Nonparametric Modeling of Respiratory Mechanics and Gas Exchange

Copyrigh t © IFAC Identification and System Pa ra m ete r Estimation 1982, Washington D.C., USA 1982

NONPARAMETRIC MODELING OF RESPIRATORY MECHANICS AND GAS EXCHANGE

v.

Z. Marmarelis and S. M. Yamashiro

Departments of Biomedical an d Electrical Engin eering, University of South ern Ca lifornia, Los A ngeles, California, USA

Abstract . A nonparametric method of modeling respiratory mechanics and gas exchange is presented and compared to existing parametric methods . The nonparametric approach is based on Wiener ' s theory of nonlinear system identification by use of white - noise test inputs . The sough t nonparametric models have the form of nonlinear functional relations between tracheal pressure and flow (in the study of respiratory mechanics) , and between tracheal flow and PC02 (in the study of gas exchange) . The motivation for this modeling study is provided by the recent emergence of high - frequency ventilation as a useful clinical tool . The obtained results increase our understanding of the quantitative relations between these physiological variables , and allow optimization studies that may lead to improved clinical applications . Keywords . Identification ; modeling; nonlinear systems ; physiological models; correlation methods ; random processes; biomedical ; time - varying systems; system analysis .

INTRODUCTION The mammalian respiratory system operates as a mechanica l bel l ows pump which produces an alternating flow of gas between the external atmosphere and the lung a l veo l i . Al veolar gas exchange is primarily accomplished by diffusion but , depending on the frequency r ange , airway gas exchange involves both diffusion and convection . In the range of normal breathing frequencies (0 - 0 . 5Hz) , airway gas movement involves largely convective mechanisms , which are associated with cyclic volume changes in excess of airway dead - space volume . Above this frequency range (0 . 5- 15Hz) , gas exchange appears to involve some type of facilitated diffusion , which requires cyclic volume changes much smaller than airway dead - space volume . ventilation at these high frequencies and small volumes is felt to offer important advantages in artificial respiration . Since 1977 , there has been increasing interest in the development of this approach for clinical purposes (Bohn and others, 1980; Slutsky and others , 1981) . However , the phenomenon is poorly understood and preliminary attempts at developing mathematical models relating mechanics and gas exchange over a broad frequency and amplitude range appear to be incomplete . The main reason for this , is the complexity of the underlying physical and physiological mechanisms, and the intrinsic nonlinearities and nonstationarities of the system , that render it invincible to most traditional modeling approaches . In this paper , we outline some potentially suitable modeling techniques and we present preliminary results in applying the white - noise excitation

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method fo r developing nonparametric models of the respiratory mechanical and gas exchange system of the cat during high- frequency venti l ation (HFV). Random test inputs have been used by several investigators (Michaelson and others, 1975 ; Williams and others , 1979) to obtain measurements of respiratory impedance in animals and the human . The use of broadband random test inputs (i . e ., white noise) is a relatively recent extension of the "forced oscillation " method , which employs sinusoidal test inputs along the traditional lines of linear system analysis . This follows a trend established by the successful application of white - noise test inputs to neurophysiology in the context of Wiener ' s theory for nonlinear system identification (MarMarelis (P . Z . ) and Marmarelis (V . Z . ) , 1978 ; McCann and !larmarelis (P . Z . ) , 197 S ; Marmarelis (P . Z . ) and Naka, 1974 ; and Stark , 1969) . The main attraction of this approach is that it allows nonlinear analysis of the system under exhaustive testing conditions over the entire range of frequencies and amplitudes , without requiring a priori postulates of specific parametric models (i . e ., nonparametric approach) . Advantages also include short test periods(owing to the information- dense input) , robustness of the testing and esti mation procedures , consistency of obtained results , and ease of application to uncoopera tive subjects . Previous use of white - noise test inputs in the study of the respiratory system was limited to small perturbations superimposed on breathing , and adhered to l inear analysis . In this study , we will use

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V. Z. Marmarelis and S. M. Yamashiro

larger amplitudes and probe into the nonlinear range. Ultimately, high-power white-noise pressure inputs will be used to maintain ventilation in absence of breathing (HFV). The aim of this paper is two-fold. On one hand, we want to present and discuss some nonlinear identification methods that appear suitable for developing mathematical models of the particular systems at hand -- but can also be used in other c a ses of nonlinear modeling. This will include a discussion of system nonstationarities, how do the y arise in this particular instance and what are the appropriate and effective means for their analysis and modeling. On the other hand, we want to demonstrate the actual use of ~1iener's theory to respiratory mechanics and gas exchange, and discuss the physiological implications of the obtained nonparametric models qS well as the virtues and deficiencies of this approach. The two systems considered are defined by their input and output variables. In the first system, the dynamic relationship between tracheal pressure (input) and tracheal flow (output) is analyzed (modeled) as an indirect measure of respiratory mechanics, to the extent the latter reflect on the formation of flow in the lungs and through the upper airways in response to tracheal pressure perturbation. This indirect assessment of the mechanical state of the lungs is appealing to the clinician because of its non-invasive nature and the relative ease of administering the test. However, it is acknowledged that the effect of the lungs themselves on the measured tracheal flow is often mitigated cons iderably by the intervening airways. In the second system, the dynamic relationship between tracheal flow (input) and tracheal PC02 (output) is analyzed (modeled) as an indirect measure of how ventilatory flow affects gas exchange in the lungs. This measurement addresses the question of ventilatory efficiency as a function of forced tracheal flow, raised by the re cent experiments and findings of HFV. The ultimate question here concerns the int rapulmonary flow patterns generated by part i cula r input (tracheal) flow waveforms and their effect on ventilation and gas exchange. A few measurements of arterial PC02 can be also taken to enlighten and verify this latter point. It should be noted that a great deal of work has been done in modeling the phenomenon of gas exchange, but none of this work is capable of explaining the quantitative behavior of the system under cond itions of high-frequency ventilation (HFV). This, along with the study of the pressure-flow relation, is the focus of our pape r.

resistance, inertance and comp liance, which characterize the mechanical state of the lungs. The pattern of pressure perturbation has evolved from the traditional pulse and sinusoidal stimuli to random forcing using broadband noise signals. In the latter case (which has been established in recent years as the most efficient and accurate one), the spectra of the input/output broadband random signals are computed to yield estimates of the system transfer function or (its inverse) impedance function. Estimates of the resistance, i nertance and compliance parameters are then obtained from the impedance mea surements via regression techniques. This last step is predicated on the assumption of a linear, timeinvariant model for the system under study. Specific parameter values are sought in order to facilitate the interpr etat ion of the obtained data. However, the values of these specific parameters are meaningful only when the employed reference model is a realisti c one for the system at hand. If the postulated model fails to rep re sent with sufficient fidelity the physical reality of the system workings, then the obtained parameter estimates ma y lead to erroneous interpretation or to apparently incon sistent results under different conditions of stimulation. This underline s the importance of postulating a realistic model for the system under test, and provides the focus of this paper. Our basic thesis is that the aforementioned physiological system exhibits substantia l nonlinearities and nonstationarities, while functioning in the full physiological r ange , as to require an accordingly nonlinear and nonstationary reference model. Very few will argue the presence of substantial nonlinearities and nonstationarities in the respiratory system (e.g., nonlinearities arising from turbulent flow in the airways, or nonstationarities due to the cyclical variation of lung-chest compl iance with lung volume -- to name just two of the most widely acknowledged). However, many will point out the lack of effective methods in dealing with such a comp li cated modeling situation. It is the purpose of this paper to propose one such method. The model, that currently summarizes our best understanding of the system in question has the form of a second-order nonlinear differential equation: I~ + R(~,v)~ + E(v)v = p

THE MODELS In the many studies of respiratory mechanics, a linear time-invariant model has been used to relate the input pressure perturbation to the ou tput flow. The model accounts for resistive and reactive elements in typically one or two compa rtments. The identification task consists of estimating system parameters representing

(1)

where p(t) is the input pressure pertu rbati on and ~(t) is the output flow, both measured in the trachea. In keeping with the phys ical realities of our system, the model of Eq. (1) acknowledges that: (a) the respirato ry resistance R depends in general on the flow ~ (because of turbulen ce in the central airways) and on the lung volume v (because of the effect of airway geometry on flow

Nonparametric Modeling of Respiratory Mechanics and Gas Exchange friction), and (b) the r espiratory elastance E (i.e., the inverse o f r espi ra to r y compliance) depends on the lung volume v (because the mechanical stiffness o f the lung ti ss u e varies as the lung s expand or contract) . The identified nonlinearities are the most com monly acknowledged, but sure l y not the o nl y ones . However , the remaining sour ces o f non linearity (i.e., mechanical cou p ling of lung with visce ral pleura , nonlinear viscoelasticity of lung tissue, nonlinear elasticity o f air passages , variable sur fa ce tension of s urfuc tant lining of alveoli , etc.) are expected to have negligible effect r e la tive to the pre viously identified sour ces . In the same spiri t, the respiratory inertance I is co nsidered constant in first approximation. Thus, the postulated model (1) , although s till an approximation, constitutes a daring departure from the traditiona l l inear mode l (where all I , Rand E are considered constant) , and certa inl y provides a c l oser representation o f the phys ical and phys i o l ogica l realities of our system . In o rder to est imate th i s model from i nput/ output data , we have to consider a par ame tric representation of Rand E in te rm s of v and v , and seek to estimate the introduced parameters. To maintain genera lity in the nonlinear representation , we must make use o f an o rthogona l or analyt i cal expans i on of Rand E in terms of v and v. The high - order term s of s uch expansions wi ll be negligible if the system nonli near ities are s mooth -- which is in fact o ur expectation for this particular system. Thu s by taking t his route, we end up with a problem of pa rametric identification, where a limited number of unknown parame t ers (that enter linearly in the model) mu st be es t i mate d from input /output data. This prob l em ca n b e so lved by use of any of the standard parame t er estimation procedures . One practical p r ob l em that may arise in th i s approach is that, since the expe rimentally recorded data are usually p(t) and v (t) , the actual eva luation of v(t ) through differentiation may introduce s i gnifican t e rr ors . Also, the presence of substantial extraneous noise wi l l requir e large amounts of data, which in turn wi ll make any lea st - squa r es f itti ng procedure computat i ona ll y burdensome. Last but not least, the universality of the obta in ed estimates would r equire extensive t esting of the system with a broad variety of input wave forms . We stress this point because of the existing tradition of using certain waveforms (e . g ., steps o r sinusoids) that give no guaranty of cover ing densely the sys tem fun ctional space . As a r esu l t , a set of parameter estimates obtained from ce rt ain input/ou tput data , is o ften proved useless in p r ed i ct i ng the system ou tput to a different input. Fo r this reas on , it is imperative that an exhaustive test input is u sed in col l ecting the key set of data, for the pa rameter estima ti on . For such an input, we suggest the use of an e rgodi c r andom process o r a proper l y des i gned sequence of deterministic waveforms. I SPt:. -I- '."

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An alternative approach is offe r ed by Wiener's theor y o f nonlinea r system identificat i on (Billings, 1 980 ; Lee , 1964; P . Z. Marmare li s and V. Z. Marmarelis, 1978; Schetzen , 1 980 ; and Wi ene r, 1 958) . In thi s , a Gau ssian white no ise (GWN) input is employed in connection with a functional representation of the inp ut-x /output-y rel ation: y(t) = dx(t') ,t',::.tJ (2)

I n the case of r espi r atory mechanics x(t) is p(t) and y( t) is v(t) , whereas in the study of gas exchange the flow signal v(t) becomes the input x(t) and the tracheal C02 tensi on is th e ou tput y(t) . The non linear depen d e n ce o f the o u tput present va l ues upon th e input past and present can be dissec t e d by considering either an o rt hogonal functional expansion of F or an orthono rmal expansion of the input past. In the former case , we end up with a nonparametric model that co nt a in s a set of unknown kernel functions const ituting the object of identification. These kerne l s must be estimated from input output data, and they offer a universal model -- in t hat it can pred i ct the system ou t put t o any given input -- since the GWN input cove rs the entire frequency and amplitude rang e . Th is genera l model , known as the Wiener series expansion , has the form:

y(t) =

L

Gn [hn;x(t'), t' '"

-

(nil)

"=0

(-l)rnn!prn

tl

L L (n _ 2m )1. m.'2rn """0 m=O

J,-. . . Jh h , .. . , 0

n

(:3)

where P is the power l eve l of the input GWN . The kernel functions {h n } character ize the nonlinear system dynamics , and they have a u nique relation with the functions Rand E of mode l (1). This exact re l ation can be studied in known, but rather involved , ways (Bar r ett , 1 963; and Ko r enberg , 1973) . The terms {G n } of the Wiener series are constructed orthogona l (in a statistical sense) when the input is GI'IN . Their orthogonality can be exp l oited to obtain r e l atively simp l e estimators for the system kernels . Th e simp l est such estimator is given in the f orm of crosscor r elation between the input and output data (Lee and Schetzen, 1 965) as: h n (11' .•• , 1 n)

=

J

'

1 n -l --,-- E { [y( t)- Z Gk(t) x(t-1 l ) ... x(t- Tn)j n.pn k =O

(4 )

Thi s formula has been used in numerous suc cess ful applications of Wi ener ' s theory to physio l og ical systems having p rima ril y loworder kernels (for r eview of such applications see P . Z . Marmarel i s and V. Z. Mar mar el i s , 1978) . Unfortunately , the esti-

V. Z. Marmarelis and S. M. Yamashiro

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mation of high-order kernels by use of Eq. (4) is hindered by the heavy computational burden associated with evaluation of highorder crosscorrelations . Actual estimation of kernels has been so far limited to the third order -- and in most applications to the second order -- making this approach suitable only to cases of "soft" nonlinearities. In the alternative Wiener approach, the input past is expanded on a complete orthonormal (CON) set in the range [0,00] (e .g., Laguerre functions) as: X(t- T) = L an(t)bn(T) n=O

(5)

00

f bn( T )x(t-T)dT =an(t) (6)

Therefore, the expansion of the input past is operationally equivalent to a CON expansion of the system kernels (cf. Eq. (3» (Lee, 1964 • The system output thus becomes a function of the values {an } at each time t . This way, we have reduced a general nonlinearity to a static (zero-memory) one : y = f(a O ,a l , a 2 ,···,a n ,··· )

1 n-l -'--pn fdtBm(t) Iy(t)- L Gk(t) ]X(t-Tl) ••• x(t-T n ) n. T k=O (8)

where, hn(t;Tl , •.• , Tn ) =

where {b n } is the CON set and {an} are the time-varying expansion coefficients (Bose, 1956; Lee, (1964). Clearly, an(t) is the output of ~ li near filter with impulse response bn(T) having x(t) as input, since from Eq . (5) we get:

o

averages (V.Z. Marmarelis, 1 981b) . This modification amounts to the use of an additional orthonormal expansion in time, employing a CON basis {~(t)} . The resulting estimator for the expansion components of the time -varying system kernels is:

(7)

The Wiener series, under this expansion of its kernels, yields a multi-dimensional orthogonal (Hermite) expansion of f in terms of the varia bles {an } ' Thus the practical question of truncating the expansion arises again. It is possible to achieve substantial computational savings (compared to the crosscorrelation scheme), if an orthogonal expansion basis can be found in each given situation to provide a parsimonious (i . e. , using few basis elements) representation of the system kernels. The number of required basis elements should be compared to the number of time-lags that must be estimated under the crosscorrelation scheme for the same accuracy. It appears likely that, in most cases , a suitable basis can be devised to reduce the number of required expansion coefficients below that of time-lags for the same mean-square accuracy of kernel representation (V . Z. Marmarelis, 1980; Watanabe and Stark , 1975). So far, the discussion has dealt with the general nonlinear but stationary case . If the system exhibits substantial nonstationarities (and we shall see that this may be the case here), then the discussed Wiener models become time-varying and their estimation is somewhat complicated. The complication results from the fact that ensembleaverages cannot be replaced by time-averages any longer. A slight modification of Wiener's approach has been proposed, which makes possible the estimation of appropriate orthogonal expansions through the evaluation of time-

L

m

Cn,m(T l , ·· ., Tn) i3m(t) (9)

The estimator of Eq. (8) yields unbiased and consistent est imates, provided that the nonstationarity is sustained and bounded, and the output process sat isfies a certain mixing condition (V. Z . Marmarelis, 1981a). All these conditions are satisfied in most actual situations. A simi lar modification is possible in the alternative Wiener approach, where the orthogonal Hermite expansion of the now time-varying function f has time-varying coefficients which must be independently expanded in time using a CON set {Bm(t)} (V.Z. Marmarelis, 1981b). Let us now examine the stationarity of the particular system at hand. When the GWN pressure perturbation is superimposed on the regular breathing pattern, the lung volume is left practically unaltered and fully dependent upon the phase of the breathing cycle. Therefore, the two components of the measured flow (viz., the one due to breathing and the other caused by the pressure perturbation -- subscripts "b" and "p"respectively) may be comparab l e in size, however their volume counterparts are vastly different: ~(t)

Vb(t)

+ vp(t)

v(t)

vb(t)

+ vp(t) (l0)

The part of the flow data that is due to breathing must be eliminated from the output data (through high-pass filtering), since it does not relate physica lly to the input pressure perturbation p(t) .The breathing variables , however, should be included in the model as to their effect on respiratory resistance and elastance. As a result, the differential - equation model descr ibing the causal relation between the input p(t) and the output ~p(t)is: Ivp + R(Vp,t)vp+ E(t)Vp = p (11)

This dins tinction is extremely important from the modeling point of view, as it transforms an ostensibl y nonlinear model into a nonstationary-non li near one for the experimentally defined and causally related input/ output variables. The nonstationarity, of course, is introduced through the presence of the breathing variables vb(t) and vb(t) in the resistive and compliant parts of the system. This nonstationarity is periodic

Nonparametric Modeling of Respiratory Mechanics and Gas Exchange (of known period) and satisfies all the conditions set for use of the extended Wiener approach. An appropriate expansion basis {Sm(t)} would be, of course , one suited for periodic signals (e.g., Fourier). The system nonstationarity can be eliminated in an experimental situation where breathing is artificially suppressed and ventilation is provided entirely by the input pressure perturbati on . Experiments have shown that this is actually possible as long as the input GWN exceeds a certain value of power level. In such a situation , the system becomes solely nonlinear, and the analysis can proceed in any of the described stationary ways. This simplified modeling situation may offer the proper paradigm for the study of the effect of respirat ory mechanics on gas exchange and, conversely, of inhaled gases on respiratory mechanics.

EXPERIMENTAL METHODS Experiments were conducted in cats both normally breathing and anesthetized with pentobarbital and paralyzed with gallamine. HFV was applied using two low-frequency 12inch l oudspeake rs connected in series and driven via d.c. power amplifiers with sinusoidal or band-limited white-noise inputs. Animals were tracheotomized prior to connection to the respirator. Fresh air was maintained at the tracheal input using a balanced air pump and vacuum system at an adjustable mean pressure level. Gas exchange was assessed through 1 ml samples of arterial blood analyzed on a radiometer blood gas system for PH, PC02, and P02. Tracheal pressure was measured using a Validyne differential pressure transducer and tracheal airflow was measured using a Fleisch pneumotachograph and Va lidyne differential transducer. Pneumotachograph frequency response was compensated using electronic filtering. Tracheal C02 was mea sured continuously with an infrared C02 analyzer.

RESULTS Tracheal p r essure , flow and C02 tension signals were recorded both in breathing and paralyzed cats , and und er white-noise and sinusoidal pressure forcing. Thus we can study four different systems under two conditions of input fo r c ing . In the following we summarize the obtained results, which in some cases are preliminary. Lack of space severely limits the amount of graphical results that can be presented here. Complete numerical and graphical results will be presented in future publications and we only hope that this brief presentation will provide preliminary evidence sufficient to instigate some intere st among our readers.

Tracheal Pressure-to-Flow System In the breathing animal, low-power white noise and sinusoidal data confirmed the findings of . previous invest igato rs, when stationary analys~s

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was performed. Cohere nce was low (-0.4) only at the low-frequency end (i.e., l ess than 1 Hz), apparentl y due to the presence of breathing and the resulting nonstationarities. Nonlinearities (primarily due to turbulence) appeared to have less of an effect reducing coherence to between 0 . 8 and 0.9 in the frequen cy ranges of 1-4 Hz and above 15 Hz. In the range between 4 and 15 Hz, the results of linear stationary analysis were highly coherent (0.9-1.0) When the power level of the input white noise was increased, coherence was decreasing over the entire fr equency range, and more drast ically at the high frequencies (above 1 0 Hz). The estimation of Wiener kernels in the time domain (through c r osscorre lation) was complicated by the fact that the system has two vastly different time-constants (separated by almost two orders of magnitude) , which increased substantially the compu tati ona l burden. Frequency-domain evaluation of the crosscorrelation functions (and kernels) is thus recommended in this case . F irst, second and third order Wiener kernels were computed from the high power wh ite -noise data. The ac cu ra cy of the obtained estimates was less than satisfactory (120 sec. long input/output r ecords were used, and two different white-noise bandwidths of 25 and 50 Hz) suggesting low signa l-to - noise ratios in our data. Nonetheless, the obtai ned estimates clearly indicated the p resence of substantial odd-order nonlinearities by virtue of the computed first and third order kernels. The computed second orde r kernel was not as significant. Nonstationary analysis yie lded substantial dynamical components (kernel expansion components -- see Eq. (9» associated with frequencies compatibl e (in the context of a Fourier expans ion) with the time - variation of the breathing pa tte r n . The accuracy of these estimates was limited , although suf ficient to manifest the presence of nonstationary components in the system kernels. Breathing irregularities may account for part of these estimation inaccuracies , and we intend to check that in the paral yzed animal under the simultaneous application of the me c hanical respirator and white-noise perturbation. This experimental arrangement will allow the necessary control ove r the critical parameters that affect the accura cy of the nonstationary analysis. In the paralyzed animal, only high - power white-noise (or sinusoidal) forcing can maintain ventilation, therefore the system is operating c learly in the non li near region and is void of nonstationarities . Linear analysi s y ielded coherence measurements that were uniformly low, and became cons iderabl y lower above 1 0 Hz. However, the l ow-fre quency range (i.e., 0-1 Hz) gave coherence values comparable to the 1-10 Hz range (which indirectly supports our previous assertion in the normal breathing data). The computed high-order kernels were more accurate than before, due to the greater nonlinear

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Z. Marmarelis and S. M. Yamashiro

contribution in the system response. The second order kernel remained insignificant compared with the third order kernel. Again, the measured odd-order nonlinearity reflects the presence of turbulence in the central airways. In terms of model (1), this implies that the resistance R is an even function of V, which is also the mathematical translation of turbulence in the generated flow.

Tracheal Flow-to-C02 Tension System

Fig. 2:

Cuts of estimated third order kernel at T3=0.06 s (left) and T3 =0.12 s (right), drawn as contour maps in (Tl,T2) •

The qualitative findings for this system were similar to those described in the previous system. However, there were remarkable quantitative differences in the form and accuracy of the obREFERENCES tained kernel estimates. The accuracy here (for Barrett, J.F. (1963). J.Elec & Contr . .!.2, the same record lengths) was much higher and 567-615. time-domaih analysis appeared more suitable. Billings, S.A. (1980). lEE PrOC. Pt. D., 127, The second order kernel was again negligible 272-285. compared to the third order one, and the coBohn, D.J., K. Miyesaka, B.E. Marchak, W.K. herence followed similar patterns. The first Thompson, A.B. Froese, and A.C. Bryan and third order kernels are shown in figures (1980). J.Appl.Physiol., 48, 710-716. 1 and 2 below. These kernel estimates elucidate Bose, A.G. (1956). Tech.Rep.309, RLE, MIT. the role of turbulence in high frequency ventiKorenberg, M.J. (1973). Proc.JACC, 597-603. lation, and allow optimization and control Lee, Y.W. (1964). In Selected Papers of N. studies of the associated clinical procedures. Wiener, M~T Press, pp.17-33. Lee Y.W. and M. Schetzen (1965). Int.J.Contr., 3., 237-254. CONCLUSIONS Marmarelis, P.Z., and V.Z. Marmarelis (1978). Effective nonparametric analysis of the respirAnalysis of Physiological Systems: The atory system is possible in the context of White-Noise Approach. Plenum. New York. Wiener's theory, using white-noise test inputs. Marmarelis, P.Z., and K.I. Naka (1974). The resulting nonlinear models (obtained from J. Neurophysiol., 36, 605-648. initial experiments) appear to confirm the preMarmarelis, V.Z. (1980). Proc. Int. Symp. on ponderance of turbulence in the nonlinear charCircuits & Systems, Houston, 448-452. acteristics of the system. Nonstationary Marmarelis, V.Z. (1981). Proc. IEEE, 69, analysis also appears feasible, opening the 841-842. possibility of effective nonlinear analysis Marmarelis, V.Z. (1981). lEE Proe. Pt. D, during normal breathing. The use of random 128, 211-214. (White-noise) forcing allows the study of the McCann, G.D., and P.Z. Marmarelis(eds.) (1975) dynamics of gas exchange, thus increasing our Proc. 1st Symp. Test. & Ident. of Nonlin. understanding of functional aspects of high-freSystems, Caltech, Pasadena. quency ventilation (HFV) and making possible the Mi c haelson, E.D., E.D. Grassman and W.R. optimization of related clinical procedures. The Pete rs (1975). J. Clin. Invest., 56,1210-30. experimental results are preliminary but corSchetzen, M. (198 0 ). The Volterra a~Wiener roborate the efficacy of the proposed methodology. Theories of Nonlinear Systems. Wiley. More work is needed to consolidate our initial Slutsky, A.S., R. Brown, J. Lehr, T. Rossing findings and broaden the scope of their interand J.M. Trazen (1981). Med. Instrum., 15, pretation. 229-233. Stark, L. (1969). Automatica, 5, 655-676. Watanabe, A., and L. Stark (1975). Math. Biosci., ~, 99-1 08 . Wiener, N. (1958). Nonlinear Problems in Random Theory. Wiley, New York. Wi11iams, S.P., J.M. Fullton, M.J. Tsai, T R.L. Pimmel and A.M. Collier (1979). J. Appl. Physiol., 47, 169-174.

--------100 msec

Fig. 1:

Estimated first order kernel of tracheal flow-to-C02 tension system.

Acknowledgement s : The authors wish to thank 11rs. S. Ghazanshahi for laboratory assistance. This work was supporte d b y Nl GMS Grant number GM23732.