Stentgraft system
A stentgraft system (Endurant® II AAA, Medtronic, Dublin, Ireland) was disassembled, and the stentgraft was removed. Then, a multicore fiber (FBGS Technologies GmbH, Jena, Germany) inserted in a metallic capillary tube (400\(\upmu \)m diameter, AISI 304L) and three Aurora Micro 6-degree-of-freedom EM sensors (length: 9 mm, diameter: 0.8 mm; Northern Digital Inc., Waterloo, Canada) were integrated into the stentgraft system (Fig. 1a). The FBG arrays written into the optical fiber were not visible, but a fiber region of \(40\, \hbox {cm}\) was marked by the manufacturer where the 38 FBG arrays are located. Thus, the first EM sensor was not placed exactly at the tip of the fiber but further inside to be sure that the sensor is within the shape sensing region. All EM sensors were fixed rigidly to the capillary tube and covered separately with a shrinkage tubing to protect them and their cables from damage. In addition, all EM sensors were placed at the front of the stentgraft system near the region where the stentgraft is placed.
Tracking systems
The optical fiber was connected to a fanout and an interrogator (FBGS Technologies GmbH, Jena, Germany) to obtain the reflected wavelength of all FBGs. Then, the shape of the 38 cm shape sensing region of the fiber was reconstructed using the method explained in Jäckle et al. [9]. The resulting shape is represented as a point set
$$\begin{aligned} \hat{S} = \{S_0,\ldots , S_n\} \end{aligned}$$
(1)
with \(n = 760\) and \(|| S_i - S_{i+1} ||_2 = {0.5}\,{mm}\) distance in between, because the fiber has 38 cm shape sensing length and 20 interpolated positions were calculated per centimeter. Moreover, the direction vectors
$$\begin{aligned} \hat{D}^S = \{\hat{D}^S_0,\ldots , \hat{D}^S_n\} \end{aligned}$$
(2)
were computed for every shape sensing point during the shape reconstruction. Each element \(\hat{D}^S_k\) is a three-element vector, which describes the direction of the shape for each point and can also be considered as a tangent vector of the reconstructed shape at each point.
The EM sensors were tracked using a Tabletop Field Generator (Northern Digital Inc., Waterloo, Canada). The current pose \(\hat{P}_{k}^\mathrm{EM}\) of each EM sensor \(k \in \{1,2,3\}\) in the EM space is defined as follows:
$$\begin{aligned} \hat{P}_{k}^\mathrm{EM} = \begin{pmatrix} \begin{array}{ccc} &{} &{} \\ &{} \hat{R}_{k}^\mathrm{EM} &{} \\ &{} &{} \\ \end{array} &{} \begin{array}{c} \\ \hat{T}_{k}^\mathrm{EM} \\ \\ \end{array} \\ \begin{array}{ccc} 0 &{} 0 &{} 0 \end{array}&1 \end{pmatrix} \end{aligned}$$
(3)
where \(\hat{R}_{k}^\mathrm{EM}\) is a \(3 \times 3\) matrix that contains the orientation information and \(\hat{T}_{k}^\mathrm{EM}\) is a three-element vector with the position of the EM sensor tip. In addition, the direction vector \(\hat{D}_{k}^\mathrm{EM}\) is given by \(\hat{R}_{k}^\mathrm{EM}\) and corresponds to the third column of \(\hat{R}_{k}^\mathrm{EM}\).
Localization model
For the calibration step and for the evaluation of the guidance methods, CT scans were made. To transform the EM sensor poses \(\hat{P}_{k}^\mathrm{EM}\) from the EM space into the CT space \(\hat{P}_{k}^\mathrm{CT}\), metallic markers were used in every measurement (Fig. 2a). A spatial calibration step was first made to find a correspondence between the shape \(\hat{S}\) and the measured poses \(\hat{P}_{k}^\mathrm{CT}\) of the EM sensors (Fig. 1b and Fig. 2a). In this step, the corresponding shape point \(\hat{S}_{i_k}\) and the correction vector \(\vec v_{k}\) for mapping each EM sensor position to its corresponding shape point were determined for each EM sensor. The calibration method was already introduced in detail in [10]. Afterwards, the shape \(\hat{S}\) can be located in the CT space with the poses \(\hat{P}_{k}^\mathrm{CT}\) of the EM sensors using the values obtained in the calibration. An overview of all processing steps is given in Fig. 2b. In the following subsections, the shape localization methods with three and two EM sensors are explained.
Three EM sensors
With the data from the tracking systems, the shape \(\hat{S}\) was reconstructed in shape space and the measured EM poses \(\hat{P}_{k}^\mathrm{EM} (k \in \{1,2,3\})\) in the EM space were obtained. Using the values obtained in the spatial calibration step, the shape points in the shape sensing space
$$\begin{aligned} \{\hat{S}_{i_1},\hat{S}_{i_2},\hat{S}_{i_3}\} \end{aligned}$$
(4)
and their corresponding points in the CT space
$$\begin{aligned} \{\hat{T}_{1}^\mathrm{CT}+\vec v_{1},\hat{T}_{2}^\mathrm{CT}+\vec v_{2},\hat{T}_{3}^\mathrm{CT}+\vec v_{3}\} \end{aligned}$$
(5)
can be determined. Using these two point sets, a rigid transformation was computed [1]. This transformation can be used to locate the reconstructed shape \(\hat{S}\) in the CT space.
Two EM sensors
In this case, the position and orientation of the first and third EM sensors are used. Moreover, the direction information \(\hat{D}^S\) along the shape is obtained during shape reconstruction. Using the information of the spatial calibration, two shape points in the shape sensing space
$$\begin{aligned} \{\hat{S}_{i_1},\hat{S}_{i_3}\} \end{aligned}$$
(6)
and their corresponding points in the CT space
$$\begin{aligned} \{\hat{T}_{1}^\mathrm{CT}+\vec v_{1},\hat{T}_{3}^\mathrm{CT}+\vec v_{3}\} \end{aligned}$$
(7)
were obtained. However, two points are not sufficient to determine a rigid transformation. For this reason, two additional points were generated by adding the direction vector with 10 mm length. The directions of the shape points \(\hat{D}^S_{i_k}\) were computed during the shape reconstruction, and the direction of the EM sensor \(\hat{D}^\mathrm{CT}_{k}\) corresponded to the third column of \(\hat{R}_{k}^\mathrm{CT}\). Then, four shape points in the shape sensing space
$$\begin{aligned} \{\hat{S}_{i_1}, \hat{S}_{i_1} + 10 \cdot \hat{D}^S_{i_1}, \hat{S}_{i_3}, \hat{S}_{i_3} + 10 \cdot \hat{D}^S_{i_3}\} \end{aligned}$$
(8)
and their corresponding points in the CT space
$$\begin{aligned}&\{\hat{T}_{1}^\mathrm{CT}+\vec v_{1}, \hat{T}_{1}^\mathrm{CT} + \vec v_{1} \nonumber \\&\quad + 10 \cdot \hat{D}^\mathrm{CT}_{1}, \hat{T}_{3}^\mathrm{CT} + \vec v_{3}, \hat{T}_{3}^\mathrm{CT} + \vec v_{3} + 10 \cdot \hat{D}^\mathrm{CT}_{3}\} \end{aligned}$$
(9)
were determined. These two point sets were used to calculate the rigid transformation [1] for locating the reconstructed shape \(\hat{S}\) in the CT space.
Evaluation
Vessel phantom
A 3D-printed vessel made of silicone and built from patient data (HumanX GmbH, Wildau, Germany) was integrated into a plastic container with size: \(40\,\hbox {cm} \times 30\,\hbox {cm} \times 19\,\hbox {cm}\) (Fig. 3a). For this purpose, access points were created and the sides of the container were covered with foam. Afterwards, the vessel was inserted and the iliac arteries were fixed with silicone to avoid leakages. In the second step, an artificial surrounding tissue was made with agar–agar. \({600}\,{g}\) agar–agar were stirred in \({11,3}\,{{\hbox {l}}}\) water while heating it up to \({63}\,{^{\circ }\mathrm {C}}\). When the agar–agar was dissolved, \({700}\,{{\hbox {ml}}}\) glycerol and \({40}\,{\mathrm{g}}\) graphite were included in the mixture. Then, the phantom was cooled down \({16}\,{\mathrm{h}}\) at room temperature and afterwards \({7}\,{\mathrm{h}}\) in the fridge. The resulting phantom is shown in Fig. 3b.
Experiments
For the spatial calibration step, the stentgraft system was fixed in a bow shape to a rigid foam placed on a CT table and six metallic markers (SL10, diameter: 1 mm; The Suremark Company, California, USA) were placed at different heights around the stentgraft system to transform the poses of the EM sensors into those in the CT space (Fig. 1b).
For the evaluation of the catheter guidance methods, the vessel phantom was placed and fixed on the CT table and five metallic markers were placed on the plastic box of the phantom. For introducing the stentgraft system, first a soft guide wire was inserted, then a standard catheter was pushed over and the soft guide wire was replaced with a stiff guide wire. After that, the catheter was removed. Finally, the stentgraft system was inserted into the phantom, moved to the aneurysm and pulled back in \(5\,\hbox {cm}\) steps using the stiff guide wire. Ultrasound gel was used to facilitate the insertion of the wires, catheter and the stentgraft system. Moreover, continuous measurements of the optical fiber and the EM sensors were made while moving the stentgraft system to the aneurysm. The data from both tracking systems were obtained at a frequency of 10 Hz.
Evaluation measures
For the spatial calibration step and for each insertion depth of the stentgraft system, the data from the optical fiber and the EM sensors were measured before and after the acquisition of a CT scan (which was used to obtain the ground truth) in order to evaluate the stability of the whole setup.
Each CT study was made with a Siemens SOMATOM Definition AS+ scanner. In the spatial calibration step, the scan was acquired with the parameters: voltage of 120 KVp, exposure of 109 mAs, image size of \({512 \times 512 \times 733}\) and voxel size of \( 0.51 \times 0.51 \times {0.40}\,{\mathrm{mm}}\). For the evaluation experiments, the following parameters were used: voltage of 120 KVp, exposure of 180 mAs, image size of \({512 \times 512 \times 1156}\) and voxel size of \(0.70 \times 0.70 \times {0.60}\) mm.
The reconstructed shapes before and after each CT acquisition were aligned by means of a point-based registration [1] and compared to evaluate the shape movement. In addition, the maximal position change \(c_{\text {p}}\) and the maximal orientation angle change \(c_{\text {o}}\) of the EM sensors before and after each CT acquisition were calculated.
Afterwards, the shapes of the stentgraft system were segmented, the EM sensor positions were obtained manually from each CT scan and both were used as ground truth for the comparison with the estimated measurements. The positions of the metallic markers, which were used for transforming the EM sensor positions into the CT space, were determined semiautomatically. For this, each marker was segmented by thresholding and the centroid of each marker segmentation was calculated, which results in a subvoxel precision for the marker localization.
For evaluation, the reconstructed shape was aligned with the ground truth shape by means of point-based registration [1]. The average and maximum errors defined as
$$\begin{aligned} e_{\text {avg}}&:= \frac{1}{m + 1}\sum _{i=0}^{m} \Vert x_i - x^{\text {gt}}_i \Vert _2 \text { and }\nonumber \\ e_{\text {max}}&:= \max ( \Vert x_0 - x_0^{\text {gt}} \Vert _2, \ldots , \Vert x_m - x_m^{\text {gt}} \Vert _2) \end{aligned}$$
(10)
where \(x_0, \ldots , x_m\) are the estimated points and \(x_0^{gt}, \dots , x_m^{gt}\) are the ground truth points were calculated. For the evaluation of the shape movement and the reconstructed and located shapes, the points were compared every \(10 \, \text {mm}\) along the shape.