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Fractional Order Models of Arterial Windkessel as an Alternative in the Analysis of the Left Ventricular Afterload

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Pith's one-line read Replacing the ideal elastic capacitor with a fractional-order capacitor in two- and three-element Windkessel models improves fits to simulated aortic input impedance, but the physiological interpretation rests on fitted parameters and in-silico data.

arxiv 1908.05239 v1 pith:OA7OLVLJ submitted 2019-08-01 q-bio.TO physics.med-ph

Mohamed A. Bahloul , Taous-Meriem Laleg Kirati This is my paper · ORCID
classification q-bio.TOphysics.med-ph
keywords modelsarterialfractional-orderwindkesselaorticimpedancemodelfractional
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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The reading

The Windkessel model treats the arterial system like a simple electrical circuit: the heart pumps blood like a current, the large arteries act like an elastic reservoir or capacitor, and the small vessels act like a resistor. The classical two-element version has one capacitor and one resistor, and the three-element version adds a characteristic impedance in series. These models are standard, but they assume the artery wall is purely elastic, while real artery walls are viscoelastic: they both store and dissipate energy, and their behavior changes with frequency.

The authors replace the ideal capacitor with a fractional-order capacitor, an element whose behavior is controlled by an exponent alpha. Alpha equal to one gives the usual elastic capacitor, alpha equal to zero gives a pure resistor, and intermediate values blend resistance and capacitance. They derive the impedance formulas for a two-element and a three-element fractional Windkessel model, then fit them to an in-silico database of 3,325 virtual subjects generated by a validated one-dimensional hemodynamic model.

The fractional models fit the simulated impedance better than the classical two-element Windkessel, and the three-element fractional model is only slightly better than the classical three-element model. The fitted alpha values differ from one, which the authors interpret as evidence of arterial viscoelasticity, and they suggest alpha could act as a biomarker for arterial stiffness. The main limitation is that all validation comes from simulated data, the models have extra free parameters, and the viscoelastic interpretation is built on fitted exponents rather than on independent measurements.

Extended reading notes

Core claim

From the abstract: "the proposed fractional-order models overcame the limitations of the standard arterial Windkessel model and captured very well the real dynamic of the aortic input impedance modulus." The paper further concludes that the fractional differentiation order alpha "may have a powerful role, as a physiological bio-marker" (Section 5). If correct, fractional-order Windkessel models would estimate frequency-dependent arterial compliance and yield a single viscoelasticity index from pressure and flow measurements.

Load-bearing premise

The entire validation treats the Willemet et al. in-silico database (Section 3.3) as ground truth for aortic hemodynamics. That database was generated by an integer-order one-dimensional model, not by a fractional-order viscoelastic wall law, so the fitted alpha is an emergent best-fit exponent rather than an independently measured tissue property. If the virtual population does not faithfully reproduce human viscoelastic arterial behavior, the physiological conclusions about fractional-order behavior and the alpha biomarker collapse.

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Assumptions & free parameters 3 free parameters · 5 assumptions · 1 invented entities

The model adds one free exponent alpha and a fractional time constant to the standard Windkessel and assumes the virtual database is a valid ground truth. The physiological interpretation of alpha is not independently tested, and the effective compliance definition includes an ad hoc sine factor.

free parameters (3)
  • alpha (fractional order) for FWK2/FWK3 = 0.3687 +/- 0.0081 (FWK2); 0.9525 +/- 0.0050 (FWK3)
    Fitted to the in-silico complex aortic input impedance via RMSE minimization (Eqs. 27-28); no independent calibration. It is then used as evidence for fractional-order viscoelasticity and proposed as a biomarker.
  • tau_alpha (fractional time constant) = Not reported explicitly; derived from C_eff via Eq. (32)
    Fitted in FWK2 and FWK3; defines the corner frequency of the fractional pole and is used to compute compliance in Eq. (32).
  • Zc (characteristic impedance) for FWK3 = 0.03911 +/- 0.000639 (units not specified)
    Fitted in FWK3 (and in WK3 as a baseline); not independently measured.
assumptions (5)
  • domain assumption The aortic input impedance Zin(omega)=Pa(omega)/Qa(omega) computed from DFT of one cardiac cycle adequately represents the arterial system.
    Section 3.4: the fit target is derived from discrete Fourier transforms of in-silico pressure and flow; assumes periodicity and stationarity over one cardiac cycle.
  • domain assumption The in-silico virtual population generated by an integer-order 1D hemodynamic model is a valid ground truth for vascular mechanical properties.
    Section 3.3: all validation is performed against this database; no in-vivo or experimental data are used.
  • domain assumption A fractional-order capacitor (constant phase element) is a valid representation of arterial wall viscoelasticity, with alpha in (0,1).
    Sections 2.3.2 and 2.5: this equivalence is asserted from the generalized Kelvin-Voigt analogy, not tested independently.
  • ad hoc to paper Effective compliance can be defined as Ceff = C_alpha * sin(alpha*pi/2).
    Section 4, Eq. (33): the sine factor is chosen so that Ceff reduces to C at alpha=1; no derivation from impedance or tissue mechanics is given. It is used to compare fractional and standard compliance.
  • standard math The Laplace transform of the fractional derivative is s^alpha assuming null initial conditions.
    Section 2.4, Eq. (6); standard for frequency-domain impedance, valid in steady state.
invented entities (1)
  • Fractional-order capacitor C_alpha as total arterial compliance
    purpose: Replaces the ideal capacitor in WK2/WK3 so the model can represent frequency-dependent, viscoelastic compliance with one element.
    A constant-phase element is a known circuit element, but its identification with human arterial compliance is not independently evidenced; alpha is fitted to the same in-silico data used for validation, and no separate experiment confirms CPE behavior of the arterial wall.

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Pith. "Pith review of Fractional Order Models of Arterial Windkessel as an Alternative in the Analysis of the Left Ventricular Afterload." pith.science (2026). https://pith.science/paper/OA7OLVLJ

@misc{pith2026190805239,
  author       = {Pith},
  title        = {Pith review of: Fractional Order Models of Arterial Windkessel as an Alternative in the Analysis of the Left Ventricular Afterload},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OA7OLVLJ}},
  note         = {Machine review of arXiv:1908.05239}
}
read the original abstract

In this paper, a new fractional order generalization of the classical Windkessel arterial model is developed to describe the aortic input impedance as an assessment of the left ventricular after-load. The proposed models embeds fractional-order capacitor to describe the total arterial compliance. In this paper, we report our investigations on fractional calculus tools and demonstrate that fractional-order impedance can be used to determine the vascular properties and studying its dynamic effects. We conceived two fractional-order lumped parametric models: the fractional-order two-element Windkessel model and the fractional-order three-element Windkessel model. We compared these models to the classical Windkessel one using in-silico ascending aortic blood pressure and flow database of 3325 virtual subjects. Results showed that the proposed fractional-order models overcame the limitations of the standard arterial Windkessel model and captured very well the real dynamic of the aortic input impedance modulus. We also demonstrated that the proposed models could monitor the changes in the aortic input impedance for various arterial physiological states. Therefore, our models provide a new tool for "hemodynamic inverse problem" solving and offer a new, innovative way to better understand the viscoelastic effect, in terms of resistive behavior of the arterial motions.

Figures

Figures reproduced from arXiv: 1908.05239 by the authors.

Figure 1
Figure 1. (a) Two-element Windkessel (WK2) analog representation. It simply describes the whole arterial system in [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Mechanical representation of Voigt Model, (b) Equivalent electrical representation of the Voigt Model, (c) [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. RC network for fractional order capacitor emulation. [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figures from the paper (12 more)
Figure 4
Figure 4. Figure 4: (a) Generalized Kelvin-Voigt viscoelastic model consisting of [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]
Figure 4
Figure 4. Figure 4: (e) illustrates the proposed model’s scheme. The [PITH_FULL_IMAGE:figures/full_fig_p014_4.png]
Figure 5
Figure 5. Figure 5: Distribution, mean value and standard deviation in (mmHg) of: (a) mean blood pressure (MBP), (b) [PITH_FULL_IMAGE:figures/full_fig_p015_5.png]
Figure 6
Figure 6. Figure 6: Distribution of the estimates fractional differentiation order parameters [PITH_FULL_IMAGE:figures/full_fig_p018_6.png]
Figure 7
Figure 7. Figure 7: (a) Comparison of goodness of fit quantified as [PITH_FULL_IMAGE:figures/full_fig_p019_7.png]
Figure 8
Figure 8. Figure 8: (a) Comparison of goodness of fit of the aortic input impedance modulus quantified as [PITH_FULL_IMAGE:figures/full_fig_p019_8.png]
Figure 9
Figure 9. Figure 9: (a) Comparison of goodness of fit of the aortic input impedance phase angle quantified as [PITH_FULL_IMAGE:figures/full_fig_p020_9.png]
Figure 10
Figure 10. Figure 10: Distribution of the effective compliance estimates for WK2 in (a) and FWK2 in (b). (c) box plots, [PITH_FULL_IMAGE:figures/full_fig_p022_10.png]
Figure 11
Figure 11. Figure 11: Distribution of the effective compliance estimates for WK3 in (a) and FWK3 in (b). (c) box plots, [PITH_FULL_IMAGE:figures/full_fig_p022_11.png]
Figure 12
Figure 12. Figure 12: Distribution of the characteristic impedance estimates for WK3 in (a) and FWK3 in (b). (c) box plots, [PITH_FULL_IMAGE:figures/full_fig_p023_12.png]
Figure 13
Figure 13. Figure 13: The aortic input impedance validation using in-silico human data for different physiological state (nor [PITH_FULL_IMAGE:figures/full_fig_p024_13.png]
Figure 14
Figure 14. Figure 14: Bar graph of the arterial parameter (SBP: systolic aortic blood pressure, MP: mean aortic blood pressure, [PITH_FULL_IMAGE:figures/full_fig_p025_14.png]

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