Evaluation of bioprosthetic heart valve failure using a matrix-fibril shear stress transfer approach

Abstract

A matrix-fibril shear stress transfer approach is devised and developed in this paper to analyse the primary biomechanical factors which initiate the structural degeneration of the bioprosthetic heart valves (BHVs). Using this approach, the critical length of the collagen fibrils l _c and the interface shear acting on the fibrils in both BHV and natural aortic valve (AV) tissues under physiological loading conditions are calculated and presented. It is shown that the required critical fibril length to provide effective reinforcement to the natural AV and the BHV tissue is l _c  = 25.36 µm and l _c  = 66.81 µm, respectively. Furthermore, the magnitude of the required shear force acting on fibril interface to break a cross-linked fibril in the BHV tissue is shown to be 38 µN, while the required interfacial force to break the bonds between the fibril and the surrounding extracellular matrix is 31 µN. Direct correlations are underpinned between these values and the ultimate failure strength and the failure mode of the BHV tissue compared with the natural AV, and are verified against the existing experimental data. The analyses presented in this paper explain the role of fibril interface shear and critical length in regulating the biomechanics of the structural failure of the BHVs, for the first time. This insight facilitates further understanding into the underlying causes of the structural degeneration of the BHVs in vivo.

Appendix 1

Plotting $\log \dot{\lambda }$ versus log η in Fig. 6 for the three experimentally quantified $\left( {\dot{\lambda },\eta } \right)$ data points, it may be observed that each two consecutive points have notably different gradients. This implies that at least two exponential terms may be required to adequately describe the behaviour of η versus $\dot{\lambda }$ via an exponential profile, over the whole range of the three existing points. This function may be mathematically expressed as:

$$\eta (\dot{\lambda }) = a\exp b\dot{\lambda } + c\exp d\dot{\lambda }$$

(A1)

where a, b, c and d are constants to be calculated by fitting this function to the data points.

Fig. 6

η versus $\dot{\lambda }$ plotted in logarithmic scale. The connecting dashed lines between each two consecutive points highlight the respective gradients. Note the ~30 % difference in the gradient values

Setting the first exponential term in equation (A1) to describe the η-$\dot{\lambda }$ behaviour of the first two points, and accepting fits with R ² ≥ 0.99 using MATLAB^®, domains for a and b over which acceptable fits may be achieved were obtained to be 740 ≤ a ≤ 850 and −200 ≤ b ≤ −140, respectively. These provided the numerical constraints for fitting equation (A1) to all three $\left( {\dot{\lambda },\eta } \right)$ points:

$$\eta (\dot{\lambda }) = a\exp b\dot{\lambda } + c\exp d\dot{\lambda },\;\quad \left\{ \begin{aligned} 740 \le a \le 850 \hfill \\ - 200 \le b \le - 140 \hfill \\ \end{aligned} \right.,$$

(A2)

However, another constraint is derived from the rheological properties of GAG. The yield stress of GAG solutions has been reported to be lower than 10⁵ Pa [42]. Assuming that τ is in the range of 10⁴ Pa ≤ τ ≤ 10⁵ Pa, the model parameters in (A2) should provide the best fit to the three $\left( {\dot{\lambda },\eta } \right)$ points such that the corresponding value of η at $\dot{\lambda } = 2.5$ s⁻¹ establishes that $10^{4} \le 2\eta \frac{{\dot{\lambda }}}{\lambda }$. Therefore equation (A2) is rewritten as:

$$\eta (\dot{\lambda }) = a\exp b\dot{\lambda } + c\exp d\dot{\lambda },\;\quad \left\{ \begin{aligned} &740 \le a \le 850 \\& - 200 \le b \le - 140 \\& 10^{4} \le 2\eta \frac{{\dot{\lambda }}}{\lambda } \\ \end{aligned} \right.,$$

(A3)

Fitting equation (A3) to the data points, the set of a, b, c and d that provided R ² = 1 while satisfying those constraints were calculated to be: a = 778.7, b = −180.3, c = 24.89, and d = −2.97:

$$\eta (\dot{\lambda }) = 778.7\exp \left( { - 180.3\dot{\lambda }} \right) + 24.89\exp \left( { - 2.97\dot{\lambda }} \right)$$

(A4)

where the lower bound for η at $\dot{\lambda } = 2.5$ was estimated to be η = 0.0145 MPa s.