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Harvard-Massachusetts Institute of Technology, Division of Health Sciences and Technology, Massachusetts Institute of Technology, Cambridge, Massachusetts 02142
Submitted 2 June 2003 ; accepted in final form 24 June 2004
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ABSTRACT |
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 TOP ABSTRACT MATHEMATICAL ANALYSIS OF THE...DETERMINATION OF TPRCARDIOVASCULAR SYSTEM...APPENDIXGRANTSREFERENCES |
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autonomic regulation; arterial and cardiopulmonary baroreceptors; local vascular autoregulation; system identification; cardiovascular regulatory physiology
The reflex regulation of the circulation by arterial baroreflexes originating in the carotid sinus and aortic arch regions has interested many physiologists for more than a century (18, 23, 35). A fall in pressure at these baroreceptors causes a reflex decrease in parasympathetic discharge and an increase in sympathetic discharge. These reflex changes lead to an increase in heart rate (HR), myocardial contractility, venoconstriction, and total peripheral resistance (TPR), all of which tend to maintain Pa. The cardiopulmonary baroreceptors, which lie mostly in the cardiac chambers, can also exert important effects on the circulation (1, 5, 24). An increase in pressure at these receptors causes reflex vasodilation, whereas a decrease causes reflex vasoconstriction. Central control of TPR by the arterial and cardiopulmonary baroreflexes, however, can be readily opposed by distally initiated control mechanisms. Metabolic modulation within active skeletal muscle attenuates sympathetically mediated vasoconstriction, so that oxygen delivery to the active muscles is preserved and local metabolic demands are met (14, 34, 36, 38, 41). Furthermore, blood flow through almost any organ changes very little if the pressure perfusing the arteries of the organ is varied (7, 17, 22). This locally controlled mechanism is termed vascular autoregulation and can contribute significantly to short-term control of TPR.
In this study, we emphasize the analytic algebraic analysis of the systemic circulation composed of arteries, veins, and its underlying physiological control mechanisms of baroreflex and autoregulatory modulation of TPR with minimal modeling. We use Frank's (10) two-element windkessel model of the arterial circulation, which successfully explains diastole as the discharging of a blood volume integrator in which Pa falls exponentially and uniformly. It is the most popular model of the arterial system for academic purposes (4, 19) and has been widely utilized to estimate arterial compliance (Ca) and stroke volume. The two-element windkessel has formed the basis of different methods to estimate Ca, such as the decay time method (10), the area method (30), the two-area method (37), and the pulse pressure method (40). The most common criticism to Frank's windkessel is that the high-frequency components of the aortic pressure wave are poorly represented. This criticism has led to many windkessel models and many concerns about their applications (6, 43, 44). We believe that more complexity is required only if the use of a simple model does not suffice to describe the properties of the system essential for determining its functioning. Stergiopulos et al. (40) showed that Frank's windkessel's unrealistic prediction of aortic pressure waveforms was due to its poor medium- to high-frequency representation of the aortic input impedance. They concluded, however, that the two-element windkessel describes the low-frequency characteristics of the arterial system correctly and can predict pulse pressure well for the right choice of Ca. Accordingly, Frank's simple windkessel model suffices for our analytic analysis of the arterial circulation where only the low-frequency characteristics are of interest to us. To include the venous circulation as part of our analysis, we modify Frank's windkessel model as to incorporate Rowell's (33) view of the properties of the venous circulation.
As a result of the aforementioned analysis, we developed a novel mathematical method able to not only track down steady-state changes in TPR very effectively but also short-term TPR fluctuations caused by baroreflex and autoregulatory modulation of TPR. Consequently, short-term Pa fluctuations can be successfully explained almost in their entirety when the contributions of 
TPR to Pa are taken into account. This method, however, does not provide any information regarding the source of TPR fluctuations. With this in mind, we complemented this method with a separate set of quantitative tools, which can be applied to delineate the actual actions of the physiological mechanisms responsible for the observed short-term TPR fluctuations in an individual subject, without altering the underlying regulatory mechanisms. A previous publication by the same group (26), which stresses HR modulation by the arterial baroreflexes and respiration, presents a cardiovascular system identification method intended for the analysis of HR derived from the surface electrocardiogram, noninvasively measured Pa, and instantaneous lung volume signals. For that purpose, Perrot and Cohen (29) proposed a model order reduction technique optimized for that particular problem. In this study, we deal with a different problem. We stress TPR modulation by the baroreflexes and autoregulation and use standard model order selection techniques (31). As a result, we propose a novel cardiovascular system identification method intended for the analysis of the independent dynamic closed-loop contributions of CO and PRA to short-term Pa fluctuations as well as the independent dynamic closed-loop contributions of Pa and PRA to short-term TPR fluctuations, which serves as a natural complement to the cardiovascular system identification method previously proposed by the same group (26), where HR fluctuations play a major role and the regulatory mechanisms responsible for short-term TPR fluctuations could not be accounted for. Furthermore, in a recent publication by the same group, Mukkamala and Cohen (25) estimated short-term TPR fluctuations simply as TPR=Pa/CO without considering the effects of arterial elasticity and were unable to directly determine the dynamic transfer relation Pa
TPR. By including an analysis on the effects of arterial elasticity, we are now able to successfully formulate a direct system identification approach capable of estimating this transfer relation.
Glossary
Pa}Pa
TPR}TPR
Pa}Pa
TPR}TPR
Pa}Pa
TPR}TPR
TPR}TPR
Pa}Pa
TPR}TPR
Pa}Pa
PaPa
TPRTPR
PaPa
TPRTPR
PaPa
TPRTPR
TPRTPR
PaPa
TPRTPR
PaPa
1Pa, KPRAPa, and KTPRPa
2Pa, KPRAPa, and KTPRPa
br
eff
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MATHEMATICAL ANALYSIS OF THE SYSTEMIC CIRCULATION |
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TOPABSTRACTMATHEMATICAL ANALYSIS OF THE...DETERMINATION OF TPRCARDIOVASCULAR SYSTEM...APPENDIXGRANTSREFERENCES |
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 | (1) |
 | (2) |
 | (3) |
 | (4) |
 | (5) |
 | (6) |
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| (7) |
Pa(s) represents the direct hydraulic effects of COPa and KPRAPa(s) represents the direct hydraulic effects of PRAPa. Figure 2 illustrates the block diagram representation of Eq. 7 with the impulse-response function representations of the linear and time-invariant (LTI) systems KCOPa(s) and KPRAPa(s) when, e.g., Ra=1.25 mmHg·s·ml–1, Ca=1 ml/mmHg, Rv=Ra/5, and Cv=16·Ca (4, 12).
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If Ra and Rv are not constant due to baroreflex modulation of TPR (TPR=Ra + Rv), one has to appropriately expand the block diagram representation shown in Fig. 2 as to account for the additional effects of TPR changes (TPR) on Pa as illustrated in Fig. 3A, where the transfer relation KPaTPR(s) represents the delayed autonomically mediated effects of PaTPR (arterial baroreflex) and the transfer relation KPRATPR(s) represents the delayed autonomically mediated effects of PRATPR (cardiopulmonary baroreflex). Accordingly, arterial and cardiopulmonary baroreflex modulation of TPR can be represented as
 | (8) |
Pa(s), KPRAPa(s), and KTPRPa(s) represent the immediate direct hydraulic effects of CO, PRA, and TPR on Pa, respectively. Thus Pa(s) can be represented as
 | (9) |
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| (10) |
Pa(s) and HPRAPa(s) represent the independent dynamic closed-loop effects of COPa and PRAPa as illustrated in Fig. 3B, which is equivalent to Fig. 3A. The solid curves in Fig. 3B illustrate the dynamic properties, composed of hydraulic and autonomic interactions, of the closed-loop transfer relations HCOPa(s) and HPRAPa(s) depicted in their time-domain form of impulse-response function representations. In contrast, the dashed curves illustrate the dynamic properties of KCOPa(s) and KPRAPa(s) in the absence of any autonomic interactions.
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TPR(s) and KPRATPR(s) can be characterized by the first order LTI systems of the form
 | (11) |
 | (12) |
br denotes the time constant of the sympathetic nervous system acting through the 
-adrenergic pathway (32); the time delay Tbr corresponds to the time it takes for the sympathetic nervous system to receive the information, process it, and start a reaction to it (8, 42); and the coefficients G{KPaTPR} and G{KPRATPR} represent the static gains of the arterial and cardiopulmonary baroreflexes, respectively. The hydraulically mediated transfer relations KCOPa(s), KPRAPa(s), and KTPRPa(s) can be characterized by the following second-order LTI systems
 | (13) |
 | (14) |
 | (15) |
1 and 2 represent the time constants characterizing the combined viscoelastic properties of the arterial and venous circulation. As a result, it is possible to show by substitution of Eqs. 11–15 into Eq. 10 that the closed-loop transfer relations HCOPa(s) and HPRAPa(s) must be characterized by the third-order closed-loop systems of the form
 | (16) |
 | (17) |
TPR(s). For a detailed discussion on this issue, please see the APPENDIX.
Steady-state analysis.
Steady-state analysis of the modified windkessel model in Fig. 1 yields
 | (18) |
 | (19) |
denotes steady-state changes in that signal around its mean value. Normalization of Eq. 19 by 
and multiplication by 100% to describe percent changes finally results in
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| (20) |
Pa}, G{KPRAPa}, and G{KTPRPa} represent the static gains of the open-loop transfer relations KCOPa(s), KPRAPa(s), and KTPRPa(s), respectively, and the denotes percent changes in that signal around its mean value. PRA, however, denotes changes in mmHg around its mean value, because, unlike the higher pressure in Pa, a 1-mmHg error in absolute PRA resulting from deviations between the precise location of the right atrium and the pressure transducer such as in the companion paper (2) can introduce a substantial degree of error if given in percent. Fluctuations in PRA given in mmHg remain unaffected and offer a solution to this problem. A simple mathematical expression can be finally attained for the steady-state relation between autonomically and hydraulically mediated interactions by evaluation of HCOPa(s) in Eq. 16 at s=0 as
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| (21) |
Pa} represents the static gain of HCOPa(s). Thus it is possible to determine G{KPaTPR} by solving Eq. 21 for G{KPaTPR} as
 | (22) |
Pa(s) in Eq. 17 at s=0 results in
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| (23) |
Pa} represents the static gain of HPRAPa(s). Accordingly, the static gain of HPRAPa(s) is said to depend on the separate static effects of two physically isolated sensory regions (arterial and cardiopulmonary baroreceptors) on a common effector region (TPR): the arterial and the cardiopulmonary baroreflexes. Consequently, it is possible to determine G{KPRAPa} by substitution of G{KPaTPR} from Eq. 22 into Eq. 23 and subsequently solving for G{KPRAPa} as
 | (24) |
Pa} is independent of G{KPRAPa} (cardiopulmonary baroreflex), whereas G{HPRAPa} depends on both G{KPaTPR} and G{KPRATPR} (arterial and cardiopulmonary baroreflexes).
Physical interpretation of the mathematical theory.
Consider the following hypothetical experiment where the right atrium is decoupled from the left ventricle and CO is maintained constant by externally controlled volume infusion into the left ventricle while the resulting venous return drains freely through the right atrium into an infinite blood reservoir. If the pressure downstream the systemic circulation (PRA) changes due to, for example, occlusion of the right atrium, the pressure upstream the systemic circulation (Pa) has to adjust itself to overcome those pressure changes while "CO remains constant," as illustrated by the upward arrows in Fig. 1. The dynamic nature of this adjustment depends on the viscoelastic properties of the system that couples PRA and Pa, which is not only composed of the arterial vasculature (Ra and Ca with RaCa=a) but also the venous vasculature (Rv and Cv with RvCv=v 
a). As a result, the dynamic open-loop relationship between PRA and Pa, represented by KPRAPa(s), may be quantitatively described in terms of the windkessel elements Ra, Ca, Rv, and Cv, as illustrated algebraically in Eq. 7 or, i.e., in terms of the "combined viscoelastic properties" of the arteries and veins. The open-loop characteristic frequencies of KPRAPa(s) may be determined from the roots of their denominator polynomials as s1,2=1/1,2, where both 1 and 2 depend on all windkessel parameters; thus 1=f(Ra, Ca, Rv, Cv) a and 2=f(Ra, Ca, Rv, Cv) v. Because TPR=Ra + Rv is not constant, but rather modulated by the centrally mediated baroreflexes KPaTPR(s) and KPRATPR(s), these reflex interactions must therefore, in addition to 1 and 2, inherently influence the dynamic closed-loop transfer relationship between PRA and Pa, represented by HPRAPa(s), as demonstrated in Eq. 10 [note the presence of KPaTPR(s) in the denominator and KPRATPR(s) in the numerator] and as illustrated graphically in Fig. 3B, where the gray area in HPRAPa(s) denotes the change in Pa due to arterial and cardiopulmonary baroreflex regulation of TPR given an impulse change in PRA. As a result, HPRAPa(s) can be said to also depend, in addition to the combined viscoelastic properties of arteries and veins, on the separate dynamic effects of two physically isolated sensory regions (arterial and cardiopulmonary) on a common effector region (TPR); hence, the arterial and the cardiopulmonary baroreflexes. Any hydraulic influence of PRA on Pa via the heart and pulmonary circulation is implicit in CO and thus already incorporated into the dynamic closed-loop transfer relationship between CO and Pa, represented by HCOPa(s). Consequently, the closed-loop transfer relations HCOPa(s) and HPRAPa(s) are meant to represent the independent dynamic closed-loop contributions of CO and PRA on Pa fluctuations. The shaded area in HCOPa(s) in Fig. 3B denotes the change in Pa due to arterial baroreflex regulation of TPR given an impulse change in CO.
Figure 4A displays HCOPa(s) and HPRAPa(s) in their time-domain form of step-response function representations, whereas Fig. 4B illustrates the dynamic properties of the arterial and cardiopulmonary baroreflexes in their time-domain form of step-response function representations. denotes the percent fluctuations in that signal around its mean value denoted by the overbar with the exception of PRA, where denotes fluctuations in mmHg. In essence, Fig. 4A illustrates how HCOPa(s) depends solely on the arterial baroreflex KPaTPR(s), whereas HPRAPa(s) depends on both baroreflexes, KPaTPR(s) and KPRATPR(s). To further elaborate on the previous statement, three cases will be individually addressed in this paragraph. First, the case where both baroreflexes are absent shall be considered, as illustrated by the dashed curves in Fig. 4A. Therefore, if CO were to increase by a step change of 1% while PRA remained unchanged, then Pa would rapidly increase G{KCOPa}% in <10 s, which is the approximate time necessary to reach a steady state in Pa given a step change in CO, i.e., about 10 times the dominant time constant of the open-loop system KCOPa(s) mainly determined by the viscoelastic properties of the arterial circulation. Likewise, if PRA were to increase by a step change of 1 mmHg while CO remained constant, Pa would increase G{KPRAPa}% in <20 s, which is the approximate time necessary to reach a steady state in Pa given a step change in PRA, i.e., about 10 times the dominant time constant of the open-loop system KPRAPa(s) determined by the combined viscoelastic properties of the arterial and venous circulation. The case where both baroreflexes are present shall be considered next, as illustrated by the solid curves in Fig. 4A. Thus if CO were to increase by a step change of 1% while PRA remained unchanged, Pa's rapid increase would be truncated and reversed by the nonimmediate negative feedback effects of the arterial baroreflex on TPR (see time delay in Fig. 4B). In this example, the arterial baroreflex kicks in at t=2 s forcing Pa to settle down in about 30 s into a steady-state value smaller than G{KCOPa} denoted as G{HCOPa}. The static gain G{HCOPa} is determined by Eq. 21, which states the mathematical relation between G{HCOPa} and the static gain of the arterial baroreflex G{KPaTPR}. Likewise, if PRA were to increase by a step change of 1 mmHg while CO remained constant, Pa's increase would be truncated and reversed by the nonimmediate negative feedback regulation of the arterial baroreflex in addition to the nonimmediate negative cardiopulmonary baroreflex regulation of TPR (see time delay in Fig. 4B). In this example, both baroreflexes kick in at t=2 s and force Pa to settle down in about 30 s into a steady-state value remarkably lower than G{KPRAPa} denoted as G{HPRAPa}. The static gain G{HPRAPa} is determined by Eq. 23, which states the mathematical relation between G{HPRAPa} and the static gains of the arterial and cardiopulmonary baroreflexes G{KPaTPR} and G{KPRATPR}. Finally, the special case where the arterial baroreflex is present but the cardiopulmonary baroreflex is absent shall be considered, as illustrated by the light gray dashed-dotted curve in Fig. 4A [present only in HPRAPa(s) because HCOPa(s) is independent of the cardiopulmonary baroreflex]. Accordingly, HPRAPa(s) resembles a scaled version of HCOPa(s). With this particular case, it becomes apparent that only the presence of the cardiopulmonary baroreflex can result in G{HPRAPa} < 0. The mathematical relation given in Eq. 23 states that G{HPRAPa} depends on the separate static effects of two physically isolated sensory regions (arterial and cardiopulmonary baroreceptors) on a common effector region (TPR). Specifically, Eq. 23 limits the influence of arterial baroreflex modulation of TPR to only yield positive values in G{HPRAPa} smaller than G{KPRAPa}, because G{KPaTPR} is always negative. As a result, a negative value in G{HPRAPa} can only be explained by cardiopulmonary baroreflex modulation of TPR because G{KPRATPR} is always negative and expressed in the numerator.
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Vascular autoregulation is characterized by local vasoconstriction after an increase of the vessel's transmural pressure and local vasodilation after a decrease. As a result, autoregulation implies a constant blood flow in the face of altered perfusion pressure, and thus, if present, autoregulation contributes to short-term control of TPR. Local vascular autoregulation is probably determined by both mechanical (P) and metabolic (Q) changes. Because the local pressures Pi (which are not in the model) do not add up to a systemic pressure P, and while all local Qi (which are not in the model either) do add up to a systemic flow Q made up of the sum of all local Qi as Q=
Qi, one can think of autoregulation as being determined by both local metabolic (Qi) and local mechanical (Pi which results in Qi 
Pi anyway) changes, where CO=Q can be viewed as a plausible measure of autoregulation. Consequently, the presence of autoregulatory modulation of TPR in addition to autonomic baroreflex regulation makes it necessary to expand the block diagram representation of the systemic circulation depicted in Fig. 3A as to account for the short-term dynamic autoregulatory effects of CO on TPR as illustrated in Fig. 5A. Accordingly, Eq. 10 becomes
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| (25) |
TPR(s) represents the dynamic autoregulatory effects of CO on TPR. Consequently, Eq. 25 describes how HCOPa(s) is affected by KCOTPR(s) while HPRAPa(s) remains independent of KCOTPR(s), which can be characterized, e.g., by the first-order LTI system of the form
 | (26) |
TPR} denotes the static gain of KCOTPR(s) and Tar corresponds to the time it takes for the local vasculature to sense a change in blood flow and subsequently react to it (22, 38). The time constant of this system can be assumed to be very similar to br, the time constant of the baroreflexes (22, 32). Likewise, from the block diagram shown in Fig. 5A, TPR can then be determined as
 | (27) |
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| (28) |
TPR(s) and KPRATPR(s) are influenced by autoregulation KCOTPR(s). As a result, the dynamic closed-loop transfer relation HPaTPR(s) describes the interaction between the centrally mediated arterial baroreflex and local vascular autoregulation, whereas the dynamic closed-loop transfer relation HPRATPR(s) describes the interaction between the centrally mediated cardiopulmonary baroreflex and local vascular autoregulation. As a result, the independent dynamic closed-loop contributions of Pa and PRA on TPR can differ considerably from the simple first-order open-loop transfer relations KPaTPR(s) and KPRATPR(s) depending on the degree of local vascular autoregulation present.
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Pa(s) in Eq. 25 represented as
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| (29) |
TPR(s) and HPRATPR(s) in Eq. 28, respectively, represented as
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| (30) |
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| (31) |
Pa} from Eq. 23 provide simple mathematical expressions for the steady-state relations between regulatory and hydraulically mediated interactions. For instance, it can be easily verified that in the presence of autoregulation, as graphically illustrated in Fig. 6B, the amplitudes of G{HPaTPR} and G{HPRATPR} must always be smaller than the amplitudes of the individual baroreflexes (G{KPaTPR} and G{KPRATPR}) due to the fact that G{KCOTPR} and G{HCOPa} must always have positive numbers and the baroreflexes negative values. Finally, solving Eq. 30 for G{KPaTPR} and consequently substituting the resulting G{KPaTPR} into Eq. 29 yields
 | (32) |
TPR}. Likewise, the solution of Eq. 31 for G{KPRATPR} and consequent substitution of the resulting G{KPRATPR} and G{KPaTPR} into Eq. 23 results in
 | (33) |
TPR}.
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Pa(s) and HPRAPa(s) in their time-domain form of impulse-response function representations in the presence of substantial local vascular autoregulation in addition to autonomic baroreflex modulation of TPR. The dashed-dotted curves represent the case where TPR is constant; thus no TPR regulation takes place. The dashed curve represents the case with substantial autoregulatory modulation of TPR but no baroreflexes [present only in HCOPa(s) because HPRAPa(s) is independent of autoregulation]. The solid curves represent the case with baroreflex and autoregulatory modulation of TPR. The dynamic properties of HCOPa(s) reveal in all three cases an immediate initial increase in Pa given a positive impulse change in CO demonstrating the direct positive hydraulic effects of CO on Pa. In this example, local vascular autoregulation kicks in at 1 s to slow down Pa's natural time decay in HCOPa(s) as a result of the longer time constant of autoregulation. Up to that point in time (t=1 s), Pa's natural time decay is mainly determined by the viscoelastic properties of the arteries. At 2 s, the arterial baroreflex kicks in to speed up Pa's natural time decay and further force Pa into negative territory as a result of its negative feedback regulation. Similarly, the dynamic properties of HPRAPa(s) reveal an initial increase in Pa given a positive impulse change in PRA demonstrating the direct positive hydraulic effects of PRA on Pa. Pa's natural time decay is determined by the combined viscoelastic properties of arteries and veins up to the time when the arterial and the cardiopulmonary baroreflexes kick in to dramatically speed up Pa's natural time decay and quickly lead Pa into negative values as a result of negative feedback regulation of the arterial baroreflex in addition to cardiopulmonary baroreflex regulation of TPR.
Figure 6 graphically illustrates with solid curves the dynamic closed-loop transfer relations HPaTPR(s) and HPRATPR(s) in the presence of substantial local vascular autoregulation in addition to autonomic baroreflex modulation of TPR. The dashed curves represent the case where autoregulation is practically absent; thus HPaTPR(s)=KPaTPR(s) and HPRATPR(s)=KPRATPR(s). Figure 6A depicts impulse-response function representations, whereas Fig. 6B displays step responses. The effects of autoregulation become especially apparent in HPaTPR(s), where the initial increase in TPR at 1 s clearly demonstrates the nonimmediate positive regulatory effects of Pa on TPR due to local vascular autoregulation. If the pressure perfusing the arteries of almost any organ is varied, blood flow through the organ changes very little. Therefore, an increase in Pa results in a likewise increase in TPR opposing the imminent increase in organ blood flow. In this example, local vascular autoregulation kicks in at 1 s and remains unchallenged until 2 s when the arterial baroreflex kicks in to oppose the increase in Pa, thereby reversing TPR and forcing it to remain negative until the end of its natural time decay. The apparent conflict between vascular autoregulation and baroreflexes results in 1) an effective shortening of the characteristic time constant compared with br as illustrated in Fig. 6A, and 2) smaller static gains G{HPaTPR} and G{HPRATPR} compared with G{KPaTPR} and G{KPRATPR}, respectively, as illustrated in Fig. 6B. Furthermore, depending on the value of G{HPaTPR}, it is possible to clearly determine the dominance of one reflex over the other as follows: if G{HPaTPR} > 0, autoregulation dominates over the arterial baroreflex. If G{HPaTPR} < 0, the arterial baroreflex dominates over autoregulation. If G{HPaTPR} 
0, the arterial baroreflex and autoregulation cancel each other out, as graphically illustrated in Fig. 6B by the solid curve representing HPaTPR with G{KCOTPR}=–G{KPaTPR}.
In conclusion, autoregulatory modulation of TPR slows down Pa's natural time decay as a result of its longer time constant, whereas arterial baroreflex modulation of TPR speeds up Pa's natural time decay as a result of its negative feedback regulation. Finally, the ongoing conflicting effects between the locally controlled vascular autoregulation and the centrally regulated autonomic baroreflexes manifest in the dynamic closed-loop transfer relations HPaTPR(s) and HPRATPR(s), which no longer represent sole control of TPR by the baroreflexes but also by local vascular autoregulation.
Closed-Loop Computational Model of Heart and Circulation
To apply the methods for TPR determination and cardiovascular system identification presented next to the analysis of experimentally acquired animal data such as in the companion paper (2), it is necessary to first evaluate the proposed methods on a theoretical basis where the TPR fluctuations and the transfer relations of interest are known a priori. To carry out an evaluation with CO, PRA and Pa signals assembled in closed loop, we implement a simple closed-loop computational model of the heart and circulation where the feedforward component characterizes cardiac function and the feedback component constitutes the peripheral vascular system properties represented by the lumped parameter model of the systemic circulation illustrated in Fig. 1 and its underlying regulatory mechanisms of baroreflex and autoregulatory modulation of TPR. Accordingly, the output variable, CO, is linked to its own input variables, Pa and PRA, by the negative feedback interaction of the vascular subdivision. As a result, other things being equal, an increase in Pa causes decreased CO in the cardiac subdivision and a decrease in CO causes a decrease in Pa in the vascular subdivision. Similarly, other things being equal, an increase in PRA causes increased CO in the cardiac subdivision and an increase in CO causes decreased PRA in the vascular subdivision. In view of our minimal model approach, we implement a rather simple pacemaker-driven impulse model of cardiac ejection, where the implemented Starling mechanism is not offered as a new model for heart dynamics but simply to close the loop between the heart and circulation and thereby provide computationally generated data sets of CO, PRA, Pa, and TPR signals assembled in closed loop, which could be subsequently used for system identification purposes and as the gold standard for TPR determination.
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DETERMINATION OF TPR |
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TOPABSTRACTMATHEMATICAL ANALYSIS OF THE...DETERMINATION OF TPRCARDIOVASCULAR SYSTEM...APPENDIXGRANTSREFERENCES |
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 | (34) |
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| (35) |
eff represents the effective time constant characterizing the dynamic behavior of the system. Adequate transformation of Eq. 35 from continuous time to discrete time (see Conversion From Continuous-Time to Discrete-Time Systems in the APPENDIX) results in the linear constant-coefficient difference equation of the form
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| (36) |
1 and 2 as
 | (37) |
 | (38) |
Important Considerations
One of the main goals of this study is to introduce a robust mathematical method able to determine from measured CO, PRA, and Pa signals not only steady-state changes in TPR but also short-term TPR fluctuations caused by baroreflex and autoregulatory modulation of TPR. With this in mind, it is necessary to demonstrate to what extent sampling effects and omission of arterial elasticity can affect the estimation of the desired short-term TPR fluctuations when the differential equation (Eq. 1) corresponding to a dynamic model of the circulation based on sound and relatively simple physical principles is further simplified (Eq. 35) and adequately transformed from continuous time to discrete time (Eq. 36). Solving Eq. 36 for TPR leads to Eq. 38, which defines the present value of TPR as a function of fs, present and past values of the sampled CO, PRA, and Pa signals, and the previously estimated Ca from Eq. 37. As a result, the relationship represented in Eq. 38 is defined over two equidistant time samples and describes TPR fluctuations as a function of fs and Ca. Figure 7A illustrates the effectiveness of Eq. 38 to accurately determine short-term TPR fluctuations from Pa, PRA, and CO fluctuations computationally generated in closed loop. The true TPR depicted in red are the actual short-term TPR fluctuations generated by the closed-loop computational model of the heart and circulation, whereas the calculated TPR depicted in blue are the short-term TPR fluctuations determined via Eq. 38 using the computational model CO, PRA, and Pa signals generated in closed loop and resampled to 0.5 Hz and the estimated Ca from Eq. 37. This method not only provides for an accurate representation of TPR but also for the correct values in 
and Ca. Despite this method's effectiveness (actual and calculated TPR in Fig. 7A are almost indistinguishable), this mathematical approach has to be taken with caution as to make sure that the measured signals are resampled to sufficiently low frequencies after appropriate antialiasing filtering (e.g., 10th-order FIR filter with cutoff frequency fc 
fs/2). The sampling frequency fs must be very carefully chosen because fc > 1/(2
eff) yields rather inaccurate results. Figure 7B illustrates the case of incorrect resampling with a comparison in time (top) and frequency (bottom) between actual (red curves) and calculated TPR determined via Eq. 38 (blue curves) using the computational model CO, PRA, and Pa signals resampled to 1 Hz (see text below for description of gray curves). The energy observed at fresp and the spectral leakage from the HR frequency component are present only in the calculated TPR and not in the actual TPR fluctuations, which have a bandwidth [f < fbr=1/(2br) 0.016Hz] limited by the much lower characteristic frequency of the baroreflexes and autoregulation, which modulate the actual TPR fluctuations and which, in this particular example, are solely responsible for their being. These artifactual fluctuations in calculated TPR arise because Eq. 38 determines TPR from signals resampled to 1 Hz, thereby allowing for the frequency peak at the respiratory frequency fresp=0.4 Hz, which is quite significant in CO, PRA, and Pa signals, and the spectral leakage from the very high energy content at the HR frequency to be transmitted into the calculation of TPR. Nevertheless, the representation of TPR at the lower frequencies [f < 1/(2eff) 0.1 Hz] remains extremely accurate and well separated from the artifactual high-frequency content, as illustrated by the almost indistinguishable difference in power spectral densities between the red and blue curves at frequencies lower than 0.1 Hz. Accordingly, resampling to sufficiently low frequencies after appropriate antialiasing filtering provides for an accurate representation of the actual TPR fluctuations by the calculated TPR determined via Eq. 38.
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TPR. The error due to Ca, mostly visible at the beginning of a sudden change, can be rightly described as an unavoidable dynamic error and remains quite evident despite aggressive low-pass filtering of the CO and Pa signals. Calculation of TPR as Pa/CO – PRA corrects for the steady-state error but does not eliminate the dynamic error clearly illustrated in Fig. 8B by the random TPR fluctuations depicted in gray. As a result, reliable short-term TPR fluctuations cannot be accurately assessed from measured CO, PRA, and Pa signals if the elasticity of the arteries is omitted in their estimation. It is not in the least surprising that the actual TPR fluctuations differ so much when components of the differential equation (Eq. 1) used to generate the actual fluctuations are omitted in the calculation of TPR=Pa/CO; however, it is essential to recognize which elements play a significant role. For example, disregarding of Cv does not seem to play any role in the calculation of TPR, whereas the disregarding of Ca clearly does. Nevertheless, it is still a common practice to estimate TPR fluctuations as TPR=Pa/CO and hence anticipated when researchers are not able to successfully formulate a system identification approach for the quantification of the baroreflexes using TPR fluctuations simply as TPR=Pa/CO.
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130 beats/min while Pa and PRA were measured via strain-gauge-based pressure transducers connected to catheters placed in the descending aorta and right atrium. Figure 9A shows a 20-s comparison between measured Pa and the predicted Pa depicted in blue. In this example, the predicted Pa assumes a constant TPR obtained from the mean value of Eq. 38 and disregards the contributions of TPR to Pa fluctuations. As a result, the predicted Pa cannot account for the very low-frequency fluctuations in measured Pa caused by changes in the actual TPR yielding a correlation coefficient of r=0.89. Nevertheless, direct visual inspection of measured and predicted Pa during a smaller time window (2 s), where TPR can be assumed to remain constant, suffices to conclude that the predicted Pa describes very well the low-frequency characteristics of the experimentally measured Pa observed during systole and diastole over a few cycles, which encompass many k data samples of the measured signals sampled at 100 Hz. In contrast, Fig. 9B displays the block diagram representation of the windkessel model depicted in Fig. 9A together with a 15-min comparison between measured Pa resampled to 0.5 Hz and predicted Pa when TPR is determined via Eq. 38 and the contributions of TPR to Pa are taken into account. The observed steady-state changes in the measured signals were induced via manipulations of pacing rate and venous return as to generate significant steady-state changes in TPR. Consequently, it is possible to clearly illustrate the difference between the blue curve depicting the case where Pa is predicted assuming TPR remains constant (r 0.7) and the black curve exemplifying the case where the contributions of TPR to Pa fluctuations are taken into account (r 1.0). As a result, we can rightly assume that the estimated model elements and Ca offer an adequate characterization for the dynamic open-loop transfer relations KCOPa, KPRAPa, and KTPRPa from Eq. 9 (depicted here in their time-domain form of step-response function representations), which respectively represent the immediate hydraulic effects of CO, PRA, and TPR on Pa fluctuations. In conclusion, the differential equation (Eq. 35) corresponding to the very basic dynamic two-element ( and Ca) windkessel model of the systemic circulation illustrated in Fig. 9A is good enough to explain the aortic pressure waveforms observed during systole and diastole and suffices to account for the entirety of the observed low-frequency (f < 0.25 Hz because fs=0.5 Hz) fluctuations in Pa when Eq. 38 is employed to determine TPR.
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CARDIOVASCULAR SYSTEM IDENTIFICATION |
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TOPABSTRACTMATHEMATICAL ANALYSIS OF THE...DETERMINATION OF TPRCARDIOVASCULAR SYSTEM...APPENDIXGRANTSREFERENCES |
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Hemodynamic System Identification
Dynamic closed-loop effects of CO and PRA on Pa can be modeled via fluctuations in CO and PRA as illustrated in Fig. 5B via the autoregressive exogenous model represented by the linear constant-coefficient difference equation of the form
 | (39) |
denotes the percent fluctuations in that signal around its mean value indicated by the overbars and ePa is the residual error. The parameter values of the ai, b1i, and b2i coefficients and the model orders n, m1, and m2 are determined by the linear least-squares minimization of the residual error in conjunction with Rissanens's minimum description length (MDL) principle (31), a model order selection criterion that evaluates a given model's performance compared with other models. Once these parameters are determined, the transfer relations HCOPa and HPRAPa are fully defined. In addition, the estimated G{HCOPa} and G{HPRAPa} can then be used to indirectly determine G{HPaTPR} from Eq. 32 and G{HPRATPR} from Eq. 33 as
 | (40) |
 | (41) |
Pa}=G{KCOPa}=TPR·CO/Pa and G{KPRAPa}=100%/Pa. Regulatory System Identification
Dynamic autonomic closed-loop control of TPR by the arterial and cardiopulmonary baroreceptors in the presence of local vascular autoregulation can be modeled via fluctuations in Pa and PRA as illustrated in Fig. 6A. The modulation of TPR can be described by the autoregressive exogenous model represented by the linear constant-coefficient difference equation of the form
 | (42) |
TPR and HPRATPR are fully defined. Monte Carlo Simulations
System identification requires the input signals to be poorly correlated and sufficiently broadband so that all the modes of the system to be identified are excited and reliable (21, 39). Consequently, the computational model of the heart and circulation was simultaneously excited by two separate sources as to simulate an orthogonal input design in which HR and venous return were independently varied with frequency band limited to 0.1 Hz about their mean values in a nearly uncorrelated fashion while CO, Pa, PRA, and TPR were measured. To evaluate the effectiveness of hemodynamic (HSI) and regulatory system identification (RSI) to quantitatively characterize the dynamic closed-loop transfer relations HCOPa, HPRAPa, HPaTPR, and HPRATPR, the uncertainty of the estimates was determined by calculating the mean and standard error for the impulse-response estimates from 100 different realizations of computer-generated data sampled at fs=0.5 Hz. Figure 10A displays a graphical comparison between the analytically derived solutions for the closed-loop transfer relations HCOPa and HPRAPa depicted in circles in their step-response function representation and mean HSI results depicted in squares. For illustrative purposes, the analytically derived solutions for the open-loop transfer relations KCOPa and KPRAPa depicted in triangles are also shown. Figure 10B displays a graphical comparison between the analytically derived solutions for HPaTPR and HPRATPR depicted in circles in their step-response function representation and mean RSI results depicted in squares. Table 1 displays a numerical comparison between the actual and estimated static gain values obtained via HSI and RSI.
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In this study, we emphasized the analytic algebraic analysis of the dynamics of the peripheral vasculature composed of arteries, veins, and its underlying physiological control mechanisms of baroreflex and autoregulatory modulation of TPR. We acknowledge the existence of far more complex and complicated models of the circulation. Our goal, however, was to enhance our understanding of the crucial functional relationships that determine the behavior of the systemic circulation and its underlying physiological regulatory mechanisms with minimal modeling. Ultimately, we hope that the presented analytic analysis simultaneously simplified and deepened our understanding of the cardiovascular system. Furthermore, we readdressed the mathematical analysis with a focus on measured data composed of sample data rather than a continuum of times, whether experimentally acquired or computationally generated, and highlight the significance of this practical issue. As a result of this analysis, we developed a novel mathematical method to determine short-term TPR fluctuations, which accounts for the entirety of observed Pa fluctuations. Finally, we directed our attention to the development and evaluation of quantitative tools, which could be subsequently utilized to delineate the actual actions of the physiological mechanisms responsible for the couplings among CO, Pa, PRA, and TPR without altering the underlying regulatory mechanisms of baroreflex and autoregulatory modulation of TPR. As a result, we proposed a novel cardiovascular system identification method, introduced as two separate model structures referred to as HSI and RSI, able to quantitatively characterize the independent dynamic closed-loop contributions of CO and PRA to Pa fluctuations as well as the independent dynamic closed-loop contributions of Pa and PRA to short-term TPR fluctuations.
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APPENDIX |
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TOPABSTRACTMATHEMATICAL ANALYSIS OF THE...DETERMINATION OF TPRCARDIOVASCULAR SYSTEM...APPENDIXGRANTSREFERENCES |
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Pa(s) and HPRAPa(s) introduced in Eq. 10 depend on the dynamic effect of the feedback system that characterizes the arterial baroreflex, KPaTPR(s), which moves the locations of the poles and zeros of the original open-loop systems KCOPa(s) and KPRAPa(s). Consequently, Eqs. 16 and 17 demonstrate that the static gain G{KPaTPR}, time delay Tbr, and characteristic frequency 1/br of the arterial baroreflex determine the poles of HCOPa(s) and HPRAPa(s). For example, if the amount of feedback in Eq. 16 were negligible, i.e., no arterial baroreflex modulation of TPR were present in HCOPa(s), the poles would simply be the characteristic frequencies of the original system, 1/1 and 1/2; however, if feedback cannot be neglected, the poles are not simply 1/1, 1/2, and 1/br because the exact pole-zero locations depend on G{KPaTPR} and Tbr as well. To further illustrate the legality of this practical matter, it is necessary to convert the continuous-time system functions HCOPa(s) and HPRAPa(s) to their equivalent discrete-time system functions HCOPa(z) and HPRAPa(z), respectively. Conversion from Continuous-Time to Discrete-Time Systems
To manipulate signals composed of a sequence of samples rather than a continuum of times, it is necessary to convert the continuous-time system function F(s) to its equivalent discrete-time system function F(z). This can be accomplished by the bilinear transform, which is an algebraic transformation between the complex frequency variables s and z that maps the entire imaginary axis in the analog s-plane onto one circumference of the unit circle in the discrete-time z-plane (27). This transformation, which also arises from applying the trapezoidal integration rule to the differential equation corresponding to F(s) (16), corresponds to replacing s by
 | (A1) |
Bilinear transformation of the continuous-time closed-loop system HCOPa(s) in Eq. 16 results in the equivalent discrete-time frequency-domain representation
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| (A2) |
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| (A3) |
TPR} embedded in 
1 determines the exact pole-zero locations of the closed-loop system HCOPa(z). Likewise, bilinear transformation of the continuous-time closed-loop system HPRAPa(s) in Eq. 17 results in the equivalent discrete-time frequency-domain representation
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| (A4) |
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| (A5) |
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| (A6) |
 | (A7) |
Bilinear transformation of the continuous-time transfer relations KPaTPR(s) and KPRATPR(s) characterized by the first-order LTIsystems in Eqs. 11 and 12 yields
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| (A8) |
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| (A9) |
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| (A10) |
 | (A11) |
Effects of Local Vascular Autoregulation
Bilinear transformation of the continuous-time transfer relation KCOTPR(s) characterized by the first-order LTI system in Eq. 26 yields
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| (A12) |
Pa(s) in Eq. 25 characterized by the third-order closed-loop system
 | (A13) |
 | (A14) |
=1G{KCOTPR}/G{KPaTPR} and the system order n depends upon the intrinsic value of the feedback parameter Tbr and the choice of fs because n=3 + fsTbr. Thus by rearranging Eq. A14 as increasing factors of z–1 and deliberately selecting for simplicity purposes the sampling frequency fs=1/Tbr=1/Tar, we arrive at
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| (A15) |
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| (A16) |
 | (A17) |
Pa(z), KPaTPR(z), KPRATPR(z), KCOTPR(z), and HCOPa(z) from Eqs. A5, A8, A9, A12, and A14, respectively, into the equivalent discrete-time frequency-domain representation of Eq. 28 expressed as
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| (A18) |
 | (A19) |
 | (A20) |
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| (A21) |
 | (A22) |
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GRANTS |
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TOPABSTRACTMATHEMATICAL ANALYSIS OF THE...DETERMINATION OF TPRCARDIOVASCULAR SYSTEM...APPENDIXGRANTSREFERENCES |
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FOOTNOTES |
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The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
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REFERENCES |
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TOPABSTRACTMATHEMATICAL ANALYSIS OF THE...DETERMINATION OF TPRCARDIOVASCULAR SYSTEM...APPENDIXGRANTSREFERENCES |
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