 5C agrees with these data. If the mass of muscle in the airway were bigger or if airway and parenchyma were less tightly coupled, the deep breath would have less effect, and closure would occur more rapidly.