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

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.


Introduction
Acute ischemic stroke (AIS), due to embolic occlusion of a cerebral artery, results in over 5.5 million deaths each year. Intra-arterial mechanical thrombectomy (MT) is a minimally invasive procedure in which the obstructing thrombus is removed using a stent retriever and/or an aspiration catheter. The success of the thrombectomy procedure can be significantly affected by the mechanical properties of the occluding thrombus (Chueh et al. 2011;Johnson et al. 2019). A thrombus or blood clot is made up of a network of fibrin, platelets, red blood cells (RBCs), and other blood components. Clot mechanical properties are strongly influenced by composition and the arrangement of components within the clot. Previous studies have carried out a variety of mechanical testing on thrombus material, including compression testing (Saldívar et al. 2002;Ashton et al. 2009;Krasokha et al. 2010;Chueh et al. 2011;Malone et al. 2018), uniaxial and biaxial tensile testing (Di Martino et al. 1998;Saldívar et al. 2002;Vande Geest et al. 2006;Gasser et al. 2008;Chueh et al. 2011), rheometry (Kim and Srinivasan 2005;van Dam et al. 2006;Malone et al. 2018), nanoindentation (Slaboch et al. 2012;Weafer et al. 2019) and friction testing (Gunning et al. 2018). The extent and composition of the occlusive clot itself influences the therapeutic efficacy of revascularisation techniques, yet clot location may also be important because of the nature and degree of collateral compensation beyond the occlusion (Liebeskind 2005).
Computational analysis is widely used for pre-clinical assessment of medical devices (Ajjan et al. 2009;Zhao et al. 2011;Rouhani et al. 2019). In this context, development of accurate biomechanical material models for blood clots is a key for accurate device simulation and analysis. Incompressibility is commonly assumed in computational modelling of soft tissue, though recent investigations have challenged such assumption (Gabbay et al. 2006;McEvoy et al. 2018;Moerman et al. 2020). Blood clots undergo large multi-axial deformations during vessel occlusion, and also during thrombectomy procedures. However, the relative contributions of volumetric and isochoric material behaviour have not previously been investigated.
Therefore, the main motivations/objectives of this study are as follows: 1. Develop and validate a new material model for the volumetric and isochoric mechanical behaviour of blood clots.
2. Investigate the relationship between clot composition/ mechanical behaviour and the location at which a clot occlusion occurs in a tapered vessel.
The paper is structured as follows. In Sect. 2, we calculate the volume change of blood clot analogues during unconfined compression tests for two clot compositions: a fibrin-rich made from a 5% haematocrit (H) blood mixture; and a RBC-rich clot made from a 40% H blood mixture. A new volumetric and isochoric hyperelastic model for clots is presented and is shown to accurately reproduce nonlinear stress-strain data and nonlinear volumetric data for both compositions. In Sect. 3 we investigate the influence of clot mechanical properties on occlusion location in a tapered vessel. Following a preliminary analysis to clearly parse the sensitivity of the final lodged position of the clot to isochoric and volumetric deformation regimes, we demonstrate that the new constitutive material model accurately replicates clot occlusion locations measured in vitro. Finally, we simulate clot occlusion in a patient-specific artery.

Constitutive law for isochoric and volumetric behaviour of blood clots
The hyperelastic model recently proposed by Fereidoonnezhad and McGarry (2020) is extended to model the volumetric and isochoric behaviour of blood clots. The isochoric strain energy density function is given as where D 1 , D 2 , E 1 , and E 2 are material parameters, i (i = 1, 2, 3) are the three principal stretches, i (i = 1, 2, 3) are the isochoric principal stretches, J = 1 2 3 is the Jacobian, and 01 and 02 are two constants which ensure the continuity of strain energy. Moreover, p, q, and r are not independent parameters; in order to maintain C 0 and C 1 continuity the following relations must be enforced: In the current study we propose an extension of this framework to account for nonlinear volumetric behaviour. Building upon equation (1), a new volumetric strain energy density function is implemented such that: in which 1 and 2 are the initial small-strain and largestrain bulk modulus, respectively, the parameters D 1v and D 2v control the transition volumetric strains, and p v , q v , and r v are obtained in a similar manner as equation (2) by using the corresponding volumetric parameters. The total strain energy density of the clot material is given as Fig. 1 The capability of the proposed model to reproduce the stress-strain experimental data of the unconfined compression of platelet-contracted blood clot analogues made from 5% H and 40% H blood mixtures . (A) undeformed and deformed blood clot during unconfined compression test; (B) FE simulation of the unconfined compression test; (C, D) Nominal Stress-nominal strain data ) and volume change for a RBC-rich platelet-contracted clot made from 40% H blood mixture, (E, F) Nominal Stress-nominal strain data ) and volume change for a fibrin-rich platelet-contracted clot made from a 5% H blood mixture. (G, H) Comparing the capability of the proposed model with the well-known hyperelastic laws in reproducing the nominal stress-strain data and the volume change, for the RBC-rich clot analogue.
= iso 1 , 2 , 3 + vol (J) . Stress-strain relationships are readily derived from (1) and (2) above, as described by Fereidoonnezhad and McGarry (2020). Sensitivity analysis of the proposed model in terms of the isochoric and volumetric parameters has been performed, and the results are presented in "Appendix" 1.

Experimental characterisation of clots and of constitutive law calibration
In Fig. 1 we demonstrate that the proposed material formulation accurately reproduces experimentally observed nominal stress-strain behaviour of blood clots subjected to unconfined compression ). Clot analogues have been fabricated from fresh ovine blood which has been found to be a suitable substitute for human blood (Duffy et al. 2017). Citrated whole blood was used to form platelet-contracted blood clots within 5 hours of blood collection from the donor animals. Test samples were then prepared by cutting the clots into cylindricalshaped samples with an approximate diameter of 10 mm, and an approximate height of 5 mm. Unconfined compression was used to determine the material behaviour of the clot analogue samples, the day after formation (minimum of N ≥15 for each clot type). The testing was performed on a Zwick uniaxial tensile machine (Zwick Z2.5, Ulm, Germany), using a customised aluminium platen. The samples were placed between two platens and the crosshead position of the machine was adjusted so that the top platen was slightly touching the top of the sample at the beginning of the test. The specimens are deformed to a nominal compressive axial strain of -0.8 at a compressive strain rate of 0.1 s −1 (Fig. 1A). The reader is referred to the recent study of Johnson et al. (2020b) for a full description of the fabrication protocols and testing method for clot analogues.
Figures 1C and E show the experimentally measured stress-strain behaviour for a RBC-rich and fibrin-rich clot subjected to unconfined compression, respectively. In the current study, using video imaging of the experimental tests of Johnson et al. (2020b), we determine the volume changes of the clot during compression using a customised image analysis programme developed in MATLAB (Nolan and McGarry 2016;McEvoy et al. 2018). The height and diameter of the cylindrical clot specimen at different time steps are obtained from the video, and the corresponding compressive axial strain ( a = a − 1 ) and volumetric strain ( v = J − 1 ) are calculated. No previous study reported the volume change of clot, and this is the first effort to measure and report the volume change in compression test for blood clots. A highly nonlinear relationship between volumetric strain and compressive axial strain is shown in Fig. 1D for the RBC-rich clot. Initial axial deformation results in a high rate of volume change (w.r.t applied axial strain, d v ∕d a ), suggesting an initially low bulk modulus. However, a significantly lower rate of volume change is observed for the applied range of axial strain of 0.1 ≤ a ≤ 0.48 . Finally, the rate of volume change reaches a higher value for a ≥ 0.48 . Full recovery of volumetric strain is observed following retraction of the loading plate.
We have then simulated the compression test by using the proposed model and fit the a − a and v − a curves to the corresponding experimental results, using the nonlinear least-square method in MATLAB. Optimised parameters for fibrin-rich clot and RBC-rich clot are provided in Table 1. Model predictions of the clot stress-strain behaviour and volume change for RBC-rich clot are superimposed in Fig. 1C-D. The model accurately reproduces the observed complex nonlinear experimental behaviour, both in terms of computed nominal stress a (R 2 = 0.997) and v (R 2 = 0.978), for the entire range of applied axial deformation. Model results suggest that effective tri-linear relationship between v and a , observed experimentally, emerges due to the interaction of material isochoric stiffening and volumetric stiffening. Similar results are presented for fibrin-rich 5% H clots in Fig. 1E-F, again with evidence of bi-linear strain stiffening (Fig. 1E). A corresponding increase in the rate of volume change is observed (Fig. 1F), such that d v ∕d a = 0.0718 if < 0.35 and d v ∕d = 0.8141 if > 0.55 . However, as shown in Fig. 1, a bi-linear, rather than tri-linear, relationship between v and a is observed experimentally for fibrin-rich 5% H clots ( d v ∕d a = 0.0718 for < 0.35 and d v ∕d = 0.8141 for < 0.55 ). Again, the computational model accurately reproduces the complex nonlinear clot behaviour over the large range of applied axial strain, both in terms of a ( R 2 = 0.986) and v (R 2 = 0.957). Best-fit model parameters are listed in Table 1. The experimental and computational investigation of volumetric behaviour of blood clots presented in Fig. 1 provides new insight into a key feature of clot behaviour. Previous studies have been limited to the simplified assumption of near incompressible behaviour.
In Fig. 1G, H we demonstrate that well established and widely implemented hyperelastic formulations are not capable of accurately predicting the nonlinear multi-axial behaviour of blood clots shown in Fig. 1 above, in contrast to the accurate and robust predictions of our proposed model. Specifically, the Yeoh, Ogden and hyperfoam formulations are investigated. Fig. 1 (G, H) presents the best-fit prediction for each model, subject to the constraint that all models are calibrated to reproduce experimental value of axial stress at the maximum value of applied axial strain ( a = −0.8 ), and the initial axial compressive stiffness ( d a ∕d a ; − 0.1 ≤ a ≤ 0.0) . Established model formulations and best-fit parameters are presented in "Appendix" 2. In contrast to our proposed formulation, Fig. 1G shows that all established models inaccurately predict the tangent stiffness and stress throughout the applied range of a . Corresponding predictions of volume change are highly inaccurate for all models for all levels of applied axial strain, as shown in Fig. 1H. The inaccuracy of each formulation is quantified in Table 2, in terms of predicted axial stress ( a ), tangent stiffness, and volumetric strain ( v ) at applied axial strains of 20%, 60%, and 80%. The superior accuracy of the proposed model suggests that blood clots transition from a low constant stiffness regime at low strains to a high constant stiffness regime at high strains, rather than exhibiting continuous power-law or exponential strain stiffening. The accuracy of the nonlinear volumetric component of our model suggests that a single parameter (bulk modulus) linear volumetric constitutive law is insufficient for the accurate simulation of multiaxial large-deformation of clots. In the following sections we demonstrate that both the isochoric and the volumetric behaviour of blood clots are key determinants of the location of clot lodgement in a tapered vessel, thus highlighting the importance of the results presented in Fig. 1.

Analysis of the lodgement of blood clots in a tapered vessel
In this section, we investigate the influence of clot mechanical properties on the occlusion location in a tapered vessel.
The motivation for such an analysis are twofold: (i) the role of clot composition on the in vivo occlusion location is a key and open clinical question that could potentially have significant implications for thrombectomy procedures; (ii) in vitro benchtop measurement of occlusion location in an engineered tapered tube can potentially be used as a quick and simple methodology to assess the mechanical properties/ compositions of excised clots or fabricated clot analogues.
In the preliminary analysis of Sect. 3.1 we initially restrict our analysis to quasi-linear material behaviour (i.e. approximately constant stiffness), for both isochoric (shape changing) and volumetric deformation in order to clearly parse the sensitivity of the final lodged position of the clot to both deformation regimes. In Sect. 3.2 we incorporate nonlinear material behaviour, using the calibrated novel constitutive law presented in Sect. 2 above. We also consider the role of clot rate-dependent visco-hyperelastic on the timescale required for a clot to reach its final lodged position in "Appendix" 3. The proposed constitutive model has been implemented into the commercial nonlinear Finite Element (FE) software Abaqus/Standard (2017) via user subroutine UMAT. Axisymmetric FE simulations have been used for all simulations in this paper except for the patient-specific simulation in Sect. 3.3 where a 3D simulation has been performed. The CAX4 elements have been used for clot and tapered tube assumed as a rigid body. A frictional contact has been considered between clot and tube.
Convergence of the results in terms of the number of elements has been performed, and it is found that element size of 0.09 mm is small enough for convergence of the results. The number of elements is different in each simulation depends on the length of the clot. More details of the mesh study are presented in "Appendix" 4.

Linear isochoric and volumetric mechanical behaviour
In this section, we consider a linear behaviour for the clot to investigate the governing parameters in the occlusion location of clot in a tapered vessel and to provide a proof of concept for the nonlinear interactions between governing parameters. Therefore, the values of parameters in this section are not necessarily in the physiological range.

Method
Figure 2 presents a schematic of the clot and vessel, indicating the key geometric parameters that describe the system and the pressure P 0 applied to the proximal surface of the clot. Several mechanisms govern the final equilibrium position of the clot in the vessel: friction between the clot and the vessel wall, the isochoric deformability of the clot (characterised by the shear modulus = 3 E∕(9 − E) , and the volumetric compressibility of the clot (characterised by the bulk modulus ). The equilibrium occlusion location is characterised by the distance u between the centre of the clot at initial contact with the vessel and centre of the clot at its final equilibrium position. It should be noted that we present our computed results in non-dimension quantities in order to efficiently demonstrate the key parameter groupings that govern the final equilibrium position of the clot. As a simple example, consider a large and a small clot, both with the same aspect ratio L∕D. Of course both clots will initially contact the vessel at different locations, and the larger clot will move a greater distance u to its final equilibrium position. However, u∕D the non-dimensional measure of the clots' final positions will obviously be identical in both cases; i.e. for a given clot aspect ratio, u scales with the size of the clot. Figure 3 shows the results of preliminary parametric study for a vessel with the taper angle of Φ =2°. As shown in Fig. 3. A1, the location of clot lodgement, u , (i.e. the equilibrium position of the clot) for a given applied pressure is strongly dependent on the clot shear modulus for nearly incompressible clots because, in this case, clot cannot change volume and the only deformation mechanism is isochoric shape change, governed by the shear modulus. However, for the case of compressible clots, as shown in Fig. 3. D1, the final lodged position is not strongly influenced by shear modulus; volume change becomes the dominant mode of deformation. In the physiological cerebral artery blood pressure range (60-100 mmHg (Pires et al. 2013)) a fivefold increase in shear modulus from 1kPa to 5kPa results in a ~48% decrease in u if a clot is nearly incompressible (Fig. 3B1), compared to a ~14% decrease if the clot is compressible (Fig. 3E1). It is noted that the range shear modulus (in the small deformation regime) for excised blood clots extends from 18.6 kPa for platelet contracted/fibrin rich clots down to 0.34 kPa for non-contracted red blood cell rich clots, based on mechanical testing and constitutive modelling of excised clots. The interaction between isochoric and volumetric parameters is further illustrated in Fig. 3C1 ( = 5 kPa) and Fig. 3F1 ( = 50 kPa) . Beyond a certain level of compressibility ( ≳ 150 kPa) the lodged position, u, is insensitive to the value of , but strongly dependent on (i.e. nearly incompressible clots behave in a similar manner to fully incompressible clots, as demonstrated by the computed value of u for = 150 kPa (nearly incompressible) and = 2500 kPa (essentially fully incompressible)). However, when the clot is highly compressible, the value of bulk Fig. 2 A schematic illustration of the occlusion of a cylindrical blood clot in a tapered vessel. Aspect ratio of clot at initial contact state (AR) 0 and at equilibrium state ( AR ) as well as the normalised aspect ratio ( Â R ) are defined. modulus significantly influences the final position; i.e. if κ is reduced from 150 kPa (nearly incompressible) to 25 kPa (highly compressible), u increases by 45% for low values of shear modulus (Fig. 3C1) and 62% for high values of shear modulus (Fig. 3F1). Figure 3D1, G1 presents a preliminary study of the influence of friction coefficient, f , between the clot and the vessel wall for compressible and nearly incompressible clots, respectively. As expected, u increases with decreasing values of f in both cases. However, even for the quasi-linear material behaviour assumed for this preliminary investigation, for any given value of applied pressure P 0 , the relationship between f and u is computed to be nonlinear. A previous experimental study (Gunning et al. 2018) suggests that the value of f depends on the clot composition; i.e., fibrin-rich blood clots have higher friction coefficient with vessel wall than the RBC-rich clots. More discussion on the friction coefficient is provided in Sect. 3.3. Influence of the shear modulus, bulk modules and friction coefficient on the length of the clots are also provided in Fig. 3B2-G2.

Influence of friction coefficient
In Fig. 4 we further explore the preliminary results by considering the computed values of u, l and Â R at a fixed physiological pressure of P 0 = 100 mmHg . Fig. 4A demonstrates the nonlinear dependence of u, l and Â R on shear modulus, while Fig. 4B demonstrates the high level of sensitivity of u, l and Â R to bulk modulus below a threshold value of compressibility ( ∕P 0 ≲4 ), and a contrasting insensitivity to bulk modulus above this threshold value. Figure 4C shows the influence of the aspect ratio of the clot (AR) 0 = L∕D (on the final equilibrium position, length and final aspect ratio of clot (AR = l∕d) . As L∕D increases (i.e. as a clot of a given diameter becomes longer), the distance travelled ( u∕D ) and the length of clot ( l∕D ) decrease while the aspect ratio ( AR ) shows a non-monotonic response. This highly nonlinear phenomenon is explained in part by the increased friction resistance for longer clots. This model predictions are consistent with the clinical observation that a shorter thrombus is more likely to have a distal location (Dutra et al. 2019;Boodt et al. 2020).

Influence of taper angle of the vessel
In Fig. 4D we assess the influence of taper angle of the vessel on u, l and Â R . As expected, the value of is highly sensitive to an increase in taper angle. These analyses can guide the design of tapered-vessel experiments so that the taper angle results in a test system sensitivity range that is capable of differentiating between physiologically relevant clot compositions.

Interactions between governing parameters
The results presented in Figs. 3, 4 take into account the influence of single system parameters on the occlusion location, assuming a base-line reference value for all other system parameters. Such preliminary analyses provide a reasonable introduction to the governing mechanism of clot occlusion in a tapered vessel. However, in reality all parameters interact with one another, requiring the computation of response surfaces to provide a more detailed understanding of the system. Such response surfaces for paired-interactions between key system parameters are shown in Fig. 5. Figure 5A1 presents a 3D surface that elucidates the − interactions introduced in Fig. 4 above. Figure 5B1 demonstrates that u is sensitive to for lower value of friction coefficient, f , while it is insensitive to shear modulus for high values of f . Similar sensitivities are observed in the − f interactions (Fig. 5C1). Fig. 5D1 shows the interaction of taper angle,Φ with f. For very high friction values, u is insensitive to Φ , i.e. once the clot initially contacts the tube, it almost immediately becomes lodged, without significant volumetric or isochoric deformation. However, as f reduces, the sensitivity to Φ increases. Similarly, the sensitivity of to the values of both and is increased if the value of Φ is reduced. In summary, a tapered-tube experimental system should implement sufficiently low values of Φ and f so that the measured value of is sensitive to the clot mechanical properties/composition. Figure 5G1 demonstrates interactions between clot aspect ratio L∕D and f , while Fig.5H1 demonstrates the interaction between L∕D and . A low level of sensitivity of u to κ is computed for extremely long clots. This is in part, due to a more dominant role of frictional resistance with increasing clot length. Therefore, low aspectratio clots should be used experimentally L∕D < 3 so that the system is sensitive to the clot material properties. The response surfaces for the influence of the parameters on the length of the clots are also provided in Fig. 5A2-H2.

Realistic nonlinear clot behaviour and vessel deformability
As demonstrated in Fig. 1, blood clots exhibit significant volumetric and isochoric strain-stiffening ).
Here, we use the proposed constitutive model in Section 0 to investigate the influence of clot stiffening on the distance travelled ( u∕D ) in a tapered tube. The baseline material parameters are chosen to be the same as those presented in Table 1 for fibrin-rich platelet-contracted clot. Figure 6 demonstrates that for the given parameter sets, strain stiffening of the clot strongly influences the occlusion location. Figure 6B demonstrates that if the stiffening ratio E 2 ∕E 1 is increased from 1 to 10, u decreases by 18%. The results of Fig. 6D show even an stronger influence of volumetric stiffening 2 ∕ 1 on u . Increasing 2 ∕ 1 from 1 to 10 results in a 38% decrease in the value of u . Such sensitivity to 2 ∕ 1 is less pronounced beyond the threshold of 2 ∕ 1 > 20 . Figure 6A,C shows that u increases with increasing values of the transition strains D 1 and D 1v . Increasing the transition strains postpones the start of clot stiffening, allowing the clot to travel further into the tapered tube. The validity of modelling the tapered vessel as a rigid body is investigated by incorporating the effect of tube compliance and thickness in the FE simulations. Silicon rubber tubes with a range of thicknesses are considered and the variation of occlusion location with tube thickness is shown in Fig. 6E. Results are sensitive to the wall thickness of a silicone tube only if the tube wall thickness is less than 5% of the clot diameter. As an example, for a clot of 5mm diameter, the tube can reliably be modelled as a rigid material provided that the tube wall thickness is greater than ~250 m . In Sect. 3.3 the nonlinear compliance of a cerebral artery is taken into account in the simulation of clot occlusion in a patient-specific vessel anatomy.

In vitro clot occlusion in a tapered tube:
An in vitro test rig is developed in which clot analogues are introduced (through a side port) into a tapered silicone vessel, as shown in Fig. 7A. The vessel is fabricated with a taper angle of Φ = 0.9854 o (BioMET, GMIT, Galway, Ireland) over a length of 250 mm. The tapered vessel is connected to a pulsatile pump (PD-1100 pulsatile pump, BDC Laboratories), which precisely controls the pulse rate, pressure and temperature of the system. 0.9% saline, which was heated to 37°C, was pumped through the model with a pulse rate of 75 BPM. Applied pressures of 90 mmHg and 30 mmHg were applied to the proximal and distal surfaces of the clots, respectively. The clot final occlusion location and deformed shape of the clot are measured. Clot specimens of length 10 mm and diameter 6 mm are fabricated in accordance with previously reported protocols . Two platelet-contracted blood clot analogues compositions are considered: made from blood mixtures with 5%H (N=5) and 40%H (N=5). Detailed experimental results for additional clot compositions and characterisation of irregular deformation of proximal clot surfaces are presented in (Johnson 2020).

Simulation of in vitro tapered vessel experiment
We have simulated the in vitro test for a fibrin-rich and a RBC-rich clot using our proposed volumetric and isochoric hyperelastic formulations and compare model predictions to the experimental measurements of Johnson (2020). Using the material parameters obtained from the compression test (Table 1), we have ran the simulations for five different friction coefficients for the contact between the silicone surface and fibrin-rich and a RBC-rich clot analogues. The results, as demonstrated in Fig. 7, reveal that friction coefficients of f = 0.07 for fibrin-rich clot and f = 0.05 for RBC-rich clot result in a reasonable agreement between computational results and the in vitro data in terms of the distance that clot travel into the tube ( u∕D ) and the deformation of the clot, characterised by the normalised aspect ratio ( Â R ). This findings are in agreement with the experimental observation of Gunning et al. (2018) that the friction coefficient for a fibrinrich clot is higher than the friction coefficient for a RBC-rich clot on a PTFE surface. Despite the fact that the fibrin-rich clot has a higher isochoric tangent modulus and friction coefficient, it travels farther into the tapered vessel than the RBC-rich clot. This is due to the higher level of compressibility (volumetric deformability) of the fibrin-rich clot. This suggests that nonlinear hyperelastic volumetric behaviour is a key determinant of the occlusion location of a clot. This further highlights the importance of the accurate characterisation of and constitutive modelling of the volumetric behaviour of a range of clot compositions. A comparison between the proposed model and Ogden model for prediction of the tapered tube experiment results is also presented in "Appendix" 5.

Simulation of in vivo patient specific clot occlusion
Finally, we simulate clot occlusion in a patient-specific cerebral artery. The vessel geometry is reconstructed from clinical CT images using the GIBBON toolbox (Moerman 2018). A portion of the M1 cerebral artery proximal to the occlusion location, based on CT images, has been used in this simulation as shown in Fig. 8A. Both ends of the artery have been fixed and outer surface of the artery is stress free.
A mesh study has been performed to find out the required mesh density for a converged solution. 57398 elements of type C3D4 for clot and 5075 elements of type C3D8 for the artery, with one element through the thickness, have been found enough for convergence of the results. The mechanical behaviour of the vessel is based on the experimental test data for cerebral arteries reported by Monson et al. (2003). Friction coefficient of 0.3 has been used for the contact between artery and clot. A cylindrical-shaped RBC-rich plateletcontracted clot, made from a 40% H blood mixture, with length of 11mm and diameter of 3.75 mm is introduced into the vessel (Fig. 8B) with a pressure P 0 = 75 mmHg applied to the proximal face of the clot to approximate in vivo physiological loading. Figure 8B (top) illustrates the clot at initial point of contact between the clot and the vessel. The final computed deformed clot at the location of occlusion is also shown in Fig. 3B (bottom row -three views shown to illustrate vessel tortuosity). The computed distribution of the maximum shear stress, maximum principal stress, and minimum principal stress in the clot at lodged position are shown in Fig. 8C, to provide a complete characterisation of the stress state within the clot. This simulation highlights the significant deformation and stress in the clot at its occluded location. Such predicted stress distributions could be included in FE simulations of clinical interventions, such as thrombectomy and aspiration (Luraghi et al. 2021) to provide improved predictions of the interaction between the clot and the intervention device and may also provide improved prediction of clot fragmentation risk during thrombectomy procedures. For example the maximum principal stress in this simulation is lower than the critical value for clot fracture recently determined by the authors (Fereidoonnezhad et al. 2020a). Finally, the strain and deformation of the arterial wall due to interaction with the clot are shown in Fig. 8D. The small values of strain and deformation suggest that modelling the artery geometry as a rigid body is acceptable.

Summary and conclusions
Mechanical thrombectomy has become the standard acute ischemic stroke treatment in patients with large vessel occlusion. The success of the thrombectomy procedure 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 blood clot. We demonstrate that the proposed model provides significant improvements over established hyperelastic models in terms of accurate prediction of nonlinear stress-strain behaviour and nonlinear volumetric behaviour of fibrin-rich and RBC-rich blood clots. We perform a rigorous parametric 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 a key and open clinical question that has significant implications for thrombectomy procedures; (ii) in vitro benchtop measurement of occlusion location in an engineered tapered tube can potentially be used as a quick and simple methodology to assess the mechanical properties/compositions of excised clots or fabricated clot analogues. Simulations demonstrate that both the isochoric and the volumetric behaviour of blood clots are key determinants of the location of clot lodgement in a tapered vessel, in addition to clot-vessel friction. The proposed volumetric/isochoric clot hyperelastic formulation is shown to provide accurate predictions of in vitro measurement so of clot occlusion location in a silicone tapered vessel, in addition to accurately predicting the deformed shape of the clot. These findings provide a significant advance in the current understanding of the relationship between clot mechanical behaviour and occlusion location in cerebral vascular network which can help to guide clinical intervention strategies and development of the next-generation thrombectomy devices Fig. 6 Influences of clot strain-stiffening on occlusion location. Effect of (A) the transition strain, D 1 ; (B) stiffening ratio E 2 ∕E 1 ; (C) volumetric transition strain D 1v , and (D) bulk modulus ratio 2 ∕ 1 are illustrated. The baseline material parameters are taken from Table 1 for fibrin-rich platelet-contracted clot and L∕D = 2, P 0 = 75 mm Hg, f = 0.2 and Φ = 2 o are used. The value of D 2 − D 1 and D 2v − D 1v is kept constant. Influences of tube thickness on the occlusion location for a silicone tapered tube and the deformed shape of the clot in a thin tube (t/D=0.01) is also shown in (E). The inset shows bulge in the tube.
The following key points should be noted: 1. The nonlinear volumetric and isochoric hyperelastic constitutive law proposed in this study provides accurate predictions of the complex patterns of experimentally measured stress and corresponding volume change for both RBCrich clots (made form 40% H blood mixture) and fibrin-rich clots (made from 5% H blood mixture). We demonstrate that established hyperelastic models cannot provide comparable levels of accuracy. Therefore, our model predictions suggests that (i) blood clots transition from a low constant stiffness regime at low strains to a high constant stiffness regime at high strains, rather than exhibiting continuous power-law or exponential strain stiffening; (ii) blood clots exhibit a volumetric strain stiffening, and traditionally implemented single parameter (bulk modulus) linear volumetric constitutive laws (Fereidoonnezhad et al. 2016(Fereidoonnezhad et al. , 2017Nolan and McGarry 2016;Rouhani et al. 2019) are not suitable for the simulation of large multiaxial large-deformation of clots. This study provides the first detailed characterisation of the volumetric change for a range of clot analogue compositions.  Table 1have been employed. Experimental results reproduced from Johnson (2020).
2. Accurate modelling of volumetric and isochoric behaviour of clots is critical in order to accurately predict clot occlusion location in a tapered vessel. The volumetric and isochoric formulation proposed in the current study is shown to provide a reasonably accurate prediction of in vitro experiments of clot occlusion in a tapered silicone vessel, both in terms of occlusion location and deformed shape of the clot. Accurate predictions are achieved for both fibrinrich and RBC-rich clots. Our simulation of clot occlusion in patient-specific vessel provides a quantitative distribution of stress throughout the clot when lodged in the vessel. This can be used as an initial condition for in silico thrombectomy simulations (Luraghi et al. 2021) .
The new understanding of volumetric behaviour of clots will be incorporated into ongoing development of in silico models for thrombectomy procedures. Future simulations will consider the multiaxial deformation of the lodged clot due to deployment of stent retrievers (Luraghi et al. 2021). The rupture risk during thrombectomy and aspiration will be simulated using cohesive zone (McGarry and McHugh 2008;McGarry et al. 2014) and extended finite element techniques (Feerick et al. 2013) based on the recent experimental investigation of the influence of clot composition on fracture resistance (Fereidoonnezhad et al. 2020a). The future work should also focus on the relationship between clot composition and occlusion location in patient specific models.

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 , 1 , 2 ); however these parameters can control the volumetric strain ( 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.    Table 1 have been employed. Experimental results reproduced from Johnson (2020).

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 where, i , i , and D i are material parameters. The Yeoh strain energy density function is given as where C i0 and D i are material parameters; I 1 is the first deviatoric strain invariant defined as The hyperfoam strain energy density function is given as 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 . 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 g(t) and the dimensionless volumetric-relaxation modulus (t) in Prony series are given as where n is the number of the terms in the Prony series, i are the relaxation time constants for each term of the series, while the parameters g i and i sets the ratio of longterm to instantaneous effective shear and bulk modulus, respectively. A two-term Prony series is implemented with g 1 = 0.15, g 2 = 0.28, 1 = 60 sec, 2 = 500 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.