INTRODUCTION

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THE EQUILIBRIUM CONFIGURATION of an elastic structure is the configuration, consistent with any constraints, for which the elastic stored energy is minimal. It appears that Agostoni et al. (1) were the first to apply this concept to the chest wall and to point out that the configuration of the chest wall during the relaxation maneuver was the configuration with minimal elastic energy for a given lung volume. Mead (13) later stated that "In terms of minimal elastic work for a given volume change, inspiration should proceed along the relaxation characteristic, for to depart from the characteristic would require additional work of distortion." The volume displacements of the abdomen and rib cage during quiet breathing in seated subjects were found to match those for the relaxation maneuver quite well (8). Thus, on the scale of the two-compartment description of chest wall shape, these data seemed to confirm the minimal-work hypothesis. However, at a finer scale, both Agostoni et al. (1) and Konno and Mead (9) observed differences between the shape of the chest wall during active inspiration and the shape of the relaxed chest wall at the same volume, and both tried to estimate the energy of the distortions they observed. Since then, others have added to the list of observed differences between active and passive chest wall shapes (5).

Mead's statement, above, is widely taken as a hypothesis that the inspiratory muscles drive the chest wall along the minimal-work trajectory, and differences between the active and passive chest wall shapes are taken as evidence that the hypothesis has limited validity. However, Konno and Mead (9) commented that "it would be remarkable, indeed, if the muscles, which by definition must act from within, would produce just the configuration obtained by a uniform pressure difference applied from without." Thus differences between active and passive chest wall trajectories may be the result of the limited repertoire of the respiratory muscles. Although the active trajectory may not be identical to the passive trajectory, it may be the trajectory, among those available to the muscles, for which the work of breathing is minimal. This paper addresses that possibility.

The chest wall is modeled as a linear elastic system that can be expanded by airway pressure and active muscle forces. The set of muscle forces that produces a given volume expansion with minimal work is computed. This computation yields equations that relate muscle force to the elastic properties of the system. Unfortunately, it would be impractical to measure the elastic properties of the chest wall in the detail that would be required to calculate muscle force from these equations, and it would be difficult to measure the force in an active muscle and test these predictions. However, the computation also yields expressions for muscle shortening, and these are very simple. For minimal work, the fractional shortening of all active muscles is a common factor times their fractional shortening during passive inflation. If the muscles can reproduce the relaxation trajectory, the factor is 1, and active shortening equals passive shortening. If not, the factor is >1, and active shortening is greater than passive shortening.

This prediction was tested by measuring the fractional shortening of the parasternal intercostal muscles in supine dogs. Muscle length was measured during passive inflation, during quiet breathing, and during more forceful inspiratory efforts against an occluded airway. Active shortening during quiet breathing was found to be loosely correlated with passive shortening, but active shortening during forceful efforts was closely correlated with passive shortening. These data are consistent with the hypothesis that, during forceful efforts, the inspiratory muscles drive the chest wall along the minimal-work trajectory.

	MODELING AND ANALYSIS

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We will begin with a simple schematic example and then go on to the general analysis. The simple example is shown in Fig. 1. It consists of three horizontal bars, representing ribs, connected by pins to a vertical bar at the left. The vertical bar can slide along the wall, and it is connected to the foundation by a spring, with spring constant k_s. At their right ends, the bars are connected to each other and to the foundation above by springs with spring constants k₁, k₂, and k₃. The container at the lower right represents the lungs. The volume of the container (lung volume or VL) changes by means of the displacement of the piston with area A. The piston is connected to the lowest rib by a rigid bar and to the foundation by a spring with spring constant k_L. Diagonal elements, labeled T₁ and T₂, represent intercostal muscles. These elements have negligible passive elastance, but they can generate active tension.

View larger version (22K): [in this window] [in a new window]   Fig. 1.   Schematic model of ribs and intercostal muscles. Volume (lung volume or VL) of the container can be expanded either by pressure P applied at the opening or by active tensions (T1 and T2) in the muscles that lie diagonally between the horizontal levers. For volume expansion by pressure at the opening (passive inflation), the piston with area A moves upward; the right ends of the levers move up with displacements x1, x2, and x3; and the springs with spring constants k1, k2, and k3 are compressed; but no force is transmitted through the levers, and the sliding bar on the left does not move. kL, Lung constant. For volume expansion by active muscle force (active inflation), sliding bar moves downward, with displacement xs, and spring with spring constant ks is extended as VL increases. The downward displacement of the bar is analogous to the caudal displacement of the sternum reported by De Troyer and Kelly (5).

The system can be passively displaced by pressure P at the opening of the container or actively displaced by tension in the muscles (T₁ and T₂). With passive inflation, the piston moves upward, and the pressure force is balanced by tension in spring k_L and compression in spring k₃. Because these springs are aligned, the springs exert no moments. The right ends of the ribs move upward, but no force is transmitted to the slide at the left, and it remains stationary. For active inflation, the muscles exert moments that must be balanced by forces at both ends of the ribs, and the slide moves downward. In this system, the muscles cannot reproduce the passive chest wall shape.

The equations of equilibrium for the levers and sliding bar can be solved to obtain the displacements that are produced by pressure or muscle tension. For simplicity, the spring constants for all springs are taken to be equal and are denoted k. The equation that describes the effects of pressure P and muscle tension on VL is

(1)

From similar equations that describe the displacements of the bars, equations that describe the shortening of the two muscles (S₁ and S₂) can be obtained. They are

(2)

(3)

For any generalized force (F), the conjugate displacement x is the displacement for which the incremental work (dW) equals Fdx. The variable VL is the conjugate displacement for P, and the displacements S₁ and S₂ are the conjugate displacements for T₁ and T₂. Thus

(4)

For conjugate displacements, the relations between displacements and forces have a special property; the matrix of coefficients is symmetric. Therefore, the matrix of coefficients in Eqs. 1-3 is symmetric. That is, the coefficient of T₁ in Eq. 1 is equal to the coefficient of P in Eq. 2, and so forth.

The objective is to compare muscle shortening (S) for passive inflation of the model lung and for active inflation with minimal work. For passive inflation, T₁ = T₂ = 0, and the relationships between the displacements and pressure are described by the first terms on the right sides of Eqs. 1-3. The first equation can be used to eliminate P from the second two and to obtain the following relations between passive muscle shortening (S^p) and VL.

(5)

For T₁ = T₂ = 0, W = 1/2 P VL, and substituting for P yields the expression for passive work (W^p)

(6)

Next, muscle tensions for volume expansion with minimal work are calculated. For P = 0, work (W) is given by the expression

(7)

By substituting for S₁ and S₂ from Eqs. 2 and 3, W is obtained as a quadratic function of the T values. Minimizing this, with the constraint that VL is fixed, yields values of T₁ and T₂. Substituting these into Eqs. 2 and 3 yields the following expressions for active shortening (S^a) and active work (W^a)

(8)

(9)

The minimal active work solution has the following remarkably simple features. The ratio of active shortening to passive shortening is the same for both muscles, and the ratio of active work to passive work is the same as the ratio of active to passive shortening.

The example can be generalized to any multiple-degree-of-freedom linear elastic system. The relations between displacements and forces are generalized to

(10)

(11)

Here, the volume displacement is treated separately in Eq. 10. The coefficient of P on the right side of that equation is denoted Crs because this coefficient describes the compliance of the respiratory system. The other terms on the right side of Eq. 10 describe the effect of muscle tension on VL. Muscle tensions are denoted T_i for i = 1 to n. The coefficients that describe the volume change per unit force in each muscle are denoted b_i. Only muscles with inspiratory effect are included in the set, and the values of b_i are all positive. The remaining n equations describe the dependence of shortening (S_i) of the ith muscle on P and T_j. The coefficients C_{i j} are components of an n × n matrix C. The component C_{i j} describes the effect of tension in the jth muscle on length of the ith muscle. The displacements have been chosen as conjugates of the forces, and, therefore, the matrix of coefficients is symmetric. That is, the coefficient of T_i in Eq. 10 and the coefficient of P in Eq. 11 for S_i are equal, and the matrix C is a symmetric matrix.

For passive inflation

(12)

Next, the forces for active inflation with minimal work are calculated. By substituting the expressions for S_i in the expression for W, the following quadratic function of the T values is obtained

(13)

We seek the values of T_i that minimize W for a given VL. The condition on volume is expressed by Eq. 10 with P = 0; namely A Lagrange multiplier () is introduced, and the function is minimized with respect to T_i. The partial derivatives of f with respect to T_i are set equal to zero to obtain the following equations

(14)

Explicit formulas for T_i are obtained by multiplying Eq. 14 by the inverse of C, denoted as C¹

(15)

This expression is substituted into Eq. 10 to obtain  

(16)

where Equations 15 and 16 constitute a formal solution to the problem. However, to evaluate T_i from these equations, the values of b_i and the values of all components of the compliance matrix C must be known. That is, the values of b_i that describe the inspiratory effect per unit force must be known for all muscles, and the values of C_{i j} that describe interaction between all muscles must be known. This seems impractical. However, by substituting P = 0 and the expressions for T_i and , given by Eqs. 15 and 16, into the equation for S_i (Eq. 11), the following simple expression for S^a_i is obtained

(17)

For minimal work, shortening of each active muscle is proportional to b_i, as it is for passive shortening, and by comparing Eq. 17 with the first part of Eq. 12, it can be seen that the ratio of active to passive shortening is (Crs/), the same for all active muscles.

The ratio of active to passive shortening is Crs/ where For a stable system, the matrix of coefficients in Eqs. 10 and 11 is positive definite, and it follows from this that Thus, in the minimal-work solution, all active muscles shorten by a common multiple of their passive shortening, and that multiple is >1.

The expression for T_i (Eq. 15) can also be substituted into the expression for work (Eq. 13) to obtain the following expression for the work done during active volume expansion

(18)

By comparing this equation with the second part of Eq. 12, it can be seen that the ratio of active to passive work is Crs/, the same as the ratio of active to passive muscle shortening.

	EXPERIMENTAL METHODS

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The experiments were carried out in five adult mongrel dogs (17-32 kg) anesthetized with pentobarbital sodium (initial dose, 25 mg/kg iv). The animals were placed in the supine posture and intubated with a cuffed endotracheal tube. A venous cannula was inserted to give maintenance doses of anesthetic. The rib cage and intercostal muscles were then exposed on the right side of the chest from the second to seventh rib by deflection of the skin and the underlying muscle layers.

The changes in parasternal intercostal muscle length were assessed by sonomicrometry (Triton Technology, San Diego, CA). This technique has previously been described in detail (3, 4). The fascia overlying the parasternal intercostal in two interspaces between the second and the fifth was carefully resected, and in each muscle thus exposed, a pair of piezoelectric crystals was inserted 6- to 11-mm apart in a well-identified muscle bundle in the vicinity of the sternum. The crystals, 1 mm in diameter, were oriented along the long axis of the muscle fibers and held in place with purse-string sutures. Airflow at the endotracheal tube was also measured in each animal with a heated Fleisch pneumotachograph connected to a Validyne differential pressure transducer (±2 cmH₂O), and volume was obtained by integration of the flow signal. Airway opening pressure (Pao) was measured with another Validyne differential pressure transducer connected to a side port of the endotracheal tube.

After instrumentation, 30 min were allowed to elapse for recovery, after which tidal volume and changes in parasternal intercostal muscle length were measured during resting breathing. Two runs were recorded over 30 min. A Hans-Rudolph valve was then attached to the pneumotachograph and the tracheal tube, and the inspiratory line of the valve was blocked during the expiratory pause for seven consecutive inspiratory efforts. Data were recorded during two successions of occluded breaths separated by a 10- to 15-min period of unimpeded breathing. The animal was then given an injection of atracurium besylate (0.3-0.5 mg/kg iv), and the relaxation length (L_r) of the two parasternal intercostal muscles was determined. The animal was connected to a calibrated syringe, and muscle length was measured as the lungs were passively inflated with 100-ml volume increments up to 1 liter above FRC. Three trials were obtained in each animal.

The changes in parasternal intercostal muscle length during unimpeded breathing were averaged over 10 consecutive breaths from each run and expressed as percent changes relative to L_r. These changes were then divided by tidal volume and expressed as percent L_r per liter increase in VL.

During inspiratory efforts against an occluded airway, VL does not change, and the data for this maneuver must be interpreted to obtain values for muscle shortening that can be compared with passive shortening. Equivalent values of active shortening per liter were obtained from the data for the occluded airway maneuver by the following procedure. It was assumed that the fractional change in length (L) is the sum of the effects of changes of VL and Pao, as described by the equation

(19)

The constants a and b in this equation can be interpreted as follows. During passive inflation, Crs Pao= VL, and the second term in this equation is zero. Thus the coefficient a describes the L per liter increase in VL during passive inflation. During spontaneous breathing with the airway open, Pao = 0, and the equation reduces to L = b VL. Thus the coefficient b describes the L per unit increase in VL. During inspiratory effort against an occluded airway, VL = 0, and L and Pao both change. Equation 19 is used to obtain a value of b from the measured values of L and Pao. That is, Eq. 19, with VL=0, is solved for b to obtain the relation

(20)

The values of a and Crs were obtained from the data for passive inflation. The values of L and Pao for the fifth and seventh occluded efforts were averaged, and these were substituted into Eq. 20 to obtain b, the equivalent muscle shortening per liter increase in VL for the occluded airway maneuver.

	RESULTS

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In agreement with previous studies (3, 4), the parasternal intercostal muscles in the five animals shortened both during quiet, unimpeded inspiration and during occluded inspiratory efforts. The values of inspiratory length change per liter increase in VL during quiet inspiration, however, were variable among the different animals. This is shown in Fig. 2, in which the individual values of fractional muscle shortening per liter increase in VL during resting breathing are plotted against the corresponding values during passive inflation. Three of the 10 muscles shortened less during active inspiration than during passive inflation, but seven muscles shortened more during active inspiration. The line, fit to all data points, has a slope of 1.7 ± 0.3, with r² = 0.36. 

View larger version (14K): [in this window] [in a new window]   Fig. 2.   Active fractional muscle shortening during quiet breathing vs. passive shortening of the parasternal intercostal muscles in 2 interspaces in dogs 1-5. Dashed line, line of identity; solid line, linear fit to all data.

The individual values of parasternal intercostal shortening per liter increase in VL, inferred from the data obtained during occluded breaths, are plotted against values of shortening during passive inflation in Fig. 3. For this maneuver, active shortening was greater than passive shortening for all muscles, and active shortening was more closely correlated with passive shortening. The line fit to the data has a slope of 1.4 ± 0.1, with r² = 0.67. 

View larger version (16K): [in this window] [in a new window]   Fig. 3.   Active vs. passive muscle shortening for forceful inspiratory efforts against an occluded airway in dogs 1-5. Symbols are same as in Fig. 2.

	DISCUSSION

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Modeling and analysis. We have modeled the respiratory system as a linear elastic system, and this entails some approximations. First, displacement is assumed to be proportional to force, and the relation between displacement and force is assumed to be independent of rate of displacement or displacement history. Thus the 15-20% hysteresis in the pressure-volume loops of the respiratory system has been neglected. Second, nonlinear effects are neglected, although there clearly are limits on the magnitudes of chest wall displacements for which the linear approximation is accurate. However, this model has advantages that compensate for these approximations. In particular, the matrix of coefficients is symmetric in the relation between displacement and force for this model. The symmetry of the matrix of coefficients follows directly from the assumption that forces and displacements are linearly related; no additional assumptions are required.

Experimental results. The ratio of active shortening during quiet breathing-to-passive shortening is quite variable, as shown in Fig. 2. The deviations from the mean ratio of active to passive shortening ranged up to ~20% per liter or ~10% per tidal volume. Previous studies have shown that the variability in active parasternal shortening among animals is large (3, 4). One of us (A. De Troyer) has observed, in dogs, that during quiet breathing, the relative expansion of the rib cage and abdomen is quite variable, and this is a source of variability in active parasternal shortening. In addition, the ratio of active to passive shortening of parasternal intercostals in different interspaces in individual dogs is also variable. The dogs were anesthetized and studied in an unphysiological posture, and these conditions may have distorted the relation between active and passive shortening. We must conclude, however, that during quiet breathing, when the work of breathing is small, the pattern of muscle shortening deviates significantly from the pattern for minimal work.

Conclusions. Chest wall distortion has been described in different terms by different investigators. These terms include compartmental volume, cross-sectional shape of the thorax, and relative displacements of ribs and sternum. The functional significance of these descriptors has been difficult to evaluate. Here we propose a new description of chest wall distortion: the ratio of active to passive shortening of active muscles. The modeling and analysis show that this measure of chest wall distortion is closely related to the work of chest wall distortion and provides a method for testing the long-standing hypothesis that the respiratory muscles drive the chest wall along the minimal-work trajectory. The analysis shows that the minimal-work trajectory available to the muscles may be different from the passive trajectory, but it has a testable characteristic: all active muscles shorten by the same multiple of their passive shortening. For this trajectory, the work of chest wall expansion is the same multiple of the work of passive expansion.