A new compressible hyperelastic model for the multi-axial deformation of blood clot occlusions in vessels

Abstract

Mechanical thrombectomy can be significantly affected by the mechanical properties of the occluding thrombus. In this study, we provide the first characterisation of the volumetric behaviour of blood clots. We propose a new hyperelastic model for the volumetric and isochoric deformation of clot. We demonstrate that the proposed model provides significant improvements over established models in terms of accurate prediction of nonlinear stress–strain and volumetric behaviours of clots with low and high red blood cell compositions. We perform a rigorous investigation of the factors that govern clot occlusion of a tapered vessel. The motivation for such an analysis is twofold: (i) the role of clot composition on the in vivo occlusion location is an open clinical question that has significant implications for thrombectomy procedures; (ii) in vitro measurement of occlusion location in an engineered tapered tube can be used as a quick and simple methodology to assess the mechanical properties/compositions of clots. Simulations demonstrate that both isochoric and volumetric behaviours of clots are key determinants of clot lodgement location, in addition to clot-vessel friction. The proposed formulation is shown to provide accurate predictions of in vitro measurement of clot occlusion location in a silicone tapered vessel, in addition to accurately predicting the deformed shape of the clot.

Appendices

Appendix

See Figures 9, 10, 11 and 12.

Fig. 9

Sensitivity analysis of the proposed model in terms of the isochoric and volumetric parameters in the unconfined compression test. Baseline parameters of $D_{1} = 0.15, D_{2} = 0.6, E_{1} = 0.3 kPa, E_{2} = 9.0 kPa, D_{1v} = 0.01, D_{2v} = 0.05, \kappa_{1} = 2 kPa, \kappa_{2} = 20 kPa,$ have been used.

Fig. 10

Influences of clot viscoelasticity on the occlusion location, ${\text{u}}/{\text{D}}$, in a tapered vessel. (A) Distribution of the maximum principal stress in clot at t=60 sec and t=200 sec, (B) Variation of the occlusion location with time for two loading profiles. The following parameters have been used: $P_{0} = 75~mmHg,~~L/D = 2,~~\Phi = 5^{o} ,~~f = 0.2$, with the material parameters of the fibrin-rich clot from Table 1 .

Fig. 11

Variation of the maximum von Mises stress in clot with the number of elements.

Fig. 12

reproduced from Johnson (2020).

Comparison of Ogden and proposed model with the in vitro experiments of occlusion of the platelet-contracted RBC-rich blood clot analogues (made from a 40% H clot mixture) in a silicon rubber vessel with taper angle of ${\Phi } = 0.9854^{{\text{o}}}$. (A) The location of clot lodgement (${\text{u}}/{\text{D}}$) and (B) aspect ratio of clot ($\widehat{{{\text{AR}}}}$). The material parameters from Table 1 have been employed. Experimental results

Table 3 Material parameters of the Ogden, Yeoh, and hyperfoam models for RBC-rich platelet-contracted clot analogues (made from a 40% H blood mixture).

Appendix 1: Sensitivity analysis of the proposed model

Sensitivity of nominal stress and volumetric strain to the material model in the unconfined compression test is shown in Fig. 9. As shown in Fig. 9 (E–H), the stress–strain response is not sensitive to the volumetric parameters ($D_{1v} , D_{2v} ,\kappa_{1} ,\kappa_{2}$); however these parameters can control the volumetric strain ($\varepsilon_{v}$). On the other hand, as shown in Fig. 9 (C, D) the stiffness coefficients $E_{1} ,{ }E_{2}$, affect both stress–strain and the volumetric behaviour.

Appendix 2: Constitutive models formulations and parameters

The model formulations and best-fit parameters for the hyperelastic models used in Sect. 2 are presented here. The strain energy density function for Ogden model is given as

$$\psi = \mathop \sum \limits_{i = 1}^{N} \frac{{2\mu_{i} }}{{\alpha_{i}^{2} }}\left( {\overline{\lambda }_{1}^{{ \alpha_{i} }} + \overline{\lambda }_{2}^{{ \alpha_{i} }} + \overline{\lambda }_{3}^{{ \alpha_{i} }} - 3} \right) + \mathop \sum \limits_{i = 1}^{N} \frac{1}{{D_{i} }}\left( {J - 1} \right)^{2i}$$

(4)

where, $\mu_{i} , \alpha_{i} ,$ and $D_{i}$ are material parameters. The Yeoh strain energy density function is given as

$$\begin{aligned} \psi &= C_{10} \left( {\overline{I}_{1} - 3} \right) + C_{20} \left( {\overline{I}_{1} - 3} \right)^{2} + C_{30} \left( {\overline{I}_{1} - 3} \right)^{3} \\ & + \frac{1}{{D_{1} }}\left( {J - 1} \right)^{2} + \frac{1}{{D_{2} }}\left( {J - 1} \right)^{4} + \frac{1}{{D_{3} }}\left( {J - 1} \right)^{6} , \\ \end{aligned}$$

(5)

where $C_{i0}$ and $D_{i}$ are material parameters; $\overline{I}_{1}$ is the first deviatoric strain invariant defined as

$$\overline{I}_{1} = \overline{\lambda }_{1}^{ 2} + \overline{\lambda }_{2}^{ 2} + \overline{\lambda }_{3}^{ 2}$$

(6)

The hyperfoam strain energy density function is given as

$$\psi = \mathop \sum \limits_{i = 1}^{N} \frac{{2\mu_{i} }}{{\alpha_{i}^{2} }}\left[ {\overline{\lambda }_{1}^{{ \alpha_{i} }} + \overline{\lambda }_{2}^{{ \alpha_{i} }} + \overline{\lambda }_{3}^{{ \alpha_{i} }} - 3 + \frac{1}{{\beta_{i} }}\left( {J^{{ - \alpha_{i} \beta_{i} }} - 1} \right)} \right]$$

(6)

The best-fit parameters of the aforementioned models for the RBC-rich platelet-contracted clot analogous, corresponding to Fig. 1 (G, H), is presented in Table 3.

Appendix 3: Influence of the clot viscoelasticity

Recent experimental data show that thrombus material exhibits rate-dependent visco-hyperelastic behaviour (Johnson et al. 2020). To investigate the influence of viscoelastic behaviour of thrombus material on the occlusion location, we have used the Kelvin–Voigt model where the behaviour of the viscos element is implemented through the specification of a non-dimensional stress-relaxation curve, parameterised through a Prony series and the proposed hyperelastic model has been used for the elastic element. The dimensionless shear-relaxation modulus $\overline{g}\left( t \right)$ and the dimensionless volumetric-relaxation modulus $\overline{\kappa }\left( t \right)$ in Prony series are given as

$$\overline{g}\left( t \right) = 1 - \mathop \sum \limits_{i = 1}^{n} g_{i} \left( {1 - \exp \left( { - t/\tau_{i} } \right)} \right)$$

(7)

$$\overline{\kappa }\left( t \right) = 1 - \mathop \sum \limits_{i = 1}^{n} \kappa_{i} \left( {1 - \exp \left( { - t/\tau_{i} } \right)} \right)$$

(8)

where $n$ is the number of the terms in the Prony series, $\tau_{i}$ are the relaxation time constants for each term of the series, while the parameters $g_{i}$ and $\kappa_{i}$ sets the ratio of long-term to instantaneous effective shear and bulk modulus, respectively. A two-term Prony series is implemented with $g_{1} = 0.15, \, g_{2} = 0.28, \, \tau_{1} = 60 \,{\text{sec}}, \tau_{2} = 500\, {\text{sec}}$ based on a previous study from our group.

In Fig. 10 two regimes of applied pressure are simulated: a single-step pressure increase, and a multi-step pressure increase. In both cases the clot eventually reaches the same final position. However, even for the case of the single-step pressure increase the clot does not reach its final position until ~400 s after the pressure application. This suggests that tapered tube experimental measurements should be executed over several minutes following pressure application. On the other end of the spectrum, this result suggests that in vivo a clot will reach its final occluded position in the vasculature at a relatively fast time-scale of several minutes, compared to the typical elapsed time (hours) prior to clinical intervention (e.g. thrombectomy).

Appendix 4: Convergence study

A typical mesh study as a proof of convergence of the results in terms of number of elements is shown in Fig. 11. The dimensions of clot and geometry of the tube are the same of the in vitro test (Sect. 3.3). The material parameters for the RBC-rich clot from Table 1 and friction coefficient of 0.09 have been used.

Based on the results in Fig. 11, the element size of 0.09 mm (33000 elements for this case) has been considered as the final mesh size and all simulations of the tapered tube in this paper have been performed with this element size.

Appendix 5: Comparison between the proposed model and Ogden model in taper tube test

We have simulated the tapered tube test by using the Ogden model, with the optimised material parameters in Table 3, for 5 different friction coefficients and the results are compared with the proposed model (Fig. 12). Ogden model was shown to replicate the stress–strain behaviour of clot with acceptable accuracy (Fig. 1G). However, the volumetric behaviour is a key determinant in tapered tube experiment and the proposed volumetric model improves the prediction of the results of tapered tube experiment, as demonstrated in Fig. 12.