Compartmental and Data-Based Modeling of Cerebral Hemodynamics: Linear Analysis.

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IEEE Access. Author manuscript; available in PMC 2016 October 19. Published in final edited form as: IEEE Access. 2015 ; 3: 2317–2332. doi:10.1109/ACCESS.2015.2492945.

Compartmental and Data-Based Modeling of Cerebral Hemodynamics: Linear Analysis B.C. Henley, Department of Biomedical Engineering, University of Southern California, Los Angeles, CA 90089 USA

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D.C. Shin [Member IEEE], Department of Biomedical Engineering, University of Southern California, Los Angeles, CA 90089 USA R. Zhang, and Southwestern Medical Center, University of Texas V.Z. Marmarelis [Fellow IEEE] Department of Biomedical Engineering, University of Southern California, Los Angeles, CA 90089 USA B.C. Henley: [email protected]; D.C. Shin: [email protected]; R. Zhang: [email protected]; V.Z. Marmarelis: [email protected]

Abstract Author Manuscript

Compartmental and data-based modeling of cerebral hemodynamics are alternative approaches that utilize distinct model forms and have been employed in the quantitative study of cerebral hemodynamics. This paper examines the relation between a compartmental equivalent-circuit and a data-based input-output model of dynamic cerebral autoregulation (DCA) and CO2-vasomotor reactivity (DVR). The compartmental model is constructed as an equivalent-circuit utilizing putative first principles and previously proposed hypothesis-based models. The linear input-output dynamics of this compartmental model are compared with data-based estimates of the DCA-DVR process. This comparative study indicates that there are some qualitative similarities between the two-input compartmental model and experimental results.

Index Terms

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Cerebral autoregulation; compartmental modeling; nonparametric modeling; vasomotor reactivity

I. INTRODUCTION Dynamic cerebral autoregulation (DCA) and CO2 vasomotor reactivity (DVR) are two fundamental aspects of cerebral vascular function that are commonly studied within the framework of mathematical modeling. The proper functioning of DCA and DVR in regulating cerebral blood flow (CBF) in the presence of blood pressure fluctuations and

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changes in metabolic activity is essential for brain health and survival. DCA involves the regulation of cerebral blood flow within physiological bounds acceptable for brain health despite changes in perfusion pressure. DVR can be defined generally as the ability of the cerebral vascular bed to increase or decrease blood perfusion in response to hypercapnia (elevated CO2 production) or hypocapnia (reduced CO2 production) respectively, due to changes in ventilation and/or metabolic activity. Because systemic arterial blood pressure and blood CO2 tension fluctuates continuously throughout our daily lives, both regulatory processes function simultaneously to maintain cerebral blood flow within appropriate bounds. The effectiveness of both regulatory processes can diminish with age, supposedly as a result of the degeneration of neurovascular components serving the neurovascular unit [3, 5]. This point is consistent with the developing evidence implicating the impairment of the DCA-DVR process in vascular dementia, Alzheimer’s Disease (AD), and mild cognitive impairment (MCI) [1, 6, 10]. These observations have motivated efforts to develop models that can aid in both advancing our quantitative understanding of the DCA-DVR process and providing effective diagnostic tools capable of reliably detecting the onset of DCA-DVR impairment [6, 16, 19, 22–27]. In input-output modeling studies, subject-specific quantification of DCA is typically based on beat-to-beat measurements of mean arterial blood pressure (MABP) as the input and cerebral blood flow velocity (CBFV) as the output [6, 17, 18]. In addition, arterial CO2 tension, approximated from measurements of end-tidal CO2 (ETCO2), is often included as a second input in order to assess DVR [7, 9, 13–14, 22– 27]. This two-input/single-output model quantifies the causal relationship between MABP, ETCO2, and CBFV. If accurate, these models are valuable because all three physiological variables can be reliably and non-invasively acquired from human subjects in a comfortable clinical setting.


Mathematical modeling studies have attempted to gain a physiological and quantitative understanding of cerebral hemodynamics through the use of parametric and nonparametric approaches. One commonly used class of parametric models is the lumped-parameter hemodynamic model in the form of RC circuits [6, 8–9, 11–14, 17–18, 21]. In this approach, the cerebral circulation is represented as a series of vascular compartments each representing the cerebral arterial, arteriolar, capillary, and venous segments within the cranium. The hemodynamics of each segment are modeled by a lumped resistance and a lumped compliance, resulting in multi-compartment Windkessel model. These lumped parameters are viewed as being related to structural and functional characteristics of the cerebral circulation. For instance, if one could represent all the vessels within a vascular segment by a single “representative” vessel, then the resistance of a vascular segment is related to the inner radius of that representative vessel. The compliance of a vascular segment accounts for the ability of the vessel to distend in response to increases in transmural pressure, thus increasing blood volume. Nonparametric modeling forms have been offered as an alternative approach to the quantitative study of cerebral hemodynamics [7, 22–27]. The main appeal of these models is that they are obtained only from the available data and do not rely on any postulates about system function or structure. Transfer function analysis was initially used to quantify the impedance characteristics of the cerebrovascular bed, and was proven useful in demonstrating that cerebral autoregulation increases vascular impedance in the lowIEEE Access. Author manuscript; available in PMC 2016 October 19.

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frequency range (< 0.1 Hz) – i.e. for relatively slow fluctuations in cerebral blood pressure and flow [7, 22–27]. While these measures are useful and easily obtained, it is often difficult to interpret them in terms of physiological mechanisms. For this reason, a number of modeling studies have combined both modeling methods as a first step towards physiological interpretation [19–21]. While it is known that both the DCA and DVR processes exhibit nonlinear and non-stationary dynamics, most studies to date focus on the linear and time-invariant relation between these two models [6, 15, 16, 19, 21]. In the studies by Gommer, et al. [6] and Zhang, et al. [16–17], a constant parameter three-element windkessel model was analyzed and compared to empirical results using transfer function analysis. Payne, et al. implemented Ursino’s flow-dependent feedback to emulate the DCA process and demonstrated a way to estimate indirectly the feedback [19]. Arterial CO2 tension was introduced as a second input to address the question of whether CO2 was the physiological basis for the reduction in low-frequency coherence [21], but the question of how the equivalent nonparametric CO2-CBFV relationship in this physiological model related to experimental results was not raised. Using a similar physiological model based on the modeling concepts of Ursino, this paper aims to examine the relation between a twoinput parametric and nonparametric model of linear and time-invariant DCA-DVR process.

II. Modeling Methods A. The Cerebral Hemodynamic Model (CHM) A “vascular compartment” is the system of equations relating the pressure, flow and volume in a particular vascular segment. The CHM can be viewed as having two main vascular compartments: arterial and venous. The system of equations describing the arterial compartment is as follows:

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Arterial volume variation and conservation

(1)

(2)

Large arterial flow

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(3)

Small arterial (microvascular) flow

(4)

Small arterial resistance-volume relationship

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(5)

Arterial compliance

(6)


In these six equations, pressure is represented by the letter p, volume by the letter v, flow by the letter q, resistance and compliance respectively by the letters r and c, and the regulatory state variables by the letter u. There are two arterial and one capillary resistance. The large arterial resistance rla,0 represents the internal carotid and large cerebral arteries, concluding the middle cerebral artery. The small arterial resistance rsa(t) represents pial and parenchymal arteries/arterioles that lie on the surface and penetrate the cerebral tissue, and rc,0 represents the cerebral capillaries. In Eq 6, it is assumed that arterial compliance ca(t) is a function of the linearly combined state of DCA and DVR (Eqs 8 & 10), where ca,0 is baseline compliance. This assumption is adopted from Ursino, et al. and is adopted in several other modeling studies [11–12, 19, 22]. The pressure variables pA(t) and pa(t) represent respectively the systemic and cerebral arterial pressure, where pA(t) is taken as the MABP input. The small arterial resistance and arterial volume vA(t) is related through geometrical and physical laws in Eq 5, where va,0 is baseline arterial volume. A compartment is assumed to be geometrically cylindrical with uniform radius and length. Assuming laminar flow in Poiseuille’s law, resistance is inversely proportional to the fourth power of the inner radius and therefore the second power of volume. As a result, an increase in volume leads to a decrease in small arterial resistance (vasodilation), while a decrease in volume to leads to an increase in resistance (vasoconstriction). A single conservation of volume equation represents the venous compartment: Cerebral venous compartment

(7)

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where rv,0 and cv,0 is baseline venous resistance and compliance, representing the cerebral venules and veins. The two pressures pv(t) and pv,0 are respectively the cerebral and systemic venous pressure. As in [11–12], it is assumed that the DCA and DVR processes can be described individually by a single first order differential equation: Regulatory Compartment

(8)


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(9)

(10)

(11)

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The DCA process is driven by the fractional change in microvascular flow from baseline xq(t), with a linear gain gq and a first order delay characterized by time constant τq (Eqs 8 & 9). The DVR process is driven by the fractional change in microvascular flow from baseline xγ(t), with a linear gain gγ and a first order delay characterized by time constant τγ (Eq 10 & 11). The two parameters of interest in this study are the two gains gq and gγ and DCA and DVR. In this paper, Ursino’s assumption is slightly amended in that the fractional changes in the stimulating variables are scaled by baseline arterial compliance ca,0 in order to relate these two stimuli to the fractional change in compliance. That is to state, during steady-state the following expression holds: (12)

B. The Nonparametric Model

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The other aspect of this study involves the linear and time invariant (LTI) relationship between MABP, ETCO2, and CBFV estimated directly from the input-output data. Inputoutput models have been used in the study of cerebral hemodynamics using Laguerre expansions of the impulse response function (IRF) to achieve high estimation accuracy [7, 23–27]. In the first-order (linear) case, the two inputs MABP and ETCO2 are transformed into the single output CBFV through two convolution integrals, as indicated in Eq. 13. The unknown functions in the two convolutions, kP(τ) and kΓ(σ), are the impulse response functions (IRF) of the respective input. These two IRFs must be estimated from input-output data. The input-output dynamic relationship of the two-input LTI system is expressed as

(13)

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where F0 is the baseline flow velocity at the insonated MCA. The variables of interest are represented by discrete samples. In other words, the two input signals ps(t) and γ(t) are defined at t = n · T, where n is the sample number and T is the sample period. This implies that the estimated IRFs are discrete. An efficient method for the estimation of the discrete IRFs is the Laguerre Expansion Technique (LET) [27]. The two input signals are convolved by a set of discrete Laguerre functions {bp(τ)} and {bγ(τ)}, referred to as the Laguerre filter bank:


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(14)

(15)

The two variables, vp,i(t) and vγ,i(t), should not be confused with the unmeasured cerebral arterial volume va(t) predicted in the compartmental model. A linear combination of the resulting set of filter bank outputs {vp(t)} and {vγ(t)} are then summed to generate a prediction of the system output F(t): (16)

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Both inputs have their own filter bank, each characterized by two key parameters, L and α. The parameters Lp and Lγ denote respectively the number of discrete Laguerre function in the filter bank for the MABP and ETCO2 inputs. The parameters αp and αγ denote the relaxation time of the filters for the inputs. The main consequence of the LET is that the modeling problem becomes the estimation of the Lp coefficients {cp} and Lγ coefficients {cγ}. Under the linear case, this means there are Lp + Lγ + 1 degrees of freedom in this nonparametric model, including the baseline term F0. C. Data Processing

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The physiological measurements of blood pressure and flow velocity were reduced to beatto-beat values by averaging the respective signal within each R-R interval (guided by the provided ECG), which were subsequently resampled via cubic-spline interpolation every 0.25 s. The end-tidal CO2 measurements were also resampled via cubic-spline interpolation every 0.25 and shifted by 4 s (16 sampling points) to account for the physiological delay between end-tidal and blood CO2 tension (approximately the length of an average breath). The resulting time-series data were high-pass filtered above 0.008 Hz to remove strong physiological components in the very low frequencies that are deemed unrelated to the dynamics of cerebral autoregulation and reactivity [23–27].

III. RESULTS A. Derivation of the apparent transfer function of the CHM

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In order to serve the main purpose in this paper of relating the linear dynamics predicted by the CHM to experimental measurements, we derive the apparent transfer function of the CHM, or the linear CHM (LCHM). This is accomplished by obtaining the first degree Taylor series approximation of Eqs 2, 4, and 5, which are nonlinear due to the variation in small arterial resistance and compliance, and the dependence of resistance on blood volume: (17)


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(18)

(19)

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The superscript (1) connotes the first degree Taylor series approximation. These equations can replace Eqs 2, 4, and 5 to give a system of linear and time-invariant equations that characterize the LCHM. As a result, the equivalent transfer function can be derived by converting each equation into the Laplace domain. Before doing this, two simplifying steps are taken. First, venous and intracranial pressure pV,0 and pic,0 are both small in magnitude relative to systemic arterial and cerebral pressures (pV,0 ≈ 6 mmHg, pic,0 ≈ 10 mmHg), and therefore are taken to be zero. Second, the constants pa,0 and va,0, representing the cerebral arterial pressure and volume, can be eliminated by considering that, under basal conditions with pV,0, = pic,0 = 0,

(20)

It then follows that

(21)

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and

(22)

After applying these simplifications, the following three equations are derived to express the LCHM in the Laplace domain: (23)

(24)

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(25)


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where . These three equations can now be used to eliminate the superfluous variables, Pa(s) and Qsa(s), to solve for the model output Qla(s) as an algebraic function of the two inputs Ps(s) and Γ(s): (26)

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The two functions HP(s) and HΓ(s) are the desired transfer functions (TF) of the MABP and ETCO2 input, respectively. Since the measured system output is flow velocity (cm/s), and the model output predicted by the LCHM is volumetric flow (ml/s), a conversion step is necessary. As in other model modeling studies, it is assumed that volumetric flow can be converted to flow velocity by diving volumetric flow by the typical cross-sectional area of one MCA, which is taken to be about Amca = 0.21 mm2 [6]. We further assume that the insonated MCA is a conduit to a third of the total large arterial flow, as in [11–12]. As a result,

(27)

where Cmca = 1/3. Cerebral venous resistance, modeled by rlv, is relatively small [19] and has little impact on the model dynamics (not shown). Therefore, the two transfer functions are simplified further by evaluating HP(s) and HΓ(s) as rlv approaches zero. Consequently, the TF of the MABP input has two poles and two zeros:

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(28)

The TF of the ETCO2 input has three poles and two zeros

(29)

The TF coefficients {bP, aP} are functions of the lumped resistances, compliance, and the gain and rate of autoregulation. The TF coefficients {bΓ, aΓ} are also functions of these parameters, as well as the gain and rate of reactivity (see Appendix). Both TFs can be expressed in terms of their poles by obtaining the partial fraction expansion of the two TFs:

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(30)

(31)


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where {R}and {λ} are respectively the set of residues and corresponding poles for each input. The TF of the MABP input is also characterized by a single direct term kp. Table 1 displays the continuous components for both TFs. It is observed that the two poles of HP(s) are complex. Among the three poles of HΓ(s), one is real and the two complex poles are equal to those of HP(s). It is also observed that the denominator coefficients of the ETCO2 input {aΓ} can be expressed as a function of the denominator coefficients of the MABP input {aP} (see Appendix). B. Comparison between the LCHM and Experimental Model

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While the LTI input-output models estimated from the eight healthy subjects are obtained in discrete time, the LCHM is expressed in continuous time. Therefore, the LCHM is converted to the discrete domain in a manner that matches the discrete and continuous poles so that the respective IRFs are equal at the sampling points. This is achieved by employing the Impulse Invariant Method [4]. In particular, the continuous TF and its discrete equivalence, H̃X(s), can be related as follows: (32)

where the independent variable z denotes the discrete complex variable. The dummy variable X = P or Γ respectively for the MABP or ETCO2 input and ΛX is the number of poles for that input. The parameter ρi,X denotes the continuous pole and Ri,X denotes the ·T corresponding residue. Thus, the discrete poles for either input are {eρi,X& } and can be evaluated using the continuous pole values in Table 1. With these parameters, the discretetime IRF of either input can be computed using the following expression

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(33)

where t = n · T and n is a non-negative integer denoting the nth sample of the discrete IRF. The variable δ(t) is the Dirac delta function. The respective FR can be computed by evaluating H̃X(z) for z = ei·2·π·f, where i is the imaginary number.

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Figure 2 and 3 shows the theoretical and experimental IRFs and FRs for the MABP and ETCO2 input, respectively. The system normalized mean square error (NMSE), or measure of discrepancy between the theoretical and experimental IRFs for the MABP and ETCO2 input is about 0.18 and 0.11. The main linear dynamics predicted by the LCHM and observed experimentally are qualitatively similar. The experimental and theoretical IRFs of the MABP input, kp(t) and hp(t), both exhibit a relatively strong positive value at zero-lag, then a subsequent undershoot. The corresponding frequency-domain representations reflect the anticipated high-pass filtering of the cerebral vasculature. Two main discrepancies between these models are the shape of the undershoot and presence of resonant peaks. In the theoretical IRF, the undershoot occurs sharply as hp(t) takes on negative lag values immediately after zero-lag. On the other hand, the undershoot in the experimental IRF


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begins after about a half second. In the frequency domain, the theoretical FR does not exhibit the resonant peak around 0.10 Hz and blunt peak 0.35 Hz observed experimentally. The experimental and theoretical IRFs of the ETCO2 input, kγ(t) and hγ(t), both exhibit an integrative response consistent with the expected physiological rise in CBFV during a rise in CO2 tension (Figure 3). In addition, both models exhibit two positive phases. The first phase has a relatively strong peak (at the zero-lag) and short duration, while the second phase has a relatively small peak with a longer duration. However, the second phase begins a little faster and has stronger peak value in kγ(t) than in hγ(t). In the frequency domain, the anticipated low-pass filtering anticipated by the mostly integrative response in the DVR process is observed in both models, with a resonant peak around 0.15 Hz that is more blunt in the theoretical model.

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C. Two-Input Dynamic Simulation Results In light of the qualitative similarity between the LCHM and experimental model, we now explore the main effects of resistance, compliance, and the status of DCA-DVR on the main features of the two-input system response. To do this, we simulate the LCHM after raising and lowering individually the model parameters of interest from their control value and record the predicted model response. The DCA process can be observed by applying a step signal at the MABP input with the ETCO2 input set to baseline. Likewise, the DVR process can be observed by applying step signal at the ETCO2 input with the MABP input set to baseline. The effect of resistance, compliance, and the gain of autoregulation on the step response to MABP and ETCO2 is illustrated in Figures 6–10. There are three main features of interest in the step response to MABP (Figure 4):

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1.

Initial rise value (IRVp), cm/s: The maximum value in the SRFp. In the LCHM, this is the value of the SRFp at the onset of the step signal (line 1)

2.

Settling time (STp), s: The elapsed time for the SRFp to reach and remain within a specific tolerance about its asymptotic value. A tolerance of ±2% is used throughout this paper (line 4).

3.

Rate of restoration (RoRp), cm/s2 (line 5): The initial slope of the restoration phase in the SRFp. This is computed by taking the ratio of the difference between the IRVp (line 1) and the minimum value (line 2) in the SRFp to time duration between these two points (line 3).

There are two main features of interest in the step response to ETCO2 (Figure 5):

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1.

Rise Time (RTγ), s: The elapsed time for the SRFγ to first reach 90% of asymptotic value (line 2).

2.

Rate of Reactivity (RoRγ), cm/s2: The slope from baseline to the point SRFγ first reaches 90% (line 1) of its asymptotic value. This is computed as the value of the SRFγ at RTγ, divided by RTγ.(line 3).

The percent change in each feature in SRFp and SRFγ due to the individual raising and lowering of each parameter by a factor of two is reported in Tables 3 & 4. In addition, the


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frequency spectra for both inputs are recorded to provide an illustration of how each parameter affects vascular impedance. It is important to note that the control value of each parameter is not determined experimentally, but set manually to reproduce the main linear dynamics observed in the experimental results. The nominal values reported in literature were initially used [19], however some of the parameters were adjusted, including the arterial resistances, compliance, and the parameters of the DCA and DVR mechanism, in order to attain a better agreement with the experimental IRFs. The values of the stead-state lumped resistances rla, rsa, and rcv are adjusted to inspect their effects on the DCA and DVR processes. Since a change in steady state resistance changes in the total cerebral vascular resistance, the steady-state CBF will also change (for the same level of systemic arterial pressure). As a result, it is assumed that the constant q0 is a function of these three resistances:

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where kq is a constant set to ensure q0 = 12.5 ml/s when the three resistances are each at their respective control values.

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For the MABP input, it is observed that raising large arterial resistance rla has the main effect of decreasing IRVp (−45.0%) and increasing STp (+10.0%). In the frequency domain, increased rla mainly results in elevated HF impedance (Figure 6). The dependence of IRVp on rla is explained in the LCHM by recalling that the limiting value of HP(s) as s approaches infinity is exactly equal to 1/rla. The dual of this in the time domain implies the zero-lag value is proportional to 1/rla. Since the LCHM is linear, the SRFp is equal to the integral of the hp(t). As a result, the initial-value in the SRFp is determined by the zero-lag value in hp(t) and thus has the same dependence on rla. It is also observed that increased rla is predicted to impair the ability of the cerebral vasculature to restore CBFV, as indicated by the decrease in SoRp. Qualitatively, this can be anticipated because the elevation in rla results in less cerebral arterial inflow, leading to less microvascular flow (arterial outflow). Since the LCHM assumes that DCA is activated by changes in microvascular flow, the DCA process is less active for the same DCA gain and the increase in cerebral vascular resistance required for flow restoration is abated.

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For the ETCO2 input, the overall response of CBFV to CO2 tension is reduced during an elevation in rla (Figure 6) and specifically causes a decrease in SoRγ. During cerebral arterial expansion as the compliance increases (Eqs 17 and 18), the reduction in arterial inflow leads to less arterial volume increase in response to increased CO2 tension. Consequently, there is a smaller decrease in small arterial resistance during vasodilation resulting in a weaker increase in flow. Unlike the large arterial resistance, the small arterial or microvascular resistance rsa does not have a significant affect on HF impedance (Figure 7). An increase in rsa has the main effect of increasing STp, SoRp, and LF impedance. In addition, a blunt resonant peak at about 0.25 Hz is observed. This somewhat paradoxical observation suggests that elevated rsa, simulating microvascular stenosis, enhances DCA. While this may or may not be IEEE Access. Author manuscript; available in PMC 2016 October 19.

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physiological, the apparent improvement in autoregulation due to increased rsa predicted by the LCHM is because the reduction in arterial outflow leads to a greater rise in the arterial pressure pa for the same step increase in MABP causing arterial inflow (Eq 25). As a result, there is a stronger subsequent increase in microvascular flow that ultimately drives the DCA compartment more for the same gain. The opposite effect is observed for a decrease in rsa, and in this case, halving rsa abolishes the undershoot in the restoration phase. Interestingly, for the ETCO2 input, an increase in rsa is observed to increase the SoRγ, and is reflected visually as a stronger initial rise in the SRFγ(Figure 7). However, the asymptotic value and STγ is only slightly decreased. In the frequency domain, an increase in rsa pronounces the blunt peak at about 0.2 Hz. The effect of the capillary-venous is qualitatively similar to, but less accentuated than that of rsa. Thus, an illustrative example is not shown to avoid redundancy.


The last vascular element we consider is the arterial compliance ca (Figure 8). Compartmental “stiffness” can be simulated by decreasing ca below its control value. For the MABP input, it is observed that a decrease in ca results in a decrease in IRVp, but to a lesser extent than an increase in rla (about 35% less). A decrease in STp is observed, and is exactly the same magnitude as the increase in IRVp due to the proportional decrease in rla(Figure 6). The SoRp is also reduced, suggesting impaired autoregulation. This observation is anticipated in light of Eq 9 in the LCHM, where a decrease in baseline ca results in a weaker DCA for the same DCA gain. Qualitatively, this is sensible, because a stiffer arterial vascular bed should reduce reactivity to changes in cerebral perfusion pressure manifested in autoregulation. In the Frequency domain, decreased ca increases in the HF and LF range. An increase in compliance results in the opposite effect, and most significantly reduces the damping in the SRFp while increasing STp such that there are longer lasting significant transient values. The SRFγ of the ETCO2 input is influenced by compliance such that a decrease in ca slightly increases STγ (+4.00%) and decreases SoRγ (−8.75%). Qualitatively, the increased STγ by 4.00% and the fall in SoRγ by about 8.75% is relatively small, compared to the effect of the other parameters. However, similar to the effect of the microvascular resistance, a more compliant arterial compartment promotes a more oscillatory initial response in the SRFγ. This is due to the increased DCA response for the same DCA gain (Eq 9). In the frequency domain, we observe a resonant peak at about the same location in the FR of the MABP input. Qualitatively, this prediction in the LCHM is anticipated, since a more compliant vessel is viewed as connoting a vessel that is more capable to expand or contract transiently during CO2 induced vasodilation.

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Finally, we consider the impact of the gain of autoregulation gq and reactivity gγ on the system dynamics. Payne, et al, using a similar model, noted the effect of gq on the form of the effective IRF of the MABP input and illustrated the main point that an increase in gq significantly increases the undershoot [19]. The initial undershoot in the IRFp corresponds to the initial restoration phase in the SRFp. Thus, Payne’s observation is consistent with the more pronounced initial restoration phase due to an increased gq (Figure 9) and is reflected in a strong increase in SoRp (+135%). Also, the restoration phase becomes more oscillatory


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with a slower STp (+90%). However, similar to rsa, the IRVp is invariant to gq. Generally, a reduction in gq, mimicking impaired autoregulation, has the opposite effect. In the case of halving gq in this model, the restoration phase is critically damped. In the impedance spectrum, it is observe that enhanced autoregulation increases LF impediance, as anticipated. In addition, the more underdamped response in the SRF is reflected by the emergence of a resonant peak at about 0.22 Hz, similar to the effect of increased rsa (Figure 7). On the other hand, impaired autoregulation (reduction in gq) reduces LF impedance, and thus weakens the high-pass behavior in which the cerebral vascular system naturally operates.

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Because the physiological response to CO2 tension increase is an increase in CBF, and CBF drives the DCA process in the LCHM, then the DVR process is anticipated to depend on autoregulation status as well. Turning to the ETCO2 input, it is observed that an increase in gq reduces the over response to CO2 tension. It is observed that both STγ and SoRγ is reduced, and the initial response become more oscillatory, similar to the effect of increased compliance and microvasuclar resistance. The resonant peak that emerged in the FR of the MABP input also appears in the spectrum of the ETCO2 input. On the other hand, impaired autoregulation (reduced gq) increases STγ and SoRγ.

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Lastly, we consider the effect of the reactivity gain gγ. In this model, the pressure-flow relationship is not affected by this parameter, so the FR and SRF of the MABP input are not included here. Qualitatively, an increase in gγ, mimicking enhanced reactivity, produces generally a stronger integrative response to a step change in CO2 tension while a decrease in gγ, mimicking impaired reactivity, produces a weaker response. Because this basic prediction in the LCHM is fairly clear and straightforward, it is not included here in the interest of space. More interesting is the question of how the DVR process if affected by different combinations of autoregulation and reactivity status. This is illustrated in Figure 10, which shows the SRFγ for controlled DCA and DVR (Plot 1), impaired DCA and controlled DVR (Plot 2), controlled DCA and impaired DVR (Plot 3), and both impaired DCA and DVR. The important observation here is that the decreased SoRγ resulting from a decrease in gγ is less severe for the same DVR impairment with a simultaneous decrease in gq. This simple simulation result illustrates generally that the anticipated reduction in the integrative strength of the DVR process during impaired reactivity can look “less impaired” through visual inspection of the obtained SRFγ if DCA is also impaired. This implies the potentially important problem of “regulatory masking” that could result in arriving at falsenegative classification of DVR impairment.

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IV. Discussion The purpose of this study was to examine the relation between a hypothesized parametric model of the DCA and DVR process with physiological parameters of clinical significance to a nonparametric input-output model quantifying the linear dynamic effects of MABP and ETCO2 on CBFV. Obtaining a relation between these two modeling approaches provides a means to understand the main system characteristics obtained experimentally in terms of physiologically meaningful parameters. This, in turn, can provide a basis upon which modelbased indices of cerebral vascular status are derived using subject-specific input-output data.


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To serve this purpose, we rely on the widely used lumped-parameter modeling concept of the cerebral circulation and it’s interaction with regulatory compartments mimicking the DCA and DVR processes [11–12, 19, 21]. In the nonparametric approach, the linear dynamics effects of MABP and ETCO2 on CBFV are estimated in eight healthy human subjects. These estimates are then averaged to obtain one representative IRF for the MABP input and one for the ETCO2 input. Thus, this study combines the mechanistic modeling viewpoint of the DCA-DVR process with the inductive point of view, providing a synergistic view of this system.


Given the relative simplicity of the LCHM, the qualitative agreement between the parametric and nonparametric models is surprising (Figures 2 & 3). For the MABP input, the IRFs in both models exhibit a strong positive initial value and subsequent undershoot and reflect generally the expected high-pass characteristic (Figures 2). However, there are two main differences between these IRFs. First, the undershoot in the theoretical IRF hp (t) occurs immediately after the initial value, whereas the experimental IRF kp (t) begins at about the half-second lag point. The sharp transition in hp(t) from the zero-lag point to the immediate lag values is due to the fact that the pressure-flow relationship is essentially an all-pass term PA(s)/rla minus a second-order low pass term Pa (s)/rla, where PA (s) is MABP and Pa (s) is the “unseen” cerebral arterial pressure (Eqs 23 & 25). The dual of this in the time domain is a non-zero term that exists only at zero-lag and two stable negated complexconjugate exponentials (because the two poles in the TF of the MABP input are complex conjugate). The smooth characteristic of the finite number of Laguerre basis functions used to facilitate the experimental IRF estimation procedure cannot produce this discontinuity. Secondly, the average experimental spectrum reveals the presence of two resonant peaks at about 0.35 and 0.10 Hz. These two peaks are not observed in the LCHM. These peaks may reflect the action of pressure-driven regulatory control mechanisms that the theoretical model does not account for. Thus, a deeper model including more “regulatory compartments” able to produce resonant behavior consistent with what is observed experimentally should be explored. The system NMSE for the ETCO2 input was about 0.11. There is general qualitative agreement between the two models, specifically an integrative IRF constituting two positive phases. The first phase is has a large peak value (at zero lag) and short duration, while the second phase has a smaller peak value and relatively long duration (Figure 3). However, the first phase in the experimental IRF is shorter and the peak value of the second phase is larger than the theoretical IRF.

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In order to examine how the change in these parameters effect the linear behavior of the DCA and DVR process predicted by the LCHM, we focus on three main features in the SRFp and two features in the SRFγ. For the MABP input, we defined the IRVp, STp, and SoRp, which represent respectively the initial rise value, settling time, and slope of restoration in the SRFp (Figure 4). For the ETCO2 input, we defined the STγ, and SoRγ, which represent respectively the settling time and slope of reactivity in the SRFγ (Figure 5). Each parameter of interest was individually raised and lowered by a factor of two and the resulting SRFp and SRFγ was recorded, along with frequency spectra in order to give a


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visual understanding of the effect of these parameters on vascular impedance. There are four main findings, summarized here: 1. Elevated resistance at the level of the MCA corresponds with a decreased IRVp The LCHM predicts that the level of resistance at the site of the carotid and large cerebral arteries, modeled by the parameter rla, has a strong correspondence with the IRVp in the SRFp. This point is illustrated Figure 6, showing that an increase and decrease in rla causes IRVp to respectively decrease and increase. In the case of raising and lowering rla by a factor of two, the IRVp respectively decreases by 45% and increases by 60% (Table 3). It is also predicted that arterial compliance effects the IRVp, but to a lesser extent (10% increase and 20% decrease due to the raising and lowering of ca). 2. Reduced arterial compliance corresponds with a decreased STp and SoRp

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The LCHM predicts that the level of arterial compliance, modeled by the parameter ca, has a strong correspondence with the STp, and SoRp (Table 3). Doubling ca increases STp by 150%, and SoRp by 8%. On the other hand, decreasing ca, simulating cerebral arterial stiffness, decreases the STp by 8%, and SoRp by 88%. While the SoRp is anticipated to be most strongly dependent on the gain of DCA, it is not surprising that it is influenced by compliance, since the CHM assumed that DCA operates through the modulation of ca. 3. Impaired DCA corresponds with a decrease in the SoRp and STp

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4. Resistance, compliance, and DCA influence the DVR process. Impaired DCA can “mask” the extent of impaired DVR

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The LCHM predicts that the level of DCA gain gq has a strong correspondence with the STp and SoRp. Doubling gq increases the STp by 90% and SoRp by 135%. Halving gq, mimicking impaired DCA, reduces the STp by 25% and SoRp by 89%. In the LCHM, halving arterial compliance reduces the SoRp by almost the same amount (−88%). In addition, elevated larg arterial resistance reduces SoRp with a similar degree of magnitude (−72%). This seems to support the notion that the autoregulatory status is not just dependent on the effectiveness of active control mechanism, but is also a function of the passive fluid dynamic properties of the cerebral vasculature [23–26].

Measurements of the ability of the cerebral vasculature to response to changes in CO2 tension may be used in a clinical context for the diagnosis of vascular-related cognitive disorders, and may potentially be used as an indicator of vascular-related cognitive impairment [23–26]. While the theoretical model used in this paper is very simple, the qualitative agreement between the predicted linear dynamic ETCO2-CBFV relationship to experimental results motivates the question of how the parameters in the CHM can affect the DVR process. In the LCHM, the DVR gain gγ is the essential parameter determining the ability of the cerebral vascular system to response to CO2 tension. That is to say, under stable conditions, gγ is the only parameter that must be non-zero for CBFV to response to a change in CO2 tension. However, the LCHM predicts that resistance, compliance, and DCA gain can also influence the ETCO2-CBFV relationship. This is exemplified by changing these three parameter types and observing the change in the form of the SRFγ (Figures 6–


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10). For example, the two forms of vascular impairment that can be simulated in the theoretical model, elevated resistance due to stenosis and reduced compliance due to vascular wall stiffness, are both predicted to impair DVR (Figures 6–7, Table 3). Specifically, doubling rla and halving ca reduces the SoRγ respectively by 69% and 9%. Visual inspection of Figure 6 shows an overall decline in the strength of the DVR process due to an increase in rla such that even the steady-state value of CBFV is reduced. However in Figure 8, decreased ca has a greater influence on the initial transient, while not changing the steady-state value.

Author Manuscript

In addition to the vascular elements in LCHM, the theoretical model predicts the dependence of DVR on the status of DCA. Raising and lowering gq by a factor of two respectively decreases and increases SoRγ by 27% and 17%, indicating that DCA has an antagonistic effect on DVR (Figure 9). Figure 10 illustrates the main point that impaired DCA can make impaired DVR (Plot 3), look less impaired (Plot 4). In the case of halving gγ, SoRγ falls by 55%. However, halving gq during this DVR impairment decreases SoRγ by 45%. Clearly, the relative simplicity and limiting assumptions in the LCHM prevents the emulation of the true complexity in the DCA-DVR process. However, the qualitative similarity between the theoretical and experimental models are encouraging, and suggest a means to develop deeper models capable of explaining more complex system behavior. To elaborate, we summarize the main modeling assumptions and their limitations that can be addressed in future work: 1. Geometrical, biophysical, and mechanical description of the cerebral circulation


The lumped-parameter modeling concept relies on the basic assumption of a vessel as a cylindrical tube with uniform diameter and fix length, referred to as a “vascular compartment.” In the context of cerebral hemodynamics, the purpose of a vascular compartment is to summarize concisely the fluid dynamic properties of a particular type of vessel along the cerebral vascular system. In the CHM and, many others, the two basic properties are hydraulic resistance and compliance that relate compartmental flow and volume to compartmental pressure [6, 8–9, 11–14, 17–18, 21]. Because of this geometrical assumption, the notion of vascular resistance can be related to inner radius and ultimately volume to model the effect of DCA and DVR on CBFV (Eq 5 & 18). Of course, the cerebral circulation is drastically more complex than the arrangement of perfect cylinders. Further, we assume that the notion of vascular compliance is mechanically analogous to electrical capacitance. However, this assumption is also challengeable, as one could image the comparison between a very elastic narrow tube with a very stiff wide tube. The narrow tube has less “capacitance” than the wider tube because it holds less volume for the same pressure, but is be more “compliant,” because it undergoes greater radial distention for the same change in transmural pressure. 2. Single-Compartment Assumption in the DCA and DVR processes The DCA and DVR process involves the aggregate effect of (at least) myogenic, neurogenic, and metabolic mechanisms that simultaneous act on the cerebral vasculature in response to


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changes in system arterial pressure and CO2 tension. The LCHM takes for granted that the “aggregate effect” in either process can be summarized by a single compartmental (differential equation) with one gain and one time constant. It was demonstrated in previous work that the single compartment assumption for the DCA process can still emulate the basic delayed counter-regulation of CBFV during a change in MABP [19], and this paper illustrates that the single-compartment assumption may be adequate to produce qualitatively similar linear dynamics observed in the DVR process as well. As a result, this comparison study can offer a means to interpret the form of the LTI system model in terms of general statements, such as “elevated resistance” and “impaired DCA or DVR,” but cannot be used to interpret more deeply the effectiveness of specific mechanisms, such as myogenic, autonomic, and local metabolic status. In order to achieve this, the single compartment assumption ought to be replaced with the interconnection of multiple compartments that can be driven by different variables (measured and unmeasured). For instance, an “autonomic compartment” driven by system arterial pressure (the measured MABP) can by posited as a model of the autonomic mechanism representing the pathway from the peripheral baroreceptors to the perivascular nerves innervating the pial arterial wall. In addition, a “myogenic compartment” driven by the unmeasured cerebral arterial transmural pressure ( pa – pic) can be hypothesized. While plausible, it will of course require the challenging task of specifying the appropriate dynamics (time constants) of each compartment. 3. Linear and Time-Invariant Constraint of the LCHM


It is widely understood that the pressure-flow and CO2-flow relation ship is nonlinear [7, 22–27]. Steady-state studies involving the forcing of MABP over the range of 40–200 mmHg consistently reveal the traditional autoregulatory curve, in which the pressure flow relationship is relatively weak between about 50–150 mmHg, but relatively strong outside this range [9, 11–12]. For the CO2-flow relationship, a soft saturating relationship can be consistently observed under physiological conditions at about 20 and 70 mmHg [9, 11–12]. In dynamic studies analyzing the relationship between time-series input-output data, it has been shown that second and third order Volterra two-input models improve the model predictive power [24–27]. The models in [11–12] and [19,21] incorporate these two static nonlinearities. However, because the analytical derivation of the equivalent second and third order kernels is mathematically unwieldy, we limit our derivation to first order [19]. At this point, it is important to recall that our main goal here is to derive model-based indices that can be used to infer the cerebral vascular and regulatory status on subject-specific basis using beat-to-beat data measured under resting conditions. This is important because the dynamic range of MABP typically does not reach 50 and 150 mmHg under these resting conditions, especially when the subjects are in supine position. For instance, the average dynamic range of the MABP data between the 8 eight subjects used in this study was about 83 to 104 mmHg. In addition, the values in the ETCO2 data typically do not reach 20 and 70 mmHg. The average dynamic range of the ETCO2 data between the eight subjects is about 28 to 43 mmHg. Therefore, we must explore the issue of whether the linear constraint is adequate for our clinical purposes, or if nonlinearities must be accounted for. It would be instructive however to examine how the nonlinearities obtained experimentally relate to that predicted in the CHM.


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Secondly, it has been suggested that the DCA and DVR processes are time-varying, possibly due to the effects of intermittent neural, endocrine, and metabolic activity [22]. It would be intriguing to test proposed parametric models of these phenomenona, however it is unclear if modeling non-stationary behavior is necessary to achieve our clinical goal.

Author Manuscript

The main conclusion here is that it is possible to relate lumped-parameter modeling of the DCA-DVR process incorporating physiologically meaningful parameters to experimental results in a linear and time-invariant context. In light of this conclusion and aforementioned modeling assumptions, the important implication is that it may be possible to infer the status of cerebral vascular function in light of clinically relevant physiological concepts defined by parametric models, if the proposed indices can be tested for accuracy and their ability to delineate significant differences between clinical cohorts. Specifically, the IRVp, STp, and SoRp in the SRFp may be used to infer the level of resistance, compliance, and effectiveness of autoregulation. The status of DVR may be measured using the SoRγ in the SRFγ, however this information should be reported with an index of DCA.

Acknowledgments This work was supported in part by NIH grant P41 EB-001978 to the Biomedical Simualtions Resource at USC.

VII. REFERENCES


1. Abeelen AS, et al. Impaired cerebral autoregulation and vasomotor reactivity in sporadic Alzheimer’s disease. Curr Alz Res. 11:11–7. 2. van Beck AH, et al. Cerebral autoregulation: overview of current concepts and methodology with special focus on the elderly. J Cereb Blood Flow Metab. 2008; 28:1071–1085. [PubMed: 18349877] 3. Viswanathan A, Greenberg SM. Cerebral Amyloid Angiopathy in the Elderly. Ann Neurol. 2011; 79:871–80. [PubMed: 22190361] 4. Lathi, BP. Linear Systems and Signals. 2. Oxford Univ Press; 2005. p. 371-72. 5. Iadecola C. Neurovascular regulation in the normal brain and in Alzheimer’s Disease. Nat Rev Neurosci. 2004; 5:347–360. [PubMed: 15100718] 6. Gommer ED, et al. Dynamic cerebral autoregulation in subjects with Alzheimer’s disease, mild cognitive impairment, and controls: evidence for increased peripheral vascular resistance with possible predictive value. J Alz Dis. 2012; 30(4):805–813. 7. Mitsis GD, et al. Nonlinaer modeling of the dynamic effects of arterial pressure and CO2 variations on cerebral blood flow in healthy humans. IEEE Trans Biomed Eng. 2004; 51:1932–1943. [PubMed: 15536895] 8. Chan GS, et al. Contribution of arterial Windkessel in low-frequency cerebral hemodynamics during transient changes in blood pressure. J Physiol. 2011; 110(4):927–25. 9. Lu K, Clark JW. Cerebral Autoregulation and Gas Exchange Studied Using a Human Cardiopulmonary Model. AJP-Heart. 2003; 286:584–601. 10. Glodzik L, et al. Cerebrovascular reactivity to carbon dioxide in Alzheimer’s disease. A review. J Alz Dis. 2013; 35(3):427–440. 11. Ursino M, et al. Interaction among autoregulatino, CO2 reactivity, and intracranial pressure: a mathematical model. Am J Physiol Heart Circ Physiol. 1998; 274:5. 12. Ursino M, et al. Cerebral hemodynamics during arterial and CO2 pressure changes: in vivo prediction by a mathematical model. Am J Physiol Heart Circ Physiol. 2000; 279:2439–2455. 13. Aaslid R, et al. Cerebral autoregulation dynamics in humans. Stroke. 1989; 20:45–52. [PubMed: 2492126] 14. Rao R. The role of Carotid Stenosis in Vascluar Cognitive Impairment. Eur Neurol. 2001; 46:63– 69. [PubMed: 11528153] IEEE Access. Author manuscript; available in PMC 2016 October 19.

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15. Zhang R, et al. Transfer function analysis of dynamic cerebral autoregulation in humans. Am J Physiol. 1998; 274:233–241. 16. Zhang R, et al. Dynamic pressure-flow relationship of the cerebral circulation during acute increase in arterial pressure. J Physiol. 2009; 587(11):2567–77. [PubMed: 19359366] 17. Daun S, Tjardses T. Modeling, analysis, and calculation of cerebral hemodynamics. Comp & Math Methods in Med. 2007; 8:205–223. 18. Dhoat S, et al. Vascular compliance is reduced in vascular dementia and not in Alzheimer’s diease. Age & Aging. 37:653. 19. Payne SJ, Tarassenko L. Combined Transfer Function Analysis and Modelling of Cerebral Autoregulation. Annals of Biomed Eng. 2006; 34(5):847–58. 20. Payne SJ. A model of the interaction between autoregulation and neural activation in the brain. Math Biosci. 2006; 204:260–281. [PubMed: 17010387] 21. Hughes TM. Pulse wave velocity is associated with β-amyloid deposition in the brains of very elderly adults. Neurology. 2013; 18(19):1711–18. [PubMed: 24132374] 22. Peng T, et al. Multivariate System Identification for Cerebral Autoregulation. Annals of Biomed Eng. 2008; 36:308–20. 23. Marmarelis VZ, et al. Time-Varying Modeling of Cerebral Hemodynamics. IEEE Trans Biomed Eng. 2014; 61(3):694–704. [PubMed: 24184697] 24. Marmarelis VZ, et al. Closed-loop analysis of cerebral flow autoregulation using Principal Dynamic Modes. Annals of Biomed Eng. 2013; 41(5):1029–1048. 25. Marmarelis VZ, et al. Linear and nonlinear modeling of cerebral flow autoregulation using Principal Dynamic Modes. Biomed Eng J. 2012; 6:42–55. 26. Marmarelis VZ. Model-based Quantification of cerebral hemodynamics as a physiomarker for Alzheimer’s disease? Annals of Biomed Eng. 2013; 41:2296–2317. 27. Marmarelis, VZ. Wiley-Interscience. 2004. Nonlinear Dynamic Modeling of Physiological Systems; p. 100-4.

V. APPENDIX Author Manuscript

The two TFs, HP(s) and HΓ(s), are derived after linearizing the nonlinear equations in the CHM (Eq 2, 4, & 5). In order to reduce the complexity of the obtained TFs, the venous compartment is ignored by taking the limit of the TFs as the venous resistance rlv approaches zero. With this simplifying step, the transfer function coefficients for the MABP input are as follows: (A.1)

(A.2)

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(A.3)

(A.4)


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(A.5)

(A.6)

The transfer function coefficients for the ETCO2 input are as follows: (A.7)

(A.8)

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(A.9)

(A.10)

(A.11)

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(A.12)

(A.13)

Note that the denominator coefficients for the ETCO2 input can be expressed as a function of that of the MABP input.

Author Manuscript IEEE Access. Author manuscript; available in PMC 2016 October 19.

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Figure 1.

The equivalent hydraulic circuit model of the cerebral circulation (top) and modular representation of the CHM (bottom).

Author Manuscript Author Manuscript IEEE Access. Author manuscript; available in PMC 2016 October 19.

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Figure 2.

Experimental (top) and theoretical (bottom) of the IRF (left) and FR (right) of the MABP input.


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Figure 3.

Experimental (top) and theoretical (bottom) of the IRF (left) and FR (right) of the ETCO2 input.


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Author Manuscript Author Manuscript Figure 4.

Author Manuscript

The main features of the SRF of the MABP input. The maximum value in the SRFp is the IRVp. In the LCHM, this is the value of the SRFp at the onset of the step signal (line 1). The elapsed time for the SRFp to reach and remain with ±2% about its asymptotic value is taken as the STp (line 4). The initial slope of the restoration phase in the SRFp is taken as the SoRp (line 5).

Author Manuscript IEEE Access. Author manuscript; available in PMC 2016 October 19.

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Figure 5.

The main features of the SRF of the ETCO2 input. The elapsed time for the SRF of the ETCO2 input to first reach 90% of its asymptotic value (line 2) is the STγ. The slope from baseline to the point the SRFγ first reaches 90% (line 1) of its asymptotic value is the SoRγ.


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Figure 6.

The FR (top) and SRF (bottom) for the MABP (left) and ETCO2 (right) inputs for controlled (blue), increased (green) and decreased (blue) large arterial resistance, rla (mmHg · s/ml).


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Figure 7.

The FR (top) and SRF (bottom) for the MABP (left) and ETCO2 (right) inputs for controlled (blue), increased (green) and decreased (blue) small arterial resistance, rsa (mmHg · s/ml).


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Figure 8.

The FR (top) and SRF (bottom) for the MABP (left) and ETCO2 (right) inputs for controlled (blue), increased (green) and decreased (blue) arterial compliance, ca (ml/mmHg).


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Figure 9.

The FR (top) and SRF (bottom) for the MABP (left) and ETCO2 (right) inputs for controlled (blue), increased (green) and decreased (blue) gain of autoregulation, gq.


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Figure 10.

SRF of the ETCO2 input for both control gain of autoregulation gq and reactivity gγ (top left), decreased gq with control gγ (top right), control gq with decreased gγ (bottom left), and both decreased gq and gγ (bottom right).


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TABLE 1

Author Manuscript

CONTROL VALUES USED TO SIMULATE THE LCHM AND LAGUERRE PARAMETERS USED FOR IRF ESTIMATION Linear Cerebral Hemodynamic Model (LCHM)

Nonparametric Model

Vascular Parameters

Control Value

MABP Input

Large arterial resistance, rla (mmHg s/ml)

3.00

Lp

5

Small arterial resistance, rsa (mmHg s/ml)

3.00

αp

0.6

Capillary resistance, rc (mmHg s/ml)

0.80

Venous resistanc, rv (mmHg s/ml)

0.60

Lγ

3

Arterial Compliance, ca (ml/mmHg)

0.50

αγ

0.8

ETCO2 Input

Regulatory Parameters

Author Manuscript

Gain of autoregulatio, gq (unitless)

1.80

Rate of autoregulation, τq (s)

3.00

Gain of reactivity, gγ (unitless)

1.50

Rate of reactivit, τγ (s)

6.00


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TABLE 2

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CONTINUOUS SYSTEM COMPONENTS OF THE EQUIVALENT TRANSFER FUNCTIONS DERIVED FROM THE LCHM ETCO2 Input

MABP Input Poles

Residue

Poles

Residue

Direct Term

0.33

-

-

-

1

−0.88 + 0.73i

−0.11 + 0.061i

−0.88 + 0.73i

−0.11 + 0.061i

2

−0.88 − 0.73i

−0.11 − 0.061i

−0.88 − 0.73i

−0.11 − 0.061i

3

-

-

−0.1667

0.0026

(The letter “i” is the imaginary number)

Author Manuscript Author Manuscript Author Manuscript IEEE Access. Author manuscript; available in PMC 2016 October 19.

Author Manuscript

Author Manuscript

Author Manuscript 5.5 0.35

18.75 0.17

Settling Time (s)

Slope of Restoration (cm/s2)

Settling Time (s)

Slope of Reactivity (cm/s2)

ETCO2 Input

−45.00

1.31

Initial Rise value (cm/s)

−68.55

1.33

−71.91

10.00

×2

MABP Input

Control Value

rla

82.76

−4

250.00

−5.00

60.00

÷2

31.35

−9.33

51.40

55.00

0.00

×2 0.00

÷2

−12.65

4

−89.32

−30.00

rsa

10.61

−9.33

7.90

150.00

10.00

×2

ca

−8.75

4

−87.98

−10.00

−20.00

÷2

PERCENT CHANGE IN THE SRF FEATURES OF THE MABP AND ETCO2 INPUTS

Author Manuscript

TABLE 3 Henley et al. Page 33


Author Manuscript

Author Manuscript

Author Manuscript 5.50 0.35

Settling Time (s)

Slope of Restoration (cm/s2)

18.75 0.17

Settling Time (s)

Slope of Reactivity (cm/s2)

ETCO2 Input

0.00

1.31

Initial Rise value (cm/s)

gq

−27.28

−6.67

+135.10

+90.00

×2

MABP Input

Control Value

+16.93

+5.33

−88.94

−25.00

0.00

÷2

0.00 −54.88

gγ

−44.72

+5.33

PERCENT CHANGE IN THE SRF FEATURES OF THE MABP AND ETCO2 INPUTS

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TABLE 4 Henley et al. Page 34


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