Load-sharing characteristics of stenting and postdilation in heavily calcified coronary artery

In this work, stenting in non-calcified and heavily calcified coronary arteries was quantified in terms of diameter-pressure relationships and load transfer from the balloon to the artery. The efficacy of post-dilation in non-calcified and heavily calcified coronary arteries was also characterized in terms of load sharing and the changes in tissue mechanics. Our results have shown that stent expansion exhibits a cylindrical shape in non-calcified lesions, while it exhibits a dog bone shape in heavily calcified lesions. Load-sharing analysis has shown that only a small portion of the pressure load (1.4 N, 0.8% of total pressure load) was transferred to the non-calcified lesion, while a large amount of the pressure load (19 N, 12%) was transferred to the heavily calcified lesion. In addition, the increasing inflation pressure (from 10 to 20 atm) can effectively increase the minimal lumen diameter (from 1.48 mm to 2.82 mm) of the heavily calcified lesion, the stress (from 1.5 MPa to 8.4 MPa) the strain energy in the calcification (1.77 mJ to 26.5 mJ), which associated with the potential of calcification fracture. Results indicated that increasing inflation pressure can be an effective way to improve the stent expansion if a dog bone shape of the stenting profile is observed. Considering the risk of a balloon burst, our results support the design and application of the high-pressure balloon for post-dilation.


Introduction
Stenting in heavily calci ed coronary arteries is challenging due to the inherent residual stent underexpansion and malapposition.Post-dilation with a shorter balloon at a higher in ation pressure has been widely adopted to improve the stent expansion.However, it increases the risk of intima dissection and vessel rupture [1,2].A retrospective study has scored stent sizing, balloon post-dilation, and pre-dilation to predict one-year adverse cardiac events following implantations of bioresorbable stent [4].There is also mounting evidence to query if pre-dilation is necessary [5][6][7].A clinical report of eight patients has demonstrated the e cacy of a noncompliant balloon with an ultra-high in ation pressure of 40 atm [3].However, the contribution of high in ation pressure to arterial mechanics was not clear.In addition, no existing publications quanti ed how the in ation pressure is transferred from the balloon to the stent-lesion system.The physical characterization of stenting and post-dilation could help to address the aforementioned controversies and shed light on better pre-clinical planning.
The Finite element (FE) method has been widely used to evaluate stent behaviors [27][28][29] and predict vessel damage and adaptations [12][13][14][15][16][17].Speci cally, three-dimensional patient-speci c artery models have been developed by coregistrating both optical coherence tomography (OCT) and computerized tomography (CT) images for performing structural and hemodynamic analyses [30].It has been shown that the pre-dilatation and lesion compositions, such as lipid pool, brous cap, and calci cation, affect the arterial responses [20,21].The abnormally high arterial stresses, induced by the stent implantation, is positively correlated with the risk of restenosis [18,19].Using patient-speci c artery models, our group has systematically inspected the in uence of the calci cation attributes on the stenting outcomes, including stent underexpansion and malapposition [22,23].Our FE results have illustrated that a larger calci cation angle constricted the stretchability of the lesion, and thus the stent expansion capacity.An in-vitro uniaxial tensile experiment also demonstrated that a larger calci cation volume leads to a reduced stretchability of the lesion [25].We conducted additional ex-vivo and in-silico experiments to further test this indicator in the post-dilation procedures [26].These observations directed our attention to the stretchability of the lesion, i.e., the amount of brous plaque along the circumferential direction, which might be the the major determinant of stent expansion.With these accumulated studies on the correlation between lesion features and stenting outcomes, now it's necessary to link the stenting outcomes in varied lesions with the required pressure for optimal stenting.Even a in ation pressure ranges from 10 to 20 atms has been adopted in the clinical practice, the amount of pressure load directly to the lesion is still a myth.Considering a pressure value of 6 atm has been widely adopted in the application of cutting beloon, the logic of the balloon design for different lesions has attracted more attentions.The quanti cations of load transfer characteristics during stenting and post-dilation procedures are in immediate needs for enhanced understandings of the stent-artery interaction and designing better clinical strategies.
The objective of this work is to quantify the load sharing among the balloon, stent, artery and lesion components (i.e., brous plaque, and calci cation, if any), during stenting and post-dilation in non-calci ed and heavily calci ed coronary arteries.The pressure-diameter relationship of the balloon, and the radial force transferred to the balloon, stent and lesion will be characterized.Further, the e cacy of high-pressure postdilation balloon will be inspected in terms of strain energy in each component and the maximum principal stresses in the artery.The fundamental understanding of load transfer and load sharing could better guide the optimal stenting in complex lesions.

Materials and methods
Model construction.An idealized coronary artery was modeled as a cylinder with a length of 20 mm, an inner diameter of 3 mm, and a thickness of 0.5 mm [31].The center-located plaque had a parabolic shape with a length of 8.5 mm and a minimum diameter of 1.2 mm, mimicking a diameter stenosis of 60% (i.e., area stenosis of 84%).The brous plaque was adopted for the case of the non-calci ed lesion.For the case of the heavily calci ed lesion, a block calci cation over a thin brous plaque was adopted, as shown in Fig. 1.The super cial concentric calci cation had a maximum thickness of 0.64 mm, and the brous plaque had a maximum thickness of 0.26 mm.A commercial Express stent (Boston Scienti c, Natick, MA, USA) was used.It has a length of 16 mm, a thickness of 0.13 mm and a nominal diameter of 3 mm at the nominal pressure of 12 atm.Non-compliant (NC) balloons with a xed length of 17 mm, and diameters of 3 mm, 3.5 mm and 4mm, were simulated to study the impact of balloon oversizing.Additional NC balloons with a xed nominal diameter of 3 mm, and lengths of 11 mm and 8.5 mm were also simulated to determine the e cacy of shortened balloons during post-dilation procedures.All NC balloons were simulated as cylinders with an initial diameter of 0.8 mm.Material properties.The hyperelastic behaviors of the artery, brous plaque, and calci cation were described by the reduced third-order polynomial strain energy density function U: The material coe cients C ij were adopted from our previous work [32], as shown in Table 1.A perfect plastic model was used to describe the tissue compaction of the brous plaque, which is necessary to capture a realistic stent expansion including stent recoil [33].The plasticity of brous plaque was initiated at a strain of 34% when its stress reached its yield strength of 0.07MPa [34].The stress-strain relationship for all these lesion components is shown in Fig. 2a.
The pressure-diameter data provided by the manufacturer was used to derive the material properties of the balloon (Fig. 2b).It is clear that the balloon exhibited a bilinear in ation behavior.The balloon diameter increased faster when the in ation pressure is below 6 atm, and much slower when the pressure exceeded 6 atm.To convert this bilinear in ation behavior of the balloon to a stress-strain relationship, the hoop strain was calculated as the relative change in diameter, and the hoop stress of a thin-walled cylinder was used to estimate the wall stress: 5 6 where D is the diameter during expansion, D 0 is the initial diameter of 0.8 mm, P is the in ation pressure, and δ is the thickness of the balloon.The pressure-diameter curve obtained from simulation was compared with the manufacturing data with maximum difference less than 5% for pressure interval from 6 atm to 25 atm (Fig. 2b).
Finite element simulations of the calci ed coronary artery in the context of idealized or patient-speci c models have been well validated in our previous work by matching the stented lumen area with ex-vivo experiments, or matching the simulated stress level with the published data [22,23,26].Mesh convergence studies were conducted in this work, and 123,100 hexahedron elements were adopted for the artery model.Symmetric boundary conditions (i.e., the displacement along the longitudinal direction is constrained, while along the transverse direction allowed) were applied to both ends of the artery and balloon such that the stenting procedure does not alter the lesion length far from the implantation site.For the stenting procedure, 10 atm was applied to the inner surface of the balloon.For the post-dilation, three pressures (i.e., 10 atm, 20 atm, and 30 atm) were sequentially applied to the inner surface of the balloon.General frictionless contacts were used for all interacting surfaces [35].Energies were monitored during the stenting and post-dilation procedures to ensure the dynamic effect (i.e., inertial forces) was acceptable.The ratio of the kinetic energy to the internal energy of the whole model was kept below 5%.The models were solved using Abaqus/Explicit 2022 (Dassault Systemes Simulia Corp., Providence, RI, USA).
Following simulation, the load transfer and load sharing were quanti ed.The load transfer refers to the actionreaction force between balloon, stent, and lesions.The load sharing refers to the strain energy stored in each component of the lesion and the stent.The pressure load is calculated as the radial force applied to the inner surface of the balloon: where P is the in ation pressure, D is the diameter of the balloon, and L is the length of the balloon.The radial forces applied on the inner surface of the stent and artery were obtained by adding all the radial component of the contact force at each node on their inner surface.

Results
Stenting in both non-calci ed and heavily calci ed lesions was compared.The load transfer from in ation pressure to the balloon, stent, and artery were quanti ed.The stress analysis and load sharing among the balloon, stent, and artery were used to further quantify the e cacy of the high-pressure balloon in improving the stent expansion in heavily calci ed lesion.
Stent expansion and pressure-diameter curve.The stent expansion in the non-calci ed lesion and heavily calci ed lesion are shown in Fig. 3.As in ation pressure increased to 10 atm, the stent expansion in the noncalci ed lesion reached an inner diameter of 2.80 mm in the non-calci ed artery, with a nearly cylindrical shape (Fig. 3a top).Stent underexpansion was observed in the heavily calci ed artery, forming a dog bone shape with a minimal diameter of 1.48 mm (Fig. 3a bottom), lead to a residual stenosis of 49% in diameter and 74% in area.The associated pressure-diameter relationship is shown in Fig. 3b.The stent expansion in the noncalci ed lesion was quite similar to the expansion in air, indicating the non-calci ed lesion has minimal resistance and load sharing.On the contrary, the stent expansion in the heavily calci ed lesion requires a much higher pressure beyond the normal pressure of 10 atm to obtain the prede ned minimal diameter.As the in ation pressure increased from 10 atm to 30 atm, the minimal diameter of the non-calci ed lesion increased from by only 19% (2.8 mm to 3.2 mm), while of the heavily calci ed lesion it increased by 90% (from 1.48 mm to 2.82 mm).
Radial force transferred from in ation pressure to balloon, stent, and lesion.The radial force transferred from the in ation pressure to the balloon, stent, and lesion are depicted in Fig. 4. The in ation pressure of 10 atm (i.e., 1.013 MPa) resulted in a cylindrical expansion with an inner diameter of 2.76 mm (Fig. 3), corresponding to a pressure load of 152 N exerted onto the balloon surface.For stent expansion in the non-calci ed lesion, a radial force of 23.8 N (i.e., 15% of the pressure load) was transferred to the stent and lesion, of which 1.4 N (i.e., 0.8% of the pressure load) was transferred to the non-calci ed lesion.For stent expansion in the heavily calci ed lesion, a radial force of 34.5 N (i.e., 22% of the pressure load) was transferred to the stent and lesion, and 19 N (i.e., 12% of the pressure load) was transferred to the lesion.We could convert the radial force transfeerred to the lesion to an equivalent pressure using Eq. 7. Therefore, a pressure of 0.092 atm (i.e.,70 mmHg) was transferred to the non-calci ed lesion, while a pressure of 1.25 atm (i.e., 950 mmHg) was transferred to the heavily calci ed lesion.It should be noted that all the above load calculation is based on cylindrical shape assumption.Considering the actual dog-bone shape in the case of the heavily calci ed lesion, the pressure load was less than 152 N, but the difference was less than 10%.
Load sharing in terms of strain energy during stenting and post-dilations.Strain energy stored in the artery, brous plaque, calci cation (if any), and stent during the stenting and post-dilation procedures are shown in Fig. 5.The stacked areas are used to show how each component contributes to the total strain energy of the stented lesion.It is clear that, regardless of the lesion type, the absorbed strain energy in all components reached its peak at each target in ation pressure, and then recoiled back following the balloon defaltion.For the non-calci ed lesion, the peak strain energy of brous plaque was 0.184 mJ at the full expansion of the stent (arrow in Fig. 5a), and was 0.182 mJ, 0.184mJ, 0.189 mJ, respectively, during three sequential postdilations with increased in ation pressures.The peak strain energy stored in both stent and lesion was 0.35mJ during stenting and increased to 0.55 mJ during the 3rd post-dilation at the in ation pressure of 30 atm.In the heavily calci ed lesion, the corresponding total strain energy stored in the stent, brous plaque and artery was 0.21 mJ during stenting and sharply increased to 0.90 mJ during the 3rd post-dilation at in ation pressure of 30 atm.The peak strain energy in the calci cation alone (labeled in the right y-axis of Fig. 5b) was 1.77 mJ, and increased to 26.5 mJ (15 times) during the 3rd post-dilation at the in ation pressure of 30 atm, making it substancially higher than the other components.The huge increase of the strain energy in calci cation indicate a potential of fracture.
Maximum principal stresses induced in lesions after stenting and post-dilations.The distribution of maximum principal stress (MPS) on the lesions at 10 atm was depicted in Fig. 6a.The time history of the peak value of MPS in the non-calci ed and heavily calci ed lesions during stenting, and the three sequential post-dilations at pressures of 10, 20 and 30 atm, were plotted in Fig. 6b.The peak value of MPS in the non-calci ed lesion, occurred in the artery, was 40 kPa during stenting at a in ation pressure of 10 atm, and increased to 70 kPa during the 3rd post-dilation at a in ation pressure of 30 atm.For the heavily calci ed lesion, the peak value of MPS, occurred in the calci cation, was 1.5 MPa during stenting at a in ation pressure of 10 atm, and increased to 8.4 MPa during the 3rd post-dilation at a in ation pressure of 30 atm.It is worth noting that the MPS time history is consistent with the load sharing analysis (Fig. 5).For the non-calci ed lesion, the arterial tissue only stores a small portion of energy and doesn't increase proportionally along with the increase of the in ation pressure.This indicates a lower risk of vessel rupture.On the contrary, strain energy stored in the calci cation increase 14 times as the in ation pressure increase from 10 atm to 30 atm, indicateing a higher probability of calci cation fracture.

Discussion
Suboptimal stenting (stent underexpansion, malapposition etc.) in complex lesions, especially in heavily calci ed lesions, has gained increasing attentions.Plaque modi cation and/or post-dilation with a highpressure balloon has been developped to improve the stent expansion and apposition.The cardiologists are under a pressure to choose e cient procedures for a target stent expansion with less risk of vessel rupture.In this work, the link between the in ation pressure to the exact force exerted on the inner surface of various lesions was inspected with computer simulations.To our best knowledge, this is the rst study to quantify the load transfer between stent-artery interaction and load-sharing capacities of different components of the during stenting and post-dilation procedures.Results have emphasized the monitoring the stent expansion during stenting and supported the design of the high pressure balloon.The stent expansion showed a nearly cylindrical shape in the non-calci ed lesion and the balloon undertook most of the increasing in ation pressure load.On the contrast, stent expansion showed a dog boen shape in the heavily calci ed lesion, and the increasing in ation pressure can effectively increase the minimal lumen area, stress in the calci cation, and potential of calci cation fracture.The e cacy of the increasing in ation pressure in the heavily calcifcaiton lesion is due to the dog bone shape of the stent expansion.
Most previous computational studies of stent-artery interactions focused on the mechanical environment change (i.e., stress and strain) in the arterial tissues [20,35] or stent fracture when exposed to cardiac wall movement following stenting [36].Our previous works have investigated the in uence of calci cation attributes on stenting expansion with stylized and patient speci c artery models [22,37], and results have shown that the calci cation will reduce the stretch capability of the lesion, further leading to stent underexpansion.In this work, load transfer analyses were conducted for stenting in non-calci ed and heavily calci ed lesions to quantify the effective load for stent expansion.Only a small portion of the pressure load (1.4 N, 0.8% of the total pressure load) was transferred to the non-calci ed lesion.This small portion of the pressure load only induced an effective pressure of 75 mmHg, which is slightly less than the tested blood pressure in a human body, which is around 100 mmHg [38].Additionally, the balloon pressure for stenting, which is usually around 12 atm (or 9120 mmHg), is much higher than the blood pressure.This drastic difference between the dilation pressure and the exact pressure required for expanding the lesion has previously been ignored in studies on stent-artery interactions.The load transfer and load sharing analysis will build a direct link between the increasing pressure and the effective force to the lesion, which will provide a rational index for optimal stenting.In addition, a radial force of 22.4 N was required to expand the stent, which is similar to the reported studies [39,40].These agreements validate the feasibility of our model and methods.For the heavily calci ed lesion, the concentrated, thick calci cation acts as a stiff ring and causes the observed dog bone shape of the stent expansion.The balloon which expands in a dog bone shape was not able to undertake as high of a pressure load as the cylindrical shaped balloon.Because of this, a higherpressure load of 19 N (1027 mmHg) was transferred to the lesion, which is more than 12 times that which is shown in the non-calci ed lesion.We can image that as the lesion changes from non-calci ed to a mild, medium, and then heavily calci ed lesion, the balloon stent expansion will change from a nearly cylindrical shape to a more irregular and/or dog bone shape, during which the pressure load transferred to the lesion will increase from a lower to a higher value.Our previous virtual bench tests also have shown that the resultant contact along the normal of one cut plane increased from 1.5 N to 2.5 N, to 3.5 N as the calci cation angle increased from 60º, to 180º, to 270º [41].
Post-dilation has long been adopted to improve stent expansion by increasing in ation pressure or using a larger balloon.One study has even shown that high pressures up to 40 atm can lead to an improved stent deployment in heavily calci ed lesions [3].Some other retrospective studies, however, have showed that a pressure of 16 atm and a balloon diameter which is 0.5 mm larger than the nominal diameter of stent are two critical cap values, and there will be more complications if the in ation pressure exceeds these [4].In this work, the load sharing analysis for post-dilation with increasing pressure was done to evaluate the e cacy of the procedure.In the non-calci ed lesion, the stent expansion has a nearly cylindrical pro le (diameter close to manufacturing data) and the balloon undertook most of the pressure load.Therefore, the increasing pressure didn't signi cantly increase the minimal lumen diameter, stress or strain energy in the lesion.For the heavily calci ed lesion, the stent expansion has a dog bone shape and the increasing pressure load can effectively increase the minimal lumen diameter, stress and strain anergy in the calci cation, and the potential of calci cation fracture.Based on this observation, increasing pressure can be an effective procedure to improve the stent expansion if a dog boen shape was observed.In addition, a larger balloon may cause the lesion to rupture if the calci cation suddenly fractures at a high pressure, which may lead to acute myocardial infarction as reported in clinics [2].These ndings indicate that the monitoring of the stenting process, especially the stenting pro le and diameter at the target in ation pressure, may help in making optimal stenting decisions for patients.Our results also support the design and application of the high pressure ballon.Currently, the burst pressure of most post-dilaiton balloon is around 20 atm, while it can not induce a calci cation fracture, and the cardiologists shift to alergar balloon, with the risk of vessel rupture.As we noticed that the increasing pressure can effective exert more force to the lesion, the high pressure balloon can be a safe way to induce the calci cation fracture by increasing the in ation pressure to a higher pressure limit of the balloon.
There are some limitations in this work, such as the selection of material model and simulation techniques, as well as simpli cations to the overall testing systems and assumptions.The isotropic hyperelastic models were adopted to describe the mechanical behavior of the arterial tissue without considering the anisotropic or viscoelastic behaviors.The plaque was simpli ed into two materials: brosis and calci cation.The complexity and eccentricity of the lesion were also simpli ed as the symmetric model for comparison with the analytical results.The balloon characteristics were also simpli ed to help better understanding of the load transfer.These simpli cations may cause a deviation in the stress analyses, but won't affect the global values such as the resulted force or strain energy, so the conclusions in this work can still be applied for a model with further speci city.The fracture behavior of the calci cation was not considered in this work since our focus is the load sharing analysis to provide the fundamental understanding, the fracture behavior will affect the evolution of the restored strain energy.We simulated the balloon with a cylindrical surface, rather than the three-folded balloon based on our benchtop test in which the balloon exhibited a cylindrical shape even at very low pressure (1 atm).Once the in ation pressure exceeded the nominal pressure, the balloon showed a larger resistance.We captured the bilinear behavior of the balloon and veri ed it with the manufacturer data.The ndings from this work will be further validated with clinical observations.

Conclusion
In this work, the diameter-pressure curve and load transfer analysis for stenting in non-calci ed and heavily calci ed coronary artery were performed.Further load sharing analysis and stress analysis were conducted to investigate the e cacy of increasing the pressure over improving the stenting expansion in heavily calci ed coronary artery.The stent expansion showed a nearly cylindrical shape in the non-calci ed lesion and the balloon undertook most of the increasing in ation pressure load.On the contrast, stent expansion showed a dog boen shape in the heavily calci ed lesion, and the increasing in ation pressure can effectively increase the minimal lumen area, stress in the calci cation, and potential of calci cation fracture.This study suggested that monitoring the stenting process with angiography imaging, especially a precise stent expansion pro le at the target in ation pressure will help optimize the stenting procedure in complex lesions.Our results also support the design and application of the high pressure balloon for post dilation.

Declarations
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