Patient-specific in silico endovascular repair of abdominal aortic aneurysms: application and validation

Abstract

Non-negligible postinterventional complication rates after endovascular aneurysm repair (EVAR) leave room for further improvements. Since the potential success of EVAR depends on various patient-specific factors, such as the complexity of the vessel geometry and the physiological state of the vessel, in silico models can be a valuable tool in the preinterventional planning phase. A suitable in silico EVAR methodology applied to patient-specific cases can be used to predict stent-graft (SG)-related complications, such as SG migration, endoleaks or tissue remodeling-induced aortic neck dilatation and to improve the selection and sizing process of SGs. In this contribution, we apply an in silico EVAR methodology that predicts the final state of the deployed SG after intervention to three clinical cases. A novel qualitative and quantitative validation methodology, that is based on a comparison between in silico results and postinterventional CT data, is presented. The validation methodology compares average stent diameters pseudo-continuously along the total length of the deployed SG. The validation of the in silico results shows very good agreement proving the potential of using in silico approaches in the preinterventional planning of EVAR. We consider models of bifurcated, marketed SGs as well as sophisticated models of patient-specific vessels that include intraluminal thrombus, calcifications and an anisotropic model for the vessel wall. We exemplarily show the additional benefit and applicability of in silico EVAR approaches to clinical cases by evaluating mechanical quantities with the potential to assess the quality of SG fixation and sealing such as contact tractions between SG and vessel as well as SG-induced tissue overstresses.

Appendices

Appendix 1: Definition of control curves and assignment of stent-graft nodes to the subsets ${\mathsf {A}}^{j}_{\mathrm {I}}$

This section provides the definition of control curves ${\mathcal {C}} \subset {\mathbb {R}}^3$ associated with the morphing algorithm that is used for the in silico SG placement. For a detailed description of the morphing algorithm, the reader is referred to Hemmler et al. (2018).

These control curves are given in the initial configuration ${\mathcal {C}}_{\mathrm {I}}$ and in the target configuration ${\mathcal {C}}_{\mathrm {T}}$. At each point $j=1,2,\ldots ,n_{\mathrm {C}}$ of the piecewise linear control curve ${\mathcal {C}}_{\mathrm {I}}$ in the initial configuration described by $n_{\mathrm {C}}$ discrete points with the coordinates ${\varvec{x}}_{\mathrm {C},\mathrm {I}}^{j}\in {\mathcal {C}}_{\mathrm {I}}$, a semi-infinite bounding box ${\mathbb {B}^{j} \subset {\mathbb {R}}^3}$ is used to assign the nodes i of the SG with the reference coordinates ${\varvec{X}}^{i} \in ({\Omega }^{{\mathrm {S}}}_{\mathrm {0}} \cap {\Omega }^{\mathrm {G}}_{\mathrm {0}})$ to one point on the control curve ${\mathcal {C}}_{\mathrm {I}}$. ${\Omega }^{{\mathrm {S}}}_{\mathrm {0}}$ and ${\Omega }^{\mathrm {G}}_{\mathrm {0}}$ describe the undeformed configurations of stent and graft, respectively. The semi-infinite bounding box ${\mathbb {B}^{j} \subset {\mathbb {R}}^3}$ is defined by two parallel, infinite planes with a distance of h (Fig. 5). All nodes i of the SG with ${\varvec{X}}^{i} \in \mathbb {B}^{j}$ are assigned to point j of the centerline ${\mathcal {C}}_{\mathrm {I}}$ and are put into the subset ${\mathsf {A}}^{j}_{\mathrm {I}}\subseteq {\mathsf {A}}_{\mathrm {I}}=\{1,2,\ldots ,n^{\mathrm {SG}}\}$ where $n^{\mathrm {SG}}$ is the number of nodes of the SG and where

$$\begin{aligned}&\overset{n_{\mathrm {C}}}{\underset{j=1}{\bigcup }} {\mathsf {A}}^j_{\mathrm {I}}={\mathsf {A}}_{\mathrm {I}}, \end{aligned}$$

(14a)

$$\begin{aligned}&{\mathsf {A}}^j_{\mathrm {I}}\cap {\mathsf {A}}^k_{\mathrm {I}}=\varnothing,\nonumber \\&\forall k \ne j,\,j=1,2,\ldots ,n_{\mathrm {C}},\,k=1,2,\ldots ,n_{\mathrm {C}} \end{aligned}$$

(14b)

holds.

Appendix 2: Control curve continuity conditions

The deformation of the SG during the in silico SG placement is fully described by the linear interpolation between two given configurations of the control curve, the initial configuration ${\mathcal{C}}_{\mathrm {I}}^{(\Pi)}\in {\mathbb {R}}^3$ and the target configuration ${\mathcal{C}}_{\mathrm {T}}^{(\Pi)}\in {\mathbb {R}}^3$. To ensure continuity between the three SG components $\varPi =\{{\mathrm {P}},{\mathrm {L}},{\mathrm {R}}\}$ during the entire SG placement, the following conditions between the initial configurations ${\mathcal {C}}_{\mathrm {I}}^{(\varPi )}$ and the target configurations ${\mathcal {C}}_{\mathrm {T}}^{(\varPi )}$ of the control curves have to be satisfied (Fig. 4IIIb):

• The distal end of the control curve ${\mathcal {C}}_{\mathrm {I}}^{{\mathrm {P}}}$ and the proximal ends of the control curves ${\mathcal {C}}_{\mathrm {I}}^{{\mathrm {L}}}$ and ${\mathcal {C}}_{\mathrm {I}}^{{\mathrm {R}}}$ have to be parallel and have to be in one plane. Same holds for the target configurations of the control curves ${\mathcal {C}}_{\mathrm {T}}^{{\mathrm {P}}}$, ${\mathcal {C}}_{\mathrm {T}}^{{\mathrm {L}}}$ and ${\mathcal {C}}_{\mathrm {T}}^{{\mathrm {R}}}$.

• The longitudinal overlap $l_{\mathrm {a}}$ of the three control curves as well as the transverse distance $l_{\mathrm {b}}$ between the three control curves has to be the same in the initial configurations ${\mathcal {C}}_{\mathrm {I}}^{(\varPi )}$ and the target configurations ${\mathcal {C}}_{\mathrm {T}}^{(\varPi )}$.

Appendix 3: Center of gravity calculation for regular stent-graft meshes

For a regular SG mesh, the mean angular distance $\bar{\theta }^{i}=\frac{1}{2}(\theta ^{i+1}-\theta ^{i-1})$ between two adjacent nodes in the set ${\mathsf {A}}_{\mathrm {I}}^{j}$ is $\bar{\theta }^{i}=\frac{2\pi }{n^{j}}$ for each node i where $n^{j}$ is the number of nodes in the set ${\mathsf {A}}_{\mathrm {I}}^{j}$. Hence, the calculation of the center of gravity of all nodes i in the set ${\mathsf {A}}_{\mathrm {I}}^{j}$ [Eq. (5)] reduces to the arithmetic mean

$$\begin{aligned} {\varvec{x}}_{\mathrm {C,De}}^{j}&=\frac{1}{2\pi }\underset{i\in {\mathsf {A}}_{\mathrm {I}}^{j}}{\sum } \bar{\theta }^{i} {\varvec{x}}^{i}=\frac{1}{n^{j}}\underset{i\in {\mathsf {A}}_{\mathrm {I}}^{j}}{\sum }{\varvec{x}}^{i}, \nonumber \\&\quad \forall j=1,2,\ldots ,n_{\mathrm {C}}, \end{aligned}$$

(15)

where ${\varvec{x}}^{i}$ are the current coordinates of all nodes i in the set ${\mathsf {A}}_{\mathrm {I}}^{j}$.

Appendix 4: Filtering of postinterventional CT data

A moving average filter with a span of

$$\begin{aligned} l_{\mathrm {span}}&=2 n_{\mathrm {postIV}} \left\lceil \frac{\varDelta z_{\mathrm {CT}}}{\bar{\varDelta s}_{\mathrm {postIV}}} \right\rceil + 1 \end{aligned}$$

(16)

is used to limit the impact of obvious artifacts in the stent diameter measurement from postinterventional CT data. In Eq. (16), $\varDelta z_{\mathrm {CT}}=1\,{\mathrm {mm}}$ is the slice thickness of the postinterventional CT data, $n_{\mathrm {postIV}}=3$ is a filtering constant that scales the length of the moving average filter. $\bar{\varDelta s}_{\mathrm {postIV}}$ is the mean edge length of the piecewise linear curve ${\mathcal {C}}_{\mathrm {De}}$, i.e., the mean distance between the centers of gravity of the sets ${\mathsf {A}}_{\mathrm {I,postIV}}^{{\mathrm {S}},j}$ defined by Eq. (5). The result of the filtering process is visualized for patient 3 in Fig. 10. Each asterisk denotes the measured average diameter $\bar{d}_{\mathrm {postIV}}^{{\mathrm {S}},(\varPi ),j}$ of one distinct set ${\mathsf {A}}_{\mathrm {I,postIV}}^{{\mathrm {S}},(\varPi ),j}$ of SG part $\varPi =\{{\mathrm {P}},{\mathrm {L}},{\mathrm {R}}\}$.

Fig. 10

Difference between measured average stent diameters $\bar{d}_{\mathrm {postIV}}^{{\mathrm {S}},j}$ from postinterventional CT data and filtered average stent diameters $\bar{d}_{\mathrm {postIV,f}}^{{\mathrm {S}},j}$ as well as visualization of the standard deviation $\sigma _{\mathrm {f}}$ for the proximal SG part (I), the left iliac SG part (II) and the right iliac SG part (III) of patient 3

Appendix 5: Quality estimation of segmented data from postinterventional CT scans

The quality of the postinterventional CT data is crucial for the reliability of a quantitative validation of the in silico EVAR results, but local artifacts have a non-negligible effect on the segmentation of the stent from postinterventional CT data. To obtain an estimation of the measurement inaccuracy due to the vagueness in the segmentation process of the stent from postinterventional CT data, we define the relative difference between the measured average diameter $\bar{d}_{\mathrm {postIV}}^{{\mathrm {S}},j}$ and the average diameter of the filtered data $\bar{d}_{\mathrm {postIV,f}}^{{\mathrm {S}},j}$ by

$$\begin{aligned} \epsilon _{f}^{j}&=\frac{\bar{d}_{\mathrm {postIV}}^{{\mathrm {S}},j}-\bar{d}_{\mathrm {postIV,f}}^{{\mathrm {S}},j}}{\bar{d}_{\mathrm {postIV,f}}^{{\mathrm {S}},j}}, \quad \forall j=1,2,\ldots ,n_{\mathrm {C}}. \end{aligned}$$

(17)

Further, the standard deviation

$$\begin{aligned} \sigma _{f}&=\sqrt{\frac{1}{n_{\mathrm {C}}}\sum _{j=1,2,\ldots ,n_{\mathrm {C}}}\left( \epsilon _{f}^{j}-\mu _{f}\right) ^{2}} \end{aligned}$$

(18)

is calculated, where

$$\begin{aligned} \mu _{f}&=\frac{1}{n_{\mathrm {C}}}\sum _{j=1,2,\ldots ,n_{\mathrm {C}}}\epsilon _{f}^{j} \end{aligned}$$

(19)

is the mean relative difference. $n_{\mathrm {C}}$ is the number of points describing the piecewise linear curve ${\mathcal {C}}_{\mathrm {De}}$ which is equivalent to the number of discrete sets ${\mathsf {A}}_{\mathrm {I,postIV}}^{j}$. In Fig. 10 we oppose the plain stent diameters from postinterventional CT data $\bar{d}_{\mathrm {postIV}}^{{\mathrm {S}},j}$, the filtered stent diameters $\bar{d}_{\mathrm {postIV,f}}^{{\mathrm {S}},j}$ and the standard deviation $\sigma _{f}$ for patient 3.

A large standard deviation $\sigma _{f}$ of the relative difference $\epsilon _{f}^{j}$ is an indicator that the measurements are strongly affected by local artifacts of the segmented stent. The standard deviation $\sigma _{\mathrm {f}}$ is very small for the proximal SG parts ($\sigma _{\mathrm {f}}^{{\mathrm {P}}}\le 2.0\%$) but more significant for the iliac SG parts (Table 6) due to two main reasons:

• The segmentation process of the CZ-Spiral SGs from postinterventional CT data is more difficult as those stent limbs are less clearly visible.

•  $\sigma _{\mathrm {f}}$ is the standard deviation of the relative difference between the measured average diameters $\bar{d}_{\mathrm {postIV}}^{{\mathrm {S}},j}$ and the filtered average diameters $\bar{d}_{\mathrm {postIV,f}}^{{\mathrm {S}},j}$. Hence, local artifacts in the postinterventional CT data of equivalent size would have a larger relative impact on $\sigma _{\mathrm {f}}$ in regions of small stent diameters such as in iliac SG parts.

Table 6 Standard deviation $\sigma _{\mathrm {f}}$ of the relative difference between the measured average diameter $\bar{d}_{\mathrm {postIV}}^{{\mathrm {S}}}$ and the average diameter of the filtered data $\bar{d}_{\mathrm {postIV,f}}^{{\mathrm {S}}}$ of the postinterventional CT data in [%] according to Eq. (18) for the three patient-specific cases and the three SG parts