INTRODUCTION

TOP ABSTRACT INTRODUCTION CARDIOVASCULAR SYSTEM... FORWARD MODEL DESCRIPTION FORWARD MODEL-BASED EVALUATION... FORWARD MODEL VALIDATION FORWARD MODEL-BASED EVALUATION DISCUSSION APPENDIX A APPENDIX B APPENDIX C REFERENCES

WHEN ONE CONSIDERS MODELING the cardiovascular system, one usually envisions constructing a model based on physical principles that is capable of generating realistic data. This type of modeling approach, which we refer to as forward modeling, is useful for enhancing one's understanding of cardiovascular physiology. One may also consider an inverse modeling approach, in which models are built from measured data. This type of modeling approach is referred to as system identification when system dynamics or memory is being considered. System identification may potentially provide a powerful means for the intelligent patient monitoring of cardiovascular function. Rather than simply recording hemodynamic signals, the signals are mathematically analyzed so as to provide a dynamical characterization of the physiological mechanisms responsible for coupling them.

To this end, we (26, 27, 29) have previously developed a cardiovascular system identification method for the analysis of short-term, beat-to-beat fluctuations in heart rate signals derived from the surface electrocardiogram (ECG) and noninvasively measured arterial blood pressure (ABP) and instantaneous lung volume (ILV) signals. The method provides a dynamical characterization of the important autonomic and mechanical mechanisms that couple the measured fluctuations specifically in terms of impulse responses. The method also provides power spectra of perturbing noise sources, which represent the residual variability in each of the signals not attributed to the variability generated through the impulse responses.

We have evaluated the cardiovascular system identification method with experimental data collected during conditions of pharmacological autonomic blockade, postural changes, and diabetic autonomic neuropathy (27, 29). We found that these three conditions altered the impulse response and power spectra of perturbing noise source estimates in a manner consistent with known physiological mechanisms. This suggests, but does not directly demonstrate, the validity of the estimated system dynamics.

To evaluate directly the system dynamics estimated by the method, it is necessary to be able to establish the impulse responses and power spectra of perturbing noise sources in a manner independent of system identification. These impulse responses and power spectra of perturbing noise sources may then be regarded as the gold standards against which the corresponding estimates from system identification may be compared. Ideally, one would validate the method based on experimental data. However, the establishment of gold standard impulse responses and power spectra of perturbing noise sources would require extreme experimental conditions that would be virtually impossible to implement in practice. Consider, for example, the establishment of an impulse response, which would require, according to mathematical definition, applying an arbitrarily narrow, unit-area input to the appropriate point of the cardiovascular system and measuring the output response of interest while all other perturbations to the output are held constant. Moreover, even if this type of experiment could be implemented, it is unreasonable to expect that the cardiovascular state and system operating point would be precisely maintained, which renders the meaning of such a comparison to be questionable.

By contrast, the method could be readily evaluated on a theoretical basis with a forward model of the cardiovascular system, because the gold standard impulse responses and power spectra of perturbing noise sources would either be precisely known or easily determined for the relevant cardiovascular state and system operating point. That is, the gold standards could be readily established according to their mathematical definitions. A forward model would also provide a powerful means to analyze the sensitivity of the cardiovascular system identification method. That is, we would be able to determine precisely how much the dynamical properties of the forward model would have to be altered before we would see a corresponding change in the estimates. The major limitation of this type of evaluation is that the results would be only as meaningful as the extent to which the forward model coincides with the actual cardiovascular system. This limitation can be attenuated by determining the robustness of the estimates over a set of models that reflect the uncertainty in the relevant properties of the forward model. We note that the general concept of analyzing inverse modeling algorithms based on forward models of the cardiovascular system is not novel. For example, investigators (9, 16) have utilized complex forward models of the systemic circulation to evaluate the estimation of lumped parameters representing systemic arterial resistance [R_a(t)] and systemic arterial compliance.

The principal goal of this study is to present a theoretical validation of our previously developed cardiovascular system identification method. To realize this aim, this study includes the following five components: 1) a brief review of the cardiovascular system identification method, which has been previously described in detail (26, 27, 29); 2) a description of a forward model of the human cardiovascular system; 3) the procedure for evaluating the system identification method against the forward model, which includes the establishment of gold standards based on their mathematical definitions; 4) the validation of the forward model in terms of power spectra of beat-to-beat variability; and 5) the evaluation of the impulse response and power spectra of perturbing noise source estimates derived by applying system identification to beat-to-beat variability generated from the forward model against the corresponding gold standards in terms of accuracy, robustness, and sensitivity. Note that the final component amounts to assessing the equivalence of the impulse responses and power spectra of perturbing noise sources determined from executing the forward model under two conditions: 1) resting conditions by applying system identification to analyze beat-to-beat variability, and 2) extreme conditions established by the literal meaning of these quantities. Because the extreme conditions cannot be implemented in practice, this study seeks to determine whether the cardiovascular system identification method is able to estimate reliably the physiologically important impulse responses and power spectra of perturbing noise sources during experimental conditions that could be readily implementable.

	CARDIOVASCULAR SYSTEM IDENTIFICATION REVIEW

TOP ABSTRACT INTRODUCTION CARDIOVASCULAR SYSTEM... FORWARD MODEL DESCRIPTION FORWARD MODEL-BASED EVALUATION... FORWARD MODEL VALIDATION FORWARD MODEL-BASED EVALUATION DISCUSSION APPENDIX A APPENDIX B APPENDIX C REFERENCES

Figure 1 illustrates the model upon which the cardiovascular system identification method is based. The model includes five physiological coupling mechanisms relating ECG-derived heart rate signals, ABP, and ILV: circulatory mechanics, heart rate (HR) baroreflex, ILVHR, ILVABP, and sinoatrial (SA) node.

View larger version (21K): [in this window] [in a new window]   Fig. 1.   Mean (solid traces) and standard deviation (dotted traces) of cardiovascular system identification estimates obtained from a standing human subject breathing randomly. Figure adapted from raw data described in Ref. 34.

Circulatory mechanics represents the mechanical feedforward effects of a pulsatile heart rate (PHR) signal [defined to be a train of unit-area impulses occurring at the times of ventricular contraction (R wave)] on the pulsatile ABP waveform. HR baroreflex represents the autonomically mediated feedback effects of fluctuations in ABP on fluctuations in an HR tachogram, which is derived from the R-R intervals (3). ILVHR represents the autonomically mediated coupling between respiration and HR, which is responsible for governing respiratory sinus arrhythmia. ILVABP represents the mechanical effects of respiration on ABP due to the alterations in venous return and the filling of intrathoracic vessels and heart chambers associated with the changes in intrathoracic pressure. SA node maps HR (the net autonomic input to the sinoatrial node) to PHR (the onset times of each ventricular contraction) through an "integrate and fire" device referred to as the integral pulse frequency modulation (IPFM) model (3). Unlike the other four physiological coupling mechanisms, SA node is predefined and not identified from the measured signals.

The model also incorporates two perturbing noise sources, N_HR and N_ABP, which are determined from analysis of the measured signals. N_HR represents the fluctuations in HR not caused by fluctuations in ABP or ILV. Such fluctuations may result, for example, from autonomically mediated perturbations driven by cerebral activity. N_ABP represents fluctuations in ABP not caused by PHR or fluctuations in ILV. Such ABP fluctuations may result, for example, from fluctuations in R_a(t) as tissue beds adjust local vascular resistance to match local blood flow to demand.

To obtain a complete characterization of each of the coupling mechanisms over their physiological range of frequencies, we employ a broadband excitation protocol, in which subjects breathe according to a random sequence of auditory tones (4). We have found that when the subjects are breathing according to this protocol and are otherwise at rest that the fluctuations are sufficiently small and stationary such that the coupling mechanisms may be characterized by linear time-invariant (LTI) transfer functions around a given system operating point (11). We specifically represent each of the coupling mechanisms with autoregressive moving average difference equations, a highly flexible subclass of LTI models, whose parameters may be conveniently identified with the analytic methods of linear least-squares estimation. Further details of the method are provided in (26, 29).

Figure 1 includes an example of the cardiovascular system identification results computed from a standing, healthy subject breathing randomly, in which ABP and ILV are noninvasively measured with a continuous blood pressure monitor (model 2300 Finapres; Ohmeda; Englewood, CO) and a Respitrace System two-belt chest-abdomen inductance plethysmograph (Ambulatory Monitoring; Ardsley, NY) (34). The identified transfer functions are presented in their time domain form of impulse responses. For example, the HR baroreflex impulse response estimate indicates that HR would immediately decrease and then return to baseline if a unit-area impulse of ABP were applied at time 0 while the other inputs to HR (ILV and N_HR) were set to 0. The estimated perturbing noise sources in the figure are depicted in their frequency domain form of power spectra. For example, the power spectrum of N_HR indicates that ABP and ILV do not fully account for HR variability within ~0.05 Hz.

	FORWARD MODEL DESCRIPTION

TOP ABSTRACT INTRODUCTION CARDIOVASCULAR SYSTEM... FORWARD MODEL DESCRIPTION FORWARD MODEL-BASED EVALUATION... FORWARD MODEL VALIDATION FORWARD MODEL-BASED EVALUATION DISCUSSION APPENDIX A APPENDIX B APPENDIX C REFERENCES

The forward model of the human cardiovascular system to which we apply the cardiovascular system identification method includes three major components: a pulsatile heart and circulation, a short-term regulatory system, and resting physiological perturbations. The pulsatile heart and circulation is a nonlinear, lumped parameter model (R. Mukkamala, D. A. Sherman, R. G. Mark, and R. J. Cohen, unpublished observations). The short-term regulatory system consists of three subcomponents: an arterial baroreflex, a cardiopulmonary baroreflex, and a direct neural coupling mechanism between respiration and HR. The resting physiological perturbations also include three subcomponents: respiratory activity, the autoregulation of local vascular beds, and 1/f HR fluctuations. We describe all of these components and subcomponents and include parameter values below. See APPENDIX A for a description of forward model implementation.

Pulsatile Heart and Circulation

View larger version (12K): [in this window] [in a new window]   Fig. 2.   Electrical circuit analog of the human pulsatile heart and circulation model. Here, current is analogous to blood flow rate (, ml/s), whereas voltage is analogous to pressure (P, mmHg). The model includes 6 compartments representing the left and right ventricles (l and r), systemic arteries and veins (a and v), and pulmonary arteries and veins (pa and pv). Each compartment consists of a linear or nonlinear resistance (R) and a linear or nonlinear and time (t)-varying compliance (C) with an associated unstressed volume. Note that each nonlinear element is denoted with a box. The reference pressure is intrathoracic (th) pressure for the ventricular and pulmonary compartments. The 4 ideal diodes represent the ventricular inflow and outflow valves.

Short-Term Regulatory System

Arterial baroreflex. The short-term regulatory system includes a baroreflex system, which is based on a previously developed model (13). This model, which is fully described here, conceptualizes the arterial (A) baroreflex arc as the feedback system illustrated in the block diagram of Fig. 3. The feedback system is aimed at tracking a setpoint (sp) pressure through the following sequence of events. The baroreceptors sense P_a(t) and then relay this pressure via autonomic afferent fibers to the autonomic centers in the brain. Here, the autonomic nervous system (ANS) compares the deviation between the sensed pressure and P with zero and then sends commands via autonomic efferent fibers to adjust four parameters of the pulsatile heart and circulation such that the ensuing P_a(t) is near P. The four adjustable parameters are instantaneous HR [F(t)], ventricular contractility [in terms of C(t), the differential compliances that the ventricles assume at their unstressed volumes during end systole] (R. Mukkamala, unpublished communications), systemic venous unstressed volume [Q(t)], and R_a(t). The ANS specifically adjusts these parameters based on the history of P_a(t)  P according to the following nonlinear, dynamical map

(1)

where X may represent any of the four controllable parameters, the atan function (parameterized by the constant S) imposes arterial baroreflex saturation, p(t) and s(t) are unit-area effector mechanisms, which, respectively, represent the fast, parasympathetic limb of the ANS and the slower, sympathetic limb (see Fig. 4), and G_p and G_s reflect the respective weighting values of the effector mechanisms.

View larger version (7K): [in this window] [in a new window]   Fig. 3.   Block diagram of the feedback system depicting the arterial baroreflex arc. The feedback system is aimed at tracking a setpoint (sp) pressure through the following sequence of events. The baroreceptors sense Pa(t). The autonomic nervous system (ANS) compares the deviation between the sensed pressure and P with 0 and then responds by adjusting 4 parameters of the pulsatile heart and circulation to keep the resulting Pa(t) near P. The four adjustable parameters are instantaneous heart rate [F(t)], ventricular contractility [C(t)], systemic venous unstressed volume [Q(t)], and systemic arterial resistance [Ra(t)]. The ANS controls these parameters through a static saturation mapping, followed by convolution with the impulse responses in Fig. 4 (see Eq. 1).

View larger version (11K): [in this window] [in a new window]   Fig. 4.   Unit-area effector mechanisms representing the fast, parasympathetic limb p(t) (A) and slower, sympathetic limb s(t) (B). These effector mechanisms characterize the dynamical properties of the block labeled ANS in Fig. 3. The signals p(t) and s(t) are derived from experimental data in Ref. 5 (figure adapted from Ref. 13).

Cardiopulmonary baroreflex. The baroreflex system also includes a cardiopulmonary (CP) baroreflex arc, which is conceptualized with a feedback diagram analogous to Fig. 3. However, the sensed pressure here is defined to be the effective right atrial transmural pressure [P(t) = P_"ra"(t)P_th(t)] of the pulsatile heart and circulation model (see Fig. 2), where P is the effective right artrial transmural pressure, P_ra is right atrial pressure, and P_th is thoractic pressure.

Direct neural coupling mechanism between respiration and HR. The short-term regulatory system model also includes a direct neural coupling mechanism between respiration and HR. This coupling mechanism is implemented in terms of an LTI impulse response, which maps fluctuations in ILV to fluctuations in F(t). In accordance with previous studies (29, 40) with experimental data, the impulse response is defined here by a linear combination of s(t) and p(t), each of which are advanced in time by 1.5 s. The model ILV signal [Q_lu(t)] that is necessary for the implementation of this mechanism is described below.

Parameter values. The numerical values of the parameters characterizing the baroreflex system are taken from (13 and T. Heldt, E. B. Shim, R. D. Kamm, and R. G. Mark, unpublished observations) (see below for exceptions) and are provided in Table 1. Note that the cardiopulmonary baroreflex control of F(t) and c(t) are both neglected in the model due to either controversial data or a lack of available data. Although the - and -sympathetic sublimbs are both characterized by s(t), the table specifies the particular sublimb corresponding to each s(t). This permits the modeling of, for example, the effects of propranolol, which selectively blocks the -sympathetic sublimb.

Resting Physiological Perturbations

Respiratory activity. The resting physiological perturbations model includes both metronome (met) and random-interval respiratory activity represented in terms of Q_lu(t). Because ILV strongly resembles a sinusoid during metronomic breathing, Q_lu(t) is mathematically represented as follows

(2)

where Q_fr is the functional residual volume of the lungs, Q_t is the tidal volume, and T_r is the respiratory period. The model here may also be considered to represent ILV during normal, spontaneous breathing, which, at least to first approximation, appears to be sinusoidal as well.

(3)

(4)

View larger version (7K): [in this window] [in a new window]   Fig. 5.   Model of ventilatory mechanics in terms of its electrical circuit analog, in which charge is analogous to air volume (Q) and voltage is analogous to air pressure (P). The resistor (R) represents the airway (air) between the atmosphere (atm) and the lungs (lu), whereas the capacitor (C) represents the lung compartment. Pth(t) and Palv(t), intrathoracic and alveolar pressures, respectively.

Autoregulation of local vascular beds. Although respiratory-induced hemodynamic fluctuations are fairly well understood, the mechanisms responsible for hemodynamic variability at frequencies below ~0.1 Hz have not been adequately elucidated. It has been previously demonstrated that ABP fluctuations are not determined by HR fluctuations at these lower frequencies, and it has therefore been postulated that fluctuations in R_a(t) caused by the autoregulation of local vascular beds are responsible for these ABP fluctuations, which, in turn, induces HR fluctuations through the baroreflex (1, 24). However, little is known about the system dynamics characterizing autoregulatory processes except that these processes are relatively slow with a characteristic time constant from ~5 to 20 s (6). We therefore include into the model of resting physiological perturbations a zeroth-order model of autoregulatory processes. This model represents these processes as a stochastic, exogenous disturbance to R_a(t) [n_Ra(t)] defined by a Gaussian white noise process with zero mean and ² variance that is bandlimited to 0.1 Hz.

1/f HR fluctuations. Figure 1 demonstrates that ABP and ILV are not the only factors responsible for perturbing HR at frequencies within ~0.05 Hz. However, the other perturbing factors, which may include cerebral activity impinging on the ANS, are essentially unknown. It is well known that HR fluctuations demonstrate fractal behavior over at least four decades of frequency, in which the highest frequency decade is within the frequency band of interest here (7, 37). We therefore include in the resting physiological perturbations model an exogenous, stochastic 1/f disturbance to F(t) [n_F(t)] The disturbance is created by passing a zero-mean, Gaussian white noise process with variance ² through a cascade combination of two LTI filters. One filter is of unit-area and is designed to have a 1/f magnitude squared frequency response over four decades of frequency starting at 10⁴ Hz (12, 20, 36). The other filter is defined by a linear combination of s(t) (-sympathetic) and p(t) (with arbitrary weighting values of 12 beats/s each), because HR fluctuations are almost exclusively mediated by the ANS (2). Importantly, n_F(t) as well as n_Ra(t) are considered to be unmeasurable quantities in the forward model.

Parameter values. The parameter values characterizing the models of Q_lu(t) and ventilation are provided in Table 2. These values are taken from (21, 41).

1

	FORWARD MODEL-BASED EVALUATION PROCEDURE

TOP ABSTRACT INTRODUCTION CARDIOVASCULAR SYSTEM... FORWARD MODEL DESCRIPTION FORWARD MODEL-BASED EVALUATION... FORWARD MODEL VALIDATION FORWARD MODEL-BASED EVALUATION DISCUSSION APPENDIX A APPENDIX B APPENDIX C REFERENCES

Gold Standard Cardiovascular System Identification Results

We establish the gold standards according to the literal mathematical meaning of the impulse responses and the power spectra of the perturbing noise sources. In particular, the impulse response is defined to represent the output response of a system to an arbitrarily narrow, unit-area input. The impulse response also completely characterizes the input-output relationship of the physiological coupling mechanisms here because of our LTI assumption. Hence, in establishing each of the gold standard impulse responses, we apply an impulse input to the desired point in the forward model and measure the output response of interest while holding all other perturbations to the output constant. For example, to establish the gold standard ILVABP impulse response, we measure the ABP response to an impulse input of ILV while holding HR constant and setting the exogenous disturbance to R_a(t) to zero.

By definition, the perturbing noise source represents the residual variability in a measured output not due to the fluctuations in the other measured inputs. So, in establishing each of the gold standard power spectra of perturbing noise sources, we first measure the output of interest in the forward model while holding the two measured input fluctuations constant and then compute the power spectrum of the output. For example, to establish the gold standard power spectrum of N_ABP, we first measure ABP in the forward model while holding HR and ILV constant and then compute the power spectrum of the resulting ABP.

The specific procedures for establishing each of the gold standard impulse responses and power spectra of perturbing noise sources in Fig. 1 are described in APPENDIX B. This description includes the signal processing necessary for the estimated and gold standard impulse responses to be sampled identically.

Root-Normalized-Mean-Squared Error

_x

(5)

_x

Monte Carlo Simulations

	FORWARD MODEL VALIDATION

TOP ABSTRACT INTRODUCTION CARDIOVASCULAR SYSTEM... FORWARD MODEL DESCRIPTION FORWARD MODEL-BASED EVALUATION... FORWARD MODEL VALIDATION FORWARD MODEL-BASED EVALUATION DISCUSSION APPENDIX A APPENDIX B APPENDIX C REFERENCES

The forward model-based evaluation of the cardiovascular system identification method is only meaningful provided that the low-frequency power spectral content of the signals analyzed by the method is realistic. Because the forward model is tuned to represent what one may think of as an "average" subject, we initially considered demonstrating that the model spectra resemble the group average spectra of the 12 healthy humans breathing at a fixed rate of 0.25 Hz (see Parameter Values in Resting Physiological Perturbations). However, averaging tended to smear out the spectral peaks, which were usually not centered at the same frequencies for each subject. We therefore demonstrate that the model spectra resemble what one may find experimentally through a comparison with the spectra from one of the healthy subjects. This does not seem unreasonable if we keep in mind that the power in three nonoverlapping frequency bands of the model spectra quantitatively match that computed from the group average of the 12 subjects (see Table 3).

Figure 6 illustrates that the power spectra for ILV, HR, and ABP at frequencies below the mean HR obtained from the forward model during fixed-rate breathing excitation at 0.25 Hz indeed resemble spectra obtained from one of the 12 subjects. Differences at the respiratory frequency in Fig. 6 and in Table 3 may be attributed to the fact that the subject here did not precisely follow the fixed-rate breathing protocol as well as discrepancies in tidal volume, which was not measured. Note that although the forward model was tuned to represent the "average" subject, it could have been tuned to match more precisely the subject here.

View larger version (19K): [in this window] [in a new window]   Fig. 6.   Power spectra of instantaneous lung volume (ILV; normalized to unity power), HR, and arterial blood pressure (ABP) at frequencies below the mean HR from the forward model and a single, standing human during fixed-rate breathing at 0.25 Hz. The model spectra represent the average over 10 different realizations of forward model-generated data, in which alv = 70 ml/s.

Figure 6 also demonstrates that the model spectra exhibit a spectral peak at ~0.07 Hz, which is near the center frequency of the "posture" peak in humans (32). Because the exogenous disturbances within 0.1 Hz are broadband (see Autoregulation of Local Vascular Beds and 1/f HR Fluctuations), this peak implies a system resonance in the forward model. This result supports the simple computer simulation of de Boer (14), which demonstrated that the "posture" peak could be due to a system resonance. de Boer specifically implicated the system resonance to the arterial R_a(t) baroreflex arc. However, based on a few simple experiments, in which the weighting values of the open-loop effector mechanisms were varied, we have found that the resonance peak is substantially diminished in the absence of the arterial baroreflex feedback pathways responsible for controlling Q(t) as well as R_a(t). That is, the arterial baroreflex controlling -sympathetic activity in general is involved in eliciting the resonance peak.

It is also necessary to demonstrate that the cross spectra between each of these signals are realistic. This essentially amounts to demonstrating that the transfer functions, which characterize the couplings between the signals are consistent with those determined from experimental data. However, this has been virtually taken care of by the very construction of the model, which was based largely on physiological findings published in the literature.

Although it is only necessary for the purposes of this study that the low-frequency spectral content of ILV, HR, and ABP be reasonable, we assume that the low-frequency content of the remaining model signals are accounted for as well by virtue of reasonably representing these three signals. Figure 7 gives some credence to this assumption by illustrating that the model spectrum of left ventricular flow rate (LVFR) at frequencies below the mean HR resembles the corresponding spectrum measured with a Doppler ultrasound technique (17) from an individual subject breathing at a fixed rate of 0.25 Hz and tilted upright (30° with respect to the supine posture).

View larger version (16K): [in this window] [in a new window]   Fig. 7.   Power spectra of left ventricular flow rate (LVFR) at frequencies below the mean HR from the forward model and a single human tilted upright 30° with respect to the supine posture during fixed-rate breathing at 0.25 Hz. The model spectrum represents the average over 10 different realizations of forward model-generated data, in which alv = 70 ml/s.

	FORWARD MODEL-BASED EVALUATION

TOP ABSTRACT INTRODUCTION CARDIOVASCULAR SYSTEM... FORWARD MODEL DESCRIPTION FORWARD MODEL-BASED EVALUATION... FORWARD MODEL VALIDATION FORWARD MODEL-BASED EVALUATION DISCUSSION APPENDIX A APPENDIX B APPENDIX C REFERENCES

Accuracy Analysis

View larger version (21K): [in this window] [in a new window]   Fig. 8.   Mean (solid traces) and standard deviation (dotted traces) of cardiovascular system identification estimates along with their respective gold standards (dashed traces), which characterize the forward model.

Robustness Analysis

Cardiopulmonary HR baroreflex. We first consider how these estimates would be altered if a cardiopulmonary HR baroreflex were included in the forward model. Inclusion of this reflex may be achieved simply by adjusting its weighting values to nonzero values (see Table 1). To proceed with this analysis, the gold standard N_HR power spectrum, which is no longer valid in the presence of a cardiopulmonary HR baroreflex, must be redefined (see APPENDIX C for the specific procedure).

View larger version (15K): [in this window] [in a new window]   Fig. 9.   Root-normalized-mean-squared error (RNMSE) results (mean ± SD) for HR baroreflex (A), ILVHR (B), NHR, (C) and ILVHR (D) with respect to the redefined gold standard (see APPENDIX C) as a function of the ratio of the static gains of the cardiopulmonary to arterial HR baroreflexes.

Arterial baroreflex saturation. The S parameter in Eq. 1 characterizes the degree of arterial baroreflex saturation in the forward model. This parameter provides an upper bound on the deviation between sensed ABP and its setpoint pressure so that the manipulated variables cannot be controlled to arbitrarily high or low values. The maximum deviation permitted by this parameter in the forward model is ~28 mmHg. This pressure is approximately equal to the range of ABP, over which the atan mapping in Eq. 1 differs from true linearity by ~10% (that is, the linear ABP range). However, experimental data from one study (25) indicate that the linear ABP range may actually be only ~10 mmHg. We therefore consider the robustness of the HR baroreflex, ILVHR, and N_HR estimates against more narrow linear ABP ranges.

View larger version (13K): [in this window] [in a new window]   Fig. 10.   RNMSE results (means ± SD) for HR baroreflex (A), ILVHR (B), and NHR (C) as a function of the linear ABP range (as defined in the text) of the arterial baroreflex.

Sensitivity Analysis

Figure 11, A and B, shows the sensitivity results to changes in autonomic function in terms of the estimated versus actual percentage of change in the static gains (sum of the weighting values) of the autonomically mediated impulse responses, where the percentage of change here is with respect to the nominal static gain values. These results indicate that a change of 25% in the static gain of the gold standard HR baroreflex impulse response is required to detect reliably a change in the static gain of the corresponding estimate. On the other hand, a 25% change in the static gain of the gold standard ILVHR impulse response is not sufficient to detect reliably a change in the static gain of the corresponding estimate.

View larger version (20K): [in this window] [in a new window]   Fig. 11.   Sensitivity results (solid trace; means ± SD) in terms of estimated versus actual percentage of change in the static gain of ILVHR (A), static gain of HR baroreflex (B), absolute peak amplitude of ILVHR (C), and absolute peak amplitude of HR baroreflex (D). The percentage of change is with respect to the nominal values and the dashed trace is the identity line.

On the basis of our experience with experimental human data (27, 29), we have found that the absolute peak amplitude of the impulse response estimates seems to be quite reliable in detecting changes in autonomic function. Figure 11, C and D, shows the sensitivity results in terms of the estimated versus actual percentage of change in the absolute peak amplitude of the autonomically mediated impulse responses. Because we varied the static gains of the impulse responses simply via scaling, the actual percentage of change in the static gain of the impulse responses is equal to the actual percentage of change in the absolute peak amplitude. The results indicate substantial improvement in sensitivity with respect to the ILVHR estimate and some improvement with respect to the HR baroreflex estimate. That is, as small as a 10% change in the absolute peak amplitude of the gold standard ILVHR impulse response is sufficient to detect reliably a change in the absolute peak amplitude of the corresponding estimate.

The substantial improvement in the sensitivity of the ILVHR impulse response may be explained by considering the ratio of the spectra of ILV to the spectra of the unmeasured HR disturbance (signal-to-noise ratio as a function of frequency) in the model. This ratio is smallest at frequencies near 0 Hz because of the 1/f character of the unmeasured disturbance as well as the reduced ILV spectral content near 0 Hz due to the probability density defining the random-interval respiratory pattern (see Respiratory Activity). The static gain of the ILVHR impulse response estimate, which reflects the 0 Hz estimate, is therefore not reliable. However, the ratio is larger at higher frequencies, and the absolute peak amplitude, which reflects the wider bandwidth parasympathetic filter, is therefore quite reliable. On the other hand, the spectral content of ABP is sufficiently significant with respect to that of the unmeasured 1/f HR fluctuations at frequencies near 0 Hz such that the static gain of the HR baroreflex impulse response is fairly reliable.

Figure 12 illustrates the sensitivity results in terms of the estimated and actual percentage of change in total, low-frequency (0-0.15 Hz), and high-frequency (0.15-0.4 Hz) power of the N_HR spectra. These results emphasize the superior reliability of the high-frequency components of the estimates due to the 1/f character of the unmeasured HR disturbance. Note that the high-frequency power of N_HR is just as sensitive a measure of parasympathetic function as the absolute peak amplitude of ILVHR. However, we believe that this absolute peak amplitude may be a better measure of parasympathetic function, because the perturbing noise source N_HR, unlike ILVHR, is not normalized for inputs, e.g., cerebral activity.

View larger version (19K): [in this window] [in a new window]   Fig. 12.   Sensitivity results (solid trace; mean ± SD) in terms of estimated versus actual percentage of change in the total power (A), low-frequency (LF) power (0.0-0.15 Hz) (B), and high-frequency (HF) power (C) (0.15-0.4 Hz) of the NHR power spectrum. The percentage of change is again with respect to the nominal values, and the dashed trace again denotes the identity line.

	DISCUSSION

TOP ABSTRACT INTRODUCTION CARDIOVASCULAR SYSTEM... FORWARD MODEL DESCRIPTION FORWARD MODEL-BASED EVALUATION... FORWARD MODEL VALIDATION FORWARD MODEL-BASED EVALUATION DISCUSSION APPENDIX A APPENDIX B APPENDIX C REFERENCES

Previous Forward Models

None of these simple models are nearly as comprehensive as our forward model in terms of accounting for pulsatility, the short-term regulatory system, and resting physiological perturbations. Our forward model thus provides a more complete means for studying the mechanisms responsible for eliciting beat-to-beat variability compared with these oversimplified models. Consider, for example, our more thorough analysis of the physiological mechanisms responsible for the system resonance phenomenon with respect to the study of de Boer (14) (see FORWARD MODEL VALIDATION).

Forward Model Limitations

Robustness to Other Forward Model Uncertainties

We also did not include an analysis of the effects of the size of the 1/f HR fluctuations on the estimates. This disturbance may be thought of as the unmeasured noise in the identification of the autonomically mediated impulse responses, HR baroreflex and ILVHR. The relative contribution of this noise term with respect to HR fluctuations due only to ABP and ILV fluctuations reflects the signal-to-noise ratio of this identification problem. This relative contribution, specifically in terms of the ratio of the standard deviation of those HR fluctuations not attributed to linear couplings with ABP and ILV to the standard deviation of all HR fluctuations, has been reported to be 0.31 ± 0.14 based on experimental data obtained from humans breathing randomly (11). However, the ratio of the standard deviation of the 1/f HR fluctuations to the standard deviation of HR fluctuations in the forward model is 0.49 ± 0.07. In fact, we have found that none of the autonomically mediated estimates in Fig. 8 are significantly altered for ratios up to ~0.6.

We also did not analyze the robustness of the mechanically mediated estimates (circulatory mechanics, ILVABP, and N_ABP) to uncertainties in the pulsatile heart and circulation model (i.e., nonlinearities in systemic arterial compliance). However, we believe that this is not too critical because linear lumped parameter models are quite reasonable when small, low-frequency hemodynamic fluctuations are being analyzed.

Sensitivity Analysis Results Supported By Experimental Data Studies

In addition to the static gain and absolute peak amplitude of the impulse response estimates, we have considered another parameter for detecting autonomic changes in experimental human data, which we refer to as the characteristic time parameter and define as follows

(6)

where h(t) is the estimated impulse response (27, 29). Changes in this parameter essentially reflect shifts in balance between parasympathetic and -sympathetic function. For example, simply scaling the impulse response, as we have done (see Sensitivity Analysis), does not change the characteristic time because there is no shift in balance in the activity of the two autonomic limbs. On the basis of our sensitivity analysis results, we expect the ILVHR impulse response estimate to be unable to measure true shifts in balance through the characteristic time parameter, because only the parasympathetic limb can be estimated accurately. However, we do expect the characteristic time parameter of the HR baroreflex impulse response estimate to be a somewhat sensitive measure of true shifts in balance between parasympathetic and -sympathetic function. The validity of these expectations is supported by our analysis with experimental human data (27, 29), in which only the characteristic time parameter of the HR baroreflex impulse response estimate was found to be sensitive to changes in autonomic function.

In conclusion, based on the results of the forward model-based evaluation of the cardiovascular system identification method, we make the following inferences about the performance of the method with respect to experimental data. First, each of the cardiovascular system identification estimates is likely to reflect the system dynamics of actual physiological mechanisms. Second, the ILVHR impulse response estimate encompasses both direct neural coupling and cardiopulmonary HR baroreflex mechanisms. Third, arterial baroreflex saturation and the relative contributions of the HR fluctuations independent of ILV and ABP are unlikely to affect significantly the autonomically mediated estimates. Fourth, the absolute peak amplitude of the ILVHR impulse response estimate is a very sensitive measure of parasympathetic function. Finally, the HR baroreflex impulse response estimate is a reasonably sensitive measure of both parasympathetic function (through absolute peak amplitude and static gain) and -sympathetic function (through static gain) and consequently shifts in balance between the two autonomic limbs (through characteristic time).

The theoretical evaluation therefore provides confidence in the performance of the method with respect to experimental data and demonstrates the power of the cardiovascular system identification method. That is, the method is able to estimate reliably physiologically important impulse responses and perturbing noise sources by analyzing beat-to-beat hemodynamic variability obtained during near normal conditions rather than extreme experimental conditions, which would be required by the literal mathematical meaning of these quantities. This study supports system identification as a powerful approach for the intelligent patient monitoring of cardiovascular function. Moreover, the forward model, on which the theoretical validation is based, provides a convenient test bed of data, which may facilitate the development of new methods that could be incorporated with the cardiovascular system identification method so as to provide a more detailed picture of cardiovascular state (26).