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RIDDLE, W., AND M. YOUNES. A modeZ for the reZation between respiratory neural and mechanical outputs. II. Meth- ods. J. Appl. Physiol.: Respirat. Environ.
A model for the relation between respiratory neural and mechanical outputs. II. Methods W. RIDDLE

Department

AND M. YOUNES of Medicine, University

of Texas Medical

RIDDLE, W., AND M. YOUNES. A modeZ for the reZation between respiratory neural and mechanical outputs. II. Methods. J. Appl. Physiol.: Respirat. Environ. Exercise Physiol. 51(4): 979-989, 1981.-In the preceding communication we developed a model for the conversion of neural output to mechanical output. We were left with two qualitative uncertainties, namely, the relation between neural output and isometric pressure, and the behavior of inspiratory muscles during expiratory flow; and two quantitative uncertainties concerning the effect of configurational pathway on pressure output, and the slope of the pressure-flow relation. For each of the above uncertainties we made certain assumptions based on indirect evidence but defined reasonable error limits. In the present communication we describe the method of implementing the model and evaluate the significance, in terms of spirometric output, of possible errors in the assumptions. Volume and flow profiles were generated from different neural output profiles. Analysis was repeated when the different assumptions were systematically altered within the limits set by the previous theoretical analysis. We conclude that the pattern of inspiratory muscle activation during spontaneous breathing and the existence of several mechanical interactions within the respiratory system combine to render spirometric output fairly insensitive to most potential errors in our assumptions. respiratory modeling; inspiratory tion; chest wall configuration

output;

force-velocity

rela-

IN THE PRECEDING COMMUNICATION (16) we developed a theoretical model that describes the translation of neural to mechanical events in the human respiratory system. The model necessarily incorporates several assumptions. These assumptions are in some cases well founded. However, in other cases the assumptions were based on indirect and at times controversial evidence, thereby creating some uncertainties. There are four basic uncertainties with the model. These are 1) whether the relation between averaged neural output and isometric pressure is linear or nonlinear; 2) the effect of different configurational pathways during expansion on the relation between neural output and isometric pressure; 3) the effect of quantitative errors in modeling flow-related pressure losses (force-velocity relation) after the behavior observed during maximal voluntary efforts; and 4) whether expiratory flow results in increases in inspiratory pressure above the isometric value; or, stated differently, whether the braking action of inspiratory muscles during expiration is limited to the isometric pressure they can generate. OlSl-7567/81/0000-0000$01.25

Copyright

0 1981 the American

Physiological

Branch,

Galveston,

Texas 77550

In the present report we describe the methods we used to implement the model. Subsequently, we address the above questions by defining the magnitude of error that may result through inaccuracies in the assumptions. Our approach will be as follows. We shall begin by employing standard assumptions for each of the above uncertainties. Thus, for the relation between neural output and isometric pressure (at a fixed configuration), we shall assume a linear relation; one unit of neural output is that which generates a pressure of 1.0 cmH20 at the relaxed functional residual capacity (FRC) configuration. For the effect of configurational pathway on isometric pressure, we shall follow the relaxed configuration. For the quantitative description of the pressure-flow relation, we shall use the relation observed during maximal voluntary efforts. For the pressure-flow relation during expiration, we shall assume that the inspiratory flow equation applies also during expiration; i.e., that expiratory flow increases inspiratory pressure output along a function described by the backward extrapolation of the inspiratory pressure-flow relation. Next we shall examine the effect of changing one assumption while keeping the others constant. The neural profiles to be used for testing, the output parameters to be examined, and the range of alternate assumptions will be selected in a way that permits a realistic evaluation of the potential errors involved in using the standard assumptions. METHODS

Calculation of Flow and Volume from Neural Output or Isometric Pressure at Passive FRC Calculations were made by use of a PDP*8/e computer. Figure 1 illustrates the basic operation and its various subroutines. The basic operation consisted of calculating instantaneous flow (VJ according to the equation ir, = [d(O.25P0v’

+ bVtRrs

+ VtErs

+ PE)~ + 4Rrs

bVt(Pit

- VrErs

- PE) (1)

- (0.25P$

+ bVtRrs

+ VrErs

+ PE)]/2Rrs

where V, is the volume at time t, expressed as difference from passive FRC, Po”lis the inspiratory muscle pressure at volume Vt, at zero flow, bVt is the flow asymptote of the hyperbolic pressure-flow equation at volume Vt, Rrs is the passive resistance of the respiratory system, Ers is the passive elastance of the respiratory system, and PE is the expiratory muscle pressure at time t. The basis for Society

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YOUNES

2)) volume and configurational losses (subroutine 3)) and flow-related losses (subroutines 4 and 5). PEAK Generation of a theoretical inspiratory neural output profile (Nt) (subroutine 1, Fig. 1). Nt was described by NEU*RAL seven parameters (Fig. 2): T1, duration of the rising PROFILE phase; TZ, duration of the inspiratory plateau; Ta, duration of the declining phase of inspiratory activity; Tq, interval between onset of declining phase and onset of the following inspiration (corresponding to neural expip FRC ration); peak, amplitude of the rising phase in arbitrary RC units (see below); and CR and CD, indexes of curvature I for the rising and declining phases, respectively. Positive values indicate that the relevant phase is concave to the time axis. The index indicates the difference between Nt and one-half the amplitude at the midpoint of the relevant phase and is expressed in percent of the amplitude. Thus a value of +20 indicates that Nt reaches 70% of the total amplitude at the midpoint of the phase. (.25 PoVt + bvt Rrs + V, . E,, + Pex,)2 + 4 Rrso bVYPoVt -V, . E,, - Pex,)The time course of Nt during the rising and declining (.25 Povt + bvt . R,, + V,. Ers + Pexp) 2 R,, phases was arbitrarily described by a parabolic function that is defined by phase duration and amplitude at the t 4 t beginning, middle, and end of the phase. Conversion of Nt to isometric pressure at passive FRC ( PfRc) (subroutine 2, Fig. 1). The relation between averaged inspiratory muscle activity and isometric pressure II was variably described as linear (4, 15) or nonlinear (7) I (see Ref. 16 for details). The second subroutine permits I j a choice between these two possibilities. With the first option (ZA), isometric pressure at FRC is treated as a I linear function of neural output. One unit of neural I L Am-------_------------output is defined as that amount of electrical activity FIG. 1. Schematic illustration of computer program. which results in a pressure of 1.0 cmHe0 when contraction is done isometrically at the configuration of passive this equation has been described previously (16). FRC. The second option (2B) was patterned after the The equation was solved for volume and flow at LO- average response calculated from the data of Grasssino ms intervals using the improved von Euler-Cauchy et al. (7) (see Fig. 1 of Ref. 16). method for “initial value” differential equations.’ For Thus single-breath calculations, the initial value for volume P0FRC = 1.319Nt (volume at the onset of neural inspiration, V,) was entered as an independent variable (Fig. 1). For sequential for Nt 5 15.0, and breath analysis, the last volume value from the preceding P0FRC= 0.743Nt + 8.642 breath was used as Vo. In both cases, calculation of V, in the interval beyond the first interval was based on the accumulated volume (V,). The latter value was also utilized to perform calculations in the various subroutines (see Fig. 1). The various subroutines were devised to permit modeling with various theoretical neural output profiles (subroutine 1) and expiratory pressure profiles (subroutine 6). Furthermore, they allowed us to test the effect of I i I different conditions that may govern the relation between neural activity and isometric pressure (subroutine DIRECT INPUT

@ -

T5,

T6

0

I il,=[J

l

’ Because the flow equation incorporates highly variable functions it does not lend itself to exact mathematical solutions. However, an approximate solution may be obtained numerically on a digital computer. We minimized the error by performing the calculations at small intervals (every 5.0 ms) and by use of the improved von Euler-Cauchy method (10) to calculate volume. At each interval we compute volume in two steps: I) Vi*,, = Vi + Atf (Vi, PPRC, PEi, Ers, Rrs, VC) ; and 2) At Vi+1 = Vi + 2 [f (vi*+lp PiE;R1’, PEi+l, Ers, Rrs, VC) + f (Vi, ERC, PEi, Ers, Rrs, VC} 1.

FIG. 2. Parameters used to model inspiratory and expiratory activity. Neural activity is plotted in terms of its pressure equivalent. Expiratory muscle activity produces downward deflections of PE. See text.

MODEL

FOR

RESPIRATORY

5o I’

NEURAL-MECHANICAL

RELATIONS.

125[

B I

-2.5 NEURAL

OUTPUT (arb. units)

FIG. 3. A illustrates output and isometric illustrates options used and flow. Positive flow

0

1

II

2.5

5.0

7.5

I, 10.0

FLOW (Vsec)

options used to model relation between neural pressure at configuration of relaxed FRC. B to model relation between inspiratory pressure values denote inspiratory flow.

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Because the value of bVt as calculated above is based on the results of maximum voluntary efforts and the quantitative relevance of the latter to spontaneous breathing is uncertain, we have included in the program three options which permit changing the value of bVt and thereby changing the pressure losses for a given flow: option 4A, bVt equals the value calculated from the above equation; option 4B, bvt equals half the calculated value; and option 4C, bVt equals twice the calculated value. Fig. 3B illustrates the pressure-flow relation using the three different methods of calculating bvt. It is clear that the above manipulations of bVt result in appreciable differences in the magnitude of flow-related pressure losses.

Pressure-flow

relation

during

expiratory

flow (sub-

for Nt > 15.0. Fig. 3A illustrates the difference between routine 5, Fig. I). When inspiratory activity continues the two functions. into the phase of mechanical expiration, the inspiratory Next we make allowance for the reaction time of the muscles lengthen while contracting. Reference has been respiratory muscles (see Ref. 16 for details). The wave made regarding the uncertainty about whether such form for POFRC is convoluted through a low-pass filter (2) lengthening is associated with increases in inspiratory having a time constant of 0.06 s to yield PtRC (RC, Fig. pressure above the isometric value (16). Options 5A and 1). Details of this process are provided in the APPENDIX. 5B permit a choice between two alternatives. With option In addition to generating theoretical PFRCwave forms, 5A the same equation is used to calculate inspiratory the program also permitted analysis of actualPrRC wave flow and expiratory flow (see Fig. 1). The net effect is the forms obtained from patients (direct input, Fig. 1). For backward extrapolation of the inspiratory flow relation the latter application, occlusion pressure was measured into the expiratory flow region (Fig. 3B). With option 5B, at 0.1-s intervals, and the values were stored on a disk. the calculator checks the value of flow. If zero or larger The data points were converted to a continuous function (i.e., inspiratory flow), calculations continue as usual. If by use of a cubic spline interpolation technique (1). less than zero (i.e., expiratory flow), the maximum inVolume and configurational losses (VCL) (subroutine spiratory pressure is considered to be the isometric value 3, Fig. 1). In the preceding manuscript (16), we calculated at the particular volume (Pit) (see Fig. 3B), and flow is the relation between diaphragmatic pressure output and recalculated according to the following equation volume (at constant electrical activity) when expansion . V t = -(&Ers - Pzt + PE)/Rrs of the respiratory system followed three different configurational pathways. The present subroutine permits a choice among these options for modeling purposes. The V, is then integrated in the usual manner. Expiratory muscle pressure (subroutine 6, Fig. 1). The result in each case is the pressure predicted at volume V and time t in the absence of flow Prl. The latter value is effect of expiratory pressure on flow and volume may be subsequently used to calculate flow according to the basic examined by generating a theoretical expiratory pressure wave form. The expiratory pressure wave form is of a flow equation outlined above. much simpler form than its inspiratory counterpart and Option 3A predicts the isometric pressure at different volumes when expansion follows the relaxation configu- is described by three parameters (Fig. 2): peak PE, the amplitude of the expiratory pressure wave form (its timration ing was made to coincide with the onset of inspiratory p;t = p;RC e-Vt/0.28VC activity); Tg, the interval between onset of inspiratory Option 3B predicts Pyt when expansion follows the activity and end of expiratory activity (Fig. 2); and Ts, pathway generally assumed during hyperpnea (see Ref. the interval between onset of expiratory activity and onset of the next inspiration (Fig. 2). PE changed linearly 16 for details) with time in both intervals. Pvt0 = 1.o~p~RCe-Vt/O*28VC Alternatively, when the actual PE wave form is known, its time course may be entered directly with a procedure Option 3C predicts Pyt when expansion proceeds along similar to that used for inspiratory pressure wave forms a vertical pathway (rib cage expansion only) beginning (see above). from passive FRC Passive properties of the respiratory system. For the PV sake of this analysis, passive Ers and Rrs will be consid0 t = PtFRC(l.O- 1.92VJVC) ered constant throughout the breath. During natural Calculation of the flow asymptote (bvt) of pressurehowever, upper and lower airway resistances flow relation (subroutine 4, Fig. 1). bVt was calculated breathing, may undergo substantial intrabreath changes, and tidal according to the following equation (see Ref. 16 for volume may be large enough to encroach on the relatively details) flat segment of the pressure-volume curve. Under these circumstances, the specific passive functions may need to b vt = 1.6VC(7.77 - e -v,/o.%j / (2.56 + e -v,/o.%vc) I

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5 0

AND

M.

YOUNES

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.**.=*’ \

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RIDDLE

::” .. ;* . ‘-----

. -t; ,A#.... t

1.5r /?

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10.00.5 -1 -------------------1 1

-0.5 ’ TIME (WC)

I 2

I 3

I 4

TIME (set)

4. Examples of graphic computer output with 2 neural output profiles (dotted lines, upper panels). Neural profile is described by numbers in first row of legend (option I). Numbers denote respectively, T1, Ta, T3, T4, CR, CD, and peak (see Fig. 2). Second row indicates that predictions were made according to options 2A, 3A, 4A., and 5A, respectively (see text). Third row describes expiratory output profile (no expiratory output in this case). Fourth and fifth rows indicate values used for passive resistance (R), passive elastance (E), volume at

onset of inspiration (V,), and predicted vital capacity (VC). SoZid line in top panel is isometric pressure at passive FRC, calculated according to option 1 (linear relation between neural output and isometric pressure). Difference between dotted and solid lines is related to muscle contraction time. The middle panels illustrate calculated dynamic pressure of respiratory muscles (dots) and flow (solid Line). Lower panels illustrate calculated volume.

be incorporated in the equations, and this should present no difficulty. Figure 4 illustrates the graphic display of two computer-generated breaths. All parameters are plotted against time from onset of inspiration. From above downward the lines re resent neural output in arbitrary units (dotted line); P? IF’ in cmH20 (solid line); PE in cmHg0 (zero in this case); dynamic pressure of the respiratory muscles (Pmus) in cmHa0 (dotted line, calculated from VtErs + V,Rrs); flow in l/s (solid line); and volume above passive FRC in liters. The numbers in the top right corner of each panel indicate the parameters used (see figure legend).

RESULTS

FIG.

Calculation of Isometric Volume and Flow

FRC Pressure from

The reverse calculation is much simpler than the above procedure. At any time t, isometric inspiratory pressure at relaxed FRC may be calculated from (16) P rRC = (VtErs

+ VcRrs + PE) . (T& + bvt)/(bvf

- 0.25~t)e-Vt/0-2avC

(3) ’ ’

As indicated in the preceding communication, this calculation is not consistent with the simultaneous presence of inspiratory and expiratory activity. The calculation is done using a PE value of zero. If the product is positive, it denotes inspiratory muscle pressure at FRC. If the product is negative, PE is calculated according to the following equation PE

With positive

= -(VtErs

+ $Rrs)

both equations inspiratory flow is assigned value and expiratorv flow a negative value.

a

AND

DISCUSSION

Relation Between Neural Pressure (Option 2)

Output

and Isometric

Analysis of the effect of the two postulated functions must necessarily remain qualitative or relative. For comparisons to be made, it is necessary to assign an arbitrary value for neural output at which isometric pressure is the same for either type of behavior (e.g., see Fig. 3A). For convenience we have chosen a neural output value of 34 as the crossing point (Fig. 3A), since the data of Grassino et al. (7) suggest that when diaphragmatic activity is 34% of maximum, isometric pressure at FRC is 34 cmH20 (see Fig. 1, Ref. 16). With this usage arbitrary values for neural output have more physiological relevance. It must be emphasized that the choice of the crossing point is not critical to interpretation. Changing this point will alter the value of neural output, in arbitrary units, at which a mechanical parameter (e.g., tidal volume) will be independent of the function used, and this is meaningless. However, a different choice will not affect the qualitative or relative differences above and below this point. The effect of this relation may be examined at two levels. The first is the relation between peak neural output and peak mechanical output (tidal volume). The second is the effect of changing the relation on the temporal behavior of volume and flow within a breath. Figure 5 illustrates the relation between peak neural output and tidal volume under two different conditions. In condition 2A, the relation between neural output and isometric pressure is linear, and one unit of neural output is equivalent to 1 cmH20 at passive FRC. In condition 2B, the relation is nonlinear and modeled after the data of Grassino et al. (7) (Fig. 3A of this paper). The effects of the two conditions are examined using two different

MODEL

FOR

RESPIRATORY

NEURAL-MECHANICAL

RELATIONS.

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2.00

1.60

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vo/

2.00

20, lo,“* 3/A &/A &A 6/0.0,0.0, 0 jt/ 4.0 E/l 0.0 0.00 vc/5.00

-L 2A28 ---

0.40 I

10.00

1

I

I

1

20.00

30.00

40.00

50.00

PEAK FIG.

volume output

5. Relation between peak neural output and calculated tidal using a linear (2A) or nonlinear (ZB) relation between neural and isometric pressure (see Fig. 3). Note that the point at which

neural profiles. In Fig. 5 (left panel), the neural profile (inset) consists of a brief (0.7 s) rising phase that is convex to the time axis, followed by a brief plateau (0.1 s). Inspiratory activity declines briskly at the end of the plateau. In Fig. 5 (right panel), neural output consists of a long (2.0 s) rising phase that is concave to the time axis. Inspiratory activity declines slowly during expiration. In this and other similar figures the legend in the top right corner indicates the various options used including predicted VC, Vo, Ers, and Rrs. The smaller legend below indicates the option that is being varied. With option ZA (linear relation, solid lines), the relation between peak neural output and tidal volume is somewhat curvilinear. The curvilinearity reflects the fact that configuration and flow-related pressure losses (and hence the effective elastance and effective resistance) increase in proportion to inspiratory activity.2*3 Figure 5 ~2 Eldridge (3) reported a linear relation between peak “averaged” phrenic nerve activity and tidal volume in cats. This finding sup&f?cially contradicts the predicted cu.&linearity between peak neural output and tidal volume (Fig. 5). It must be pointed out, however, that Eldridge’s fmding is not directly relevant to model predictions. In his study peak phrenic nerve activity was altered by hypercapnia, hypoxia, hyperventilation, vagotomy, and different levels of anesthesia. With these manipulations changes in peak inspiratory activity are inevitably associated with changes in timing and shape of neural output, and these variables influence tidal volume for a given peak activity (see later). Our model, on the other hand, predicts the relation between peak neural activity and tidal volume when shape and timing remain constant. Additionally, for Eldridge’s data to be comparable, one must assume that expiratory activity, as well as the relative contribution of phrenic activity to total activity, were the same under alI conditions. In other words expiratory activity and the relative contribution of intercostal muscles to total output were independent of the level of anesthesia, Pco~, POT, or vagal feedback. This assumption can hardly be justified. One may also note that the curvilinearity predicted by the model (with option ZA) is not excessive and the linearity reported by Eldridge is only an approximation. Some of the illustrated figures display distinct curvilinearity (e.g., Figs. 20,40, and 7B, Ref. 3). 3 It may also be argued that the constancy of the relation between Pocc and VT (effective elastance), reported by some investigators (llXi), conflicts with model predictions. However, as we pointed out earlier (16), peak neural activity in spontaneous and occluded breaths is not the same, and the range of inspiratory activity over which observations were made was very limited.

10.00

20.00

30.00

40.00

50.00

PEAK

the 2 lines cross is different with the 2 neural in legend indicate that the missing variable output) is the independent variable.

profiles (in this

(insets). Asterisks case peak

neural

also illustrates what may be expected from a system with resistive and elastic components; i.e., that the amplitude of the mechanical output (tidal volume) is not only a function of the amplitude of the forcing function (neural output) but also of the latter’s duration and shape. For any peak neural output, tidal volume is larger - with neural prbfiZe B (Fig. 5, right panel). The difference between the responses predicted with the two options (Fig. 5) is the result of two mechanisms. The first relates to the difference in the amplitude of isometric pressure wave form resulting from a given neural activity. Inspection of Fig. 3A reveals that the difference between the linear and nonlinear functions increases as inspiratory activity increases, reaching a maximum at a neural output value of 15. Beyond this point the difference decreases. The second mechanism relates to the effect of option ZB on the shape of the isometric pressure wave form within each breath. Where neural output is convex to the time axis (e.g., Fig. 5, profiZe A), pressure output will be less convex, and where it is concave (e.g., Fig. 5, profile B), it will be more so. As discussed earlier this effect will tend to increase tidal volume at any given pressure amplitude. Unlike the predictable effect of option ZB on pressure amplitude, however, the increase in tidal volume as a result of the intrabreath shape mechanism is highly variable and is expected to depend on amplitude of neural output (larger increases with higher amplitudes), inspiratory duration (smaller increases with longer inspirations), and shape of neural output (larger effect with convex shapes). The net effect of the two mechanisms on the relation between neural output and tidal volume is, accordingly, highly complex. One consequence of this complexity is illustrated by the point at which tidal volume is the same regardless of the option used. With both neural profiles, this point occurs at a neural value in excess of the value associated with equal peak FRC pressures (34.0). The iso-tidal volume points are, however, not the same for the two profiles (48.0 for profile A and 35.0 for profile B), thereby emphasizing the higher sensitivity of profiZe A

984

W. RIDDLE

(short

TI, convex

rising

phase)

to the shape

effect

of

option 2B. Another consequence of option 2B is to render the relation between peak neural output and tidal volume distinctly more curvilinear. This effect is fairly significant. Thus the left panel of Fig. 5 indicates that, using option ZB, it would be necessary to increase neural output amplitude 350% in order to double tidal volume. This is to be contrasted with the 250% increase required with option 2A. The effect of the two options on the temporal behavior of flow and volume, within a breath, will be discussed next. To illustrate this point more clearly we selected the neural output that results in the same tidal volume with either option. For profiLe A this is a neural output of 48.0; forprofize B it is 35.0. Fig. 6 illustrates the time course of isometric pressure (top), flow (middle), and volume using the two neural profiles illustrated in Fig. 5. The actual neural profdes were deleted from the top panels for clarity. Solid and dotted lines describe the response with options A and B, respectively. Option 2B almost eliminated the convexity of profiZe A (dotted line, top panel, left); it increased the concavity of profiZe B (right). Flow was modifed in that with option 2B it was higher in the early part of inspiration and lower later on. However, it is evident that the overall effect, particularly with respect to the shape of the volume lines, is minimal. From the above discussion it is clear that errors in modeling the relation between neural output and isometric pressure may be highly significant with respect to the relation between neural output amplitude and tidal volume but not so significant with respect to the time course of volume within an isolated breath. At present it is not possible to make a choice between the different options. For simplicity, we shall use a linear relation between total neural output and isometric pressure with the limitation that a given relative increase in calculated isometric pressure may underestimate the true relative increase in neural output.

50r

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AND

M.

YOUNES

Effect of Different Configurational Pathways on the ReZation Between NeuraZ Output and Isometric Pressure (Option 3) To examine the effect of different configurational pathways we shall assume that, while breathing at rest, a subject begins inspiration at passive FRC and proceeds along the relaxed configuration pathway. As ventilation increases, the subject’s end-expiratory level decreases and the configurational pathway shifts to the left of the relaxed configuration (5,B). The bottom solid line of Fig. 7 illustrates the relation between peak neural output and tidal volume (VT) when inspiration begins at FRC (V, equals zero) and proceeds along the relaxed configurational pathway (same as in Fig. 5). The top two lines represent the relation when inspiration begins 0.5 liter below FRC. In one case, calculations were made as if the entire expansion proceeded along the relaxed configuration (option 3A, solid line); in the other, we used the equation for the hyperventilation pathway [option 3B, see METHODS and the preceding communication (16) for details]. The difference between the two solid lines simply reflects the effect of beginning inspiration below passive FRC, whereas the difference between the top solid and dashed lines represents the effect of different calculations that take into account configurational pathway. The latter difference amounts to approximately 3% of tidal volume. The diagonal lines in the left panel predict the error if one were to ignore the progressive change in configurational pathway as ventilation increased. At rest, end-expiratory level is passive FRC, neutral output is 12 arbitrary units (i.e., isometric pressure at FRC 12.0 cmHgO), VT is 0.5 liter, and expansion proceeds along the relaxed configuration. The point corresponding to this state falls on the bottom solid line. As ventilation increases, end-expiratory level progressively decreases, inspiratory activity increases, and the configurational pathway progressively deviates from the relaxed configuration. The last point is one in which end1_/2.0,0.1,0.8,1.9, 2/ -

50 r

20, lo,35 a/A &/A &/A 6/0.0,0.0, 0 fy To E/l 0.0 yyo.ooo V~/5.00

FIG. 6. Effect of option 2 on time course of flow and volume during two breaths with different neural profiles (left and right panels). SoLid Lines: option ZA. Dots: option ZB. Only isometric pressure waveforms, with two options, are shown in top panels. Neural output profiles were deleted for clarity.

TIME kec)

TIME bed

MODEL

FOR

RESPIRATORY

NEURAL-MECHANICAL

RELATIONS.

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1

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I

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7. Relation in normal subjects FIG.

1

1

40.00

50.00

PEAK

between peak neural output and tidal volume when expansion follows during increased levels of ventilation (3B). See text for explanation.

relaxed

(3A) and configuration

configuration

1/2.0,0.1,0.8,1.9, -2/A

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8. Relation between peak neural output configuration (rib cage expansion only, 3C). FIG.

I

I

40.00

50.00

PEAK

and

tidal

volume

when

expiratory level is 0.5 liter below FRC, neural output is 50.0, and the configurational pathway is as illustrated in the bottom right inset. In one case we assume that expansion always proceeded along the relaxed configuration (dot-dash); in the other we take into account the changing configuration (dots). In both casesthe relation between neural output and tidal volume is much steeper than when the end-expiratory level does not change. However, the difference between the two treatments is extremely small. The error is likely to be even smaller if one considers the increase in elastic work associated with configurational pathways other than the relaxed pathway (6, 9). The same conclusions also apply to neural profiZe B (right panel). Figure 8 illustrates the relation between neural output and tidal volume when the relation between neural output and isometric pressure (PHI) is modeled after the

expansion

follows

relaxed

configuration

(3A)

and

“straight

up”

relaxed configuration (option 3A) or after the “straight up” configuration (option 3C), respectively. The difference, while still small, is more substantial than in the previous analysis and amounts to approximately 13% of tidal volume. Again, the error is likely to be smaller if one considers the increased elastic work associated with this vertical pathway [estimated at 10% by Goldman et al. (al. It is evident that, while the relation between neural output and isometric pressure is sensitive to configurational pathway, the relation between neural output and tidal volume appears to be fairly immune. Stated differently, if one used the relaxed pathway to predict tidal volume from the isometric pressure at passive FRC (or vice versa), not much error should result over a fairly wide range of physiologically relevant configurational pathways. Likewise, it may be expected that within-

986

W. RIDDLE

breath changes in configurational pathways, brought about by temporal shifts in distribution of activity among muscle groups, will have little effect on spirometric output. It must be recalled that the relation between pressure output and volume (and configuration) for the whole respiratory system was modeled after the response of the diaphragm. It is possible that in the presence of inspiratory intercostal activity the response of the system will continue to be weighted toward that of the diaphragm or, if not, that the relation between intercostal muscle pressure and volume and configuration is not much different. This expectation appears to be justified, though indirectly, by the results obtained in human subjects (17) . Effect of Quantitative Errors in Modeling Flow-related Losses After the Behavior During Maximal Voluntary Efforts (Option 4) In the preceding communication (16), we indicated that the major potential sources of error in estimating flow-related losses from maximal voluntary efforts are I) different configurations (at the same volume) at different flow rates and 2) different levels of inspiratory activity at different flow rates. The true pressure-flow relation is one established when flow is varied while configuration and inspiratory activity remain constant. We believe that the options used in this program (4A-4C) encompass the range of errors that may reasonably be expected as a result of changes in configuration and inspiratory activity at different flow rates (Fig. 3B) (see APPENDIX, Ref. 16). With option 4A, we assume that the “true” pressure-flow relation is similar to that observed during maximal voluntary efforts; i.e., that configurational and activity differences at different flow rates did not materially influence pressure output. With option 4B (bottom line, Fig. 4A), we presume that configurational and activity differ-

AND

M.

YOUNES

ences at different flow rates were such that pressure measured at maximal flow (10 l/s) was spuriously increased by 75%. Option 4C, on the other hand, allows for a 37% underestimation of the pressure (at maximal flow) as a result of configurational and activity differences. At a flow rate of 4.0 l/s, flow-related pressure losses are equivalent to 6.5, 10.8, and 3.9 cmHz0 1-l. s, respectively (Fig. 3B). Figure 9 illustrates the time course of flow and volume using the three flow options. The same two neural profiles are utilized, and the amplitude of neural output (50.0) is selected to amplify the difference. With profile B (Fig. 9, right) the differences in flow and volume, as a result of using the three options, are so small as to simply cause some thickening of the lines. With profile A (Fig. 9, left) the differences are more pronounced although they remain very small. Figure 10 illustrates the effect of the three options on the relation between neural output and tidal volume over a wide range of neural outputs. The effect is negligible with profile B and very small with profile A. The different sensitivity of the two profiles to the three options is explainable on two grounds. First, for the same amplitude, the average flow with profile A is higher (see Fig. 9). Pressure output is, accordingly, more vulnerable to which flow option is used. Second, withprofde A peak flow is reached later in inspiration, at a time when inspiratory activity is high. By contrast, with profile B, peak blow is reached early in inspiration when inspiratory activity is low. It may be recalled that flow-related losses are proportional to inspiratory activity. In other words, the same flow will result in a smaller pressure drop when inspiratory activity is low. Agai .n, this analysis illustrates that the mechani Cal output of the respiratory system is qu ite insensitive to substantial errors in estimating flow-related losses. We shall henceforth use the relation obtained during maximal voluntary efforts (option 4A) for modeling purposes. l

l/2.0,0.1,0.8,1.9, -2/A

20, lo,50 3_/A 4/ 5/A S/0.0,0.0, 0 R_/-4.0 E/ 10.0 vyyo.ooo vyJ5.00

lJO.7,0.1,0.2,2.0,-20,-20,50 2_/A 3_/A &/ 5/A iy0.0.0.0, 0 R_/ 4.0 E_/lO.O yyo.ooo V~/5.00 50

FIG. 9. Effect of different assumptions regarding slope of pressure-flow relation on predicted volume and flow. Upper panels are not affected by the option and describe neural output (dots) and isometric pressure (soLid lines). See text.

TIME (set)

TIME (set)

MODEL

FOR

RESPIRATORY

NEURAL-MECHANICAL

RELATIONS.

-l/2.0,0.1,0.8,1.9,

~/0.7,0.1,0.2,2.0,-20,-20,** 2/A z/A 6. &./A 6/0.0,0.0, 0 R/-4.0 E/ 10.0 vc/s.OO -v”/o.oo

y/

2.00

1.60

0.80

4A49 . . .. . .. . . 4c ---.

e

0.80

4A46 . . . . . . . 4c --

0.40

1

10.00

I

20.00

1

30.00

I

40.00

50.00

10.00

20.00

30.00

PEAK FIG.

10.

20, 10,” 3/A &/ s/A fyo.o,o.o, 0 R/ 4.0 E/l 0.0 0.00 v//5.00

2/A

1.60

co

987

II

Relation

between

peak

neural

40.00

50.00

PEAK

output

and tidal

volume

using

different

assumptions

for the pressure-flow

relation

(see text

and Fig.

3B). 1/0.7,0.1,0.2,2.0,-20,-20,50 1/A

3_/A

&/A

1_/2.0,0.1,0.8,1.9, 5/

6/0.0,0.0, 0 @i.O E/ 10.0 yyo.ooo vc/s.OO

l&4i3oo

z

lo,50 5/ 0

E/l 0.0 vc/5.00

50

x

)

ii 25

&/A s/0.0,0.0R/ 4.0

50 $

20, 3/A

-2/A

2

25

z

FIG. 11. Calculated flow and volume when inspiratory pressure is allowed to increase in presence of expiratory flow (option 5A, solid lines) and when maximal permissible inspiratory pressure is isometric pressure (option 5B, dots).

Effect of Expiratory Flow on Pressure Output of Inspiratory Muscles (Option 5) Figure 11 illustrates the time course of flow and volume when inspiratory pressure increases (above its isometric value) in the presence of expiratory flow (option 5A) and when the maximal permissible inspiratory pressure is the isometric value (option 5B). The response with option 5A is illustrated by the solid volume and flow lines; the response with option 5B is given by dotted lines. The dots can hardly be separated from the solid lines (in the volume and flow panels),4 indicating that the difference in response with the two options is negligible. The almost complete insensitivity of volume and flow 4 The dotted and solid lines in the top panel represent and the equivalent isometric FRC pressure, respectively. influenced by the option under consideration.

neural activity These are not

to this option may be explained as follows. Inspiratory flow is not subject to the effect of this option and is accordingly unchanged. With profile A, inspiratory activity declines precipitously during expiration. There is very little for the options to exercise their influence upon. Expiration is almost entirely passive. With profile B, inspiratory activity is high early in expiration thereby tending to promote substantial pressure gains with expiratory flow. This tendency, however, is moderated by three factors. 1) Early in expiration volume is the highest. Isometric inspiratory pressure is reduced on account of volume and configurational losses. The intercept of the pressure-flow is low (despite the high inspiratory activity), and the slope of the relation is decreased accordingly. 2) The high volume at this point decreases the slope of the pressure-flow relation independent of its effect on the intercept (see Ref. 16). 3) On account of the

988

W. RIDDLE

braking action of inspiratory muscles at this stage expiratory flow is low. As expiration progresses, the influence of these factors declines but so does inspiratory activity. As a result, the maximal increase in inspiratory Pmus during expiration with option A was less than 0.5 cmHz0 (not shown). The apparent insensitivity of spirometric output to errors in modeling the effect of volume, configuration, and flow on pressure output does not imply that these effects are individually insignificant. Figures 3 and 9 of the preceding communication (16) indicate that these losses can be substantial. The insensitivity, however, is related to the presence of several mechanical interactions within the respiratory system and to the pattern of inspiratory muscle activation during spontaneous breathing. Mechanical interactions. If one overestimates the pressure losses associated with a particular factor, e.g., flow (force-velocity relation), calculated flow will be smaller. As a consequence, volume will increase at a slower rate. Pressure losses due to volume and configuration and the pressure expended against the passive elastance and resistance will be less, thereby making more pressure available for flow. The same may be said for errors in estimating volume- and configuration-related losses. This mechanical negative feedback operates in a similar manner to reduce the effect of added loads (14, 15). Pattern of inspiratory muscle activity. The pattern of inspiratory muscle activity during spontaneous breathing tends to particularly minimize the effect of errors in modeling flow-related losses. Inspiratory activity generally increases in a ramplike fashion. Early in inspiration, activity is low, and flow-related losses (in absolute terms) are, accordingly, small Late in inspiration, inspiratory activity is higher but volume is also higher. The effect of the increased volume in moderating flow-related losses has been discussed earlier. Even though the above analysis demonstrates considerable tolerance of mechanical output to errors in individual assumptions, one cannot be sure that combinations of these errors will not result in more substantial deviations. Furthermore, the possible influence of intercostal muscle activity on the response of the system remains uncertain. These and the ever-present possibility of unforeseen complicating factors make it desirable to test the equation on real data. This is discussed in the following communication (17).

AND

M.

YOUNES

FIG. 12. Method of convoluting N with impulse response function for RC circuit, to account for muscle contraction time. Neural output profile (N) is treated as a series of trapezoids. See text.

trapezoid as a time function circuit gives

is a ramp.

Convoluting

pFRC = (Bt + A) * & Taking

the Laplace

transform

[$+$I

e(-r/Rc)

[-&I

the above

equation

Z(B + As) s2(2 + s)

=

8

by partial

FRC B/Z-A PS =- s+z Taking

an RC

2 = l/RC. pFRC

Solving

with

gives

P:Rc= where

a ramp

the inverse

For equal

Laplace

gives

B A-B/Z + -+s2 S

transform

PFRc=

(i-A)e-“+

PBRC =

Bt +

intervals

fractions

Bt+(A

-$,t,

A and B may be defined

of At,

by (Fig.

12)

A = Ni - pFRC

Ni+l - Ni B =At and

(eszAt - 1) + pFRC

P L!‘=BAt+ APPENDIX The mechanical response of respiratory muscles may be characterized by an exponential function with a time constant of 0.06 s (see Ref. 16 for details). The effective pressure generated at FRC (PFRC) in response to an increasing input (activity, N) that has no well-defined form can be obtained by convoluting N with the impulse response function for an RC circuit [(l/RC) e(-“Rc)]. This process may be expressed as (2)

p tFRC The convolution differential equation

=

Nt

may be included by approximating

* -

1

et-t/RC)

RC with the stepwise N by trapezoids

solution of the (Fig. 12). The

= Ni+l - Ni + -At/RC (e

- 1)

(Ni+l - Ni) At

[(

RC - N. 1 + PFRC 1

1

+ PiFRC

The authors thank B. Goodgame and L. Odbert for secretarial work and M. Powell for technical assistance. This research was supported by National Heart, Lung, and Blood Institute Grant HL-22910. Received

1 October

1980; accepted

in final

form

5 May

1981.

MODEL

FOR

RESPIRATORY

NEURAL-MECHANICAL

RELATIONS.

II

989

REFERENCES 1. CONTE,

S. D., AND C. DEBOOR. Approach (2nd

An Algorithmic 233-240. 2. COOPER,

G. R., AND C. D. MCGILLEM

and System Analysis. New York:

3. 4.

5.

6.

7.

EZementary NumericaL Analysis, ed.). New

Holt

York:

(Editors). Rinehart

McGraw,

1972, p.

Methods of Signal

& Winston, 1967, p. 59-61. ELDRIDGE, F. L. Relationship between phrenic nerve activity and ventilation. J. Physiol. London 221: 535-543, 1971. ELDRIDGE, F. L. Relationship between respiratory nerve and muscle activity and muscle force output. J. AppZ. Physiol. 39: 567-574, 1975. GOLDMAN, M. D., A. GRASSINO, J. MEAD, AND T. A. SEARS. Mechanics of the human diaphragm during voluntary contraction: dynamics. J. AppZ. Physiol.: Respirat. Environ. Exercise Physiol. 44: 840-848, 1978. GOLDMAN, M. D., G. GRIMBY, AND J. MEAD. Mechanical work of breathing derived from rib cage and abdominal V-P partitioning. J. AppZ. Physiol. 41: 752-763, 1976. GRASSINO, A., M. D. GOLDMAN, J. MEAD, AND T. A. SEARS. Mechanics of the human diaphragm during voluntary contraction: statics. J. AppZ. Physiol.: Respirat. Environ. Exercise Physiol. 44:

829-839, 1978. 8. GRIMBY, G., M. GOLDMAN, AND J. MEAD. Respiratory muscle action inferred from rib cage and abdominal V-P partitioning. J. AppZ. Physiol. 41: 739-751, 1976. 9. KONNO, K., AND J. MEAD. Static volume-pressure characteristics

of the rib cage and abdomen. J. AppZ. Physiol. 24: 544-548, 1968. 10. KREYSZIG, E. Advanced Engineering Mathematics (2nd ed.). New York: Wiley, 1967, p. 86-89. 11. LYNNE-DAVIES, P., J. A. BOWDEN, AND W. GILES. Effect of hypoxia on immediate ventilatory load response in dogs. J. AppZ. Physiol. 39: 367-371, 1975. 12. LYNNE-DAVIES, P., J. COUTURE, L. D. PENGELLY, AND J. MILICEMILI. Immediate ventilatory response to added inspiratory elastic loads in cats. J. AppZ. Physiol. 30: 512-516, 1971. 13. MCCLELLAND, A. R., G. W. BENSON, AND P. LYNNE-DAVIES. Effective elastance of the respiratory system in dogs. J. AppZ. Physiol. 32: 626-631, 1972. 14. MILIC-EMILI, J., AND J. D. PENGELLY. Ventilatory effects of mechanical loading. In: The Respiratory Muscles, edited by E. J. M. Campbell, E. Agostoni and J. Newsom-Davies. Philadelphia, PA: Saunders, 1970, p. 271-290. 15. PENGELLY, L. D., A. M. ANDERSON, AND J. MILIC-EMILI. Mechanics of the diaphragm. J. AppZ. Physiol. 30: 797-805, 1971. 16. YOUNES, M., AND W. RIDDLE. A model for the relation between respiratory neural and mechanical outputs. I. Theory. J. AppZ. Physiol.: Respirat. Environ. Exercise Physiol. 51: 963-978, 1981. 17. YOUNES, M., W. RIDDLE, AND J. POLACHECK. A model for the relation between respiratory neural and mechanical outputs. III. Validation. J. AppZ. Physiol.: Respirat. Environ. Exercise Physiol. 51: 990-1001, 1981.