Computer Networks and ISDN Systems 29 ( 19%‘) 167% 1683

Deformable models for laparoscopic surgery simulation

Konstantinos Moutsopoulos, Duncan Gillies * Department of Computing, Imperial College of Science Technology and Medicine, 1 SO, Queen’s Gate, London, UK SW7 282

Abstract

This paper introduces a method for handling deformation in interactive, real time computer graphics simulations which involve deformable objects and require a high degree of visual realism. Our proposal, the oirrua[ structure, is a “divide and conquer” approach, which combines a novel physical model with a geometric modelling utilizes the theory of elasticity and Newtonian mechanics, applied by a numerical method, the finite element method. Using different levels of structural resolution for global and local or collision with other objects. The geometric modelling uses the physical structure as a set of control points and produces a fine polygonal mesh generated by a B-spline surface. In order to demonstrate the method we have modelled the gallbladder as a spherical membrane containing liquid, in an interactive simulated environment for laporoscopic ch,olecystectomy. Q 1997 Published by Elsevier Science B.V.

Keywords: Deformable models; Simulation; Laparoscopic surgery; Finite element method

1. Introduction and procedures, and a scoring scheme to measure each individual’s performance.

The main objective of our research is to devise a The requirements for the simulation of physio- physical modelling technique which will form the logical processes, and surgical intervention training, basis for the development of a real-time, computer place considerable demands on the current level of simulator for teaching laparoscopic cholecystectomy. technology being used in the field of computer The simulator will help the trainee surgeon to be- graphics and simulated dynamics. Furthermore, the come familiar with the technical aspects of the oper- complex nature of the human soft tissue presents ation and acquire a high level of skill with the researchers with a very complex and challenging manipulation of the laparoscopic instruments. Unlike task. Geometric deformation techniques [I ,2,9,14], artificial, mechanical simulators (such as Semm’s are static modelling methods which have been used Pelvitrainer [l 11) which are relatively expensive, mainly by designers as sculpting tools. These meth- computer-based simulation couId ultimately provide ods are quite efficient but have limited interactive practice with a variety of different human anatomies capabilities and cannot animate non-rigid objects in

an interactive simulated environment without explicit user input. For that reason, physics-based methods have been developed, which make deformable mod-

* Corresponding author. Email: dfg@doc.ic.ac.uk els active, by simulating a variety of physical prop-

0169-7552/97,/$17.00 8 1997 Published by Elsevier Science B.V. All rights reserved. PII SOl69-7552(97)0008X-3

1676 K. Moutsopoulos, D. Gillies/ Computer Nehvorks and ISDN Systems 29 (1997) 1675-1683

erties in order to provide dynamic behavioural real- ism in addition to static visual realism.

Deformable models based on physical principles, such as those described by Terzopoulos et al. [22-251 are able to simulate a wide range of material be- haviours such as elastic, inelastic, plastic, thermoe- lastic, melting, and fractures. Furthermore, they have the ability to interact with any kind of interference, and can react in an automatic way to colliding objects and imposed constraints. These dynamically deformable objects are represented as mesh struc- tures made from nodal points or nodes which are used to transfer forces to and from the simulated object. The resulting discrete models are described by systems of differential equations which are solved numerically through time and space according to boundary and initial conditions. However, the high computational complexity of this approach makes it inapplicable for real-time applications. Other physi- cal modelling methods use different approaches such as vibration-mode analysis [20], or energy minimiza- tion constraints [31] which yield similar results at reduced complexity and generality.

Hybrid methods combine simple physical compo- nents, usually springs and hinges, with geometric deformation techniques, in order to provide a solu- tion with an optimum combination of efficiency and realism. For example, Miller [17] used a spring lattice and bi-cubic patches to model snake and worm locomotion, while Chen et al. [6] developed a biomechanical model of skeletal muscle using a coarse finite element structure and free form defor- mation lattices. Weil [31] modeled the appearance of a piece of cloth, which is suspended at certain con- straint points, as a coarse grid of catenaries while a finer mesh is created by fitting splines to the grid.

2. Laparoscopic cholecystectomy and training methods

Laparoscopic cholecystectomy is a relatively new operation that was developed to substitute open cholecystectomy and was first performed in France in 1987. Surgeons use endoscopic surgery techniques to manipulate the gallbladder and the surrounding vital structures. The operation is viewed on a video- screen with magnification, through a telescopic rod- lens system linked to a charge-coupled device (CCD)

camera, which is inserted through a 1Omm trocar (a feed-through like valve which prevents the CO, es- caping from the abdomen and allowing at the same time the microsurgical instruments to slide through it>. The surgical operation usually requires general anesthesia and is conducted inside the pneumoperi- toneum which is established with the insufflation of carbon dioxide (CO,) gas into the peritoneal cavity. Further tiny incisions are made for more trocars through which miniaturized, long-handled microsur- gical instruments are inserted. The surgeon operates by remote surgical manipulation of the instruments via the camera and the video monitor.

The operative steps, which include identification, isolation, and division of the cystic duct and artery, with subsequent removal of the gallbladder from its attachment to the liver, require a high degree of skill. Once free, the gallbladder is pulled through one of the small incisions to the exterior, the laparoscope and instruments are removed, and the incisions are closed with sutures and covered with small ban- dages. Patients have little pain after the operation, and hospital stays (l-2 days) and convalescence (1-2 weeks) are usually shorter than after open cholecystectomy. The benefits are both human and economic. Patients can leave the hospital and return to normal daily activities sooner, thereby reducing both direct health-care costs and indirect costs of lost worker productivity.

However, although the financial appeal is undeni- able, there are risks involved. For example, if some- thing goes wrong, it can be more difficult to fix the problem than in “open” surgery, Comparative re- sults (according to data from a consensus conference convened by the Office of Medical Applications of Research and the National Institutes of Diabetes and Digestive and Kidney Diseases of the National Insti- tutes of Health (USA) [191) between open and la- paroscopic cholecystectomy treatments reveal an in- crease on common bile duct injuries, although this rate is still sufficiently small to justify the use of laparoscopic cholecystectomy in the treatment of symptomatic gallstones. These problems are at- tributed to difficulties associated with the familiar- ization of completely new techniques such as: trocar access through the abdominal wall, loss of stereo- scopic vision, loss of tactile sense and manipulation of miniaturized long-handled instruments

K. Moutsopoulos, D. Gillies/ Computer Network and ISDN Systems 29 11997) 167.5-1683 1611

3. The dynamics of physically deformable models

The high computational cost involved with mod- elling deformable objects using physical methods is a result of the large stiffness, mass, and damping matrices found in the discrete form of Lagrange’s equation of motion (Eq. (1)) which is used by the majority of physical methods [4,20,30].

[M](d*U/dt*‘) + [D]{du,/dt} + [K](u) = {“f},

where u is a 3n X 1 vector of the (x, y,z) displace- ments of the n nodal points (of the structure of the discrete model), f is a 3n X I vector of the external forces applied to the nodal points, and M (the mass matrix), D (the damping matrix) and K (the stiffness matrix) are 3n by 3n matrices as described by Terzopoulos [2:3].

All these matrices contain non-redundant informa- tion which cannot be discarded in favour of a poly- nomial spline curves and surfaces in the way that some heuristic methods do. If non-linear treatment is required (and real objects exhibit non-linear elastic behaviour), then Eq. (1) has to be solved in a stepwise manner in which every step is small enough that we can assume linear behaviour. The size of each of these matrices must then be of the order of one thousand, in order to obtain useful results. If we now consider that these equations must be solved in a fraction of a second for a real-time application, it appears that the problem is beyond the reach of even a supercomputer.

4. Idealization of the problem

During a laparoscopic cholecystectomy the sur- geon’s actions and attention is focused on the gall- bladder and that is where we focus ours. Dynami- cally it is the most complex and challenging de- formable object in the peritoneal cavity because of its anatomy. The anatomy of the gallbladder, reveals a deformable object consisting of two interacting deformable media, the membrane and the contained fluid. A full. dynamic physical simulation of the gallbladder will involve a fluid-structure finite ele- ment vibrational analysis [7,8,15]. Fluid-structure vi-

brational analysis requires the solution of the eigen- value problem which is at least one order of magni- tude more expensive in terms of computations than elimination. The problem is exacerbated by the large number of nodal points required for the discretiza- tion of the volume of the bile, which is substantially higher than the number of nodal points required for modelling the surface structure of the membrane. Furthermore, because of the small tips of the micro- surgical instruments, their interactions with the vari- ous organs demand physical models with very re- fined physical structures (with hundreds or thousands of nodal points) in order to generate deformations with an acceptable level of detail.

Fortunately, the study of videos of laparoscopic cholecystectomies showed no noticeable vibrational behaviour of the gallbladder. Structural analysis says that if forces shake a structure at less than roughly one-third its lowest natural frequency of vibration, then the problem can be treated as static [7]. More rapid shaking makes inertial behaviour important, and the problem must be considered dynamic.

This means that we can concentrate on finding the shape of static equilibrium configurations of the gallbladder as it is being deformed by the actions of the surgeon. Even if the gallbladder had oscillatory strong tendencies this would not invalidate our ap- proach which would still give the correct shape at distinct instances (after the vibrations have ceased). So we treat the problem in a “pseudostatic” manner whereby the effect of the mass of the contained bile is simulated through the action of the hydrostatic forces it applies to the surrounding membrane. A similar approach has been used by Christie and Medland [12] in their “quasi-static” solution, whereby all dynamic terms have been neglected, and the deformation of the heart valves is given as a function of pressure ignoring pulsatile conditions. Gourret et al. [ 131 also used a “pseudostatic” model for the physical simulation of the hand which ignores all inertial terms and the configuration of their struc- ture results from statically applied loads from the hand muscles and the grasped objects.

The absence of oscillatory tendencies in our model means that the inertial behaviour of the contained liquid is negligible and that only the gravitational forces (expressed through the hydrostatic forces) are important. In the interest of brevity, we will not

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describe how the hydrostatic forces are calculated, but we can briefly say that all we need to know in order to calculate them, is the highest point of the structure and the specific weight of the bile. Compu- tationally, the number of calculations involved scales linearly to the number of nodes-triangles of the tessellated surface which contains the liquid. After our approximations, Eq. (1) becomes:

(2) Assuming that separate issues, like collision de-

tection and constraint satisfaction, are solved by standard methods [l&2], we can focus onto the algorithmic details of our hybrid modelling method called the virtual structure.

5. The concept of the virtual structure

5.1. Coarse finite element structure

A coarse finite element structure is the first stage in our modelling approach. It propagates stress across the model and is responsible for the global shape of the simulated object. This low resolution mesh model allows for fast responses, which is a critical feature for a real-time application. The number of nodal points that can be used in this stage is limited by the available hardware. The computational complexity for solving the stiffness matrix Eq. (2) is roughly proportional to the cube of the number of nodes, so that halving the number of nodes results in a speed-up of roughly one order of magnitude.

5.2. Structural extensions

A finite element structure interacts only through its nodes. However, models in interactive, physical simulations need to have near-continuous boundaries if partly penetrated objects and jerky reactions are to be avoided. Using a fine-mesh finite element struc- ture does not solve the problem, it just makes it less intense.

We introduce the second modelling stage which is a fine-mesh analysis around the area of interest, the area of interaction with the instruments. There are a number of different microsurgical tools and each one of them serves a different purpose. The distribution

Fig. I. IA micro-surgical tool interacting with the virtual structure.

of stress varies with each different instrument, and the modelling of the end effects in the area of interaction with the membrane requires a very de- tailed structure. This stage is again based on finite elements and is tightly linked to the collision detec- tion module, and a new physical structure is intro- duced as soon as the collision is detected. This new structure or structural extension is attached to the coarse structure of the first stage and exists for as long as the gallbladder is interacting with the instru- ment. The extra elements are placed on the surface of the object arranged around the area of interaction with the laparoscopic instrument (Fig. 1). The resolu- tion of this structure is sufficiently high so that the modelling detail in this region is such that the struc- ture resembles the deformed shape of the real gall- bladder. The new structure is built with boundary conditions set by the coarse-mesh finite element structure of the first stage.

Fig. 1 shows a part of the coarse finite element structure. PI, P2, and P3 are the nodes which belong to the initial structure and are nearest to the point PO of contact between our model and another body (the laparoscopic instrument). We introduce the following new nodes: Pl’, P2’, P3’ and PO, the dashed lines represent the extra elements which connect the new nodes with the old ones and between them. The nodes PI’, P2’, and P3 are placed in the middle of POPl, POP2, and POP3 respectively. The node PO is placed at the point of contact between the gallblad- der and the instrument. Other instruments may have a different shape and more points of contact, which should be treated as PO, although the extra elements must be carefully arranged to allow for possible formations of creases. This way we expand the structure dynamically for all points of contact be- tween our model and objects which collide with it.

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The structural expansion can transfer forces to and from the object through the extra elements PlPl’, P2P2’, and P3P3’. If the initial structure is very coarse or we need to have a closer view of the area near the structural expansion, then more nodes and elements can be generated by subdividing the newly introduced elements.

Our two-stage approach, separating local and global deformations is widely used in the theory of elasticity. It has. been known, for a long time, that a change in the distribution of the load on an end of an elastic body, without change of the resultant, alters the stress significantly only near the end. The change of distribution of a load is equivalent to the superpo- sition of a system of forces statically equivalent to zero force and zero couple. The expectation that such a system, applied to a small part of the surface of a body, would give rise to localized stress and strain only, was enunciated by Saint-Venant in 1855 and came to be known as Saint-Venant’s principle [28]. It accords with common experience in a variety of circumstances not confined to small strains in elastic materials obeying Hooke’s law. For example, the application of a small clamp to a length of thick rubber tube causes appreciable strain only in the immediate neighborhood of the clamp.

5.3. B-spline srrlface interpolation

In the final stage of our method, a B-spline surface [3] uses the nodes of the physical structure as control points in order to provide a smooth polygo- nal mesh, The node structure and the structural extensions form the “skeleton” of our model while the B-spline :surface is the “skin”. The control points of the B-spline surface are the finite element mesh nodes except around the area of the structural extensions. In our example shown in Fig. 1, the nodes Pl, P2, and P3 have no control of the skin. The control points of the B-spline surface in this area are the structural extension points Pl’, P2’, P3’ and PO.

B-spline surfaces have been combined with a variety of diff&ent physical models in several hybrid methods [10,26,30;28,16]. The reason for their popu- larity in hybrid methods is because they guarantee C2 continuity, have local control, and are quite efficient. Alternatively we can use free form defor-

mations with direct manipulation [14] in those areas which deform under a complex distribution of exter- nal forces. Such situations arise when the gallbladder is manipulated by the grasping forceps or the semm endocoagulation forceps. Each of these instruments deforms the gallbladder locally with a distinct shape. Free form deformations with direct manipulation can provide a mould for a pre-determined shape and simulate the effect of the grasp of the appropriate microsurgical instrument. However, the number of points in the lattice must be kept small because the inversion of the pseudoinverse can be computation- ally expensive. Simple free form deformations [21], or extended free form deformations 193 have been used in other hybrid methods [5,6], but they are unsuitable for our application because they cannot interpolate a given point.

We must emphasize here that the intersection tests for collision detections use the B-spline surface and not the physical structure, because otherwise there would be a visible difference between the interacting surface and the rendered surface, resulting in instru- ments partly penetrating the surface of the organ.

6. Real time simulation

The high complexity of simulating a physical model in real-time demands a large amount of com- putational resources even by today’s standards. In this section we examine how we can set the level of structural resolution of the two stages of a model based on the virtual structure in order to achieve solution times achievable by the available hardware.

6. I. Performance

In order to evaluate our method, we compare two models, one based on the standard finite element method and one based on the virtual structure. We now look at the computational savings that can be achieved using our staged modelling approach. If we assume that the first stage finite element structure has n (where n is of the order of 100) nodes or 3n degrees of freedom, then the cost for solving for the displacement vector is proportional to the cube of the number of degrees of freedom which is approxi- mately (3~2)~. (Using Gaussian elimination (not the

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most efficient but a very reliable method) the solu- tion time is proportional to N (N = number of degrees of freedom, b = semibandwidth), and b is roughly proportional to N.) In our comparisons we used a finite element model with 245 nodal points. For comparable levels of structural resolution the virtual structure model, would require a coarse struc- ture with 61 nodes and three structural extensions with a total of 12 nodes. This would need a total computational cost for solving all matrices of (36j3 + (183j3 = 5 X lo4 + 6 X lo6 G 6 X lo6 floating point operations. In comparison the computational cost for solving an initial stiffness matrix of 245 nodes and 735 degrees of freedom would be propor- tional to (735j3 P 4 X lo*. A high powered WOI sta- tion can simulate in real-time a structure with around 60 nodal points (response time for an average dis- placement in l/25 of a second).

The structure with the 245 nodes will have smaller elements, about half the size of the elements of the structure with the 61 nodes, and according to the finite element formulation the maximum allowable displacement (stemming from the small angle hy- pothesis in the finite element analysis) will be ap- proximately half of that for the structure with the fewer nodes. This adds to our comparison a factor of 2 in favor of our staged approach which uses larger elements and therefore can make larger (double) displacement steps. As we can see there is a compu- tational gain of approximately two orders of magni- tude in favor of the two-stage approach, not to mention the much reduced memory requirements which indirectly contribute to faster execution times, depending on hardware architecture.

6.2. Error

The two models give different results because they operate on different structures. The standard approach based on the finite element model with the uniform higher resolution structure has smaller over- all error, compared to the model based on the virtual structure. Our approach produces slightly higher lev- els of error because the second-stage stiffness matri- ces carry the discretization error of the first-stage coarse finite element structure which is higher than that of the standard finite element model with a uniform structure which has more nodal points.

However, the ratio of speed-up over error justifies the small compromises of our method, and in our tests, for an average speed-up of about 40 times the resulting mean-value of error for our two-stage ap- proach is of the order of 10% of the maximum displacement.

6.3. Stability

Unlike geometric models which require a single step solution method of simple parametric equations or the manipulation of small matrices, physical mod- els based on numerical methods require a large number of computations and many iterations whereby intermediate results are fed into the subsequent stages of the solution method.

High structural resolution is often considered as a very desirable attribute in every modelling approach mainly due to the closer resemblance with the simu- lated continuum. However, having a liner mesh re- duces discretization error but promotes truncation and rounding errors, so that a profusion of elements can make answers worse because of instability and consequently misconvergence. Our method, thanks to the small number of nodal points of the first stage structure of our models creates no problems to our matrix equation solver (we use Cholesky decomposi- tion to solve our matrix equations which is more

Fig. 2. Rendered image of the virtual structure handled by two instruments.

K. Moutso~oulos. D. Gillics /Computer Networks and ISDN Systems 29 11997) 1675-1683 1681

Fig. 3. Image of .a real gallbladder handled by two instruments.

efficient and stable than most matrix equation solv- ing techniques:).

6.4. Test results

Figs. 2 and 4 show rendered images of the virtual structure, while Figs. 3 and 5 show images of the gallbladder recorded from a laparoscopic cholecys- tectomy. Figs. 2 and 3, show corresponding configu- rations of a simulated and a real gallbladder as it is being manipulated by two surgical instruments, the atraumatic gra.sping forceps and a bipolar electrode with channel. The undeformed shape of the virtual

Fig. 4. Rendered image of the virtual structure handled by the atraumatic grasping forceps.

Fig. 5. Image of a real gallbladder handled by the atraumatic grasping forceps.

structure model is spherical, and is constrained by a horizontal supporting plane. The real gallbladder has a rather different undeformed shape and resembles the shape of a pear. In spite of the different unde- formed shapes, and the fact that boundary condi- tions, and constraints as well as instrument displace- ments are quite different in the two cases, we can see that in both occasions the deformation patterns are similar.In Figs. 4 and 5, we can see the deformed shape of a simulated and a real gallbladder respec- tively, handled by the atraumatic grasping forceps, which is the instrument used most during the opera- tion. Again, although the undeformed shapes and the boundary conditions and constraints are very differ- ent, the deformed area of the membrane around the instrument looks similar in both images.

7. Future work

In developing a model for the simulation of la- paroscopic cholecystectomy it is essential to offer a robust method capable of integrating many different objects together. Currently we are working to extend our approach for condensed modelling on a higher le-/el. The virtual structures which describe separate deformable objects are condensed and joined to- gether in a graupstnrcture. This approach resembles a similar technique called substructuring [7] which is used in some finite element applications that deal with structures which can be “broken” into distinct

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parts. The groupstructure is responsible for transfer- ring stresses between objects. For example, if during the operation the surgeon pulls the gallbladder so that it is lifted from its supporting “ground”, or it is shifted across it, then its behaviour and shape are controlled by the reaction of the cystic duct which in turn is connected to other organs.

At this stage, each virtual structure condenses into a single element, all of whose nodes are on interface boundaries. Other virtual structures are similarly treated. The condensed virtual structures are assem- bled into a single stiffness matrix (K groupstructure). At this stage, all groupstructure degrees of freedom (D groupstructure) lie on the interfaces between the virtual structures. In this stage of structural analysis we can “afford” to solve Lagrange’s equation of motion (Eq. (1)) to its full extent and include inertial terms which are necessary in order to simulate each individual organ’s rigid body motion.

8. Conclusions

The price for simulating physics with numerical methods has always been quite high, prohibiting the use of physical modelling in real-time systems. Our proposal is a robust, scaleable solution which man- ages to reduce substantially the computational com- plexity of the problem at the price of a controlled level of error. More importantly, the computational requirements of our method for a non-trivial resolu- tion structure model are within the reach of a medium priced workstation making it a feasible solution, and an attractive alternative to a mechanical simulator for laparoscopy. Finally, the dynamic scheme of struc- tural extensions offers infinite boundary resolution, essential for detailed and precise interactions.

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Konstantinos Moutsopoulos graduated from the Nationai Techni- cal University of Athens with a degree in Computer Science in 1990. He subsequently obtained the Ph.D. Degree in the ama of bio-medical modelling for computer simulation from Imperial College. He is currently doing national service in Greece, after which he plans to return to his research interests in graphics and

Duncan Gillies graduated from Cam- bridge University with a degree in Engi- neering Science in 1971. He subse- quently obtained the M.Sc. Degree in Computing and a Ph.D. in the area of artificial intelligence from London Uni- versity. After teaching for six years at the Polytechnic of the South Bank, he moved to the Department of Computing in Imperial College where he is now a Reader. His research interests are in graphics and computer vision.