Mathematical Models of Physiological Responses to Exercise

Transcription

1 Mathematical Models of Physiological Responses to Exercise Somayeh Sojodi, Benjamin Recht, and John C. Doyle Abstract This paper is concerned with the identification of mathematical models for physiological responses to exercise. We first find single-inpt single-otpt models describing the responses of heart rate (HR), ventilation (V E), oxygen (O) and carbon dioxide (CO) to workload (W ) and then identify a single-inpt mlti-otpt model from W to HR, V E, O and CO. To this end, a novel method is proposed to identify a Hammerstein system from measrement data. We then investigate the possibility of the existence of a niversal model for physiological variability in different individals dring treadmill rnning. Simlations based on real data sbstantiate that the proposed methods are able to obtain accrate models of physiological response to workload variation. In particlar, it is observed that (i) different physiological responses to exercise can be captred by low-order linear or mildly nonlinear models; and (ii) there may exist a niversal model for O that works for different individals. I. INTRODUCTION The area of system identification deals with bilding mathematical models for physical processes based on experimental measrements [] []. Several methods and algorithms have been developed in the literatre for the identification of a linear time-invariant (LTI) system in both time and freqency domains [], [6]. However, nonlinear system identification is a more difficlt problem and there are many challenging isses that are yet nsolved [7], [8]. If the strctre of the nonlinear system to be identified is known a priori, the identification process amonts to a parameter estimation problem. The main difficlty associated with a nonlinear parameter estimation problem is the inability to find a globally optimal soltion in general. It is often the case that no a prior information is available on the strctre of the system being identified. To address this problem, different partially-sccessfl methods have been proposed in the literatre, which are based on Volterra series, Wiener series, or basis fnction representation of an approximate model of the system [9], []. The existing system identification methods have been extensively sed to find biological models. For example, the identification of genetic networks sing a convex optimization method is stdied in [], or the fast orthogonal search method is deployed in [] to analyze gene expression sojodi@berkeley.ed, brecht@berkeley.ed, and doyle@caltech.ed Somayeh Sojodi is with the Departments of Electrical Engineering and Compter Sciences and Mechanical Engineering, University of California, Berkeley. Benjamin Recht is with the Department of Electrical Engineering and Compter Sciences, University of California, Berkeley. John C. Doyle is with the Department of Control and Dynamical systems, California Institte of Technology. profiles. Linear and nonlinear identifications of the cardiovasclar system are treated in [3] and [], which are beneficial for monitoring the physiological state of a patient with cardiac disease or for the evalation of cardiac fnction. The identification of physiological variability dring exercise is another important problem whose goal is to obtain simple physiological models. Sch models can be sed for a wide variety of biomedical applications, sch as the prediction of the most appropriate changes in treatment, diet, or exercise program. In particlar, the model identification of heart rate variability, oxygen consmption, etc., dring exercise has been the center of attention in several researches []- [6]. For instance, the paper [] models the heart rate response dring and after treadmill walking exercise as a feedback interconnected nonlinear system. The work [7] also stdies the feasibility of atomatic feedback control of oxygen ptake dring treadmill exercise, which reqires the parameter identification of both the plant and the feedback controller. By combining system identification with basic physiological models, paper [8] shows that robst efficiency and actator satration tradeoffs can explain healthy hman heart rate variability. The primary objective of this work is to introdce a simple, yet practical, system identification method that can be sed to find an appropriate model for the physiological variability in response to exercise. We show that most of the physiological variabilities can be modeled by low-order linear (or mildly nonlinear) dynamical systems. The proposed algorithms are based on iterative techniqes, and cold in principle converge to a sb-optimal soltion that is a non-global local soltion or a saddle point. However, or simlations on real data confirm that these algorithms indeed converge to satisfactory soltions. This cold, in part, be jstified by the reslts of the recent paper [9] that shows that gradient descent efficiently converges to a global optimizer of the maximm likelihood objective of an nknown single-inpt single-otpt system. The remainder of the paper is organized as follows. Theoretical reslts of this work are developed in Section II. The proposed method is tested on a nmber of real experiments in Section III, and several important observations are made accordingly. Conclding remarks are drawn in Section IV. II. MATHEMATICAL MODEL OF EXERCISE The objective is to find a mathematical model describing the physiological response to different levels of exercise intensity. It is assmed that this model takes workload (W ) as the inpt and generates for otpts, namely heart rate (HR), ventilation (V E), oxygen (O) and difference of

2 carbon dioxide and oxygen (CO O). This paper proposes a system identification method to find a satisfactory model of exercise based on the data collected from biking or rnning on treadmill. One of the main challenges associated with the nderlying problem is that the strctre of the nderlying system from W to HR, V E, O, CO O is not known a priori, and indeed this system may be highly nonlinear. To simplify the problem, we first restrict or attention to the system identification of linear single-inpt single-otpt (SISO) systems, which can be sed to identify the best model from W to each of the otpts HR, V E, and O. The method will later be generalized to the system identification of nonlinear single-inpt mlti-otpt (SIMO) models, as the objective of the present work demands. A secondary goal of this paper is to stdy the possibility of the existence of a niversal model that fits the physiological variability of different individals. A. System identification of linear SISO systems Given a positive discrete time instant τ and two seqences of scalar nmbers û := {û[k]} τ k= and ŷ := {ŷ[k]}τ k=, it is desired to obtain a discrete-time SISO system with the known order n N sch that when the inpt û is applied to the system, its otpt on the time interval [, τ] becomes as closely to ŷ as possible. The model being soght is assmed to be of the form x[k + ] = Ax[k] + B[k] y[k] = Cx[k] + d, k =,,,... where x[k] R n, [k] R and y[k] R denote the state, inpt and otpt of the system, respectively, and d R is an offset term. The nknown parameters of this system are A, B, C, d and x[]. Note that the nmber of scalar parameters involved in the system identification problem is eqal to (n+) +n in general. However, one can realize the system in a canonical form to redce this nmber to 3n +. More precisely, assme with no loss of generality that A = a... a a n... + I () (a) C = [... ] (b) where a, a,..., a n are some nknown parameters to be fond. It is important to note that a variant of the observability canonical form is considered here for the pair (A, C), with the difference that the matrix A has an extra term I whose role will become clear later in this work. For the sake of the system identification, define a cost fnction as follows: τ g(y, ŷ, û; A, B, d, x[]) := y[k] ŷ[k] (3) k= where denotes the matrix -norm operator and y := {y[k]} τ k=. Note that this cost fnction measres the discrepancy between the experimental data ŷ and the system s otpt y when the inpt û is applied to the system. The goal is to minimize the cost fnction g(y, ŷ, û; A, B, d, x[]) for the parameters A, B, d, x[]. To this end, one can write: y[] C y[] CA y[] = CA x[] + d... y[τ] CA τ û[] CB û[] + CAB CB û[] CA τ B CA τ B CB û[τ ] () It can be conclded from this eqality that the minimization of g(y, ŷ, û; A, B, d, x[]) may not be a convex problem de to the existence of the prodct term AB. In what follows, a local search method is proposed, which will be later shown to work satisfactorily for the physiology application. The key idea is that the minimization of g(y, ŷ, û; A, B, d, x[]) amonts to a simple least-sqare problem provided the matrix A is known (in light of the above relation). Hence, the following descent algorithm can be sed to find a sboptimal set of parameters A, B, d, x[]. Algorithm : Step ) Pick an initial Schr matrix A () R n n in the form of (a), and set i =. Step ) Solve a least-sqares problem to minimize the cost fnction g(y, ŷ, û; A (i ), B, d, x[]). Denote the globally optimal vales of B, d and x[] with B (i), d (i) and x[] (i), respectively. Step 3) Use a local search algorithm to minimize g(y, ŷ, û; A, B (i), d (i), x[] (i) ) for a Schr matrix A in the form of (a). Denote a locally optimal vale of A with A (i). Step ) If A (i) A (i ) + B (i) B (i ) + d (i) d (i ) + x[] (i) x[] (i ) > ε for a given positive tolerance ε, increase i by and then jmp to Step. Step ) The system () with the parameters A = A (i), B = B (i), d = d (i) and x[] = x[] (i) is a sb-optimal SISO model for the inpt-otpt data sets û and ŷ. B. System identification of a nonlinear SISO system Since the nknown real system associated with the inptotpt data sets û and ŷ may be a nonlinear system, finding the best linear model fitting the data can be potentially too conservative. It is known that every LTI discrete-time system with fading memory can be approximated arbitrarily precisely by an LTI model followed by a static nonlinear part. Sch a nonlinear system is referred to as Weiner system. This implies that althogh the space of nonlinear fading-memory systems is tterly complex, the nicely characterizable set of Weiner systems is dense in this space []. Yet, it is still difficlt to find the best Weiner system associated with the inpt-otpt seqences û and ŷ.

3 This work aims to deal with the class of Hammerstein systems, where each system is composed of a static nonlinear part on the inpt followed by an LTI model []. Althogh the class of Hammerstein systems is a conterpart for the class of Weiner systems, no reslt has been reported in the literatre to date abot the denseness of this class in the space of nonlinear fading-memory systems. Hence, the best Hammerstein model for fitting the data sets û and ŷ may not be able to approximate the real-world system satisfactorily. In what follows, we will discss how to modify Algorithm to find a locally optimal Hammerstein model. It will be later verified on real experimental data that the physiological responses to exercise can be modeled by Hemmerstein systems satisfactorily. Assme that the nknown SISO system being fond for the inpt-otpt data sets û and ŷ is reqired to be of the form x[k + ] = Ax[k] + f([k]) () y[k] = Cx[k] + d, k =,,,... as opposed to the affine form (), where f( ) : R R n is an nknown fnction to be identified. The space of the nknown parameters of this system is infinite dimensional (de to the presence of f()), which is an obstacle to the nderlying system identification problem. To alleviate this isse, one can search for the best f() that is representable as a linear combination of a set of pre-specified basis fnctions. Given the basis fnctions f (), f (),..., f µ (), we write f() as B f(), where f() = [ f () f () f µ () ] T and B R n µ is the matrix of nknown coefficients. The data fitting problem now amonts to finding the best model in the form of x[k + ] = Ax[k] + B y[k] = cx[k] + d, k =,,,... for the inpt-otpt seqences { f(û[k])} τ k= and {ŷ[k]}τ k=. One can adopt Algorithm to find a sb-optimal set of the parameters A, B, d, x[]. C. System identification of a niversal nonlinear SISO system Given a natral nmber p, consider p inpt-otpt seqences û j := {û j [k]} τ k= and ŷ j := {ŷ j [k]} τ k=, j =,,..., p. The objective of this part is to find a matrix A R n n, a fnction f : R R n, scalars d, d,.., d p and vectors x [], x [],..., x p [] R n for which the otpt of the system x[k + ] = Ax[k] + f([k]) y[k] = cx[k] + d j, k =,,,... with the initial state x[] = x j [] in response to the inpt û j is as closely as possible to ŷ j over the time period [, τ] for every j {,,..., p}. The primary application of this problem is in finding a niversal model for different athletes doing exercise. To address this problem, the first step is (6) (7) (8) to make the parameters space finite dimensional sing the aforementioned basis fnction techniqe. Ths, a model in the form of (7) is soght. A variant of Algorithm can now be sed for this prpose. Algorithm : Step ) Select an initial Schr matrix A () R n n in the form of (a), and set i =. Step ) Solve a least-sqares problem to minimize the cost fnction p g(y, ŷ j, û j ; A (i ), B, d j, x j []) (9) j= Denote the globally optimal vales of the relevant parameters with B (i), d (i) j and x j [] (i), for j =,,..., p. Step 3) Use a local search algorithm (sch as a gradient method) to minimize: p g(y, ŷ j, û j ; A, B (i), d (i) j, x j[] (i) ) () j= for a Schr matrix A in the form of (a). Denote a locally optimal vale of A with A (i). Step ) If A (i) A (i ) + B (i) B (i ) + p j= d(i) j d(i ) j + p j= x j[] (i) x j [] (i ) > ε for a given positive tolerance ε, increase i by and then jmp to Step. Step ) The system (8) with the parameters A = A (i), f() = B (i) f(), d j = d (i) j and x[] = x j [] (i) is a sb-optimal SISO model for the inpt-otpt data sets û j and ŷ j, for every j {,,..., p}. The analysis provided in [9] can be sed to partially jstify the convergence of the above algorithm to a global soltion. D. System identification of a linear SIMO system Given a natral nmber r, consider an inpt seqence û := {û[k]} τ k= and r otpt seqences ŷ j := {ŷ j [k]} τ k=, j =,,..., r. Define: ŷ[k] = [ ŷ [k] ŷ [k] ŷ r [k] ] T, k =,,,... () and ŷ := {ŷ[k]} τ k=. The objective is to find a single-inpt r-otpt system of the form: x[k + ] = Ax[k] + B[k] y[k] = Cx[k] + d, k =,,,... () sch that y := {y[k]} τ k= is as closely as possible to ŷ if the inpt û is applied to the system. The nknown parameters of the system identification problem are A R n n, B R n, C R r n and x[] R n. Algorithms and provided in the preceding sbsections were based on the fact that finding the optimal vales of the parameters is a leastsqares problem if the matrix A is known. However, this idea breaks down for the identification of a SIMO system. More precisely, there does not exist an observable canonical form for a SIMO system to start with. Since the nknown system being identified has a single inpt, one may realize

4 it in the controllable canonical form. Nonetheless, this does not help becase the corresponding optimization problem is not convex in that case even if A is known a priori. A new techniqe will be introdced in the seqel to resolve this isse. Find r SISO systems: x j [k + ] = Ax j [k] + B j [k] y j [k] = [ ] x j [k] + d j, k =,,... (3) (where j =,,..., r) with the parameters A R n n, B,..., B r R n, d,..., d r R and x [],..., x r [] R n sch that the following cost fnction is locally minimized: r g(y j, ŷ j, û; A, B j, d j, x j []) () j= where y j := {y j [k]} τ k=. Algorithm can be easily modified to find a sb-optimal soltion of this problem. Ths, r SISO systems can be obtained for the r individal inpt-otpt data sets and ŷ j, j =,,..., r, so that they all have the same matrix A. The next step is to combine these SISO systems to obtain a single-inpt r-otpt system corresponding to the inpt-otpt data sets and ŷ. To this end, we form a parallel connection of these r systems as follows: where: x[k + ] = Ã x[k] + B[k] y[k] = C x[k] + d () Ã R rn rn is a block diagonal matrix with the (j, j) th block A for every j {,,..., r}. B R nr is a colmn vector obtained from B, B,..., B r by stacking them p. C R r rn is a block diagonal matrix with the (j, j) th block entry [ ] for every j {,,..., r}. d R r is a colmn vector obtained from d, d,..., d r by stacking them p. The obtained system () is a near-optimal model for the data sets and ŷ, bt its order is nr rather than the desired vale n. De to the particlar strctre of the matrix Ã, it can be observed that the system (Ã, b, C) is ncontrollable and indeed it has a realization (A, B, C) where A R n n, B R n and C R r n. So far, a single-inpt r-otpt system of order n is obtained whose transfer fnction from its inpt to its j th otpt is eqal to the transfer fnction of the j th SISO system (3) fond earlier, for every j {,,..., r}. The only isse that may case a problem is that the r separate initial states x [], x [],..., x r [] obtained by solving r SISO system identification problems cannot always be incorporated into a single initial state for the redced-order model. III. SIMULATION RESULTS This section targets the identification of appropriate mathematical models for the physiological responses to exercise. The algorithms proposed in the previos section will be deployed to find satisfactory models from the data gathered from for different experiments of rnning on treadmill. The inpt watts corresponding to these experiments are given in Figre. The first goal is to identify a model of oxygen variability ( V O ) with respect to the workload intensity (W ). Algorithm is sed to stdy the possibility of the existence of an accrate first-order linear model for each of these experiments. Figre shows the otpt of the first-order linear model of VO for Experiment. It can be observed that the model obtained by the proposed heristic method fits the data very well. An important implication of this reslt is that the V O response to W is governed by a simple firstorder differential eqation (this phenomenon was observed for other experiments as well). The V O signals for Experiments to are very similar in natre. Hence, one may speclate that there exists a niversal model for the response of VO to W. To stdy this conjectre, we first se Algorithm to find three separate first-order linear models corresponding to V O for Experiments to 3 and then deploy Algorithm to obtain a single niversal first-order linear model. The corresponding plots are given in Figre 3, which clearly demonstrate that a simple system obtained by Algorithm for VO works satisfactorily for different experiments. In contrast to the V O signal, the experimental data sggests that the ventilation variability ( V V E ) is more complex and probably more individal. For each of the nderlying experiments, Algorithm failed to find a satisfactory firstorder linear system modeling the V V E response to W. This can be de to the nonexistence of sch a model. Nonetheless, an acceptable second-order linear model can always be fond for Experiments to. For Experiment, the model s otpt is compared to the experimental data in Figre. The plots clearly show that the response of V V E to W can be modeled via a second-order linear system. The next qantity of interest is the difference between CO prodction and O consmption, i.e., the CO O variability. Or stdy on a nmber of different data sets reveals that VCO O is far more complex than V O, V V E, V HR and, therefore, its governing model is nlikely to be a low-order linear system. The best second-order linear model fitting the V CO O data is compared with the real data in Figre for Experiment to demonstrate the poor performance of a linear system. To find a nonlinear model for the same experiment, define the following basis fnctions: where: f i () = max(, τ i ), i =,, 3 (6) τ =, τ =, τ 3 = 9 (7) Using only three simple basis fnctions as given above, the method proposed here is sed to find a second-order nonlinear system modeling the V CO O response to W. The corresponding plots are given in Figre 6 to show how accrately the fits describe the data. The signals g () and g () in the figre are the first and second entries of the nonlinear vector fnction f(). Note that these nonlinear fnctions are

5 Watt (a) Watt (b) Watt (c) Watt Fig. : (a)-(d): The inpt signals for Experiments -. (d) VO Fig. : The otpt of the first-order linear model corresponding to V O for Experiment. very simple and a low-order system with mild nonlinearities can model the nderlying response satisfactorily. We noticed that a sperior fitting can be attained by considering more basis fnctions, which leads to a complicated nonlinearity on the inpt. It is desirable to find an integrated single-inpt for-otpt system modeling all responses V O, V CO, V V E, and V HR to W. By applying the method proposed in Sbsection D to Experiment, it can be observed that there exists a secondorder linear model captring these for individal responses. The corresponding plots are given in Figre 7 to illstrate how well a low-order linear model can predict for different qantities VO, V CO, V V E and V HR from W. Similarly, there exist integrated linear single-inpt for-otpt systems of order for other experiments, and Figre 8 shows the reslts obtained for experiment. As stated earlier, the V HR response to W can be modeled by a low-order linear system, whereas the V CO O response to W reqires a nonlinear system. The qestion arises as to whether the models for VHR and V CO O can be governed by the same dynamics (i.e., an identical A matrix). Algorithm is sed for Experiment to find the best thirdorder nonlinear models for VHR and V CO O when their dynamics are forced to be identical. The models signals are compared with experimental signals in Figre 9. The fnctions g (), g (), g 3 () and ḡ (), ḡ (), ḡ 3 () in the figre denote the inpt nonlinearities of the models for V HR and V CO O, respectively. Remark : It is expected that a physiological model corresponding to each of the signals V HR, VO, VCO, and V V E has a non-decreasing inpt fnction f(). This implies that the coefficient matrix B being fond shold be constrained to have only nonnegative entries. This constraint can be easily incorporated into Algorithm or Algorithm by imposing to solve a nonnegative least-sqares problem as opposed to a standard least-sqares problem. The corresponding simlations given above are all obtained based on imposing a nonnegativity constraint on B. Remark : It can be observed in Figres,, and 7 that the experimental data is simply a line over three distinct time intervals. The reason is that the data cold not be measred in those intervals and therefore a simple line interpolation method was sed to predict the missing data. Hence, the model s otpt may not fit the experimental data very well on these intervals as the nmeasred real data is not expected to be a line. IV. CONCLUSIONS This work introdces a system identification method to find mathematical models for the physiological responses to exercise. In particlar, the variability of heart rate (HR), ventilation (V E), oxygen (O) and carbon dioxide (CO) to workload intensity (W ) is stdied. Based on simlations performed on different experimental data, we observed that: i) the response of each of the signals HR, V E, O and CO to W can be modeled by either a first- or a second-order

6 VO Common fit Data Best fit 3 6 VO VO Fig. 3: A niversal first-order linear model corresponding to V O for Experiments, and 3 is compared with three individal first-order linear models for these experiments. VV E Fig. : The otpt of the second-order linear model corresponding to V V E for Experiment. VCO O g() g() x 3 3 Fig. 6: The relevant signals of the second-order nonlinear model corresponding to V CO O for Experiment. VCO O Fig. : The otpt of the second-order linear model corresponding to V CO O for Experiment. linear system; ii) there exists a single (niversal) first-order linear model for the O variability of different individals; iii) the effect of W on all of the signals HR, V E, O and CO can be captred by a single-inpt for-otpt second-order linear model. Besides, this stdy shows that the response from W to O CO cannot be modeled by a first- or second-order system nless mild nonlinearities are incorporated in the inpt of the system. REFERENCES [] T. Söderström and P. Stoica, System identification, 989. [] L. Ljng, System identification. theory for the ser. prentice, 987. [3] G. Pillonetto, F. Dinzzo, T. Chen, G. De Nicolao, and L. Ljng, Kernel methods in system identification, machine learning and fnction estimation: A srvey, Atomatica, vol., no. 3, pp ,. [] P. Eykhoff, Trends and progress in system identification: IFAC Series for Gradates, Research Workers & Practising Engineers. Elsevier,, vol.. [] R. Pintelon and J. Schokens, System identification: a freqency domain approach. John Wiley & Sons,. [6] T. Katayama, Sbspace methods for system identification. Springer Science & Bsiness Media, 6. [7] O. Nelles, Nonlinear system identification.. [8] J. Sjöberg, Q. Zhang, L. Ljng, A. Benveniste, B. Delyon, P.-Y. Glorennec, H. Hjalmarsson, and A. Jditsky, Nonlinear black-box modeling in system identification: a nified overview, Atomatica, vol. 3, no., pp. 69 7, 99. [9] W. J. Rgh, Nonlinear system theory. Johns Hopkins University Press Baltimore, 98. [] K. Li, J.-X. Peng, and G. W. Irwin, A fast nonlinear model identification method, IEEE Transactions on Atomatic Control, vol., no. 8, pp. 6,. [] A. Jlis, M. Zavlanos, S. Boyd, and G. J. Pappas, Genetic network identification sing convex programming, IET systems biology, vol. 3, no. 3, pp. 66, 9. [] M. J. Korenberg, Applications of nonlinear system identification in moleclar biology, in Engineering in Medicine and Biology Society, 6. EMBS 6. 8th Annal International Conference of the IEEE. IEEE, 6, pp. 6 9.

7 VHR th order model fit nd order model fit 6 8 VO 6 8 Time(sec) VCO VV E Fig. 7: The otpts of the integrated second-order linear model for Experiment. VHR VCO 6 nd order model fit VO VV E 6 Fig. 8: The otpts of the integrated second-order linear model for Experiment. VHR g() g() g3() 3.. x VCO O ḡ() ḡ() ḡ3().. 3 x 3 x 3 3 Fig. 9: Different signals for the third-order nonlinear models corresponding to V HR and V CO O for Experiment.

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