WO2014009294A1

WO2014009294A1 - Method for determining contact forces between a deformable implant and surrounding tissue

Info

Publication number: WO2014009294A1
Application number: PCT/EP2013/064328
Authority: WO
Inventors: Michael GESSAT; Volkmar Falk; Edoardo Mazza; Raoul HOPF
Original assignee: Universität Zürich; ETH Zürich
Priority date: 2012-07-09
Filing date: 2013-07-05
Publication date: 2014-01-16

Abstract

The invention relates to a method for determining reaction forces (F) exerted by a material surrounding an deformable implant (1) after the implant (1) was implanted into said material leading to a deformed state of said implant (1). According to the invention, the method comprises the steps of: generating a shape model of the shape of the implant in an undeformed state of the implant (1), determining the displacements (u) of points (P) of the shape model in the deformed state with respect to the undeformed state using at least one image of the implant (1) in the deformed state, generating a finite element model of the implant (1) in its undeformed state using said shape model of the shape of the implant (1) in the undeformed state and a constitutive model for modelling the material properties of the implant, imposing said displacements (u) as kinematic boundary conditions on points (BP) of the finite element model associated to said points (P) of said shape model, and numerically calculating a reaction force (F) exerted by said surrounding material on each of said points (MP) of the finite element model using said boundary conditions (u), particularly by means of a finite element simulation.

Description

Method for Determining Contact Forces between a Deformable Implant and

Surrounding Tissue

Specification

The invention relates to a method for determining reaction forces exerted by a material (tissue) surrounding an e.g. elastically deformable (medical) implant after the implant was implanted into said material leading to a deformed state of said implant. Furthermore, the invention relates to an implementation of said method in a computer program.

Knowledge about the contact (reaction) forces between an implant and its surrounding tissue can provide essential information on the overall performance of the implant. There exists no method to measure these forces for individual patients.

In particular, in the case of a transcatheter aortic valve implantation (TAVI), insufficient contact forces between implant (stent) and tissue might cause paravalvular leaks¹ or valve embolization On the contrary, excessive contact forces may cause annulus ruptures and are suspected to impair the conductibility of the atrioventricular (AV) node, the bundle of His, and/or the left bundle branch. All three result in left ventricular arrhythmia or bradicardia, which are a common complication after TAVI, usually requiring pacemaker implantation². The two main risk factors which could be identified as significant in clinical studies are (i) the type of valve selected for implantation and (ii) valve oversizing (resulting in high radial loads)^2,3. These findings indicate a connection between mechanical stress induced on the tissues and the necessity for a pacemaker after TAVI. Nevertheless, there exists no quantitative investigation of this relationship, which would allow predicting or help reducing the onset of conduction abnormalities after TAVI.

Thus, the problem underlying the present invention is to provide for a tool for quantitative analysis of implant (stent)-tissue interaction, for TAVI as well as for other applications (implants).

This problem is solved by a method having the features of claim 1 . According thereto, the method according to the invention comprises the steps of: automatically generating or providing a (3D) shape model (being defined by at least a set of 3D coordinates of points of the implant as well as eventually geometric structures such as curves or areas connecting said points) of the shape of the implant in an undeformed state prior to said implantation, automatically determining the displacements of points (or other geometrical structures) of the shape model in the deformed state with respect to the undeformed state using at least one image of the stent in the deformed state, i.e., when the implant is implanted as intended, automatically generating a (3D) finite element model of the implant in its undeformed state using said shape model of the shape of the implant in the undeformed state and a constitutive model for modelling the material property of the implant (i.e. the finite element model is particularly defined by a number of finite elements and their material properties) due to which the implant e.g. tends to return into its undeformed state when no reaction forces are exerted onto the implant/stent, for instance, the individual finite elements (constituents of the finite element model) may be modeled as linear elastic, isotropic hookean solids, imposing said displacements as kinematic boundary conditions on points of the finite element model corresponding to said points of said shape model, i.e., the measured displacements of every point of the shape model are applied to the points of the finite element model, and numerically calculating a reaction force exerted by said surrounding material (tissue) on each of said points of the finite element model using said boundary conditions, particularly by means of a finite element solver/simulation.

Particularly, implanting the implant does not form part of the method according to the invention. Rather, the method is performed after the implant has been implanted and uses information of the undeformed implant before implantation.

In the present case, the kinematic boundary conditions, i.e., said displacements described above and the constitutive model (material properties of the individual finite elements) represent input quantities and the reaction forces are the output of the simulation (numerical calculation by means of a computer). By means of the numerical calculation, these forces are determined such that the finite elements are in force equilibrium.

In other words, the method according to the invention is based on two key elements: force balance at the implant (stent)-tissue interface (i.e., said points of the implant) and the availability of high-resolution geometrical data of the implant (stent) before and after implantation. In fact, in order to obtain the stress field at the (contact) points, it is sufficient to determine the boundary conditions corresponding to the measured state of deformation of the implant. Modern medical imaging systems, such as CT, are preferably employed for acquiring image data of sufficient resolution for reconstructing the deformed implant shape/geometries in three dimensions.

According to an aspect of the invention, said points of the implant/material (tissue) interface of the finite element model are allowed to relax within a pre-defined region, particularly in the form of a spherical volume, corresponding to a measurement uncertainty in said points of the deformed implant, e.g. due to a means for generating said at least one image, the movement of the implant upon generating said at least one image, and/or a (automatic image processing) detection method applied for identifying said points of the deformed implant. In other words, depending essentially on the measurement uncertainty of the particular (medical) imaging system used to record the deformed shape of the implant inside said material (e.g. tissue(s) of a patient's body), additional measures may be used to reduce the resulting spurious stresses and strains, which would otherwise form a noise covering the desired (radial) reaction forces.

Particularly, to each of said points of the finite element model a boundary point is associated, wherein the respective boundary point is spaced apart from the associated point, particularly towards a center of the undeformed implant (e.g. in a radial direction) by a distance, wherein each of said points is coupled by a nonlinear spring element to its associated boundary point, which nonlinear spring element has a relatively low stiffness for all displacements lower than a measurement uncertainty in said points of the deformed implant and a relatively high stiffness for displacements above said measurement uncertainty, and wherein particularly the determined displacement of each of said points of the implant in the deformed state with respect to the undeformed state is applied to the associated boundary point.

Preferably, the calculated reaction forces are decomposed into radial reaction forces pointing towards a longitudinal axis of the implant/stent.

According to an aspect of the invention, the implant to be analyzed comprises a plurality of elements, particularly longitudinally extending strings, being (multiply) connected to each other, e.g. along their respective longitudinal extension direction, at said points, so as to form a mesh-like pattern, wherein particularly said elements are made out of Nitinol, a metal alloy of nickel and titanium, where the two elements are present in roughly equal atomic percentages. Particularly Nitinol alloys exhibit shape memory and superelasticity. Preferably said implant is a stent, particularly for TAVI.

According to a further aspect of the invention, for generating said shape model of the implant in its undeformed state, at least one image is generated from the implant in its undeformed state, particularly by means of MicroCT imaging (i.e. a CT method that has a resolution in the micrometer range). Other imaging methods can also be used. Particularly, the coordinates of said (intersection) points of the mesh-like structure formed by the (Nitinol) elements are extracted e.g. by means of automatic image processing from said at least one image (or a plurality of images). Furthermore, said elements are particularly detected, and centerline of each of the imaged elements is preferably approximated by a curve, particularly a Bezier curve, particularly of third order, to create a smooth, noise-less, and uniformly sampled representation of the centerline of each element.

According to a further aspect of the invention, the coordinates of said (intersection) points of the implant in the deformed state are extracted from the at least one image by means of automatic image processing, wherein particularly said at least one image is generated by means of ECG gated cardiac Computed Tomography (CT) after implantation of the implant (stent), i.e., an electrocardiogram taken from the patient during image acquisition on the beating heart is used to factor out motion artefacts. Alternatively, 3D angiography or fluoroscopy could be used, where a three- dimensional image of the implant an the surrounding material (tissue) is generated, for instance by means of rotational x-ray angiograms.

According to yet another aspect of the present invention, said (intersection) points of the implant (stent) in the deformed state are assigned with the corresponding points of the implant in the undeformed state. Preferably, the coordinates of said points in the deformed state and in the undeformed state of the implant are transformed into a common coordinate system for determining said displacements (e.g. by automatically calculating a difference of vectors in this common coordinate system pointing to said points).

For generating said (3D) finite element model of the implant in its undeformed state said curves are used to determine (calculate) discrete nodes on the center line of each element (of the implant in its undeformed state), wherein particularly each node is provided with a local coordinate system comprising a tangent vector of the respective curve and two perpendicular vectors forming an orthonormal, right handed trihedron, wherein particularly neighboring nodes on each element form a start point and an end point of a beam element, particularly in the form of a three-dimensional first order two point Timoshenko beam element, wherein particularly the end point of the previous beam element yields the start point of the following beam element, wherein particularly, each start point and end point of the respective beam element is provided with a tangent vector and two perpendicular vectors. The tangent vector at a node is computed as the normalized average (mean) of the axis along the inbound and outbound beam element. The first perpendicular vector is a unit vector normal to the tangent vector and points away from the global longitudinal axis of the implant/stent. The second perpendicular vector is the unit vector orthogonal to the tangent vector and the first perpendicular vector.

Such beam elements are useful to discretize slender structures subjected to bending, tension and torsion. There exists a large number of different formulations for these element types. In the case of an implant in the form of the stent, the structure is non- planar and therefore the beam elements are preferably formulated in three dimensions, i.e., a cross section geometry is assigned to each beam element, which is particularly taken from measurements on the implant. The cross section may have a rectangular shape with a height and a width. Said tangent vectors and perpendicular vectors at the start and endpoint are used to define the orientation of the respective cross section with respect to a global coordinate system. These beam elements consist of two nodes, on which the orientation of the cross section has to be defined. In order to avoid twist or curvature of the beam element along its main axis in the undeformed configuration, both nodes have to be assigned with the same tangent and perpendicular vectors, which then ensure a constant cross section for every beam element.

In order to model said intersections (connections) of the individual elements, each of said (intersection) points correspond to two adjacent nodes (where the elements are connected or intersect) in the finite element model, which two adjacent nodes are coupled in all six degrees of freedom to undergo the same kinematics.

Further, middle points are preferably generated in the middle of each distance between (each pair of) said coupled adjacent nodes.

Furthermore, to each middle point one boundary point is preferably (uniquely) associated, wherein the respective boundary point is spaced apart from the associated middle point towards a center of the (undeformed) finite element model of the implant in a radial direction by a distance that is orders of magnitude smaller than said displacements (particularly Ι μηι), and wherein each middle and associated boundary point is coupled by a nonlinear (force) spring element having a relatively low stiffness for all displacements being lower than a measurement uncertainty in said points of the deformed stent and a relatively high stiffness for displacements being above said measurement uncertainty.

According to a further aspect of the present invention, the determined (measured) displacement of each of said points of the implant in the deformed state with respect to the undeformed state is applied to the corresponding boundary point (as kinematic boundary condition).

Preferably, for the constitutive model, a linear elastic, isotropic hookean solid is used, i.e., the finite elements (e.g. beam elements) forming part of the finite element model are modeled as such hookean solids. Such a constitutive model used here consists of two independent material parameters, the E-modulus £ and the Poisson ratio v. These material parameters determine an invertible linear mapping between the stress tensor and the strain tensor

a=C(£, v )e,

where C represents the fourth order stiffness tensor. This is the most basic three dimensional constitutive model that covers all linear elastic, isotropic, homogeneous materials.

Finite element discretization of the beam equation (elliptic Partial Differential Equation (PDE)) - in case the above-mentioned beam elements are used - yields the linear mapping between nodal forces and displacements for each beam element. This involves numerical integration steps in order to obtain the element stiffness matrix K_e which depends on the cross section (see above) and the constitutive law

where u_e holds the nodal displacements for all six degrees of freedom (3 translations, 3 rotations), and F_e holds the according nodal forces (and moments). The end point of each element represents the starting point of the next element until the end of the structure is reached, which imposes kinematic transition conditions. Based on the boundary conditions (displacements of said points) and the implemented geometry, a global system of linear equations can be assembled (e.g. by a finite element software/method such as a commercial PDE-solving system)

Ku = F,

which is then solved for the reaction forces F using said numerical methods/software. Other methods may also be used.

Further, additional connector (spring) elements may be employed to account for a measurement uncertainty concerning the displacements of said points of the implant (see above).

A further embodiment of the present invention may also be formulated as follows:

Method to create a local estimation of the contact forces (F) between a deformable medical implant and surrounding tissue(s), wherein this method makes particularly use of the availability of high resolution medical imaging systems to set up a finite element simulation of the implanted implant (1 ). The contact forces (F) between tissue(s) and implant (1 ) are equal in magnitude to the reaction forces on the implants boundary, the method comprising the steps:

(a) Modeling of the unloaded shape of the implant (1 ) in order to obtain geometric data for a reference model.

(i) An appropriate means of obtaining the 3D shape of the unloaded implant (1 ) is applied.

(ii) The unloaded shape of the implant (1 ) is expressed in the form of a 3D computer model.

(b) Extraction of the deformed shaped of the implant (1 ) using medical images.

(i) postinterventional medical images are acquired from the patients.

(ii) geometric feature are extracted, which provide sufficient information about the morphology of the deformed implant (1 ).

(c) Setting up a finite element model using the data obtained from steps (a) and (b).

(i) a finite element mesh of the implant in its unloaded reference configuration is created, using the geometric data of step (a).

(ii) a constitutive model is added, in order to provide the required material properties of the implant. (iii) kinematic boundary conditions are set up using the data obtained in step (b). Depending on the measurement uncertainty of the particular medical imaging system used to record the deformed shape of the implant inside a patient's body, additional measures may be used to reduce the resulting spurious stresses and strains.

(iv) the simulation is solved using a commercial finite element solver. External reaction forces on the implant form the output of the simulation.

(v) the reaction forces may be decomposed into desired components in a post processing step.

Yet another embodiment of the present invention may also be formulated as follows:

Method, particularly according to said latter embodiment, for implants in the form of aortic nitinol stents (1 ), wherein this method applies to nitinol stents (1 ) that feature a substructure consisting of slender elements, particularly in the form of strings (S), wherein the stent (1 ) is regarded as a composition of strings (S), which intersect with each other, thus forming a mesh-like pattern, the method comprising the steps:

(i) MicroCT imaging is applied to obtain a 3D dataset containing the stent's (1 ) 3D shape and structure.

(ii) The 3D coordinates of the intersection points (P) of the mesh-like structure formed by the Nitinol strings are extracted from these images. 3rd order Bezier interpolation is applied to create a smooth, noise-less, and uniformly sampled representation of each string's centerline (S').

(b) Extraction of the deformed shaped of the stent (1 ) using medical images.

(i) ECG gated cardiac Computed Tomography images are acquired after stent (1 ) implantation.

(ii) Automatic image processing is applied to extract the 3D coordinates of all intersection points (P) of the deformed stent (1 ) from the images.

(iii) Automatic point labeling and registration is applied to assign the intersection points (P) in the deformed stent with the corresponding intersection points in the unloaded stent model (P) and to transform the coordinates into one mutual coordinate system. (c) Setting up a finite element model using the data obtained from steps (a) and (b)

(i) the Bezier control polygons of the strings (S), obtained in step (a), are used to calculate discrete points (N) on the center lines (S') of each string (S) in its undeformed configuration. The resolution of this discretization is selected to provide adequate accuracy of the approximated curved shape and the linear interpolation between two neighboring points (N) in space. Two neighboring points (N), which lie on the same string (S), form the start- and end point (121 , 122) of a one dimensional beam element (12) in three dimensional space. The end point of the previous element yields the starting point of the following element (12). The normal for each point (121 , 122) in each beam element (12) is calculated according to the point's location in a predefined, global coordinate system, in order to provide the orientation of the cross section and avoid initial twist and curvature of the beam element (12). Every intersection point on a string (S) corresponds to another intersection point on a neighboring string (S). They are coupled in all six degrees of freedom to undergo the same kinematics. All beam elements (12) are assigned a cross section geometry which is taken from measurements on the stent.

(ii) For the constitutive model a linear elastic, isotropic hookean solid is used.

(iii) Middle points (MP) representing the physical center of the intersection volume are added. Furthermore, to every middle point (MP) an additional boundary point (BP) is added. These points are obtained from a translation of the middle points (MP) towards the center of the stent (1 ) in radial direction. The distance between the middle point (MP) and the boundary point (BP) is kept in orders of magnitude smaller than the measured local displacements (u), so that it remains negligible relative to those displacements (u). Between each middle- and boundary point (MP, BP) a nonlinear spring element (K) is added. This spring element (K) exhibits low stiffness for all displacements (u) lower than the measurement uncertainty and very high stiffness for displacements (u) above the measurement uncertainty. The measured displacement (u) of every intersection point (P) is applied to the according boundary point (BP).

(iv) the simulation is solved using a commercial finite element solver. External reaction forces (F) on the implant form the output of the simulation.

(v) radial components of the reaction forces (F) are extracted in a post processing step. Further, the problem underlying the present invention is solved by a computer program (software), which is particularly stored on a computer readable medium or downloadable (e.g. from a computer via the internet), and which is particularly designed to be loaded into the memory of a computer, wherein the computer program product is designed to conduct - upon being executed on a computer - the afore-described method (algorithm) according to the.

Said computer program may be designed to read from a database a pre-generated 3D shape model of an undeformed stent, which may have generated by means of the computer program in beforehand or may form an input to the computer program.

Further, images generated in the present method to extract the shape of the undeformed/deformed implant may not be generated with help of the computer program, but may simply be inputs to the computer program.

Further features and advantages of the present invention shall be explained by means of the following description of embodiments with reference to the Figures. Therein,

Fig. 1 shows an implant in the form of a stent including leaflets (not modelled) with a highlighted element (string) and according intersection points, as well as Bezier-control polygons and discretized center lines;

Fig.2 shows a solid model (left) with a close up view of the mesh (middle), as well as a beam element model (right) with rendered beam profiles and element orientation vectors;

Fig. 3 shows mode 1 and results of the corresponding analysis for all mesh resolutions;

Fig. 4 shows mode 1 : FE-model, experiment, measured curve and simulation;

Fig. 5 shows an intersection middle point, a boundary point, a connector element and the corresponding nonlinear force law; and

Fig. 6 shows a three dimensional plot of the force field and nodal force plot.

An exemplary implementation of the invention is shown in the following for the determination of contact (reaction) forces between implant (stent) 1 and tissue after TAVI. The procedure uses geometrical data obtained from high-resolution medical imaging systems and the (reaction) force field F is reconstructed through finite element simulations. The present analysis is performed for a commercially available implant (stent) 1.

In a first step, a micro-CT scan of the implant 1 is acquired from which a finite element mesh is generated corresponding to its undeformed reference configuration (cf. Fig. 1 ). Experimental validation is used to evaluate the reliability of the stent model. To obtain the actual in-vivo displacement field after implantation,

postoperative CT images are analyzed. With help of this data a discrete

displacement map for all junction (intersection) points P of the stent is obtained. This displacement field is used to impose kinematic boundary conditions at the

corresponding points P of the stent model. Due to the measurement uncertainty, directly applied boundary conditions lead to high noise in the reconstructed force field F. Therefore, a system using nonlinear connector elements is introduced (cf. Fig. 5), in order to apply boundary conditions with tolerances corresponding to the limits given by measurement uncertainties. All meshing and simulation is performed by a software according to the invention, but may also be done by means of commercial software such as MATLAB (MathWorks), which can be used to generate a finite element input file (meshing and simulation set-up), which is then exported in form of a finite element input file. Subsequently, simulations can be solved using the program ABAQUS/standard (Dassault Systemes Simulia Corp., Providence, Rl, USA), which was used here. In post processing steps (e.g. MATLAB), the reaction force field F is visualized. Lower bound thresholds are set to determine zones that can be regarded as stress-free due to weak or no contact between stent 1 and tissue.

Stent model implementations found in current literature are mostly set up using full three dimensional solid element formulations^4,5. Meshes using a variety of element formulations, different resolutions and integration schemes can be found. Due to the slender substructure which is largely subject to bending, tension and torque states of loading, three dimensional element formulations can yield numerical problems, such as shearlocking and hourglassing modes. In order to compare with the performance of the beam model formulation, first a full solid meshing system is implemented. This model then is used to perform a convergence study for three different states of deformation.

The geometry of the stent 1 as extracted from CT images consists of 30 elements in the form of strings S and 165 intersection points P (cf. Fig.1 ). The strings S have a rectangular profile with an average width and height of 0.2mm and 0.48mm. From micro-CT imaging data, Bezier splines defining the center line S' of each string of the undeformed stent 1 can be fitted, as described in detail in [6], which may also be employed in the present context. These splines (curves) are defined piecewise, using a 4-point control polygon 1 1 . The first step in the meshing process is to discretize the string center lines S' as shown in Fig 1. The sampling rate to generate the center line nodes N is adjustable.

One of the main difficulties in meshing three dimensional structures with beam elements 12 is to maintain correct orientation for all elements 12 with respect to the geometry and the norm of the finite element solving system that is used. For this purpose, every node N on the center line S' is provided with a local coordinate system 13 (cf. Fig. 2 (right)), that holds the tangent vector of the spline and two perpendicular vectors which form an orthonormal, right-handed trihedron. The tangents can directly be obtained from the definition of the Bezier splines. The outward normal is calculated from a global cylindrical coordinate system and the tangent, whereas the second normal is obtained from the first two vectors.

The beam element model requires the discretized center lines S' of the strings S and the definition of a cross section. The chosen elements for this model are three dimensional first order two point Timoshenko beam elements (e.g. element type B31 in ABAQUS) 12. In order to avoid initial twists or curvatures, both starting- and endpoint of the element 12 need to be provided with the corresponding tangent- and perpendicular vectors. These are obtained by interpolating the vectors which were used to define the nodal coordinate systems 13. At each intersection point P, both corresponding nodes N of the neighboring strings are coupled using kinematic equations in all six degrees of freedom.

For the solid model (cf. Fig. 2 (left)) the cross section is discretized in a plane reference system and then transformed into the according local nodal systems. The nodes are connected to form first order, hexahedral solid elements 14 and a reduced integration scheme is applied (e.g. element type C3D8R in ABAQUS).

The material used in stent 1 is a nickel-titanium alloy (Nitinol), which exhibits super elastic properties to be able to fully recover from strains of up to 10% as a result of the stents 1 crimping process to the catheter. The constitutive model used for the present study is a linear elastic Hookean solid. In fact, the crimping process is not included in the analysis and, the final deformed state of the stent 1 may include finite rotations but deformation is expected to remain in the small strain regime. Overestimation of stiffness due to choosing a linear elastic model was observed when comparing with experimental data at large deformations.

The convergence study (cf. Fig. 3) is performed using four mesh resolutions ranging from orders of 10Ό00 (grade 1 ) to 600Ό00 (grade 4) elements for the solid model. Approximately 4Ό00 elements are used for the beam model. Three different global loading states are tested. Fig. 3 shows the results of the analysis of mode 1 loading (global tension) for the four mesh configurations of the solid model. Mode 2 was defined as a global compression and mode 3 a torsional state. The results of modes 2 and 3 are consistent with Fig. 3. An important remark concerns the result of grade 1 . Due to the low mesh resolution the profile has only one element in the cross section. The reduced integration scheme for first order elements in states of bending therefore lead to zero-energy deformation modes (hourglassing), which yields completely unrealistic results as seen in Fig. 3. The same deformation modes were applied to the beam element model which provides results for all modes comparable with grades 3 and 4, but with only around 25Ό00 degrees of freedom.

The experimental validation of the beam model is performed with one dimensional two point tensile tests. Three configurations were investigated: One symmetrical mode and two nonsymmetrical modes (mode 1 shown in Fig. 4).

Such experiments are relatively easy to perform and mechanically well defined in terms of boundary conditions. In spite of the one dimensional boundary conditions, the coupled grid structure of the stent 1 leads to complex states of deformation, which test both structural as well as constitutive model reliability.

Simulations show good predictive capabilities for low to moderate deflections but a clear overestimation of stiffness for larger deformations. This is deemed to be a consequence of the chosen linear elastic constitutive model. The super elastic material behavior of Nitinol shows lower stresses at the corresponding level of deformations imposed in the experiments.

In order to obtain a patient-specific measurement for the deformation of the stent 1 after implantation, a sparsely sampled deformation field is generated. Therefore, the 165 intersection points P which define the stent's 1 grid structure are automatically extracted from CT images. With a high spatial resolution (< 0.4 mm in all axes), these intersection points P are well visible in CT images as small, well-defined clusters of high-intensity voxels due to the high X-ray contrast of Nitinol. Nevertheless, calcium clusters and image artifacts can create similar clusters or create enough background noise to merge neighboring clusters. A combination of threshold filtering and model- based generation of hypotheses for likely positions of these intersection points is applied in order to identify the intersection points P. In practical tests with 25 patient datasets, the proposed method has proved reasonably robust to image noise and calcium. Due to sampling effects and image blurring, the spatial accuracy of the landmark localization was evaluated to be in the range of two voxel lengths, i.e. 0.8 mm.

If this deformation map was be used directly to impose kinematic boundary conditions at the intersection points P, the high ratio of tensile to bending stiffness in beamlike structures would yield large fluctuations in the values of calculated external forces F, due to inaccuracy in intersection point position measurement. Therefore, an intermediate step was introduced in order to relax these spurious forces. The proposed method is illustrated in Fig. 5. A middle point MP is calculated for each intersection of to strings S (the centerlines S' do not touch each other). An additional boundary point BP in radial direction towards the center of the stent 1 is created. The distance between MP and BP in the undeformed configuration is by orders of magnitude smaller than the applied displacements. The middle point MP and the boundary point BP are connected with a nonlinear force (spring) element K. The force law for the connector element K is set up to have a cut-off at approximately d_c=0.8 mm, which corresponds to the uncertainty of the CT-data. The measured displacement is now applied at the boundary point BP, instead of the middle point MP (corresponding to the intersection point P) directly. This allows the actual physical middle points MP (P) to position themselves within a spherical volume around the nominal position to account for measurement uncertainty, thus leading to a global minimum of potential energy in the stent 1 . This approach avoids artificially high reaction forces and provides a lower limit for the actual force field acting on the stent 1 in-vivo.

The discrete vector field F containing the nodal reaction forces of each intersection point P is imported into MATLAB for post processing. The radial components of the forces can be represented as arrows in a three dimensional plot, or their values reported for all points P along selected paths, as shown in Fig. 6. The points P correspond to the plotted arrows at points (junction nodes) P 1 -7 in the diagram. The sign of the nodal plot was chosen to indicate whether the force points radially inward or outward. Reaction forces F in outward direction are considered as artifacts, since the surrounding tissue will not exhibit any adhesion properties in comparable magnitudes. These negative contact forces possibly indicate a local underestimation of the measurement uncertainty in the post-op CT data.

The average value of radial force divided by the corresponding aortic tissue area, typically yields a radial pressure in the range of 30-40 mbar. These stress levels seem plausible when compared with the average values of blood pressure fluctuation between systolic and diastolic pressure (approx. 50 mbar).

An effective beam model of the aortic stent 1 has been created, compared to solid element models and validated experimentally. A method to extract a patient specific nodal deformation map from medical imaging data and a corresponding procedure to deal with measurement uncertainty have been developed. The model performed well in terms of efficiency as well as robustness and showed good predictive capabilities in low to moderate ranges of displacement corresponding experiments.

The feasibility of the novel procedure for reconstruction of stent-tissue interaction from medical images of in-vivo stent deformation has been demonstrated. Previous studies using finite element simulation for post-operative outcomes have used medical imaging systems to obtain patient specific data^4,5,7,8. This data was then used to either model patient specific vascular geometries, or to obtain global information of the implanted stent. However, to the best of our knowledge so far no procedure exists to obtain contact forces between stent and tissue from post-op data.

Further, considering future applications that involve the crimping and releasing processes, a non-linear constitutive model has to be implemented able to reproduce the super elastic properties of Nitinol.

References

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Claims

1 . Method for determining reaction forces exerted by a material surrounding a deformable implant (1 ) on said implant (1 ) after the implant (1 ) was implanted into said material leading to a deformed state of said implant (1 ), the method comprising the steps of:

- generating a shape model of the shape of the implant (1 ) in an undeformed state of the implant (1 ),

- determining the displacements (u) of points (P) of the shape model in the deformed state with respect to the undeformed state using at least one image of the implant (1 ) in the deformed state,

- generating a finite element model of the implant (1 ) in its undeformed state using said shape model of the shape of the implant (1 ) in the undeformed state and a constitutive model for modelling the material properties of the implant,

- imposing said displacements (u) as kinematic boundary conditions on points (MP) of the finite element model associated to said points (P) of said shape model, and

- numerically calculating a reaction force (F) exerted by said surrounding material on each of said points (MP) of the finite element model using said boundary conditions (u), particularly by means of a finite element method.

2. Method according to claim 1 , characterized in that said points (MP) of the finite element model are allowed to relax within a pre-defined region, particularly in the form of a spherical volume, to account for a measurement uncertainty in said points (P) of the deformed implant.

3. Method according to claim 1 or 2, characterized in that to each of said points (MP) of the finite element model a boundary point (BP) of the finite elements model is associated, wherein the respective boundary point (BP) is spaced apart from the associated point (MP) of the finite element model, particularly towards a center of the undeformed implant (1 ) in a radial direction by a distance (d₀), wherein each of said points (MP) is coupled by a nonlinear spring element (K) to its associated boundary point (BP), which nonlinear spring element (K) has a relatively low stiffness for all displacements (u) lower than a measurement uncertainty in said points (P) of the deformed implant and a relatively high stiffness for displacements (u) above said measurement uncertainty, and wherein particularly the determined displacement (u) of each of said points (P) of the implant (1 ) in the deformed state with respect to the undeformed state is applied to the associated boundary point (PB).

4. Method according to one of the preceding claims, characterized in that said reaction forces (F) are decomposed into radial reaction forces pointing towards a longitudinal axis of the implant (1 ), respectively.

5. Method according to one of the preceding claims, characterized in that the implant (1 ) comprises a plurality of elements (S), particularly in the form of longitudinally extending strings (S), connected with each other at said points (P), wherein particularly said elements (S) are made out of Nitinol, wherein particularly said implant (1 ) is a stent.

6. Method according to one of the preceding claims, characterized in that for generating said shape model of the implant (1 ) in its undeformed state at least one image is generated from the implant (1 ) in its undeformed state, particularly by means of MicroCT imaging, wherein particularly the coordinates of said points (P) are extracted from said at least one image, and wherein particularly a centerline (S') of each of the imaged elements (S) is approximated by a curve, particularly a Bezier curve, particularly of third order.

7. Method according to one of the preceding claims, characterized in that the coordinates of said points (P) of the implant (1 ) in the deformed state are extracted from the at least one image by means of automatic image processing, wherein particularly said at least one image is generated as a 3D Image, particularly by means of ECG gated cardiac CT after implantation of the implant (1 ), or particularly by means of 3D angiography or fluoroscopy after implantation of the implant (1 ).

8. Method according to one of the preceding claims, characterized in that said points (P) of the implant (1 ) in the deformed state are assigned with the corresponding points (P) of the implant (1 ) in the undeformed state, and wherein particularly the coordinates of the points (P) in the deformed state and in the undeformed state of the implant (1 ) are transformed into a common coordinate system for determining said displacements (u).

9. Method according to claim 6 or one of the claims 7 to 8 when referred back to claim 6, characterized in that for generating said finite element model of the implant (1 ) in its undeformed state said curves are used to determine discrete nodes (N) on the center line (S') of each element (S), wherein particularly each node (N) is provided with a local coordinate system (13) comprising a tangent vector of the respective curve and two perpendicular vectors forming an orthonormal, right handed trihedron, wherein particularly neighboring nodes (N) on each element (S) form a start point (121 ) and an end point (122) of a beam element (12), particularly in the form of a three-dimensional first order two point Timoshenko beam element, wherein particularly the end point of the previous beam element yields the start point of the following beam element, wherein particularly, each start point (121 ) and end point (122) of the respective beam element (12) is provided with a tangent vector and two perpendicular vectors.

10. Method according to claim 9, characterized in that a cross section geometry is assigned to each beam element (12), which is particularly taken from measurements on the implant (1 ).

1 1 . Method according to claim 5 or one of claims 6 to 10 when referred back to claim 5, characterized in that each of said points (P) correspond to two adjacent nodes (N) in the finite element model, which are coupled in all six degrees of freedom to undergo the same kinematics.

12. Method according to claim 1 1 , characterized in that middle points (MP) are generated in the middle of each distance between said adjacent nodes (N).

13. Method according to claim 12, characterized in that to each middle point (MP) one boundary point (BP) is associated, wherein the respective boundary point (BP) is spaced apart from the associated middle point (MP) towards a center of the undeformed implant (1 ) in a radial direction by a distance (d₀), and wherein each middle and associated boundary point (MP, BP) is coupled by a nonlinear spring element (K) having a relatively low stiffness for all displacements lower than a measurement uncertainty in said points of the deformed implant and a relatively high stiffness for displacements above said measurement uncertainty.

14. Method according to claim 12, characterized in that the determined displacement (u) of each of said points (P) of the implant (1 ) in the deformed state with respect to the undeformed state is applied to the associated boundary point (PB).

15. Method according to one of the preceding claims, characterized in that for the constitutive model a linear elastic, isotropic hookean solid is used.

16. Computer program, wherein the computer program is designed to conduct, when executed on a computer, the method according to one of the preceding claims.