Research Papers

Sensitivity of Arterial Hyperelastic Models to Uncertainties in Stress-Free Measurements

[+] Author and Article Information
Nir Emuna

Faculty of Aerospace Engineering,
Technion—Israel Institute of Technology,
Haifa 32000, Israel
e-mail: emuna@campus.technion.ac.il

David Durban

Faculty of Aerospace Engineering,
Technion—Israel Institute of Technology,
Haifa 32000, Israel
e-mail: aer6903@technion.ac.il

Shmuel Osovski

Faculty of Mechanical Engineering,
Technion—Israel Institute of Technology,
Haifa 32000, Israel
e-mail: osovski.technion@gmail.com

Manuscript received November 9, 2017; final manuscript received May 4, 2018; published online July 2, 2018. Assoc. Editor: Seungik Baek.

J Biomech Eng 140(10), 101013 (Jul 02, 2018) (13 pages) Paper No: BIO-17-1516; doi: 10.1115/1.4040400 History: Received November 09, 2017; Revised May 04, 2018

Despite major advances made in modeling vascular tissue biomechanics, the predictive power of constitutive models is still limited by uncertainty of the input data. Specifically, key measurements, like the geometry of the stress-free (SF) state, involve a definite, sometimes non-negligible, degree of uncertainty. Here, we introduce a new approach for sensitivity analysis of vascular hyperelastic constitutive models to uncertainty in SF measurements. We have considered two vascular hyperelastic models: the phenomenological Fung model and the structure-motivated Holzapfel–Gasser–Ogden (HGO) model. Our results indicate up to 160% errors in the identified constitutive parameters for a 5% measurement uncertainty in the SF data. Relative margins of errors of up to 30% in the luminal pressure, 36% in the axial force, and over 200% in the stress predictions were recorded for 10% uncertainties. These findings are relevant to the large body of studies involving experimentally based modeling and analysis of vascular tissues. The impact of uncertainties on calibrated constitutive parameters is significant in context of studies that use constitutive parameters to draw conclusions about the underlying microstructure of vascular tissues, their growth and remodeling processes, and aging and disease states. The propagation of uncertainties into the predictions of biophysical parameters, e.g., force, luminal pressure, and wall stresses, is of practical importance in the design and execution of clinical devices and interventions. Furthermore, insights provided by the present findings may lead to more robust parameters identification techniques, and serve as selection criteria in the trade-off between model complexity and sensitivity.

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Grahic Jump Location
Fig. 1

Fit (solid lines) to data (dots) from multiple biaxial testing protocols of a human common carotid for (a) the phenomenological seven-parameter Fung model and (b) the structure motivated eight-parameter, four-fiber family HGO model. The arrows indicate the force ratio Pz: Pθ applied in the longitudinal and circumferential directions, respectively. The fit quality (overall, axial, and circumferential) of each model is displayed by the error measures e2D, ez, and eθ, respectively.

Grahic Jump Location
Fig. 2

Simulated inflation–extension biaxial test data in four axial stretch protocols (Λz = 1.20, 1.25, 1.30, 1.35) using (a) the seven-parameter phenomenological Fung model and (b) the structure motivated eight-parameter, four-fiber family model. The data were generated based on (1) constitutive parameters identified from a real planar biaxial data and (2) measurements of the UL and SF configurations geometry (ρo, Ri, Ro, α, and Λ0) from Table 4 of Ref. [64] for specimen #9R.

Grahic Jump Location
Fig. 3

Constitutive sensitivity versus measurement uncertainty in the opening-angle ((a) and (b)) and axial prestretch ((c) and (d)) of the phenomenological seven-parameter Fung model ((a) and (c)) and the structure motivated eight-parameter, four-fiber family HGO model ((b) and (d)). The left panel of each subfigure shows the value of the error measures ePi, eFz, and eIE (in the luminal pressure, axial force, and overall response, respectively) versus ϵα ((a) and (b)) and ϵΛ0 ((c) and (d)), the relative errors in the true values of α and ϵΛ0, respectively. The right panels show the relative errors in the fitted constitutive parameters ϵc of Eq. (14) as a function of ϵα and ϵΛ0.

Grahic Jump Location
Fig. 4

Summary of maximal and average relative absolute constitutive sensitivities for SF measurements uncertainties in opening angle (α) and axial prestretch (Λ0): (a) The phenomenological seven-parameter Fung model and (b) the structure motivated eight-parameter, four-fiber family HGO model

Grahic Jump Location
Fig. 5

Propagation of measurement uncertainty of ±10% in the opening-angle ((a) and (b)) and axial prestretch ((c) and (d)) to errors in the inflation–extension response of the phenomenological seven-parameter Fung model ((a) and (c)) and the structure motivated eight-parameter, four-fiber family HGO model ((b) and (d)). Solid lines represent the simulated experimental curves, and dots and dashes indicate positive (ϵα=+0.1, ϵΛ0=+0.1) and negative (ϵα=−0.1, ϵΛ0=−0.1) deviations, respectively. The level of axial stretch (Λz = 1.20, 1.25, 1.30, and 1.35) is indicated above each curve.

Grahic Jump Location
Fig. 6

Summary of the predictive sensitivity of the phenomenological seven-parameter Fung model ((a) and (c)) and the structure motivated eight-parameter, four-fiber family HGO model ((b) and (d)) to ±10% errors in the opening-angle ((a) and (b)) and axial prestretch ((c) and (d)). The average (circle), maximal (upper bar), and minimal (lower bar) relative margins in the internal pressure ΔPi¯ (left panels) and axial force ΔFz¯ (right panels) are displayed in different axial stretch protocols (Λz = 1.20, 1.25, 1.30, and 1.35). The value of the circumferential stretch λo for which the maximal margin was obtained is indicated next to its associated upper bar.

Grahic Jump Location
Fig. 7

Propagation of measurement uncertainties of ±10% in the opening-angle ((a) and (b)) and axial prestretch ((c) and (d)) to errors in the stresses predicted by the phenomenological seven-parameter Fung model ((a) and (c)) and the structure motivated eight-parameter, four-fiber family HGO model ((b) and (d)). Solid lines represent the “real” stresses, dots and dashes indicate the positive (ϵα=+0.1, ϵΛ0=+0.1) and negative (ϵα=−0.1, ϵΛ0=−0.1) deviations, respectively. The left (right) panel display the stresses and the associated errors under low (high) deformation, indicated by values of the circumferential (λo) and axial (Λz) stretches.



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