Fiber Optical Shape Sensing of Flexible Instruments for Endovascular Navigation
Sonja Jäckle, Tim Eixmann, Hinnerk Schulz-Hildebrandt, Gereon Hüttmann, Torben Pätz
NNoname manuscript No. (will be inserted by the editor)
Fiber Optical Shape Sensing of Flexible Instruments for EndovascularNavigation
Sonja J¨ackle · Tim Eixmann · Hinnerk Schulz-Hildebrandt , , · Gereon H¨uttmann , , · Torben P¨atz September 10, 2019
Abstract : Purpose:
Endovascular aortic repair procedures are currently conducted with 2D fluoroscopy imaging. Trackingsystems based on fiber Bragg gratings are an emerging technology for the navigation of minimal-invasiveinstruments which can reduce the x-ray exposure and the used contrast agent. Shape sensing of flexiblestructures is challenging and includes many calculations steps which are prone to different errors. Toreduce this errors, we present an optimized shape sensing model.
Methods:
We analyzed for every step of the shape sensing process, which errors can occur, how the error affectsthe shape and how it can be compensated or minimized. Experiments were done with a multicore fibersystem with 38 cm sensing length and the effects of different methods and parameters were analyzed.Furthermore we compared 3D shape reconstructions with the segmented shape of the corresponding CTscans of the fiber to evaluate the accuracy of our optimized shape sensing model. Finally we tested ourmodel in a realistic endovascular scenario by using a 3D printed vessel system created from patient data.
Results:
Depending on the complexity of the shape we reached an average error of 0 .
35 to 1 .
15 mm and maximalerror of 0 .
75 to 7 .
53 mm over the whole 38 cm sensing length. In the endovascular scenario we obtainedan average and maximal error of 1 .
13 mm and 2 .
11 mm, respectively.
Conclusions:
The accuracies of the 3D shape sensing model are promising and we plan to combine the shape sensingbased on fiber Bragg gratings with the position and orientation of an electromagnetic sensor system toobtain the located shape of the catheter.
Keywords
Fiber Bragg grating (FBG), shape sensing, flexible instruments, endovascular navigation.
Cardiovascular diseases are the main cause of death in western industrial nations [9]. Some of thesediseases like abdominal aortic aneurysms can be threaded by an endovascular aortic repair (EVAR)procedure, in which a stent is placed in the region of the aneurysm under 2D fluoroscopy. To reduce theX-ray exposure time and to supersede the angiography a three-dimensional navigation is needed. Fraunhofer MEVIS, Institute for Digital Medicine, L¨ubeck, Maria-Goeppert-Straße 3, 23562 L¨ubeck, Germany Medical Laser Center L¨ubeck GmbH, Peter-Monnik-Weg 4, 23562 L¨ubeck, Germany Institute of Biomedical Optics, Universit¨at zu L¨ubeck, Peter-Monnik-Weg 4, 23562 L¨ubeck, Germany German Center for Lung Research, DZL, Airways Research Center North, 22927 Großhansdorf, Germany Fraunhofer MEVIS, Institute for Digital Medicine, Bremen, Am Fallturm 1, 28359 Bremen, Germany a r X i v : . [ phy s i c s . m e d - ph ] S e p Sonja J¨ackle et al. Fiber Bragg grating (FBG) based systems are used for shape sensing, which enables three-dimensionalnavigation. FBGs are interference filters, which reflect a specific wavelength and are inscribed into thecore of a optical fiber. Therefore, the change in reflected wavelength can be used to calculate strain.Combining multiple FBGs at the same longitudinal position allows to calculate curvature and directionangle. The most common configuration are three fibers arranged triangular around the structure to bemeasured [4,13]. This introduces significant errors due to possible changes in the core geometry [4], whichcan be overcome by multicore fibers, where several cores are integrated into one fiber [10]. In additionother FBG types with different geometrical configurations have been introduced, as for example helicallywrapped [17].Most research groups use FBG systems for shape and force sensing of medical needles [11]. These areshort instruments, which have a simple bending profile allowing shapes with low bending and typically notorsion. For example Park [12] applied FBGs for shape sensing of biopsy needles and Roesthuis used it toreconstruct the shape of a nitinol needle [13]. A few works using optical fibers for flexible instruments havebeen reported in the literature: Shi [15] used FBG systems together with EM-tracking and ultrasoundto achieve a vasculature reconstruction and catheter modeling. Also Khan [7] used four multicore fibersto reconstruct the first 118 mm of a catheter. However, to our knowledge, there are currently no studieson the accuracy of fiber optical shape sensing for very long and flexible medical instruments.In general shape reconstruction of flexible structures is more challenging, because higher deflectionsand torsion can occur. Thus the error analysis of the shape reconstruction from measured wavelengthsto the reconstructed shape becomes more important. Also the accuracy of the measurement has to bevery accurate, since the shape error accumulates along the instrument.Therefore we introduce our optimized model for shape sensing of flexible instruments. Then weevaluated our model with 3D experiments. Finally, we tested it in a realistic endovascular scenario byinserting our fiber in a 3D printed aortic vessel system.
Left: A FBG system with center-to-center distance d and sensor length (cid:96) . The different cores arerepresented by numbers; Right: The cross section of a FBG system with triplet configurationWe consider a multicore fiber with n FBG arrays along a flexible instrument, as shown in Fig. 1 Eacharray contains seven FBGs, one center core and six outer cores. All FBGs have fixed length (cid:96) and thearrays are uniformly distributed with center-to-center distance d .2.1 Shape Sensing ModelWe analyzed every shape sensing step and optimized it by minimizing the errors. The result is ouroptimized shape sensing model with the following steps: iber Optical Shape Sensing of Flexible Instruments for Endovascular Navigation 3
1. Wavelength shift calculation with reference wavelength.2. Strain computation for every core.3. Strain interpolation for every core.4. Curvature and angle calculation by solving equations.5. Curvature and angle correction.6. Shape reconstruction with circle segments.Every step is described in more detail in the following sections.
FBGs are interference filters inscribed in short segments of the core of an optical fiber, which are ableto reflect a specific wavelength of the incoming light [8]. The Bragg wavelength of a FBG is defined as λ B = 2 n e Λ , where n e is the effective refractive index of the grating in the fiber core and Λ the grating period.Mechanical strain or temperature change the reflected wavelength, which is the basic idea for shapesensing. This results in a wavelength shift ∆ λ = λ − λ B of the current measured wavelength λ in comparison to the reference wavelength λ B of the FBG. If thereference wavelengths of the FBGs are unknown, they have to be determined by a separate measurementwhere no strain is applied to the fiber system. The measured wavelength shift ∆ λ B , which can be caused by an applied strain ε or by a temperaturechange ∆ T in the Bragg gratings, is given by∆ λ = λ B (cid:0) (1 − p e ) ε + ( α Λ + α n )∆ T (cid:1) , where p e is the photo-elastic coefficient and α Λ and α n are the thermal expansion coefficient and thethermo-optic coefficient of the fiber system [5]. Assuming a constant temperature ∆ T = 0 simplifies theformula and allows to calculate the mechanical strain of the fiber with the measured wavelength shift:∆ λ b = λ b (1 − p e ) ε. (1)The photo-elastic coefficient p e is directly related to the gauge factor GF = 1 − p e . Photoelasticityis defined as the change in reflected wavelength depending on the mechanical strain applied in axialdirection. For FBG systems the photo-elastic coefficient p e ≈ .
22 can be found in the literature [16].Additionally, experiments have been described for determining the photo-elastic coefficient of any FBGsystem [2].
When curvature and angle direction are calculated for every FBG array, the required intermediate valuescan be determined by interpolation. This method assumes that the determined values of one FBG arrayare the values for one specific position, usually the center of the array.Henken [4] compared common interpolation methods for shape sensing and concluded that cubic splineinterpolation is the best solution, which is currently the state-of-the-art interpolation. Interpolating thecurvature is straight forward since it is continuous for any shape, whereas the direction angle interpolationis challenging for flexible structures, which may have discontinuous direction angle.Thus, we suggest to interpolate the strain, since it is continuous. Also, we use the averaged cubicinterpolation, as introduced in [6]: This yields a realistic interpolation based on the spatial properties ofthe FBGs by modeling the measured sensor values as an average over the sensor range.
Sonja J¨ackle et al. The calculation of the curvature and direction angle depends on the fiber system. The most common oneis a triplet configuration [4,13]: Here the FBG system has three fiber cores with specific angles (typically120 ◦ ) in between, as illustrated in Fig. 1.For this configuration the relation between the strain and the curvatures and direction angles isdescribed by the following equation system: ε a = − κr a sin( ϕ ) + ε ε b = − κr b sin( ϕ + γ a ) + ε ε c = − κr c sin( ϕ + γ a + γ b ) + ε , where ε x is the strain, r x the radius and γ x the angle of the corresponding fiber x . By solving the equationsystem we obtain the strain bias ε , the curvature κ and the direction angle ϕ . The equation system canalso be extended for four or more fibers.The equation shows, that the curvature is influenced by the radii r x in a similar way as by the photo-elastic coefficient. The strain bias ε includes a couple of effects: For the strain calculation we assumeda constant temperature, but conducting a measurement with another temperature than in the referencewavelength measurement, results in a bias. Also axial strain and pressure are part of the strain bias. The determined curvatures and direction angles are influenced by various variables. To get the right valueswe suggest the following parameters for correction: The curvature values are scaled by the photo-elasticcoefficient p e and the center-to-core distances r x . Since both parameters can be biased, we determine ancorrection parameter c to get the right curvature values κ real = c · κ. (2)Also the fiber can be twisted during production or storage. But these twists are not contained in ε .Thus we obtain a measured direction angle ϕ = ϕ real + ϕ twist , (3)which does not equal the real angle ϕ real because it is distorted by the twist angle ϕ twist . Since it isan offset of the real angle ϕ real for every fiber it cannot be determined for FBGs in this geometricalconfiguration without a measurement, where κ (cid:54) = 0. Helically wrapped fibers include torsion in theirmodel and the twist error can be compensated. For short and stiff instrument, this error is negligibly,whereas for flexible instruments the twist angles must be determined. In the last years three different algorithms have been proposed for shape reconstruction: Moore [10]presented a method based on the fundamental theorem of curves, which states that the shape of anyregular three-dimensional curve, which has non-zero curvature, can be determined by its curvature andtorsion [1]. It should be noted that the torsion of curves in mathematical contexts corresponds to thechange of direction angle. The shape is obtained by solving the Frenet-Serret equations: dTdt = κN, dNdt = − κN + τ B, dBdt = − τ N, where κ is the given curvature, τ the given torsion, T the tangent vector, N the normal vector and B thebinormal vector of the given curve at the length position t . The integration of the determined tangentvectors yields the shape of the curve. This method fails at points with κ = 0 and consequently thedirection angle is undefined. Thus, this algorithm is not suitable for shape sensing of flexible structures. iber Optical Shape Sensing of Flexible Instruments for Endovascular Navigation 5 Cui [3] suggested a method based on a parallel transport approach to overcome this. The equationsto be solved are: dTdt = κ N + κ N , dN dt = − κ T, dN dt = − κ T. where κ and κ are the curvature components corresponding to the normal vectors N and N , whichare orthogonal to the tangent vector T . The shape reconstruction is conducted in the same way as withFrenet-Serret.Roesthuis [13] proposed another method based on circle segments: The shape is reconstructed byapproximating it with elements of constant curvature. So for every element a circle segment of curvature κ and length l is created. Afterwards this segment will be rotated by the direction angle ϕ . By repeatingthis procedure for every given set ( κ, ϕ ) we obtain the whole shape.2.2 Experimental methodsFor all experiments described below we used a multicore fiber system (FBGS Technologies GmbH)consisting of 7 cores, one center core and six outer cores each with an angle of 60 degree in between,as shown in Fig. 1. It has 38 FBG arrays each with 5 mm length and 10 mm center-to-center distance,which are chains of draw tower gratings (DTG (cid:114) ).In the next sections we used the following parameters and algorithms if they are not analyzed orspecified there: We fixed our covered fiber to a precise ruler and used the measured wavelength asreference wavelengths, we used a photo-elastic coefficient p e = 0 .
22 for strain calculation, we madeaveraged cubic interpolation of the strain values, we used four outer cores of our FBG system and wereconstructed the shape with circle segments.For matching reconstructed and ground truth shape we used the iterative closest point algorithm[14]. For evaluation we calculated the average and the maximum error defined as e avg := 1 n n (cid:88) i =0 (cid:107) x i − x gt i (cid:107) and e max := max( (cid:107) x − x gt0 (cid:107) , . . . , (cid:107) x n − x gt n (cid:107) ) , where x , . . . , x n are the reconstructed points and x gt , . . . , x gtn are the measured ground truth pointslocated every 10 mm along the shape. For our multicore fiber we had no reference Bragg wavelengths given. Thus, we had to determine thesewavelengths with a measurement without any strain. Therefore we analyzed the effect of the Braggwavelength estimation: At different times we fixed the fiber in a straight line, measured the wavelengths,used it as reference Bragg wavelengths and reconstructed various types of shapes.
The photo-elastic coefficient influences the shape by scaling the curvature. To analyze the effects of thisparameter, we bent our fiber to varying degrees and reconstructed the shape using different p e values. For interpolation evaluation we formed our fiber to a snakelike shape, which has a few singularity points.Then we interpolated the measured strains as proposed in section 2.1.3 and compared the resultingcurvature and direction angles with the common interpolation methods.
Sonja J¨ackle et al. Since we have a multicore fiber with 6 outer cores and one center core and an interrogator, where wecan connect 4 cores, we do not have to use a triplet configuration with 120 degrees in between. Thuswe analyzed the effect of different combinations of 3 or 4 outer cores on the resulting curvatures anddirection angles.
To determine the twist angle ϕ twist we bent our fiber to 2D-shapes, where every position should have thesame angle, as for example a bow shape, determined the direction angles and used it as twist angles, asdescribed in Equation (3). To get the curvature scale factor c we made several bow shapes with differentradii, determined the best value assuming a photo-elastic coefficient p e = 0 .
22 and used it for curvaturecorrection, as described in Equation (2).
The shape reconstruction quality depends completely on the measured curvature and direction angles.When the measured values are correct, the proposed algorithms are able to reconstructed the correctcorresponding shape. Therefore we analyzed the following two aspects:First we looked at the convergence, i. e. how fine the segments in each iterative step of the algorithmsmust be to obtain the correct shape. Second we analyzed the noise handling of the three algorithms, i. e.how the results of the algorithms change with increasing gaussian noise. In both cases we simulated anarc shape with torsion, calculated the average curvature and median direction angle for every segmentand reconstructed the shape.
To evaluate our optimized shape sensing procedure we recorded 3D measurements: we covered our opticalfiber (diameter: 200 µ m) with a metallic capillary tube (inner diameter: 300 µ m and total diameter:400 µ m), fixed it in a specific shape, reconstructed the shape and compared it with the segmentedground truth from the CT images. For the endovascular experiment we inserted the FBG system into a3D vessel system, which was created from a CT scan from a real patient. iber Optical Shape Sensing of Flexible Instruments for Endovascular Navigation 7Shape Error First Reference Second Reference Third ReferenceStraight Line e avg .
36 0 .
16 2 . e max .
00 0 .
30 5 . e avg .
70 1 .
56 1 . e max .
92 4 .
53 4 . e avg .
80 1 .
76 1 . e max .
58 4 .
62 3 . Table 1:
Results of the Bragg wavelength study: Measured error e avg and e max in mm for differentshapes using various Bragg wavelengths Fig. 2:
Effect of the photo-elastic coefficient for different bending strengths (green = ground truth,yellow = reconstruction with p e = 0 .
21, red = reconstruction with p e = 0 .
22, blue = reconstruction with p e = 0 . Sonja J¨ackle et al. Fig. 3:
Results of interpolation study: the images show the resulting curvatures and direction angles ofdifferent interpolations methods along the fiber3.5 Curvature and angle correctionFirst we bent the fiber into a circular shape and used the determined angle as twist angle, as describedin Equation (3). The result of this experiment is displayed in the left image of Fig. 4: the reconstructionwithout angle correction is twisted whereas the corrected shape lies in the plane of the ground truth.Afterwards we made several circular shapes with various radii to determine the curvature scale factorof our FBG system. The results are shown in the right part of Fig. 4. We found that a scale factor of ≈ .
026 achieves the best results and used it for curvature correction, as described in Equation (2).
Fig. 4:
Left: Results of the twist angle study: The reconstructed shape with twist correction (blue) andwithout (orange) is shown. Ground truth from the CT scan is displayed in white; Right: Results of thecurvature scale study: The average error (straight line) and maximum error (dashed line) are plotted forthree different circles iber Optical Shape Sensing of Flexible Instruments for Endovascular Navigation 9
Fig. 5:
Shape reconstruction study results: The images show the average error as a function of segmentlength and of noise3.7 3D shape reconstruction accuracy
Fig. 6:
3D experiment with the fiber: the segmented shapes from the CT scan of the circle, s-curve andhelix measurements are shown et al. For the 3D shape experiments we integrated the results of the previous experiments in our model.We made several measurements bending our optical fiber to different 3D shapes. The segmented shapesfrom the CT scan of the circle, s-curve and helix measurements, as shown in Fig. 6 were used as groundtruth. The accuracies, shown in Tab. 2, depend on the complexity of the forms: For the circular shapeswe obtain average error e avg ≈ . e max ≈ Shapes \ Errors (mm) e avg e max circle 1 0 .
35 0 . .
50 1 . .
50 1 . .
70 1 . .
57 1 . .
15 7 . .
00 4 . .
13 2 . Table 2:
Results of the 3D experiment: Measured errors e avg and e max in mm for different 3D shapemeasurementsIn the last experiment we evaluated our model in a realistic endovascular scenario and insertedour fiber into a 3D printed vessel phantom, as shown in Fig. 7. Here we obtained an average error e avg = 1 .
13 mm and maximum error e max = 2 .
11 mm, which indicates an accurate shape reconstruction.This is also visible in the right image of Fig. 7: The reconstructed shape, represented by the blue line,fits almost perfectly to the ground truth of the CT scan.
Fig. 7:
The first image shows the vessel phantom with the fiber inside on a CT bed, the second imagethe corresponding CT scan with the reconstructed shape (blue) and ground truth (white) iber Optical Shape Sensing of Flexible Instruments for Endovascular Navigation 11
In this work we presented an optimized model for shape sensing with multicore fibers for flexible in-struments. We conducted a detailed error analysis for every step of the shape reconstruction procedure.The main error sources of shape sensing with multicore fibers are corrupted reference wavelengths forthe wavelength shift computation, direction angles changed by the twist present in the multicore fiberand curvature values, which are distorted by using a wrong photo-elastic coefficient or wrong radii. Thisindicates that two calibration measurements for every FBG fiber need to be done. The first one is used todetermine the Bragg wavelength λ b with ε = 0, the second one to get the twist angle ϕ twist where κ (cid:54) = 0.Further factors influencing the shape are the equation system defined by the used fiber configuration,the interpolation of curvature and angle values and the chosen reconstruction algorithm.Furthermore we evaluated the accuracy of our model with 3D measurements in a CT scanner. Wereceived an accuracy around e avg ≈ .
35 to 1 .
15 mm and e max ≈ .
75 to 7 .
53 mm. Finally we testedour fiber system in a real endovascular scenario and obtained high accuracies ( e avg = 1 .
13 mm , e max =2 .
11 mm). These experiments show promising results for using multicore fibers for shape sensing ofcatheters.In future work we aim to enable a full endovascular catheter navigation. For this purpose we plan tocombine the reconstructed shape obtained by the multicore fiber with the position and orientation of aelectromagnetic tracking system.
Acknowledgements
We thank Armin Herzog, Institute for Neuroradiology, University Hospital Schleswig-Holstein, L¨ubeckfor support when using the CT scanner and the Department of Surgery, University Hospital Schleswig-Holstein, L¨ubeck for providing the 3D vessel model printed by Fraunhofer EMB. This work was fundedby the German Federal Ministry of Education and Research (BMBF, project Nav EVAR, funding code:13GW0228C).
Compliance with ethical standards
Funding:
This work was funded by the German Federal Ministry of Education and Research (BMBF, project NavEVAR, funding code: 13GW0228C).
Conflict of interest:
The authors declare that they have no conflict of interest.
Ethical approval:
All procedures performed in studies involving human participants were in accordance with the ethicalstandards of the institutional and/or national research committee and with the 1964 Helsinki Declarationand its later amendments or comparable ethical standards. This article does not contain any studies withanimals performed by any of the authors.
Informed consent:
Informed consent was obtained from all individual participants included in the study.
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