Model-based Quantification of Cerebral Hemodynamics as a Physiomarker for Alzheimer’s Disease?

Abstract

Previous studies have found that Alzheimer’s disease (AD) impairs cerebral vascular function, even at early stages of the disease. This offers the prospect of a useful diagnostic method for AD, if cerebral vascular dysfunction can be quantified reliably within practical clinical constraints. We present a recently developed methodology that utilizes a data-based dynamic nonlinear closed-loop model of cerebral hemodynamics to compute “physiomarkers” quantifying the state of cerebral flow autoregulation to pressure-changes (CA) and cerebral CO2 vasomotor reactivity (CVMR) in each subject. This model is estimated from beat-to-beat measurements of mean arterial blood pressure, mean cerebral blood flow velocity and end-tidal CO2, which can be made reliably and non-invasively under resting conditions. This model may also take an open-loop form and comparisons are made with the closed-loop counterpart. The proposed model-based physiomarkers take the form of two indices that quantify the gain of the CA and CVMR processes in each subject. It was found in an initial set of clinical data that the CVMR index delineates AD patients from control subjects and, therefore, may prove useful in the improved diagnosis of early-stage AD.

Appendices

Appendix I: Basics of Volterra and Open-Loop PDM-based Modeling

The general nonparametric Volterra model is applicable to all finite-memory dynamic nonlinear systems, which covers almost all physiological systems (with the exception of chaotic systems or non-dissipating oscillators).26,29 In the proposed methodology, the modeling task commences with the estimation of a second order Volterra model of the two-input CHAR system using Laguerre expansions of the kernels24:

$$ \begin{aligned} y(t) = & {k_0} + \int\limits_0^\infty {{k_p}(\tau )p(t - \tau )} d\tau + \int\limits_0^\infty {{k_x}(\tau )x(t - \tau )} d\tau \\ & + \int {\int\limits_0^\infty {{k_{pp}}({\tau _1},{\tau _2})\;p(t - {\tau _1})p(t - {\tau _2})} d{\tau _1}d{\tau _2}} + \int {\int\limits_0^\infty {{k_{xx}}({\tau _1},{\tau _2})\;x(t - {\tau _1})x(t - {\tau _2})} d{\tau _1}d{\tau _2}} \\ & + \int {\int\limits_0^\infty {{k_{px}}({\tau _1},{\tau _2})\;p(t - {\tau _1})x(t - {\tau _2})} d{\tau _1}d{\tau _2} + \varepsilon (t)} \\ \end{aligned} $$

(A1)

where p(t) denotes the MABP input, x(t) denotes the ETCO2 input, y(t) denotes the mean cerebral blood flow velocity (MCBFV) output and ε(t) denotes possible measurement or modeling errors. The dynamic characteristics of this system/model are described by the kernels: k _p , k _x , k _pp , k _xx , k _px , which are estimated using given input–output data: p(t), x(t), and y(t), by means of Laguerre expansions and least-squares fitting as described below. Consider, for instance, the Laguerre expansion of the rth order kernels corresponding to two inputs:

$$ \begin{gathered} {k_{p, \cdots p,r}}({\tau _1}, \cdots ,{\tau _r}) = \sum\limits_{{j_1} = 1}^L { \cdots \sum\limits_{{j_r} = 1}^L {{a_r}({j_1}, \cdots ,{j_r}){b_{{j_1}}}({\tau _1}) \cdots {b_{{j_r}}}({\tau _r})} } \hfill \\ {k_{x, \cdots x,r}}({\tau _1}, \cdots ,{\tau _r}) = \sum\limits_{{j_1} = 1}^L { \cdots \sum\limits_{{j_r} = 1}^L {{c_r}({j_1}, \cdots ,{j_r}){b_{{j_1}}}({\tau _1}) \cdots {b_{{j_r}}}({\tau _r})} } \hfill \\ \end{gathered} $$

(A2)

where {b _j (τ)} denotes the orthogonal Laguerre function basis. Then, we have the following input–output relation which involves linearly the Laguerre expansion coefficients {a _r } and {c _r }:

$$ \begin{aligned} y(t) = & {c_0} + \sum\limits_{r = 1}^Q {\sum\limits_{{j_1} = 1}^L { \cdots \sum\limits_{{j_r} = 1}^{{j_{r - 1}}} {{a_r}({j_1}, \cdots ,{j_r}){v_{{j_1}}}(t) \cdots {v_{{j_r}}}(t)} } } \\ & + \sum\limits_{r = 1}^Q {\sum\limits_{{j_1} = 1}^L { \cdots \sum\limits_{{j_r} = 1}^{{j_{r - 1}}} {{c_r}({j_1}, \cdots ,{j_r}){z_{{j_1}}}(t) \cdots {z_{{j_r}}}(t) + \varepsilon (t)} } } \\ \end{aligned} $$

(A3)

where the signals v _j (t) and z _j (t) are the convolutions of the Laguerre basis function b _j with the respective input. Note that in the model equation (A3) the cross-kernel contribution has been suppressed in the interest of simplifying the model expression. The cross-kernel effects are accounted by expansion terms involving products of the signals v _j (t) and z _j (t). The fact that the Laguerre expansion coefficients enter linearly in the nonlinear input–output model of Eq. (A3) allows their estimation via least-squares fitting (a simple and robust numerical procedure). Following estimation of the Laguerre expansion coefficients, we can construct the Volterra kernel estimates using Eq. (A2) and compute the model prediction for any given input using Eq. (A1) or (A3).

Although the Laguerre expansion technique brings considerable model estimation efficiencies, it does not remove the “curse of dimensionality” associated with the multi-dimensional structure of high-order kernels. In order to overcome this practical limitation, we have introduced the concept of Principal Dynamic Modes (PDM), which aims at identifying an efficient “basis” of functions (distinct and characteristic for each system) that are capable of representing adequately the system dynamics (i.e., provide satisfactory expansions of the kernels). The computation of the PDMs for each input is based on Singular Value Decomposition (SVD) of a rectangular matrix composed of the first order kernel estimate (as a column vector) and the second order self-kernel estimate (as a block matrix) weighted by the standard deviation of the respective input.

The resulting PDMs form a filter-bank that receives the respective input signal and generates (via convolution) signals that are subsequently transformed by the “Associated Nonlinear Function” (ANFs), which represents the nonlinear characteristics of the system for the respective PDM dynamics, to form additively the system output, as depicted schematically in Fig. 2. Thus, the PDM-based model separates the dynamics (PDMs) from the nonlinearities (ANFs). Since the “separability” of the system nonlinearity cannot be generally assumed, we include “cross-terms” in the PDM-based model that are properly selected on the basis of a statistical significance test on the computed correlation coefficient between each cross-term (i.e., the pair product of PDM outputs) and the output signal, using the w-statistic.26 Three PDMs for each input and up to 6 cross-terms were found to be adequate in this application.

The structure of the PDM-based model of the two-input/one-output CA-CVMR system is shown in Fig. 2. The employed “global” PDMs represent a common “functional basis” for efficient representation of all kernels of the CHAR system for all subjects. These “global” PDMs are obtained via SVD of a rectangular matrix containing the PDMs of all subjects in a selected reference group (16 control subjects in this case). Although the global PDMs are common for all subjects, the estimated ANF for each global PDM and the coefficients of the cross-terms are subject-specific and can be used to characterize uniquely the CHAR process for each subject. The use of PDMs allows us to write the output Eq. (A3) as:

$$ y(t) = {c_0} + \sum {{f_h}\left[ {{u_h}(t)} \right]} + \sum {{f_m}\left[ {{w_m}(t)} \right]} + {\text{Cross - Terms}} + \varepsilon (t) $$

(A4)

where {u _h } and {w _m } are the PDM outputs (i.e., convolutions of the input with the respective PDM) for the MABP and ETCO2 inputs, respectively, and {f _h } and {f _m } are the ANFs associated with each PDM. The ANFs are typically polynomials (cubic in this application). The “Cross-Terms” in Eq. (A4) are pair products of {u _h } and {w _m } that have significant correlation with the output. The coefficients of the selected Cross-Terms are estimated, along with c ₀ and the coefficients of the (cubic) ANFs via least-squares regression of Eq. (A4).

Appendix II: Closed-Loop PDM-based Modeling

The closed-loop analysis requires the estimation of two open-loop PDM-based models A and B, following the approach described in Appendix I, which are placed in the configuration of Fig. 3. The disturbance signals F _d (t) and P _d (t) are computed in each case as the model-prediction residuals and they are viewed as the physiological “disturbances” or “drives” of the closed-loop system. In accordance with Fig. 3, we have two equivalent closed-loop equations:

$$ \begin{aligned} F(t) = & A\left\{ {P,C} \right\} + {F_{\text{d}}}(t) \\ & = A\left\{ {B\left[ {F,C} \right] + {P_{\text{d}}}(t),C} \right\} + {F_{\text{d}}}(t) \\ \end{aligned} $$

(A5)

$$ \begin{aligned} P(t) & = B\left\{ {F,C} \right\} + {P_{\text{d}}}(t) \\ & = B\left\{ {A\left[ {P,C} \right] + {F_{\text{d}}}(t),C} \right\} + {P_{\text{d}}}(t) \\ \end{aligned} $$

(A6)

which are nonlinear stochastic integral equations, since we have for each open-loop model:

$$ F(t) = {F_0} + \sum {f_i^{PF}\left[ {u_i^{PF}(t)} \right]} + \sum {f_j^{CF}\left[ {u_j^{CF}(t)} \right] + \sum {c_{k,\ell }^{PC}u_k^{PF}(t)u_\ell^{CF}(t)} + {F_{\text{d}}}(t)} $$

(A7)

$$ P(t) = {P_0} + \sum {f_i^{FP}\left[ {u_i^{FP}(t)} \right]} + \sum {f_j^{CP}\left[ {u_j^{CP}(t)} \right] + \sum {c_{k,\ell }^{FC}u_k^{FP}(t)u_\ell^{CP}(t)} + {P_{\text{d}}}(t)} $$

(A8)

where the signals u(t) are convolution integrals of the input signals with the PDMs, the functions f [·] are polynomials (cubic in this case), and F ₀, P ₀ are the baseline values of MABP and MCBFV respectively (i.e., their values when there are no systemic disturbances). In the case of discretized data, Eqs. (A5) and (A6) are tantamount to nonlinear auto-regressive equations in F(t) or P(t) with stochastic coefficients and an exogenous variable C(t). Simulations with broadband (e.g., band-limited white-noise) systemic disturbances can reveal the spectral characteristics of the closed-loop model for various power levels under isocapnic (C(t) = 0), hypercapnic (C(t) > 0) or hypocapnic (C(t) < 0) conditions.

It is instructive to examine the particular case where the operators A and B are linear, i.e., described by Transfer Functions in the frequency domain. Then, Eq. (A5) yields the following relation in the Laplace domain (i.e., the variables are described by their Laplace Transforms):

$$ F = \frac{{{H_{PF}}{H_{CP}} + {H_{CF}}}} {{1 - {H_{PF}}{H_{FP}}}} \cdot C + \frac{{{H_{PF}}P{}_d + {F_d}}} {{1 - {H_{PF}}{H_{FP}}}} $$

(A9)

where H _XY denotes the Transfer Function from input X to output Y. Thus, the dynamics and stability of the closed-loop model are determined by the poles (i.e., the roots) of the expression: (1 − H _PF H _FP ), i.e., the product H _PF H _FP in the complex Laplace domain must remain away from 1 in order to maintain stability. The resonances of the closed-loop system are defined by the troughs of the function |1 − H _PF H _FP | expressed in the Fourier domain.