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 centerlocated 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 fibrous plaque was adopted for the case of the noncalcified lesion. For the case of the heavily calcified lesion, a block calcification over a thin fibrous plaque was adopted, as shown in Fig. 1. The superficial concentric calcification had a maximum thickness of 0.64 mm, and the fibrous plaque had a maximum thickness of 0.26 mm. A commercial Express stent (Boston Scientific, 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. Noncompliant (NC) balloons with a fixed 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 fixed nominal diameter of 3 mm, and lengths of 11 mm and 8.5 mm were also simulated to determine the efficacy of shortened balloons during postdilation procedures. All NC balloons were simulated as cylinders with an initial diameter of 0.8 mm.
Material properties. The hyperelastic behaviors of the artery, fibrous plaque, and calcification were described by the reduced thirdorder polynomial strain energy density function U:
$$U=\sum _{i,j=1}^{3}{C}_{ij}{({I}_{1}3)}^{i}{({I}_{2}3)}^{j}$$
1
$${I}_{1}={\lambda }_{1}^{2}+{\lambda }_{2}^{2}+{\lambda }_{3}^{2}$$
2
$${I}_{2}={\lambda }_{1}^{2}+{\lambda }_{2}^{2}+{\lambda }_{3}^{2}$$
3
The material coefficients Cij 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 fibrous plaque, which is necessary to capture a realistic stent expansion including stent recoil [33]. The plasticity of fibrous plaque was initiated at a strain of 34% when its stress reached its yield strength of 0.07MPa [34]. The stressstrain relationship for all these lesion components is shown in Fig. 2a.
The pressurediameter 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 inflation behavior. The balloon diameter increased faster when the inflation pressure is below 6 atm, and much slower when the pressure exceeded 6 atm. To convert this bilinear inflation behavior of the balloon to a stressstrain relationship, the hoop strain was calculated as the relative change in diameter, and the hoop stress of a thinwalled cylinder was used to estimate the wall stress:
$${\epsilon }=\frac{{DD}_{0}}{{D}_{0}}$$
5
$${\sigma }=\frac{PD}{2\delta }$$
6
where D is the diameter during expansion, D0 is the initial diameter of 0.8 mm, P is the inflation pressure, and δ is the thickness of the balloon. The pressurediameter 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).
Table 1

C10(MPa)

C01(MPa)

C11(MPa)

C20(MPa)

C02(MPa)

C30(MPa)

C03(MPa)

Artery

0.10881

0.101

0.1790674

0.0885618

0.062686



Fibrous plaque

0.04




0.003


0.02976

Calcification

0.49596

0.50661

1.19353

3.6378


4.73725


Finite element simulations of the calcified coronary artery in the context of idealized or patientspecific models have been well validated in our previous work by matching the stented lumen area with exvivo 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 postdilation, 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 postdilation 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 quantified. 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 inflation 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.