New approaches to catheter navigation for interventional radiology simulation

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For over 20 years, interventional methods have improved the outcomes of patients with cardiovascular disease. However, these procedures require an intricate combination of visual and tactile feedback and extensive training periods. In this paper, we
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  New Approaches to Catheter Navigation forInterventional Radiology Simulation C. Duriez 3 , S. Cotin 1 , 2 , J. Lenoir 1 , P. Neumann 1 , 2 1 Sim Group, CIMIT, Cambridge, MA 02139, USA 2 Harvard Medical School, Boston MA 02114, USA 3 INRIA-Futurs, Universit´e de Lille, 59655 Villeneuve d’Ascq, FRANCE Abstract.  For over 20 years, interventional methods have improved theoutcomes of patients with cardiovascular disease. However, these proce-dures require an intricate combination of visual and tactile feedback andextensive training periods. In this paper, we describe a series of novelapproaches that have lead to the development of a high-fidelity simu-lation system for interventional neuroradiology. In particular we focuson a new approach for real-time deformation of devices such as cathetersand guidewires during navigation inside complex vascular networks. Thisapproach combines a real-time incremental Finite Element Model, anoptimization strategy based on substructure decomposition, and a newmethod for handling collision response in situations where the number of contacts points is very large. We also briefly describe other aspects of thesimulation system, from patient-specific segmentation to the simulationof contrast agent propagation and fast volume rendering techniques forgenerating synthetic X-ray images in real-time. 1 Introduction Stroke, the clinical manifestation of cerebrovascular disease, is the third lead-ing cause of death in the United States. Each year, more than 700 , 000 strokesresult in over 200 , 000 deaths [2]. Ischemic strokes can now be treated usinginterventional neuroradiologic therapies which rely on the insertion and navi-gation of catheters and guidewires through a complex network of arteries torestore blood flow. Because the treatment is delivered directly within the closedbrain, using only image-based guidance, the dedicated skill of instrument navi-gation and the thorough understanding of vascular anatomy are critical to avoiddevastating complications which could result from poor visualization or poortechnique. This was the foundation for a recent decision by the FDA requiringall physicians who wish to treat carotid disease using catheter-based techniquesto train to proficiency before performing high-risk procedures in the cerebralcirculation. However, while most publications in the field of medical simulationhave addressed issues related to laparoscopic training, few aspects of interven-tional radiology simulation have been explored. Aside of modeling the soft tis-sue deformation of arteries, other complex problems need to be solved to enablereal-time, accurate simulation of such procedures. The one we address in this  paper concerns the simulation of non-linear deformations of wire-like structuresunder a large number of non-holonomic constraints, and the definition of suchconstraints to confine the catheter inside the vascular network. We also brieflymention other results in the areas of real-time fluid flow computation, real-timesynthetic X-ray rendering, and patient-specific segmentation. Although some of theses problems have been addressed in previous work [4,11,8,1], in particularin the areas of visualization and catheter modeling, many challenging problemsremain, especially when trying to reach a higher level of fidelity and accuracy inthe simulation, while maintaining real-time computation performance. 2 Modeling of wire-like structures To control the motion of a catheter or guidewire within the vascular network, thephysician can only push, pull or twist the proximal end of the device. Since suchdevices are constrained inside the patient’s vasculature, it is the combination of input forces and contact forces that allow them to be moved toward a target. Themain characteristics of wire-like structures that current models attempt to cap-ture include geometric non-linearities, high tensile strength and low resistanceto bending. We previously proposed a multibody dynamics model [6] where a setof rigid elements are connected using spherical joints [4], thus mimicking the ba-sic behavior of such devices. Another interesting approach to modeling wire-likestructures was introduced by Lenoir  et al.  [9]. In this model, a one-dimensionaldynamic spline model is used, providing a continuous representation. Differentconstraints can be defined to control the model, such as sliding through fixedlocations. Although real-time computation is possible, this model does not in-corporate torsional energy terms. A virtual catheter model, based on a linearelasticity, was introduced by Nowinski  et al.  [11], using a set of finite beam ele-ments. The choice of beams for the catheter model is natural since beam equa-tions account for cross-sectional area, cross-section moment of inertia, and polarmoment of inertia, allowing solid and hollow devices of various cross-sectionalgeometries and mechanical properties to be modeled. The main issue with sucha linear model is its inability at representing the large geometric non-linearitiesthat occur during navigation of the catheter or guidewire within the vascular net-work. Another approach also directly targeted at virtual catheter or guidewiremodeling was proposed by Alderliesten [1]. In this model, only bending energiesare computed, assuming no elongation and perfect torque control. The modelhas characteristics similar to a multi-body dynamics model but integrates morecomplex bending energies, as well as local springs for describing the intrinsiccurvature of the catheter. Although based on a more ad-hoc representation, agood level of accuracy is obtained using this model. The main drawbacks arehow collision response is handled during contact with the walls of the vessel,and computation times that are not compatible with real-time requirements.To improve the accuracy of previously proposed models, and handle geomet-ric non-linearities while maintaining real-time computation, we have developeda new mathematical representation based on three-dimensional beam theory,  combined with an incremental approach that allows for highly non-linear be-havior using a linear model, thus guaranteeing real-time performance. Furtheroptimizations based on substructure decomposition are introduced. Finally, anew method for correctly handling contact response in complex situations wherea large number of nodes are subject to non-holonomic constraints, is presented. 2.1 Incremental Finite Element Model To model the deformation of a catheter, guidewire, or any solid body whosegeometry and mechanical characteristics are similar to a wire, rod or beam,we use a representation based on three-dimensional beam theory [12], where theelementary stiffness matrix  K e  is a 12 × 12 symmetric matrix that relates angularand spacial positions of each end of a beam element to the forces and torquesapplied to them: K e  =  E l  A 0  12 I  z l 2 (1+ Φ y ) 0 0  12 I  y l 2 (1+ Φ z )  Symmetric 0 0 0  GJ E  0 0  − 6 I  y l (1+ Φ z )  0  (4+ Φ z ) I  y 1+ Φ z 0  6 I  z l (1+ Φ y )  0 0 0  (4+ Φ y ) I  z 1+ Φ y − A  0 0 0 0 0  A 0  − 12 I  z l 2 (1+ Φ y )  0 0 0  − 6 I  z l (1+ Φ y )  0  12 I  z   l 2 (1+ Φ y ) 0 0  − 12 I  y l 2 (1+ Φ z )  0  6 I  y l (1+ Φ z )  0 0 0  12 I  y l 2 (1+ Φ z ) 0 0 0  − GJ l  0 0 0 0 0  GJ E  0 0  6 I  y l (1+ Φ z )  0  (2 − Φ z ) I  y 1+ Φ z 0 0 0  6 I  y l (1+ Φ z )  0 (4+ Φ z ) I  y l (1+ Φ z ) 0  6 I  z l (1+ Φ y )  0 0 0  (2 − Φ y ) I  z 1+ Φ y 0  − 6 I  z l (1+ Φ y )  0 0 0  (4+ Φ y ) I  z 1+ Φ y  with  G  =  E  2 ∗ (1+ ν  )  where  E   is the Young’s modulus and  ν   is the Poisson’s ratio; A  is the cross-sectional area of the beam, and  l  its length;  I  y  and  I  z  are cross-section moments of inertia;  Φ y  and  Φ z  represent shear deformation parametersand are defined as  Φ y  =  12 EI  z GA sy l 2  and  Φ z  =  12 EI  y GA sz l 2  with  A sy  and  A sz  the sheararea in the the  y  and  z  directions.In order to determine the stiffness property of the complete structure, a com-mon reference frame must be established for all unassembled structural elementsso that all the displacements and their corresponding forces will be referred toa common (global) coordinate system. Since the stiffness matrix  K e  is initiallycalculated in local coordinates, oriented along the frame of first node of the beam(to minimize the computing effort), it is necessary to introduce transformationmatrices changing the frame of reference from a local to a global coordinatesystem. The first step in deriving such a transformation is to obtain a matrix  relationship between the element displacement  u  in the local coordinate sys-tem and the element displacement  u  in the global coordinates (see Fig 1). Thisrelationship is expressed by the matrix equation: u − u 0  =  Λu  (1)where  Λ  is a matrix of coefficients obtained from the direction cosines of an-gles between the local and global coordinate systems and  u 0  reflects the initialconfiguration of the beamAs the resulting virtual work must be independent of the coordinate system, itfollows that: δ  u T  f   − δ  u T  f   = 0 ⇔ δ  u T  ( Λ T  f   − f  ) = 0 ⇔ Λ T  f   − f   = 0 (2)By use of previous equations, the following element force-displacement equationis obtained in global coordinates: f   = [ Λ T  K e Λ ]      K e u  (3)  L o c a l  f r a m e u 0  Initial configuration Global frame(a)  L o c a l  f r a m e uuu  Deformed configuration   Global frame(b) Fig.1.  Initial (blue) and deformed (red) configurations. (a) The initial configurationof a catheter could be curved. Thus, for each beam, we store an initial displacement u 0  based on the position of the tip node in the local frame of the base node. (b)Actual deformations can always be measured locally in the local reference frame using u . Displacements from initial configuration to deformed configuration are measuredusing  u  in the global reference frame and by  u − u 0  in the local frame. To model a wire-like structure such as a catheter or guidewire, we serially linka series of beam elements (see Fig 1). As a result, for the entire structure de-scribing a catheter or guidewire, the global stiffness matrix  K  is computed bysumming the contributions of each element, thus leading to the following equi-librium equation: Ku  =  f   (4)  where  K  is a band matrix due to the serial structure of the model (one nodeis only shared by one or two elements) and  u  represents a column matrix of displacements corresponding to external forces  f  .The matrix  K  is singular unless some displacements are prescribed throughboundary conditions. Such boundary conditions are naturally specified by settingthe first node (base node) of the device model to a particular translation orrotation imposed by the user. The motion and deformation of the model is thenentirely determined by its set displacement at the base node and the forces  f  applied at different nodes as the result of contacts with the vessel wall (seesection 2.3). There is, however, one main drawback in using directly such amodel: it is linear and as such cannot represent the geometric non-linearitiesthat a typical wire-like object exhibits. Therefore we propose to update  Λ  foreach beam element, and at every time step, by using the solution obtained at theprevious time step. The new set of local stiffness matrices are then assembledin [ K ]. This approach is different than other methods such as the co-rotationalmethod [7] or other techniques that remove the rigid body transformation froma given configuration to remain in the linear domain [10]. Here, we do not usethe initial configuration as the reference state, but instead use the previouslycomputed solution. By controlling when each new  K  is going to be computed,we can ensure we remain in the linear domain for each incremental step, leadingto a correct, global deformation.When using this approach, however, the model could exhibit a inelastic be-havior, i.e. in the absence of forces or torques, the model would only return tothe previous state, not the reference configuration. We overcome this problemby computing an elastic force  f  e  on each beam, defined as: f  e  = − γ   K e  ( u − u 0 ) (5)with 0  < γ   ≤  1. This force is added to the external forces  f   in the globalcoordinate system ( f  e  =  Λ T  f  e ) before solving the linear system, and it can beshown that it acts as a damping force, where  γ   relates to the damping coefficientof the model.To simulate accurately a device such as a guidewire or catheter in the con-text of interventional neuroradiology, we need to discretize the device using alarge number (from 100 to 200) of beam elements. Although solving linear sys-tems with about 1,000 unknowns can be done in real-time using iterative meth-ods, when integrating non-holonomic constraints, real-time computation on asingle-processor workstation is no longer possible. To improve speed and handleaccurately collision response, we propose the following optimizations. 2.2 Optimization using Substructures Analysis The inverse matrix  K − 1 of the stiffness matrix gives a measure of the flexibil-ity of the model. For a force applied on one node, the induced displacementof all nodes can be obtained through the flexibility matrix. Using the flexibil-ity representation of the behavior of the wire-like object provides also a mean
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