Optimization CHI-SANG

character of inspiratory POON,

SHYAN-LUNG

LIN,

AND

ORLIN

neural drive

B. KNUDSON

Harvard University-Massachusetts Institute of Technology Division of Health Sciences and Technology, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139; and Department of Electrical and Electronics Engineering, North Dakota State University, Fargo, North Dakota 58105 POON, CHI-SANG,SHYAN-LUNGLIN, AND ORLIN B. KNUDSON. Optimization character of inspiratory neural drive. J. Appl. Physiol. 72(5): 2005-2017, 1992.-A previous optimal chemical-mechanical model (C.-S. Poon. J. Appl. Physiol. 62: 24472459,1987)suggested that the normal ventilatory responses to CO, and exercise inputs and mechanical loading can be predicted by the minimization of a controller objective function consisting of the total chemical and mechanical costs of breathing. In this study the model was generalized to include a description of the inspiratory neuromuscular drive as the control output. With a mechanical work rate index for both inspiration and expiration, the general optimization model accurately reproduced the observed responses in the waveshape of inspiratory drive, breathing pattern, and total ventilation under differing conditions of CO, inhalation, exercise, and inspiratory/ expiratory mechanical loads. The simulation results are in general agreement with a wide range of respiratory phenomena, including exercise hyperpnea, CO, chemoreflex, and postinspiratory (postinflow) inspiratory activity, as well as respiratory neural compensations for mechanical loading, respiratory muscle fatigue, and muscle weakness. respiratory optimization; exercise hyperpnea; carbon dioxide chemoreflex; respiratory pattern; respiratory load compensation; postinspiratory inspiratory activity; respiratory muscle fatigue; respiratory muscle weakness

TRADITIONALLY, studies of respiratory

control are often divided into two organizational levels: 1) the control of ventilation and 2) the control of breathing pattern. Such dichotomy stems from the general belief that each level of control is governed by different physiological mechanisms. In ventilatory control, specific stimuli of humoral and neural origins have been variously suggested to be responsible for the precise automatic regulation of ventilatory output (9,X). This control scheme is exemplified by the classic chemostat model of Gray (16) and Grodins (18) in which ventilatory output is reflexly driven by chemosensory feedbacks to the brain stem respiratory controller in a closed loop. On the other hand, the control of breathing pattern is generally thought to be effected by a central “pattern generator” that determines the rate and depth of breathing for any given ventilation (13). The control mechanism is known to be influenced by various respiratory-mechanical factors and reflexes (7,9, 14,49, 0161-7567/92 $2.00 Copyright

0

53). From this perspective, therefore, ventilation and breathing pattern appear to be regulated independently of each other under a hierarchy of discrete commands. In 1965, Priban and Fincham (43) proposed an alternative view of the respiratory control system in which ventilation and breathing pattern are regulated in an integrated fashion. They postulated that the object of the control is to “keep the operating point of the blood at the optimum while using a minimum of energy.” In this way, the control of ventilation and breathing pattern seem to share a common goal: to maintain arterial blood gas homeostasis with the minimum respiratory effort. It has long been appreciated that the control of breathing pattern is consistent with the minimization of some measures of respiratory power output (2, 31, 32, 34, 45, 47,54,55). Recently, it has been suggested that the control of ventilation, per se, may also be subject to similar optimization (35-37). None of these models considers the simultaneous optimization of ventilation and breathing pattern as a possible mode of integrative respiratory control. In this paper, we extend an earlier model of ventilatory optimization (35-37) to include an explicit description of inspiratory neural drive. A basic hypothesis is that all tidal respiratory responses, including breathing pattern and ventilation, may be direct consequences of the optimization of the instantaneous respiratory neural output. Thus rather than assume a hierarchy of control mechanisms for the various levels of respiratory control, one obtains a hierarchy of control outputs (neural drive + breathing pattern + ventilation) that stems from the same optimization mechanism. Simulation results showed that many well-known steady-state in vivo responses in ventilation, breathing pattern, and the waveshape of inspiratory neural drive can be simultaneously predicted by respiratory optimization. Glossary A

a,, a,, a2

1992 the American

Amplitude of inspiratory driving pressure Shape constants of inspiratory driving pressure Physiological

Society

2005

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2006

OPTIMAL

Ers f I,, I,, I,

J, J,, J, n

Pa co29PI co2 P(t)

Pmax Pmax Rrs TT by t2 TI, TE vco

2

VD

V.O VE VT V(t) . V(t) .. V(t) \iv, WI,

\jVE

7 rs

THE

CONTROL

OF

INSPIRATORY

Total (active) respiratory system elastance Respiratory frequency Central, peripheral, and total chemical drive Total, chemical, and mechanical costs of breathing Power index of efficiency factor Arterial and inhaled PCO, Isometric inspiratory pressure (driving pressure) measured at functional residual capacity (FRC) Maximum inspiratory pressure Maximum rate of increase of inspiratory pressure Total (active) respiratory system resistance Respiratory cycle period Neural inspiratory and expiratory duration Mechanical inspiratory and expiratory duration Metabolic CO, output Respiratory dead space End-expiratory lung volume Total ventilation, or ventilatory output Tidal volume Instantaneous lung volume above FRC Instantaneous airflow Instantaneous volume acceleration Total, inspiratory, and expiratory me chanical indexes Chemoreceptor sensitivity Chemoreceptor response threshold I Efficiency factors for Pmax and Pmax Weighting factors Time constant of postinspiratory (postinflow) inspiratory activity Time constant of respiratory mechanical system

MODEL

We model the respiratory control system as a closedloop feedback control system comprising four major functional blocks: the controlled system, the feedback paths, the controller, and the actuator/effecter. The mathematical descriptions of all functional blocks but the effector system have been detailed elsewhere (35-37) and are again briefly outlined below. A detailed description of the inspiratory wave shape and the optimization indexes of the effector system are then given in the succeeding sections. The Glossary lists all symbols used in the model, and Table 1 gives the nominal parameter values. Controlled System: Pulmonary Gas Exchanger

For simplicity, we only consider the regulation of Pa co2, assuming the conditions of normoxia and normal acid-base balance in blood. For an ideal lung with uniform gas exchange and complete blood-gas equilibration,

TABLE

NEURAL

DRIVE

1. Parameter values

Parameter

Value

Reference

0.0934 37.78 Torr 150 cmH,O 1,000 cmH,O/s 3.02 cmH,O 1-l. s, control 8.0 cmH,O 1-l s, loaded* 21.9 cmH,O/l, control 3 1.9 cmH,O/l, loaded*

p”

Pmax Pmax Rrs

37 37 4 4 42 42 42 42

l

l

Ers

l

See Glossary for definitions of abbreviations. ration and/or expiration, depending on type

* Values of load.

apply

to inspi-

the steady- state exchange of CO, is given by the following well- kn own equation Pa co2 = PI co2

863irc0, +

VE(~ - VDIVT) l

(1)

where Pa co2 is assumed to be identical to mean alveolar Pco,. Equation 1 describes the steady-state effect of ventilation (the control signal) on Pac02 (the controlled variable) subject to any disturbances in the inhaled and metabolic production of CO,. To account for the changes in anatomic dead space with airway caliber, we employ the following empirical relation suggested by Gray and Grodins (17) VD

=

0.037vc(1

+ VT/a)

(2)

where VC is vital capacity. For simplicity, we neglect the relatively minor effect of airway smooth muscles on the optimization of VD (54). Feedback Paths: Chemoreceptors and Mechanoreceptors

The feedback loop for Paco2 comprises two sensory structures, the central and peripheral chemoreceptors. We assume that each chemoreceptor has a linear response with respect to Pace, and that their effects are additive. The resultant chemical drive may therefore be expressed as (11, 18) I o = I, + I, = a(Pa,,, - p) (3) In addition to chemical feedback, the optimization model also assumes the existence of certain mechanical feedbacks that modulate the respiratory neural wave shape. The mechanical feedback- signals may be mediated by various mechanoreceptors in the respiratory system (9,14,49,53) or by corollary disch .arge from respiratory motoneurons (50). These neuromechanical reflexes are known to play an important role in the generation and shaping of the respiratory neural pattern (13). A fundamental assumption of the optimization model is that these mechanical feedback signals may also influence the control of venti .lation by way of an optimal controller that integrates the chemical a.nd mechanical feedbacks. Their effects are implicit in the optimization criterion given below, which includes a mechanical component. Controller: Medulla Oblongata

The respiratory controller in the medulla oblongata of th .e brain stem is the integrating station for neural activi-

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OPTIMAL

CONTROL

OF

INSPIRATORY

ties coming from the chemical and mechanical feedback paths. As before (3%37), we assume that the object of the control is to minimize an overall objective function that reflects the balance of the chemical and mechanical costs of breathing J = J, + J,

(4)

where J = cu2(Paco, - fl)2 C

(5 a )

J ,=lnW

(5b) The square law for chemical feedback and logarithmic law for mechanical feedback conform to Steven’s law (33, 60) and the Weber-Fechner law (33) of sensory reception, respectively. The summation in Eq. 4 reflects the conflicting requirements of minimizing both the chemical and mechanical costs of breathing. For example, for any given PIcog and VCO,, an increase in respiratory effort is needed to reduce the chemical challenge of hypercapnia and vice versa. The optimal control output is determined by the minimum of J. We have previously shown that when the mechanical index W is expressed in terms of VE alone, one may predict the normal ventilatory responses to a variety of inputs (35-37). Furthermore, when W is expressed explicitly in terms of the tidal breathing pattern, then the latter can also be accurately predicted in a similar fashion (36). In the present study, we assume that W is an explicit function of the respiratory neural drive and is parameterized by the mechanical constraints of the respiratory system. This requires a quantitative description of the mechanical effector system and the corresponding mechanical cost function. We present a detailed account of these various system components in the following sections. Effector: Respiratory Mechanical System

The description of the effector system is based on a lumped-parameter model proposed by Younes and Riddle (56, 57) for the relation between respiratory neural and mechanical outputs. In this model, the equation of motion is given by the following first-order dynamical equation P(t) = V(t)Rrs + V(t)Ers

(6) The parameters Rrs and Ers represent, respectively, the total flow-resistive and volume-elastic components of the respiratory system. These include the passive resistance and elastance of the lung, chest wall, and airways. To account for any pressure loss due to the intrinsic properties of the respiratory muscles, we assume that these parameters represent the “active resistance” and “active elastance,” respectively, which include the effective impedance of the respiratory muscles (32a). Thus the nonlinear pressure-flow and pressure-volume characteristics are linearized about the relaxation pressure. In addition, any pressure loss due to inertial effects is neglected. These simplifications, as adopted also by other investigators, are probably justified for the ranges of respiratory airflow and volume being considered. The isometric pressure waveshape is divided into an

NEURAL

2007

DRIVE

inspiratory and expiratory phase. The inspiratory pressure is a monotone increasing function of time. In human subjects the inspiratory waveform can be approximated by a quadratic function of the form (56, 61) P(t) = a, + a,t + a2t2 0 5 t 5 t, (7) The parameters a, and a, represent, respectively, the net driving pressure and its rate of rise at the onset of the neural inspiratory phase. The parameter a2describes the shape of the pressure wave. Specifically, the inspiratory driving pressure is concave upward when a2 > 0, convex upward when a2 < 0, and is linear when a2 = 0. During quiet breathing, inspiratory activity does not cease abruptly after reaching the peak value but usually decays nearly exponentially throughout much of the expiration phase (1,42). In the present model, such “postinspiratory inspiratory activity” (or postinflow inspiratory activity; PIIA) is represented by an exponential discharge function P(t) = P(tI) exp[-(t

- tl)h]

t, 5 t < t, + t,

(8)

where P(tl) is the peak inspiratory pressure at time t, and 7 defines the rate of decline of inspiratory activity. The exponential pressure profile for the expiratory phase does not include any active expiratory pressure that may gradually develop toward end expiration. During quiet breathing, such expiratory activities should be negligible. To make the simulation tractable, we simply omitted any expiratory pressure at this stage of investigation. We reasoned that any recruitment of expiratory activity in vivo should be preceded by a reduction or cessation of PIIA. Thus the latter response obtained in the model simulation should be a useful indication of the possibility of expiratory muscle recruitment. With the above mathematical descriptions of P(t) over a complete respiratory cycle, we obtained the following analytical solutions for the time profile of respiratory volume within a breath. For 0 5 t 5 t, V(t) = {A,t + A2t2 + A,[1 - exp(-t/7,,)]}7,,/Rrs

+ v, expW7,,)

(9)

For t, 5 t < t, WI) V(t) = Rrs(l/T,, - l/7)

x {exp[(t, - WI - exp[(t, - Wrsl) + v(t,) exp[(t, - W,,l

(10)

where V 0 = V(0) = V(t, + t2)

(10

and A1 =

a1

A 2 ZZa2 A 3 = a0

-

2a27rs (12)

-

%7,,

+

2a27fs

Equation 11 is the boundary

condition for V(t) that must be satisfied in the ideal steady state on a breath-tobreath basis. Similarlv. the instantaneous flow and accel-

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2008

OPTIMAL

CONTROL

eratio n profiles can be obtained by differentiating above volume functions with respect to time. OPTIMIZATION

OF

INSPIRATORY

the

INDEXES

The general optimization equation (Eq. 4) requires an explicit description of respiratory power. If J, is held constant, then the optimal respiratory drive will simply be the one that minimizes J, or, equivalently, IV. Under this circumstance, the model is analogous to previous optimization models in which only the breathing pattern and/or airflow profiles are optimized subject to constant alveolar ventilation. However, in a closed-loop system where J, is free to vary in a direction opposite to that of J,, all respiratory variables will be simultaneously optimized. Nevertheless, inclusion of the J, term in the optimization should not affect the form of the mechanical cost. Thus similar indexes for W as in previous studies were adapted to the present model. Several mechanical indexes have been previously proposed for respiratory optimization. Some common indexes include inspiratory work rate, inspiratory pressure-time integral, and volume acceleration (20, 21). These mechanical functions are intended to represent various aspects of respiratory power output, such as mechanical work rate, energy cost, and avoidance of possible rupture and general overstraining of lung and chestwall tissues due to rapid acceleration. Most early investigators considered only inspiratory mechanical costs. Hamalainen and Viljanen (21) and Hamalainen and Sipila (20) modified the mechanical index by decoupling the costs of the inspiratory and expiratory phases of breathing. For the inspiratory index, they combined a weighted average of mechanical work rate and volume acceleration. For the expiratory index, they used a weighted average of volume acceleration and the square function of the driving pressure. In the present study, we evaluated the performance of various mechanical indexes in predicting the inspiratory pressure wave shape and overall ventilatory responses to various types of system inputs. Specifically, we compared the following alternate mechanical cost functions for the inspiratory phase and expiratory phase, respectively. Inspiratory

Phase . WI

.

WI

Expiratory

=

=

1

TT

T1 ‘tt)‘@) s

0

1

&

P(t)

n + 441 2

TT

(134

n n 4E 12

Xlti(t)2

-

WE

=

where the mechanical fined as

1

TT

TT s T I 1

m..

TT s T I

dt

P( t)v( t)dt

V( t)2dt

DRIVE

& = 1 - P(t)/Pmax (154 t2 = 1 - P(t)lPmax (W Equation 13a represents the mechanical work rate of inspiration. Equation 13b is a weighted sum of the inspiratory pressure-time integral, a measure of the oxygen cost of breathing during isometric contraction (46), and the average square magnitude of volume acceleration. The latter was included since the pressure-time index alone does not yield a feasible optimal solution (28,47,58). The square function ensures that rapid acceleration and deceleration are equally penalized. The efficiency factors & and t2 account for the effects of respiratory-mechanical limitation and the decrease in neuromechanical efficiency with increasing effort. We have previously shown that the efficiency of neuromechanical coupling is an important determinant of the effectiveness of mechanical load compensation (37). In the present model, the overall efficiency is dependent on two factors: the maximum isometric pressure (Pmax) and the maximum rate of rise of the isometric pressure (Pmax). These indexes are closely related to the capacity of the respiratory muscles to perform mechanical work. In particular, in subjects with respiratory muscle fatigue or muscle weakness, these indexes are expected to decrease, reflecting the reported lowering in maximum pressure and maximum velocity of shortening of the respiratory muscles under these conditions. In analogy with the efficiency factor previously proposed for TjE (37), the efficiency factors for Pmax and Pmax are assumed to decrease proportionately from 1 to 0 with increasing magnitudes of the corresponding output variables. The parameter n describes the nonlinear variation of the efficiencies over the corresponding dynamic ranges. For any given Pmax and Pmax the efficiencies of neuromechanical coupling decrease with increasing n. Equations 14a and 14b correspond, respectively, to the mechanical work rate and volume acceleration indexes during expiration. In this instance, airflow reverses in direction and the inspiratory muscles therefore perform negative work; i.e., the energy stored in the respiratory system elastance is returned to the inspiratory muscles. Equation 14a postulates that a portion of the energy is reabsorbed by the inspiratory muscles rather than dissipated as heat. This is based on recent experimental findings that the mechanical efficiency of skeletal muscles during concentric (miometric) contraction is considerably enhanced if it is preceded by an eccentric (pliometric) contraction caused by an active stretch (19, 22, 23). The mechanism for this potentiation effect is not clear, although various factors have been proposed, such as the recovery of elastic energy stored in stretched tendons and cross bridges (22, 51) and the augmentation of the force exerted by each cross bridge (5). It has been suggested that the reabsorption of negative work during active prestretch and its release during the positive work phase may underlie the relatively high mechanical efficiency noted in vertebrate locomotory muscles in vivo (3,

1 Wb)

Phase WE

NEURAL

(1-m

efficiency factors in Eq. 13 are de-

22, 44).

Other than the maximization of negative work, Eq. 14a also causes the work done on the flow resistance of the respiratory system to be minimized. This is because the

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OPTIMAL

CONTROL

OF INSPIRATORY

total potential energy stored in the respiratory system elastance at end inspiration is fixed and is determined only by the tidal volume. Consequently, an alternate form of Eq. 14a is the work rate integral WE

=

1 TT Rr&( TT s l-1

Q2dt

This results in an expiratory airflow profile that is most uniform subject to other optimization constraints. This is analogous to Eq. 14b, which requires the volume acceleration during expiration to be uniform. Accordingly, the total mechanical index is assumed to be a weighted sum of the inspiratory and expiratory indexes w

= WI

+ X,\iVE

(16)

The factors X, and X2 (with appropriate dimensions) determine the relative weights of the corresponding index terms in the optimization. Thus different weights would result in different sensitivities of the overall response to the corresponding indexes. COMPUTER

SIMULATION

From the above, the mechanical cost function may be expressed as an explicit function of the isometric inspiratory pressure profile J m = Jm(al)9 al9 a2779 t19 &2) (17) From Eq. 5a, the chemical cost is an explicit function of Pa co2, which, in turn, is given by Eq. 1 in the steady state. By substituting Eq. 1 into Eq. 5a, one may express J, as an explicit function of the ventilatory pattern J, =

J,(VT,

f; hco2,

ko,)

In the above equation, the terms P1C02and VCO, indicate the influence of these disturbance variables on J,. For any given total respiratory resistance and elastance, VT and fare completely determined by the inspiratory pressure profile VT

=

v(t,)

-

v,

f = ll(t, + t2)

uw

U8b)

One may therefore regard P(t) as the primary control variable that determines both the chemical and mechanical costs of breathing and hence the total cost

J = Jb,, a,, a297, t,, t,; PIcop Vco2)

(19)

The optimization problem is to minimize Eq. 19 with respect to a vector of six parameters of the inspiratory pressure profile P = [a,, a,, a297, t,9 t21

(20)

subject to the boundary condition (Eq. 1I). Alternatively, one may eliminate a, by substituting Eq. 11 into Eq. 19, which may then be minimized with respect to five independent parameters. The inspiratory neural optimization model was simulated on a digital computer in FORTRAN codes. A quasiNewton algorithm (27) was used to carry out numerical optimization, with a convergence criterion that resulted in

ico2

0.6

0.8

1.0

p L/min.

7. Predicted ventilatory responses during eucapneic and hypercapnic exercise under inspiratory resistive load (top) and inspiratory elastic load (bottom). VCO,, metabolic CO, output. FIG.

DISCUSSION

I

,,I’,I’ I4

!

35

r

40

0~ Exercise naaa-4 Inhaled 1

45

CO2

1

I

1

50

55

60

PaCOz 9 Torr 6. Predicted ventilation inhalation and exercise. FIG.

0.4

tory time and decay time constant were found to decrease, contrary to experimental observations (42).

10 0

0.2

vs. arterial

PCO* (Pa,,*)

during

CO,

range of empirical data. The pressure-time integral for the inspiratory mechanical index (Eq. 13b) was discarded because it always led to an impulsive inspiratory pressure profile with an extremely small inspiratory duty cycle, an artifact also noted by other investigators (31, 47, 58). Furthermore, the ventilatory responses to exercise and CO, inputs were characterized by a rapid increase in f with little or no change in VT. The behavior of the model was not significantly improved by increasing the values of X, and X,. When the chosen model (&s. 21 and 22) was modified by replacing Eq. 22 with a volume acceleration index (Eq. 14b), the model responses to CO, and exercise inputs remained largely unchanged. However, the modified model failed to reproduce the observed responses to mechanical loading in that the parameters t, and t, were found to decrease (rather than increase) under ERL and IRL, respectively. Furthermore, under ERL both expira-

A fundamental question being addressed in this modeling study was, To what extent could the automatic regulation of breathing in humans be predicted by an optimal control model? In this regard, our simulation results appear encouraging. We have demonstrated that many well-known observations in respiratory control can be described by the general optimization model in a unified and coherent manner. Specifically, the model predicts the following: 1) the characteristic wave shapes of the normal inspiratory neural drive and PIIA under varying CO, and exercise inputs, 2) compensatory changes in the amplitude and shape of the respiratory neural drive in response to different mechanical loads and neuromuscular disorders, 3) changes in the cyclic breathing pattern in response to different mechanical loads and stimulus inputs, and 4) the overall ventilatory responses to CO,, exercise, and mechanical loading. Most important, the wide spectrum of system behaviors is derived from a single optimal controller that receives only the conventional chemical and respiratory-mechanical feedbacks, without the need for any exercise or other inputs. The predicted responses are in good qualitative agreement with those observed at various hierarchical levels of control system outputs. Comparison With Other Models

The general predictive power of the present model is in contrast to the limited applicability of previous models

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2014

OPTIMAL A

/\

/ /A-‘I

’

~-\ \

Time

CONTROL

-----

Pmax=

- -

Pmax= 150 Pmax=250

OF

INSPIRATORY

NEURAL

lambda= 1.25 lambda= 1 .O lambda=O.i’5

\

Time FIG.

--

i)max= i)max=

weighting

1500 1000

10. Predicted factor

/ \ d’, ------- i)max= 500 / ,/i’ I:, 1 ,I’ \ ‘,\ / ,/’ \ ‘\\\ /,Y” \ ‘\\ r \ ,‘\\‘-. --3

driving

(second) pressure

profiles

with

various

values

of

X,.

6

(second)

8. Predicted driving pressure profiles under various force amplitude limitation (top) and force rate limitation Pmax, maximum inspiratory pressure; Pmax, maximum crease of inspiratory pressure.

.2

degrees of (bottom). rate of in-

FIG.

UI3XJ M %*s+z+$ xxx-m

--

----

50

(second)

Time

DRIVE

30b

T2

48.

(second)

NL Eucapnia NL H percapnia IRL H ucapma IRL Hypercapnia

bV 1

OI 00b I

0.0

0.2

1

I

1

1

0.4

0.6

0.8

1.0

Tau FIG.

ix0

(t2; top) bottom).

L/min.

2)

UIZUI NL Eucapnia AMAA NL H percapnia WZ+G+> IRL %ucapma

.>

Es o!

0.0

T

0.2

I

0.4

I

0.6

1

0.8

1

1.0

PC0 2 FIG.

index

9. Similar of efficiency

plots as in Fig. factor.

9 L/min. 7, top, with smaller

values

06b

for power

(second)

11. Sensitivities

of cost functions to neural expiratory duration and time constant of postinspiratory inspiratory activity (7; J, J,, J,, total, chemical, and mechanical costs of breathing.

aration of ventilatory variables from respiratory pattern variables precludes any detailed analysis of the total respiratory response to complex inputs. Another disadvantage of the hierarchical approach is that separate mechanisms of chemoreflex or mechanoreflex must be invoked to explain the observed response at each level. It is as yet not clear how the variously proposed mechanisms at different levels of control might be integrated by the respiratory center to produce an overall response under varying conditions. By contrast, the present model offers a general and unified theoretical framework for predicting the total response at all hierarchical levels of respiratory control under a wide range of conditions. Significance

that were intended to describe only certain aspects of the respiratory control system. Thus most previous models are restricted to the control of ventilation alone, or the cyclic/instantaneous respiratory patterns alone, under specific experimental conditions. Such a hierarchical sep-

12.

of Model

Assumptions

Prediction of exercise hyperpnea. A major assumption in previous models of ventilatory control is that the response to exercise is reflexly driven by a certain “exercise stimulus.” Our previous work suggested that such a presumed stimulus is not needed if the controller follows an

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OPTIMAL

IRL

CONTROL

OF

INSPIRATORY

z;, >tx~t)~( Jm

Tl

(second)

P

0

T2 tion

(second)

12. Sensitivities of cost functions (tJ and t, under IRL (top) and ERL

FIG.

to neural (bottom).

inspiratory

dura-

op Smization criterion (37). This quality is preserved in the present model where the optimization criterion is generalized to include an explicit description of the inspiratory neural wave form. Thus the proposed form of chemical-mechanical optimization is a general criterion that remains valid even when the instantaneous inspiratory drive is used as the output variable. Prediction of inspiratory ruaue shape. The inspiratory cost function (Eq. 21) is a generalization of our previous mechanical cost function (37), which *was simply assumed to be a logarithmic function of VE. Because the P urpose of the present mo de1 is to predic t inspiratory n eural wave shape, it iS necessa ry to express *I as an explicit function of P(t). The proposed form of WI, which is related to the mechanical work rate of inspiration, is similar to those assumed by other investigators (2, 20, 21, 31, 47, 55). The mechanical work rate was found to be a satisfactory index of the energy cost of breathing particularly under resistive loading (10). In the present study, this in dex satisfactorily p redicted the changes in inspiratory wave shape under various mechanical loading. The parameters [I and & are generalizations of the efficiency factor introduced previously (37) to describe the mechanical constraints on respiratory neuromuscular output. This factor accounts for the attenuation of the ventilatory response during mechanical loading (cf. Figs. 7-9). Because P(t) may be limited in terms of peak activity as well as rate of rise of activity, we used two separate indexes to describe these events. This permits a quantitative prediction of P(t) when Pmax and/or Pmax are decreased as a result of inspiratory muscle weakness or muscle fatigue (Fig. 10). The variations of these efficiency factors over the assumed dynamic ranges of P(t) and P(t) are determined by the power index n. In this study the value of n was estimated indirectly from

NEURAL

DRIVE

2015

the predicted ventilatory responses under mechanical loading (Figs. 7 and 9). This estimate is not unique; with different assumed values for Pmax and Pmax the estimated value for n may be different. Prediction of PIIA. An intriguing aspect of the inspiratory neural drive is the appearance of PIIA. Because any inspiratory effort during the expiratory phase would tend to delay lung emptying, such activities have been thought to be counterproductive in terms of ventilation. Thus other factors conducive to respiratory optimization must be involved in the generation and maintenance of PIIA. The proposed expiratory mechanical index (Eq. 22) is the only function that was found to successfully reproduce the responses in PIIA under varying conditions of CO, and exercise inputs as well as inspiratory/expiratory loading. According to this index, PIIA may represent an optimal response in that the negative work done during expiration is beneficial for the minimization of the overall mechanical cost of breathing. Although expiratory airflow will be somewhat retarded, the resting ventilation is unlikely to be significantly compromised, since t, (and hence respiratory frequency) is independently optimized. This is clearly demonstrated in Fig. 11, which shows that, in the resting state, J, is much more sensitive to t, than to 7. This may explain the predicted and observed insensitivity of the time course of PIIA to small increases or decreases in expiratory load (see Fig. 11 and Refs. 42, 48). On the other hand, an inordinately large value of 7 relative to t, (or, equivalently, an inordinately small t, relative to 7) may lead to an increase in end-expiratory lung volume that, in turn, would place the inspiratory muscles at a mechanically disadvantageous position, thereby increasing the work of breathing. Thus for overall optimality, the value of 7 should not be too large relative to t,. This may account for the predicted optimal value of 7 at rest and its corresponding decrease during hyperpnea. Limitations

of the Model

Expiratory activity. In this study we did not include any expiratory muscle activity. Thus the predicted responses are strictly valid only for situations where expiratory activity is inhibited [e.g., in the resting state or during CO, inhalation (6) or supine exercise (24)]. Although inclusion of expiratory drive may improve the accuracy of model predictions in some cases, it also further increases model complexity. For this reason, we chose to first limit our analysis to inspiratory drive at this stage to better delineate the determinants of the inspiratory wave shape. The present results will be a useful basis for further analysis of the expiratory wave shape in future. For example, the predicted decrease in the time constant of PIIA during hyperpnea suggests a possible tendency for recruitment of expiratory muscles, a response that is typical of healthy human subjects on light to heavy exercise in the upright position (6, 24). Model performance. It is possible that the performance of the model could be further improved by inclusion of other factors (e.g., nonlinearities in Rrs and Ers) that are relevant to respiratory optimization. Also, for the inspiratorv index, inclusion of a small term corresponding to the

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2016

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pressure-time integral and/or volume acceleration may account for the energy expended in maintaining the inspiratory driving pressure or the defense against lung rupturing, respectively. These strategies may help to fine tune the instantaneous airflow pattern predicted by the model (2, 20, 21). Because the pressure-time integral favors a shortening of inspiratory time, this may also be useful in keeping the predicted inspiratory duty cycle to within the normal range, which would complement the present results. Further experimental and simulation studies are needed to determine the precise form of IVI and \jVE under other critical conditions. Variability of breathing pattern. Although the predicted optimal responses in VE are relatively insensitive to the choice of model parameters, the predicted responses in cyclic and instantaneous breathing patterns may be influenced to some extent by the various simplications in the formulation of the model. Thus any changes in model parameters would lead to substantial variations in the predicted breathing pattern. The proposed chemical (Eq. 5a) and mechanical (Eqs. 21 and 22) indexes are the simplest functions that were found to conform with most empirical data under consideration. Because the defining parameters may vary substantially between individuals, the model is useful in predicting only the directional changes in breathing pattern variables in a typical subject. Physiological Significance of Optimization Principle

The proposed optimization criterion is an empirical model of the respiratory controller in that it is intended solely to reproduce observed data under a consistent operating principle. As such it is analagous to the classic feedforwardifeedback model that assumes reflex control as the operating principle. Three important questions must be addressed in evaluating these empirical models: I) To what extent is the model useful in predicting observed data? 2) Is the presumed model structure parsimonious and consistent under different test conditions? 3) Can the model be related to any known physiological mechanisms? The present results demonstrate that the optimization model, despite its many inherent simplifications, is far more predictive than the reflex model. Specifically, the optimization model offers a more general and accurate description of the phasic respiratory output under different chemical, mechanical, and exercise challenges. The optimization model is also more parsimonious than the reflex model in that it accurately predicts the isocapnic exercise hyperpnea response without the need for a postulated exercise stimulus. Also, the same optimization principle is consistently applied to the description of the phasic inspiratory drive, cyclic respiratory pattern, and total ventilatory output in a coherent fashion. A major drawback of the optimization model is that presently there is little knowledge of the underlying neural mechanism, if any, that may explain the presumed optimization behavior. Thus one may consider the optimization model as a hypothesis of a possible neural mechanism. Brain stem respiratory neurons are known to be endowed with many computational abilities,

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such as memory and temporal gating (12). It is of interest to investigate whether these neural mechanisms play a role in respiratory optimization. Although the optimization principle will remain a general empirical model of the integrated respiratory response, the hypothesis of respiratory optimization may be a useful guiding principle for further experimental investigation of the mechanisms of neural information processing in the brain stem respiratory regions. This investigation was supported by National Heart, Lung, and Blood Institute Grant HL-30794. Address for reprint requests: C.-S. Poon, Harvard-MIT Div. of Health Sciences and Technology, Rm. 2OA126, Massachusetts Institute of Technology, Cambridge, MA 02139. Received 2 October 1990; accepted in final form 14 November 1991. REFERENCES 1. AGOSTONI, E., G. CITTERIO, AND E. D’ARGELO. Decay rate of inspiratory muscle pressure during expiration in man. Respir. Physiol. 36: 269-285, 1979. 2. BATES, J. H. T. The minimization of muscular energy expenditure during inspiration in linear models of the respiratory system. Biol. Cybern. 54: 195-200, 1986. 3. Bosco, B., J. TIHANYI, P. V. KOMI, G. FEKETE, AND P. APOR. Store and recoil of elastic energy in slow and fast type human skeletal muscles. Acta Physiol. Stand. 116: 343-349, 1982. 4. CAMPBELL, E. J. M., E. AGOSTONI, AND J. NEWSOM DAVIS. The Respiratory Muscles. Mechanics and Neural Control. Philadelphia, PA: Saunders, 1970. 5. CAVAGNA, G. A., M. MAZZANTI, N. C. HEGLUND, AND G. CITTERIO. Storage and release of mechanical energy by active muscle: a nonelastic mechanism? J. Exp. Biol. 115: 79-87, 1985. 6. CHA, E. J., D. SEDLOCK, AND S. M. YAMASHIRO. Changes in lung volume and breathing pattern during exercise and CO, inhalation in humans. J. Appl. Physiol. 62: 1544-1550, 1987. 7. CHERNIACK, N. S., AND M. D. ALTOSE. Respiratory responses to ventilatory loading. In: Regulation of Breathing, edited by T. F. Hornbein. New York: Dekker, 1981, pt. II, p. 905-964. (Lung Biol. Health Dis. Ser.) 8. CLARK, F. J., AND C. VON EULER. On the regulation of depth and rate of breathing. J. Physiol. Lond. 222: 267-295, 1972. 9. COLERIDGE, H. M., AND J. C. G. COLERIDGE. Reflexes evoked from tracheobronchial tree and lungs. In: Handbook of Physiology. The Respiratory System. Control of Breathing. Bethesda, MD: Am. Physiol. Sot., 1986, sect. 3, vol. II, pt. I, chapt. 12, p. 395-430. 10. COLLETT, P. W., C. PERRY, AND L. A. ENGEL. Pressure-time product, flow, and oxygen cost of resistive breathing in humans. J. Appl. Physiol. 58: 1263-1272, 1985. 11. CUNNINGHAM, D. J. C., P. A. ROBBINS, AND C. B. WOLFF. Integration of respiratory responses to changes in alveolar partial pressures of CO, and O2 and in arterial pH. In: Handbook of Physiology. The Respiratory System. Control of Breathing. Bethesda, MD: Am. Physiol. Sot., 1986, sect. 3, vol. II, pt. II, chapt. 15, p. 475-528. 12. ELDRIDGE, F. L., AND D. E. MILLHORN. Oscillation, gating, and memory in the respiratory control system. In: Handbook of Physiology. The Respiratory System. Control of Breathing. Bethesda, MD: Am. Physiol. Sot., 1986, sect. 3, vol. II, pt. I, chapt. 3, p. 93-114. 13. EULER, C. VON. Brain stem mechanisms for generation and control of breathing pattern. In: Handbook of Physiology. The Respiratory System. Control of Breathing. Bethesda, MD: Am. Physiol. Sot., 1986, sect. 3, vol. II, pt. I, chapt. 1, p. l-67. 14. FRAZIER, D. T., AND W. R. REVELETTE. Role of phrenic nerve afferents in the control of breathing. J. Appl. Physiol. 70: 491-496, 1991. 15. GALLAGHER, C. G., V. IM HOF, AND M. YOUNES. Effect of inspiratory muscle fatigue on breathing pattern. J. Appl. Physiol. 59: 1152-1158, 1985. 16. GRAY, J. S. The multiple factor theory of the control of respiratory ventilation. Science Wash. DC 103: 739-744, 1946. 17. GRAY, J. S., F. S. GRODINS, AND E. T. CARTER. Alveolar and total

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