The analysis of arterial tissue remains a pivotal topic of research in the field of cardiovascular pathologies. It was well established that a plethora of cardiovascular diseases find their origin within the mechanics and the biology of the vessel tissues (Humphrey and Schwartz. (2021)). Attention was focused on both large and small vessels including different types of pathologies like dissections, stenosis/atherosclerotic arteries and aneurysms (Celi et al. (2013); Gültekin et al. (2019); Vignali et al. (2020)). An aneurysm is defined as a local dilatation in the aortic wall, that is usually asymptomatic up to the sudden rupture which may be linked with patient’s death (Ramanath et al. (2009)). The current clinical practice is to define a critical aortic size criterion to determine the necessity of surgical implantation. The aneurysm pathology remains an open clinical challenge and it still requires a deep insight in terms of formation and progression mechanisms. Different studies reported that the critical state of an aneurysm case arises from the status of the tissue biomechanics and its degradation (Vignali et al. (2020); Vignali et al. (2021c)). The usage of the mechanical analysis principle could ideally provide an improved understanding of the aneurysm nature and, in general, of the arterial behavior under given pathological conditions.

Following this analysis principle, different groups have provided mechanical insights concerning the arterial tissues. In the current state of the art, various experimental testing procedures have been proposed, like uniaxial/biaxial traction tests (Vignali et al. (2021a); Peña et al. (2015)) and bulge inflation approaches (Duprey et al. (2016)). It is also worth noting that several studies were focused on the investigation of correlation of biological and mechanical features of the arterial tissue (Vignali et al. (2020; 2021c)), given their important link. This literature field presents a shared flaw, which is the necessity of ex-vivo tissue samples to be tested. Given this, the mechanical analysis is necessarily limited to post-operative cases, in which the surgical procedure has already been performed. Consequently, it is impossible to have a direct mechanical characterization of the arterial tissue without an invasive procedure.

Obtaining mechanical features of the arterial tissue non-invasively still represents an open research topic. Nevertheless, research efforts towards this direction have been made recently. Several image-based approaches have already been explored to estimate mechanical properties of soft tissues in general (Fanni et al. (2020); Di Lascio et al. (2014)). These in vivo estimation methods are made possible thanks to the recent advances in terms of clinical imaging (Celi et al. (2017)), which allow high-resolution reconstruction of cardiovascular structures at different cardiac phases. The dynamic nature of imaging techniques like ECG-gated Computed Tomography (CT), echography and 4D Magnetic Resonance Imaging (4D-MRI) opens the possibility to reconstruct the displacement fields of cardiovascular structures.

The main focus of non-invasive mechanical analysis resides mainly in strain estimation on vessels like the ascending aorta section. The reported in vivo strain evaluation techniques are usually based on mapping algorithms aimed at reconstructing the aortic kinematics along the cardiac cycle. Among the different mapping techniques, iterative registration approaches (Liu et al. (2019b); Narayanan et al. (2021)), projections along the normal of the aortic surface (Pasta et al. (2017)) and centerline-based decompositions with parametric templates (Farzaneh et al. (2019b; Farzaneh et al. (2019a)) were proposed. Beyond the knowledge of aortic strain, stress is still required for a stiffness estimation. Nevertheless, the in vivo evaluation of stress remains a difficult task. For this reason, different studies were limited to strain-only analyses (Pasta et al. (2017)), or presented assumptions on the load to infer simplified stress distributions (Duprey et al. (2016); Martin et al. (2013)). Some groups proposed iterative Finite Element (FE) approaches for the direct estimation of aortic stiffness (Krishnan et al. (2015)), but the requirement for multiple numerical simulations can be computationally onerous. Other groups also proposed aortic volumetric distensibility as a surrogate for stiffness estimation (Trabelsi et al. (2018)), but the global nature of the parameter did not allow for a local estimation of the material properties.

With the current study, a new method for the strain and stiffness estimation of the aortic vessel is proposed. The method aims at providing local information, in opposition with volumetric/global approaches (Trabelsi et al. (2018); Danpinid et al. (2009)) by relying on CT data, centerline calculation and mesh-morphing based mapping. Other previous studies relied on centerline-based information (Zeinali-Davarani et al. (2011)), nevertheless their focus was more centered on the abdominal aorta district and they used the centerline to map wall thickness interpolations along the vessel. To our knowledge, this is the first approach proposing a mesh-morphing sequence to allow for the mapping of the aortic surface along the different cardiac cycle phases. The morphing of the baseline mesh on the different deformed surfaces can allow for a fast method for mapping and, consequently, it gives a great potential for local strain evaluation. Concerning the stress, a further step to go beyond the literature relying on the Laplace hypothesis (Liu et al. (2019a); Martin et al. (2013)) could be imposed by relying on inverse FE simulations.

The aim of the current work is to propose a new approach to provide local distribution of arterial stiffness, based on the knowledge of in vivo patient-specific imaging data. The method is based on the calculation of aortic strain on the basis of contour length ratios for each section. For the estimation of stress, two approaches were tested: an approach based on Laplace hypothesis, and an approach based on an inverse engineering FE simulation to estimate the distribution of wall tension on the aortic geometry. The entire procedure was first validated on a set of FE simulations. The two estimation methods were first compared for the validation phase. Finally, the approach was applied on three patient-specific ECG-gated CT data to estimate the local stiffness distribution in vivo. The results are then presented and discussed to assess the new method performances and future applications.

In this section the three main steps of the entire pipeline are presented: method description (Subsection 2.1), validation (Subsection 2.2) and application (Subsection 2.3).

The results from the FE validation procedure are presented first. Concerning the simplified geometries, the curvature effect factor is reported first. According to Eq. 3, values of m = 1, m = 0.93 and m = 0.86 were calculated for the cylinder and the elbows at 45° and 90°, respectively. The results in terms of Young’s modulus maps for cylinder and elbows geometries are presented in Figures 5A–F. The relative error maps, calculated according to the reference value imposed at simulation level, are reported as well (Figures 5G–L). The corresponding RMSPE values for all the idealized geometries are reported in Table 1.

Before presenting the patient-specific validation cases, the negligibility of the curvature was checked by evaluating the curvature effect factor. The geometry revealed an m factor suitable for the condition of Eq. 3 (m = 0.89). In Figures 6A, B, Figures 7A, B and Figures 8A, B the Young’s modulus maps of the patient-specific validation cases with homogeneous material properties distributions are reported for the imposed values of E ̄ θ H = 0.5 MPa, 1.75 MPa, 3.0 MPa. The RErr maps for are reported as well in Figures 6C, D, 7C, D, 8C, D. The results are presented for both LH and IE method. The distributions along the centerline coordinate for the patient-specific validation cases with homogeneous distributions case are reported in Figure 9. Highlights concerning the location of the centerline coordinate, including aortic root, ascending aorta and aortic arch, are represented in figure as well. The distributions according to both LH and IE methods exhibited approximately a constant trend for all the simulated cases of stiffness. The three reference values of E ̄ θ H = 0.5 MPa, 1.75 MPa, 3.0 MPa are also reported in the graph with dashed lines. The corresponding RMSPE values are all reported in Table 1, with values ranging from 8.5% and to 10.1%.

In Figure 10 the Young’s modulus maps of the second validation case with proximal/distal material properties distribution are reported. The different maps of E θ L H and E θ I E can be evaluated to determine the performances of both the Laplace-hypothesis-based (Figure 10A) and inverse-engineering-based (Figure 10B) estimation methods in the proximal/distal validation case. The RErr maps for are reported as well in Figures 6C, D for both LH and IE methods. The distributions along the centerline coordinate for the proximal/distal validation case are reported in Figure 11. The distributions according to both LH and IE methods exhibited a step-like behavior, with a higher stiffness in the proximal section, as expected. Similarly to the first validation case, the reference values and the highlights concerning the location of the centerline coordinate are reported in the plot. The corresponding RMSPE value are reported in Table 1, with values of 16.3% and 16.1% for the LH and IE methods, respectively.

The results from the patient-specific cases analyses are then presented. Given the equivalent performances of the estimation methods from the first FE validations, the IE method was chosen for the patient-specific analysis. The values of SNR for each of the patient-specific CT datasets were the following: 35.5 for case 1, 36.1 for case 2 and 37.0 for case 3. The circumferential strain maps at the different cardiac phases from clinical data cases are presented in Figure 12 for all three cases. By inspecting the maximum strain value for each case, it was possible to select the systolic peak phase for each dataset: phase 20% for Case 1, phase 40% for Case 2 and Case 3. The only phase chosen as systolic peak was adopted for the stiffness calculation.

After selecting the systolic peak phase for each case on the basis of the strain, the results in terms of Young’s modulus were calculated. The maps for the different patient-specific cases are reported in Figure 13. The Young’s modulus trends as a function of centerline coordinate are also presented for all the patient-specific cases, as showed in Figure 14. For Case 1 and Case 2, a more homogeneous trend was reported. On the contrary, the behavior of Case 3 appeared to be less homogeneous, as stiffer values were encountered in the ascending aorta in proximity of the aortic arch section.

The results presented in the previous section demonstrate the effectiveness of the proposed technique for the stiffness estimation in the ascending aorta from in vivo data. The performances of both Laplace-hypothesis based and Inverse-engineering based approaches have been presented. In particular, the methods were first tested on FE-based validation cases in which homogeneous and heterogeneous Young’s modulus distribution were imposed on both idealized and patient-specific geometries. The proposed approach was successfully implemented also with patient-specific data thanks to shape morphing techniques, revealing the strain and stiffness distribution of three real cases of ascending aortic sections.

The results from the validation on the idealized geometries are presented in Figure 5. From the resulting maps it is possible to assess that the produced errors are always below 10% for all the chosen cases. In particular, it is interesting to notice that for the ideal cylinder there is no substantial difference between the LH and IE estimation methods, as in both cases the results presented correspond to the Laplace theory. In both cases, the cylinder produces negligible errors, below 2% (see also Table 1). By introducing a curvature, the differences between the two methods emerge. In fact, by inspecting the results from the elbow cases, it is evident that the effect of curvature influences the performances of the LH method. This effect is particularly evident for the elbow at 90°, where the curvature factor equals to 0.86, in which the RErr values reach a maximum of 7% for the LH method, while the IE method produces maximum errors of 3%. These results demonstrate that the curvature effects the method performances, nevertheless both approaches produced satisfactory estimations of stiffness distribution with errors always remaining below the 10% threshold.

The results obtained in the patient-specific validation cases in which stiffness distribution was imposed as homogeneous are presented in the maps of Figure 6, Figure 7 and Figure 8. From the maps, it is possible to observe that according to both the estimation techniques that the homogeneity of the stiffness was correctly coped for all the imposed values of Young’s modulus. This behavior is confirmed also by the Young’s modulus trend as a function of centerline coordinate from Figure 9. For all the imposed values of Young’s modulus, the same trends have been encountered. In particular, for the LH cases, a wider variation of stiffness values is encountered only within the aortic root section. This behavior can be assumed as a consequence of stress direct dependence on the section radius. This oscillation in the aortic root section is absent instead according to the IE method for all the three values of imposed Young’s modulus. Underestimations are instead encountered within the aortic arch section for the E θ I E calculation. It is reasonable to assume these underestimations linked with E θ I E can be caused by imposed boundary conditions at FE simulation level. Nevertheless, it is interesting to highlight that both methods reveal a similar and constant trend within the ascending aorta section, as reported by both the maps of Figure 6, Figure 7 and Figure 8 and also the plots of Figure 9. The similarity of LH and IE performances is also confirmed by the RMSPE values reported in Table 1. In all patient-specific validation cases, the RMSPE percentages were similar, with values around 10%. In particular, a maximum of 10.1% for the LH method with E ̄ θ H = 0.5 MPa and a minimum of 8.5% for IE method with E ̄ θ H = 3.0 MPa were experienced. Concerning homogeneous stiffness distributions for the patient-specific validation, these error values make plausible to assume that both LH and IE methods are comparable in terms of performances.

Similar trends were encountered also for the second validation case with heterogeneous distribution. From the maps of Figure 10, the underestimation area in the aortic root zone of the LH method map remains evident, as observed also in the previous validation case. Concerning the IE method, the same underestimation area in the aortic arch area, already observed in the homogeneous validation case, can be highlighted on the heterogeneous validation case. Nevertheless, both LH and IE methods correctly cope the zone distribution of the Young’s moduli in the proximal and distal sections of the ascending aorta. In fact, the transition from the high stiffness ( E ̄ θ P = 2.0 MPa) area in the proximal section to the low stiffness ( E ̄ θ D = 0.5 MPa) area in the distal region is correctly marked in both E θ L H and E θ I E maps, as showed in Figures 10A, B. The same transitions can also be detected in the graphs of Figure 11, where the Young’s modulus trend according to centerline coordinate is reported. The E θ L H is oscillating in the aortic root zone, as already observed in the first validation case. Additionally, the E θ I E underestimation in the aortic arch section remains even in this validation case. It is safe to assume that the underestimation of E θ I E in the aortic arch section remains linked with the boundary conditions imposed in the FE simulation, as already observed in the first validation case. It is interesting to observe that the estimations remain approximately equivalent, regardless of the method used, within the ascending aorta section. This aspect is confirmed by the presence of the same outliers, encountered in both E θ L H and E θ I E trends. Additionally, for both LH and IE case, it was possible to observe the transition along the centerline. The RMSPE are reported in Table 1. The errors are, in fact, equivalent for both LH (16.3%) and IE (16.1%) method. Higher percentage of errors are encountered for the heterogeneous case in comparison with the homogeneous case. This behavior can be motivated by considering that the sudden change in Young’s modulus cannot be completely coped by the strain estimation approach, which necessarily introduces a smoothing action by considering cross sectional planes.

With these validation results, it was possible to assess the performances of the workflow. It was safe to assume that both LH and IE method revealed in general satisfactory performances, with similar errors for both validation cases. The IE method was chosen to proceed with the patient-specific cases. The strain maps at the given cardiac phases were calculated first (Figure 12). The evaluation of strain maps at the different phases allowed for the individuation of the systolic peak for each case, by evaluating the strain maximum. In all cases, a strain below 10% was encountered. The range reported was in accordance with previously observed data calculated on in vivo aortic cases (Bell et al. (2014); Satriano et al. (2018); Wilson et al. (2019)). It is also possible to observe heterogeneity from case to case. In particular, Case 1 exhibited the highest values of circumferential strain in comparison with the two other patient-specific cases. In addition, Case 3 revealed an area with high strains at the proximal section of the ascending aorta. The individuation of the systolic peak phase made possible the estimation of the Young’s modulus (Figures 13, 14). The calculated stiffness distributions are in line with the reported strain maps. In fact, while Cases 1 and 2 revealed mainly an homogeneous distribution of Young’s modulus, with average values of 0.6 MPa and 1.0 MPa respectively, Case 3 exhibited a marked heterogeneity, with a stiffer section close to the aortic arch and ranging from 0.7 MPa to 2.9 MPa. It is evident from both the maps of Figure 13 and the trends of Figure 14 that Case 3 revealed an increased stiffness in comparison with the other two cases, with peaks below 3.0 MPa. This phenomenon is in line with the already established connection between arterial stiffness and age, as Case 3 data are associated with the older patient case (Qiu and Onuh (2020)). It was plausible to expect this behavior, also considering the lower strain values encountered from the analysis of the different cardiac phases. Considering all three cases, the estimated values of Young’s modulus were always contained within the 0.5–3.0 MPa range, which is in line with already reported physiological values from the state of the art (Lin et al. (2022); Vignali et al. (2021b)).

Further points of development and limitations of the current workflow can be highlighted. Both the proposed LH and IE do not take into account of inertial loading, as in both cases the assumption of quasi-static solicitations was made. A limitation is given by the reduced number of patient-specific cases tested after validation. To better define the approach outcomes, a wider number of cases will be analyzed and presented in the future. The current method is limited on the estimation of circumferential strain only and it assigns a single value for each centerline-based slice of the aorta. Thus, a limitation of the estimation method in its current state is the impossibility to calculate the inner/outer curvature strain difference on the aorta surface by considering the whole contour length. Moreover, even if the mapping of the surface accounts for the axial displacement of sections, the approach based on the contour length cannot estimate the longitudinal strain. Further developments of the algorithm will allow in the future to estimate the axial deformation of the aorta and they will open up the path for the adoption of more complex constitutive models including anisotropy. Concerning hyperelasticity, it is well established that the aortic tissue has a non-linear mechanical response, given the presence of collagen fibers and the presented method does not account for these phenomena. Nevertheless, it was also established that deformations occurring between diastolic and systolic phase can be assumed as small (Gundiah et al. (2008); Vignali et al. (2021b)). This aspect confirms the hypothesis of linear material behavior assumed in the presented method. To further assess the effectiveness of the proposed method, it would be interesting to test the approach even on a complete aortic geometry, including epiaortic vessels and descending aorta. As an additional point of development, the method can be tested also with CT with ECG-gating with finer time sampling. Concerning the CT image quality, to assess the influence of noise on the presented procedure’s outcomes a full uncertainty quantification process would be required. Nevertheless, the SNR assessment for the processed images of the three patient-specific cases confirmed that the image quality was satisfactory and in line with diagnostic standards (Shen et al. (2015)). In this way the method could have the potential to obtain a more accurate estimation of the systolic peak cardiac phase.

In summary, the current study presents a new method for local strain and stiffness estimation from in vivo ECG-gated CT aortic images, on the basis of mesh-morphing mapping and inverse engineering methods. The method was first validated on two test cases, obtained from FE simulations of aortic structures with different material properties local distributions. After a successful validation, the method was applied on three patient-specific aorta cases. The results demonstrated the successful obtainment of a regional in vivo characterization of patient-specific aortas in terms of deformations and stiffness.