Assessment of Fractional-Order Arterial Windkessel as a Model of Aortic Input Impedance

Goal: Fractional-order Windkessel model is proposed to describe the aortic input impedance. Compared with the conventional arterial Windkessel, the main advantage of the proposed model is the consideration of the viscoelastic nature of the arterial wall using the fractional-order capacitor (FOC). Methods: The proposed model, along with the standard two-element Windkessel, three-element Windkessel, and the viscoelastic Windkessel models, are assessed and compared using in-silico data. Results: The results show that the fractional-order model fits better the moduli of the aortic input impedance and fairly approximates the phase angle. In addition, by its very nature, the pseudo-capacitance of FOC makes the proposed model's dynamic compliance complex and frequency-dependent. Conclusions: The analysis of the proposed fractional-order model indicates that fractional-order impedance yields a powerful tool for a flexible characterization of the arterial hemodynamics.

simplest WK configuration, with two physiolog-ically interpretable parameters, this model fails to reproduce the natural spectrum of the aortic input impedance in the frequency domain.Additionally, the estimated blood pressure is unlike the real pulse over the cardiac period.To overcome these limitations, an improved WK model has been proposed, referred to as three-element Windkessel (WK3).As shown in Fig. S1 (B), an additional series resistance (Zc) has been connected in the inlet of WK2 to construct WK3.Despite its improvement in both frequency and time domains, WK3 repre-sents a fundamental issue that deserved additional attention in subsequent works.Indeed, the additional element, (Zc), does not have a unique explicit physiological interpretation.The three most relevant interpretations that were attributed to it are: 1) the characteristic impedance, 2) the aortic valve resistance, and 3) the internal resistance of the left ventricle.Conceptually, these three different determinations lead to different paradigms which do not necessarily reproduce the global behavior of the arterial after-load.For instance, based on the second and third interpretations, it is obvious that Zc belongs to the source part (heart) and hence, should not be a part of the arterial tree described by WK [S2].To avoid this discrepancy and by the fact that arteries are more likely viscoelastic rather than pure elastic, a modified Windkessel model has been proposed, referred to as viscoelastic Windkessel (VWK).As depicted in Fig. S1 (C), VWK consists of a complex and frequency-dependent compliance (Cc) parallel with the total peripheral resistance (Rp).Cc is based on the electrical analogue of the Voigt mechanical cell [S3, S4].It consists of a small resistance (Rd) and an ideal capacitor (Cvw) connected in series and accounting for the viscous losses of the arterial wall motion and the static compliance, respectively.It is worth to note that, mathematically, WK3, and VWK are equivalent in terms of data fitting performance; however, physiologically, they lead to different interpretations.Even though, many studies have argued that the Voigt representation is a very poor configuration of the vascular viscoelasticity, since it does not account for the stressrelaxation experiment, yet, this concept is commonly recognized as an acceptable global description of the dynamic of the arterial tree.This is related to the fact that, even if higher-order viscoelastic models would provide a more natural and realistic representation, real data cannot depict sufficient information to identify all their parameters [S3].

S.II. FRACTIONAL-ORDER CAPACITOR
The fractional-order capacitor (FOC), defined as a constant phase element [S5, S6], is an electrical component that represents a fractional-order derivative relationship between the current (i), passing through, and the voltage (v), across it, with respect to time (t), that is: with the aortic input impedance modeling concept, FOC can be considered as a suitable candidate for the complex and frequency-dependent compliance, which might overcome the discrepancies stemming from integer-order limitation.In fact, • The proportionality constant Cα (pseudo-capacitance) is expressed in unit of [Farad.sec 1−α ] that makes, by its ) very nature, the equivalent capacitance, in the unit of [Farad], frequency-dependent as shown in (4), hence FOC where Cα is a constant called pseudo-capacitance, expressed in units of [Farad / second 1−α ].The capacitance in unit of [Farad] of a FOC, at a particular frequency ω0, can be expressed as: Assuming a null initial condition and applying Laplace transform to (S1), the fractional-order impedance (ZC) of FOC can be written as: ) provides physical foundation in representing the complex and frequency dependence of the arterial compliance.• Based on the fractional differentiation order α, the storage and the dissipation parts of the resultant FOC's impedance can have different levels as illustrated in fig.S2, thus FOC might offer a key advantage in modeling complex systems, that is the whole spectrum of dissipation and storage mechanisms may be included in a single parameter (the fractional differentiation order).• As depicted in fig.S3, the equivalent analogue circuit of

S
., where φ represents the phase shift given by the formula: φ = απ/2 [rad] or φ = 90α [degree or • ].As illustrated in Fig. S2, it is clear that as α goes to 0, the imaginary part (ZS) of ZC vanishes to 0 and hence the FOC characteristic becomes more like that a pure resistor, whereas as α approaches to 1, the real part (ZD) converges to 0 and hence, FOC operates as a pure capacitor.Furthermore, it has been demonstrated that the characteristics of FOC can be approximated using the RC ladder structure [S7] similar to the one shown in Fig. S3.
Based on the above properties and in comparison to an integer order model where α is strictly fixed to an integer (0 or 1), the parameter α offers extra flexibility for a fractional order lumped-element modeling.In connection

Fig. S1 :
Fig. S1: Schematic representation of the electrical analog of the ordinary arterial Windkessel models (A, B, and C) along with the proposed fractional-order arterial Windkessel model (D) (adopted from Fig. 1 in [S4]).(A) Two-element Windkessel, (B) Three-element Windkessel, (C) Viscoelastic Windkessel, and (D) Fractional-order two-element Windkessel models.CW 2 and CW 3 account for the total arterial compliance in WK2 and WK3, respectively.CC represents the frequencydependent compliance that consists of small resistance Rd in series with an ideal capacitor CV W accounting for the viscous losses and static compliance of the arterial wall, respectively.Rp represents the total peripheral resistance, and Zc accounts for the characteristic impedance.Qa is the arterial blood flow pumped from the left-ventricle of the heart, and Pa refers to the aortic blood pressure.
viewed as an infinite number of voigt cells connected in parallel, hence FOC might lead to a reduced order representation of the mechanical properties of the Z F OC (w) = Cαω α cos(φ) −j Cαω α sin(φ), (S4) arterial network by using only two parameters (α and Cα).