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Non-invasive, energy-based assessment of patient-specific material properties of arterial tissue

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Abstract

The mechanical properties of human biological tissue vary greatly. The determination of arterial material properties should be based on experimental data, i.e. diameter, length, intramural pressure, axial force and stress-free geometry. Currently, clinical data provide only non-invasively measured pressure-diameter data for superficial arteries (e.g. common carotid and femoral artery). The lack of information forces us to take into account certain assumptions regarding the in situ configuration to estimate material properties in vivo. This paper proposes a new, non-invasive, energy-based approach for arterial material property estimation. This approach is compared with an approach proposed in the literature. For this purpose, a simplified finite element model of an artery was used as a mock experimental situation. This method enables exact knowledge of the actual material properties, thereby allowing a quantitative evaluation of material property estimation approaches. The results show that imposing conditions on strain energy can provide a good estimation of the material properties from the non-invasively measured pressure and diameter data.

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Abbreviations

\( {\Psi } \) :

Strain energy density function

\( {\Psi _\mathrm{mat}}\) :

Isotropic contribution of the extracellular matrix material to the strain energy density function

\( {\Psi _\mathrm{col}} \) :

Anisotropic contribution of the collagen fibre families to the strain energy density function

\( I_1 \) :

First invariant of the right Cauchy–Green strain tensor

\( I_4 \) :

Invariant related to the orientation of the collagen fibre families

\( \lambda _i \) :

\( i = r,\theta ,z \), three principal stretch ratios in the radial, circumferential and the axial directions of the artery, respectively

\( \mu \) :

Stress-like parameter representing the stiffness of the matrix material

\( k_1 \) :

Stress-like parameter representing the stiffness of the collagen fibres

\( k_2 \) :

Dimensionless parameter of the collagen fibres

\( \alpha \) :

Angle between the mean collagen fibre direction and the circumferential direction of the artery

\( \kappa \) :

Parameter related to the dispersion of the collagen fibres

\( L \) :

Initial axial sample length

\( l \) :

Deformed axial sample length

\( R_\mathrm{o} \) :

Outer radius in the stress-free configuration

\( R_\mathrm{i} \) :

Inner radius in the stress-free configuration

\( \rho _\mathrm{o} \) :

Outer radius in the unloaded configuration

\( \rho _\mathrm{i} \) :

Inner radius in the unloaded configuration

\( r_\mathrm{o} \) :

Outer radius in the loaded configuration

\( r_\mathrm{i} \) :

Inner radius in the loaded configuration

\( R \) :

Initial radius at a certain position

\( r \) :

Deformed radius at a certain position

\( {\Theta _0} \) :

Opening angle in the stress-free configuration

\( \varvec{F_1} \) :

Deformation gradient relating the stress-free and the loaded configuration

\( \varvec{F_2} \) :

Deformation gradient relating the unloaded and the loaded configuration

\( {\mathcal {R}}_0 \) :

Stress-free configuration

\( {\mathcal {R}}_1 \) :

Unloaded configuration

\( {\mathcal {R}}_2 \) :

Loaded configuration

\( R,\theta ,z \) :

Radial, circumferential and axial cylindrical coordinates in the stress-free configuration

\( \rho ,\phi ,\xi \) :

Radial, circumferential and axial cylindrical coordinates in the unloaded configuration

\( r,\theta ,z \) :

Radial, circumferential and axial cylindrical coordinates in the stress-free configuration

\( \varLambda \) :

Axial stretch ratio between the stress-free and the unloaded configuration

\( \lambda \) :

Axial stretch ratio between the unloaded and the loaded configuration

\( \lambda _z \) :

Total stretch in the axial direction

\( H \) :

Wall thickness in the stress-free and unloaded configuration

\( h \) :

Wall thickness in the loaded configuration

\( j \) :

Index going from 1 to \(n\), \(n\) being the total number of data points considered

\( k \) :

Index representing different points throughout the arterial wall, going from 1 at the inner wall to \(m\) at the outer wall

\( \varvec{F}^\mathrm{FEM} \) :

Reduced axial forces extracted from the simulation

\( \varvec{r_\mathrm{o}} \) :

Outer radii extracted from the simulation

\( \varvec{P}^\mathrm{FEM} \) :

Pressures extracted from the simulation

\(\textit{pars}\) :

Vector of fitted parameters

\( w_\mathrm{p} \) :

Weighting factor for the pressure in the objective function

\( w_\mathrm{f} \) :

Weighting factor for the force in the objective function

\( \varvec{P}^\mathrm{mod} \) :

Intraluminal pressures predicted by the function \({\Psi }\)

\( \varvec{F}^\mathrm{mod} \) :

Reduced axial forces predicted by the function \({\Psi }\)

\( \varvec{A}^\mathrm{mod} \) :

Cross-sectional area predicted by the function \({\Psi }\)

\( \lambda _{\mathrm{o}} \) :

Circumferential axial stretch at the outer wall

\( \lambda _{\mathrm{i}} \) :

Circumferential axial stretch at the inner wall

\( \varvec{\sigma }_{\theta \theta }^\mathrm{mod} \) :

Circumferential stresses predicted by the function \({\Psi }\)

\( \varvec{\sigma }_{\theta \theta } \) :

Circumferential stress calculated by enforcing the equilibrium

\( \varvec{\sigma _{zz}}^\mathrm{mod} \) :

Axial stresses predicted by the function \({\Psi }\)

\( \varvec{\sigma }_{zz} \) :

Axial stresses calculated by enforcing the equilibrium

\( F^\mathrm{est} \) :

Estimated reduced axial force

\( \bar{P} \) :

An arbitrary pressure, e.g. 100 mmHg

\( \bar{r}_{\mathrm{i}} \) :

Inner radius related to the pressure \(\bar{P}\)

\( \bar{h}_{\mathrm{i}} \) :

Wall thickness related to the pressure \(\bar{P}\)

\( \gamma \) :

Ratio of the axial to the circumferential stresses at the pressure \(\bar{P}\)

\( w_{{\Psi 1}} \) :

Weighting factor for the strain energy density across the wall thickness

\( w_{{\Psi 2}} \) :

Weighting factor for the strain energy density at diastole

\( F^\mathrm{average} \) :

Average of \(\varvec{F}^\mathrm{mod}\)

\( \varvec{{\Psi }}^{\mathrm{dias, mod}} \) :

Strain energy density at diastole

\(\varvec{{\Psi }}^\mathrm{average} \) :

Average of \({\varvec{{\Psi }}}^\mathrm{dias,mod}\)

\( \varvec{{\Psi }}^{\mathrm{dias, mod}}_{\mathrm{coll}} \) :

Collagen strain energy density at diastole

\( \varvec{{\Psi }}^{\mathrm{dias,mod}}_{\mathrm{mat}}\) :

Matrix strain energy density at diastole

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Acknowledgments

This research was supported by Research Foundation Flanders (FWO Vlaanderen).

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Authors and Affiliations

  1. Biomechanics Section, Faculty of Mechanical Engineering, KU Leuven, Celestijnenlaan 300, 2419, Leuven, Belgium

    M. Smoljkić, J. Vander Sloten & N. Famaey

  2. IBiTech-bioMMeda, iMinds Medical IT, Ghent University, Ghent, Belgium

    P. Segers

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  1. M. Smoljkić
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  2. J. Vander Sloten
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  3. P. Segers
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  4. N. Famaey
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Correspondence to M. Smoljkić.

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Smoljkić, M., Vander Sloten, J., Segers, P. et al. Non-invasive, energy-based assessment of patient-specific material properties of arterial tissue. Biomech Model Mechanobiol 14, 1045–1056 (2015). https://doi.org/10.1007/s10237-015-0653-5

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