WO2004081899A1

WO2004081899A1 - Method of generating a computer model of a deformable object

Info

Publication number: WO2004081899A1
Application number: PCT/AU2004/000307
Authority: WO
Inventors: Ian Brown; David Healy; Zorana Mayooran; Cory Seligman
Original assignee: Monash University
Priority date: 2003-03-13
Filing date: 2004-03-15
Publication date: 2004-09-23
Also published as: AU2003901155A0

Abstract

The present invention provides methods for modelling the deformation of a computer generated object in response to a collision in a virtual environment. The methods are applicable to many types of simulation, but are particularly applicable to surgical simulations. The methods allow simulations to run in approximately real time without the need for a particularly fast host computer.

Description

METHOD OF GENERATING A COMPUTER MODEL OFA DEFORMABLE

OBJECT

FIELD OF THE INVENTION

The present invention relates to computer generated models of physical objects. More specifically the invention provides an improved method of generating a deformable model that is able to approximate real-time performance on a computer.

BACKGROUND

Computers have become an indispensable tool in modelling and simulation. As computational power increases, users and applications are demanding ever increasing levels of realism in these domains. This trend is particularly apparent in computer graphics where more sophisticated geometric shapes and physical objects are being modelled in the context of complex physical environments

In particular, the ability to model and manipulate deformable objects is essential to many applications. Approaches for modelling object deformation range from non-physical methods - where individual or groups of control points or shape parameters are manually adjusted for shape editing and design - to methods based on continuum mechanics - which account for the effects of material properties, external forces, and environmental constraints on object deformation.

Deformable object modelling has been studied in computer graphics for more than two decades, across a range of applications. In computer-aided design and computer drawing applications, deformable models are used to create and edit complex curves, surfaces, and solids. Computer aided apparel design uses deformable models to simulate fabric draping and folding. Deformable models have been used in animation and computer graphics, particularly for the animation of clothing, facial expression, and human or animal characters. Finally, surgical simulation and training systems demand both real-time and physically realistic modelling of complex, non-linear, deformable tissues.

A constraint in the development of useful computer-based deformable models has always been the limitations imposed by the computer hardware of the day. This is especially problematic where the model is a dynamic model that is to be represented in real-time.

In the prior art, early works were restricted to pure geometric deformations, mostly carried out directly on parametric surface models. However, with physically based modelling paradigms, more realistic models arose. The prior art describes two major types of physical models. One is based on mass-spring system and uses finite difference schemes to solve dynamics. The other applies Finite Element Method (FEM) for the solution of the inherent partial differential equations. These two methods have common aspect, they all minimise the energy function of the model subject to deformation.

Mass-Spring Models

Non-physical methods for modelling deformation are limited by the expertise and patience of the user. Deformations must be explicitly specified and the system has no knowledge about the nature of the objects being manipulated. Using these tools alone, modelling an object as complex as the human face, for example, is a daunting task. As desktop computing power and graphics capabilities increased during the 1980's, the graphics community began exploring physically based methods for animation and modelling. These methods use physical principles and computational power for realistic simulation of complex physical processes that would be difficult or impossible to model with purely geometric techniques.

Mass-spring systems are one physically based technique that has been used widely for modelling deformable objects. An object is modelled as a collection of point masses connected by springs in a lattice structure. The spring forces are often linear, but nonlinear springs can be used to model tissues such as human skin that exhibit inelastic behaviour. In a dynamic system, Newton's Second Law governs the motion of a single mass point in the lattice. The equations of motion for the entire system are assembled from the motions of all of the mass points in the lattice.

The behaviour of the spring-mass model is highly dependent on the topology of its 3D lattice. These models suffer from stability problems which must be accounted for by constraint equations where possible, which add to the complexity of the system, leading to problems in real-time performance.

Finite Element Models

Mass-spring models start with a discrete object model. More accurate physical models treat deformable objects as a continuum: solid bodies with mass and energies distributed throughout. In making this distinction, it is important to separate the model from the method used to solve it. Models can be discrete or continuous but the computational methods used for solving the models in computer simulations are ultimately discrete. In the analysis of dynamic systems, numerical integration techniques approximate the system at discrete time steps. Furthermore, even a continuum model must be parameterised by a finite state vector. For deformable object modelling, this state vector often comprises the positions and velocities of representative points within the material. However, unlike the discrete mass-spring models, continuum models are derived from equations of continuum mechanics.

The full continuum model of a deformable object considers the equilibrium of a general body acted on by external forces. The object deformation is a function of these acting forces and the object's material properties. The object reaches equilibrium when its potential energy is at a minimum.

The work done by applied loads is due to three sources: concentrated loads applied at discreet points, loads distributed over the body, such as gravitational forces, and loads distributed over the surface of the object, such as pressure forces. The system potential energy reaches a minimum when the derivative of II with respect to the material displacement function is zero. This approach leads to a continuous differential equilibrium equation that must be solved for the material displacement.

Because it is not always possible to find a closed-form analytic solution of these differential equations, a number of numerical methods are used to approximate the object deformation. As discussed previously, mass-spring methods approximate the object as a finite mesh of points and discretize the equilibrium equation at the mesh points. Finite element methods, FEM, divide the object into a set of elements and approximate the continuous equilibrium equation over each element.

FEM is used to find an approximation for a continuous function that satisfies some equilibrium expression such as the deformation equilibrium expression described above. In FEM, the continuum, or object, is divided into elements joined at discrete node points. A function that solves the equilibrium equation is found for each element. The solution is subject to constraints at the node points and the element boundaries so that continuity between the elements is achieved. Unlike mass-spring methods, where the equilibrium equation is discretized and solved at finite mass points, in FEM, the system is discretized by representing the desired function within each element as a finite sum of element-specific interpolation, or shape, functions.

In the FEM techniques, the finite element equations of motion describe the complete (rigid and non-rigid) motion of an object in a single system of equations. Typically these equations are very complex and their solution computationally expensive; as a result, it is less compatible with real-time applications. In the past few years, some global deformation models have been proposed for interactive animation by restricting deformations to the combination of a given set of vibration modes or of a specific class of global deformations. However, such restrictions on the behaviour considerably affect the realism of the animation. To reduce computation time, other approaches appeared, allowing real-time simulation of elastic bodies by preinverting the stiffness matrix or applying superposition theorem to use a linear combination of the precomputed displacement of each external nodes. Unfortunately, these methods used quasi-static models, thus losing the dynamic behaviour.

A common approach to Finite Element modelling has- been to appropriately subdivide the model into elements and perform a pre-processing step to derive the mass and stiffness matrices of the system. This pre-processing step can take hours to complete and is a major disadvantage when the topology of the model changes mid-way through the simulation, such as when cutting is performed. As soon as a model is changed, new system parameters must be calculated to continue with the simulation, leading to a break-down in the realtime performance.

In light of the above it will be appreciated that a significant of problem of both mass-spring and FEM is that these approaches require solving a number of complex equations in order to generate a new image. Accordingly, when a virtual force is applied to a mass-spring or finite element model, there will be a delay in the generation of the deformed model as a result of the time taken for the computer to process the large number of instructions required. Where the model is represented graphically, this delay results in a decreased frame rate. Accordingly, the real-time representation of the deforming model is simply not achievable.

Minimally Invasive Surgery. While the real-time implementation problem applies to many applications for computer modelling, one area that has received considerable attention is in the modelling of human anatomy for the purposes of implementing surgical simulations. Flight simulators have been used to train pilots in complete safety for fifty years. By contrast, surgical simulators are now only being used to train surgeons in techniques used across a range of procedures. While traditional open surgery requires a great deal of skill to master, minimally invasive surgical techniques are even more difficult to learn. Surgery in any form is a stress on the body. With traditional open surgery, the incision itself imposes additional risk of infection, trauma, and recovery time upon a patient, beyond that imposed by the condition giving rise to need for surgery. Advances in miniaturization of tools and implements, as well as video systems to view the inside of patients, have given rise to minimally invasive surgical techniques. In this epoch of surgery, small incisions are made in and the surgical implement is inserted into a vein, artery, or space between tissue. Tactile sensation imparted to the surgeon by the implement as it is inserted into the patient and visual display images from x-ray, television monitor or other system allowing an internal view of the body are then used to position the implement and complete the necessary task of the operation, be it repair of an organ such as the heart, removal of blocking tissue, the placement of a pacemaker lead, endoscopic surgery or other procedure. Due to the minimally invasive nature of this type of surgery, operations may be performed in a very brief period of time with less anaesthesia and hospitalization. Given the nature of this type of operating procedure there are a number of special considerations. Failure to properly orient the implement within the patient, or properly recognize the tissue through which the implement is passing, may result in the implement puncturing or rupturing a vein, artery, organ, or other internal tissue structure. Such an accident will almost certainly result in subjecting the patient to immediate emergency invasive surgery, morbidity, and perhaps death.

It is difficult for inexperienced surgeons to obtain a desired level of familiarity and competence leading to requisite certifications. Additionally, there are procedures that are performed infrequently. Without performing the operation repeatedly, the practitioner has no way of maintaining the high degree of skill that is obtained only through regular performance of the procedure. Further, it is not possible to implement new methods, operations, and procedures except on live patients. Accordingly, there is a need for an effective means to simulate real-life operations, so as to develop and maintain skill, and implement new techniques. Minimally invasive surgical simulators based on computer technology have been known in the art for a number of years. The vast majority are not able to run in real-time, so that there is some lag between what the trainee does and what he sees on the computer screen. This is clearly less than optimal. Some simulators have the ability to run in real-time, but require vast processing power or parallel processing techniques. Such equipment is expensive and is not therefore widely available.

Prior art primarily models 3D anatomical environments as deformable meshes using mass-spring models and/or finite element models. Mass-spring models have certain stability problems, especially with the cutting action, which may be solved with constraint equations. Finite element modelling cannot simulate cutting of anatomical objects in real-time since this changes mesh topology and requires further pre-processing time while the simulation is running. Both of these methods have not achieved adequate real-time performance.

In light of the above, there is a clear need for a surgical simulator that is able to run in real-time on a computer that is inexpensive and easily available. To this end, the present invention overcomes or alleviates a problem of the prior art by providing an improved method for modelling the dynamics of a deformable object. The method provides the ability to simulate deformation of an object in approximately real-time with an acceptable degree of realism.

The discussion of documents, acts, materials, devices, articles and the like is included in this specification solely for the purpose of providing a context for the present invention. It is not suggested or represented that any or all of these matters formed part of the prior art base or were common general knowledge in the field relevant to the present invention as it existed in Australia before the priority date of each claim of this application. BRIEF DESCRIPTION OF THE FIGURES

Figure 1 shows a computer model of a left uterine tube, ovarian ligament and ovary, segmented into N parts, each with its own x, y, z coordinate system. This representation is of the organs in an equilibrium state.

Figure 2 shows a cantilever mass-spring model of a segmented uterine tube, m = mass of each segment alpha = deflection of each segment axis from the horizontal beta = deflection of each segment axis from its resting state when a load is applied.

Figure 3 shows a cantilever mass-spring system.

F = applied force k = spring stiffness coefficient

I = length of the beam theta = deflection of the beam from its resting state when a load is applied d = distance by which the load end of the link is displaced when a load is applied.

Figure 4 shows a simulation flowchart

Figure 5 shows a minimum energy algorithm. The figure demonstrates the resting position of a fallopian tube, and the position after it is grasped and lifted with a virtual instrument. The broken lines represent the movement of the origin of the co-ordinate system for each fragment from the initial position to the new position.

Figure 6 shows an iterative process start in the minimum energy algorithm

Figure 7 shows completion of one cycle of an iterative process in the minimum energy algorithm. Figure 8 shows a solution of the minimum energy algorithm. Figure 9 shows an intersection line and bounding cylinder of a virtual laparoscopic instrument .

Figure 9 shows an intersection line and bounding cylinder of a virtual laparoscopic instrument.

Figure 10 shows rotation of a linked segment in response to intersection with a virtual instrument. This diagram is not to scale. B = pivoted 3D body

I = intersection line representing the rigid instrument

I' = position of I in the following frame

Pi = pivot point of I

P₂ = pivot point of the first segment above B P₃ = first point of contact between I and the geometry of B (collision point) θι = angle traversed by I from the point of collision to the next frame θ₂ = angle I should move B by around P₂

SUMMARY OF THE INVENTION In a first aspect, the present invention provides a method for modelling the deformation of a first object in response to collision with a second object, the method including the steps of

providing a segmented model of the first object, providing an algorithm governing the deformation of the first object in response to collision with the second object

wherein the algorithm provides that the main parameter used to calculate the shape and/or position of the first object after collision is the spatial displacement of the second object.

The present invention further provides a computer executable program embodying a method described herein. The present invention also provides a computer including a computer executable program described herein.

The present invention also provides a method for training surgeons including a computer as described herein.

In another aspect the present invention provides a method for modelling the deformation of a first object in response to collision with a second object, the method including the steps of

wherein the algorithm does not consider a parameter in an equation of motion.

Preferably, the algorithm does not consider a parameter selected from the group including velocity or acceleration.

DETAILED DESCRIPTION OF THE INVENTION In a first aspect, the present invention provides a method for modelling the deformation of a first object in response to collision with a second object, the method including the steps of

wherein the algorithm provides that the main parameter used to calculate the shape and/or position of the first object after collision is the spatial displacement of the second object. Many virtual reality situations require two objects to appear to physically interact ("collide"). It has previously been thought that for adequate realism to be achieved, sophisticated equations of motion for each segment of the model must be implemented. While the use of such equations no doubt improves reality of motion, the result is that simulations run very slowly and with frame rates that do not give acceptable real-time performance. This is a particular problem for simulations that must run in real-time such as surgical simulators. In order to maintain adequate realism a trainee surgeon must get visual and haptic feedback simultaneously. The Applicants have discovered an effective compromise between realism of the simulation and speed of the simulation. This has been achieved by considering the spatial displacement of the object exerting the virtual force as the main parameter in determining the resultant deformation of the deformable object.

The inventors have discovered that a model having acceptable realism is achievable by us ing displacement only. There is no need to consider equations of motion, includ ing the parameters of mass and velocity. While the simulation is not perfect it is acceptable for many applications in virtual reality settings. The approach used in the present invention is therefore a clear departure from the prior art methods using a mass-spring or finite element methodology.

As used herein the term "spatial displacement" is intended to include the distance moved by a virtual object within a two-dimensional virtual plane, or within a three-dimensional virtual space. In the context of a surgical simulator, the first object is a virtual surgical instrument, and the second object is a deformable organ. The virtual surgical instrument moves (or is spatially displaced) in response to the physical manipulation of a dummy instrument by the operator. The method of the invention uses this displacement data from the instrument to define the new shape and position of the virtual organ after collision with the virtual instrument. Of course, the methods of the present invention are not limited to use in surgical simulations, and are equally applicable to any dynamic modelling problem requiring an increase in performance. The algorithm used to solve the shape and position of the second object after collision with the first object can include any suitable algorithm known in the art. It should be understood that the present invention is not limited to the use of any particular algorithm described herein.

As used herein the term "main parameter" means the parameter that has the single most important influence on the shape and/or position of the second object after collision. Also included is the situation where displacement is the only parameter considered.

Preferably each segment of the first object has an independent system of coordinates. Thus, each segment in the object has its own point of origin from which other boundaries of the segment are defined. This has the advantage of quickly determining at which point along the object a collision occurred as well as moving each segment independently of the others. Transformation of each segment will therefore produce a deformable shape, rather than a unique transformation of every point in the object.

The first object may be any two- or three-dimensional shape. Preferably the first object is substantially linear. While the object may be segmented in any way suitable for modelling the object, where the object is linear it is preferable that the object is segmented laterally.

Preferably the origin of the co-ordinate system is positioned at the upper end of the segment, in the geometric centre of the upper face. For example, if the segment is a cylinder that is fixed at one end and "hanging" under the influence of a virtual gravitational force, then the origin of the co-ordinate system for that segment is at the end distal to the virtual gravitational source. More particularly, the origin is at the centre of the circular plane at the end of the upper face of the cylinder.

In one embodiment of the invention the first object is fixed at one end. While the first object is deformable, it is contemplated that it may be fixed at one or more point. Using the example of a surgical simulator, the first object may be a fallopian tube that is flexible along its length, but is fixed at the point where the fallopian tube attaches to the uterine wall. Lifting the end of the fallopian tube with an instrument (ie colliding with the second object) will result in deformation of the fallopian tube as a result of movement of the individual segments. However the first segment of fallopian tube adjacent to the uterine wall will be restricted in movement due to its attachment at one end. It is contemplated that this restriction could be removed from the model by "cutting" the tube at the fixed end. This would allow the entire tube to be freely manipulated, and even removed from the surgical field if necessary. The present invention further contemplates that the first object may be fixed at both ends, or across an entire face, or faces of the object.

Preferably, at least two segments of the first object are connected in a cantilevered arrangement. In a general cantilever mass-spring system, F= -k*d, where F is an applied force, k is the spring stiffness coefficient, and d is the distance by which the load end of the link is displaced. Figure 1 shows a cantilever mass-spring system for a segment of a tube. In this tube system k is determined as follows:

kd = mg

M sin θ =mg

Preferably the spring stiffness for each segment is defined by the equation

N

k - ^ω

/,. sin θ, In a model of a segmented tube as shown in Figure 1 , the spring stiffness for each segment (i) can be found by the above equation.

The second object will generally be a substantially rigid object and is preferably substantially linear.

Movement and/or deformation of the first and/or second objects is not limited in any way within the virtual environment. Preferably, the spatial displacement of a segment of the second object is limited to rotation about a single point.

Preferably the angle of rotation by which the segment needs to move is found by trigonometry.

Preferably, the angle of rotation by which the segment moves is found by the equation:

^f c θ₂ = sin^-1 2--1 smf + γ

\ a )

The derivation of the above equation is as follows:

/^^ si ^•n— ^θl

l₂ - 2r₂ sm-~- _z

, ~ l₂ cos or

. θ. . θ₂ r_j sin — = r₂ sm-r- cosαr

from the following trig identities

sin(x + y) = sin x cos y + cos x sin y

cos(x + y) = cos x cos y + sin x sin y

r_! sin- r₂ cos Hi M⁺'^ιsfaHM) sm-

tef

0, c = sm

δ = r₂ cos

I 2 y

a = r₂ sm p + -~

c = ό cos— + a sin ^■ ^^²- sin — 0,

2

— = — cos-² sm— + sm — α α 2 2 2

substituting for - = t .γ where γ -\ β + -~ b 2

c _ cos^ sinff₂ π cos ft_; a siny 2 1,2 2 c _ cos sin ft₂ + sin^ (l - cosft₂) a 2saιγ

2— sinf = si + cos sin#₂ -sin^ cos#₂ a

2--1 s .γ = s (θ₂ ~γ) a J

-s Λ

6^ = sin-¹] j 2 — 1 smγ + γ

The pronumerals used in the above derivation are explained in reference to Figure 10.

Preferably, the algorithm provides that deformation of the first object is effected by displacement of a segment adjacent to that which has been collided with by the second object.

It is contemplated that the method of the present invention is embodied in the form of a computer executable program. The skilled person will be able to implement the methods described herein in one of a number of many programming languages known in the art. Such languages include, but are not limited to Fortran, Pascal, Ada, Cobol, C, C++, Eiffel, Visual C++, Visual Basic or any derivative of these. The program may be stored in a volatile form (for example, random access memory) or in a more permanent form such as a magnetic storage device (such as a hard drive) or on a CD-ROM.

Preferably the first and/or second object are modelled using a software package selected from the group including Wavefront/Alias Maya, 3DstudioMax or any other software package suitable for modelling 3D organic/curved shapes known to the skilled artisan. The present invention also provides a computer including a computer executable program described herein. Preferably the computer has a central processing unit having a central processing unit with a clock speed higher than approximately 200 MHz. More preferably the clock speed is higher than about 100 MHz. The skilled person will understand that the selection of central processing unit will depend on the complexity of the simulation to be implemented. Preferably the central processing unit is selected from the group including Pentium 1 , Pentium 2, Pentium 3, Pentium 4, Celeron, MIPS RISC R10000 or better.

In a preferred embodiment of the invention, the computer executable program can run in approximately real-time on a computer.

The realism of the visual component of a virtual reality computer simulation is reliant on the ability of the modelling method to refresh the visual display at a sufficiently high number of frames per second. Preferably the method provides frame rates of at least 24 frames per second. More preferably the method provides frame rates of at least 30 frames per second.

Preferably the method is a component of a virtual reality system. Virtual reality systems based on computer technology are well known in the art. Such systems generally include a central processing unit containing all computer hardware and software required to effect the simulation. Also included are input devices such as motion sensors and output devices such as a visual display unit.

Preferably the virtually reality system is used for training in surgical techniques. The virtual reality systems of the present invention may be used in the training of a range of surgical techniques. However, in preferred embodiments of the invention the virtual reality systems may be used in the training of gynaecological surgery, gall bladder surgery, neurosurgery, thoracic surgery, eye surgery, and orthopaedic surgery. It is contemplated that the methods and/or virtual reality systems described herein may include other features such as a hierarchical segmented implementation of visual and tactile features including interactive touch whereby virtual objects can be felt when touched with virtual instruments.

The methods and/or virtual reality systems described herein may also include anatomical structures having pathological features that can be seen in the visual display unit, and felt via haptic feedback from the instruments. It is also anticipated that interactive movement of different anatomical organs could be implemented by segmenting the anatomical field into anatomical objects, each with different dynamic attributes.

The methods and/or virtual reality systems described herein may also incorporate interactive movement of different parts of an anatomical object by allowing a virtual instrument to interact with a segment of the anatomical object at the point of contact and then allowing neighbouring segments to move according to prescribed rules.

In another embodiment of the present invention the methods and/or virtual reality systems described herein may further include interactive touch - haptic feedback of different part of the anatomical object is achieved by allowing the virtual instrument to interact with a segment of the anatomical object at the point of contact and allowing the model to define the appropriate haptic feedback vector at that point.

The methods and/or virtual reality systems described herein may also include tissue pathology attributes applied to a group of segments of each anatomical object, with pathology providing an input to both the visual and tactile models as described above.

A highly preferred form of the invention provides a virtual reality system representing the female pelvic anatomical field (as viewed by an endoscopic camera during surgery). This complex anatomical field, consists of a number of organs and structures, each with different visual, movement, tactile and disease characteristics. The simulation represents this complex field by segmenting the anatomy into anatomical objects (organs and other structures), and sub objects or segments of organs and structures. Visual, movement, tactile and disease properties are then attributed to a segment of the anatomy as represented by that segment of the model. Properties of model segment can also be derived from the position or movement of adjacent segments.

As the segment of anatomy is moved (and felt) by endoscopic instruments the anatomical model will only need to move (or be felt) when the haptic instruments move into the region of the model in which that segment lies. Thus when an instrument approaches a particular segment of the model the movement and tactile attributes of that segment will become active. It is unnecessary for the whole organ or the whole model to move or be felt. Movement can however be conveyed from one segment to an adjacent segment if the rules enabling movement in that segment allow this to happen.

In relation to motion every segment has its bounding volume used to test against intersections with an instrument and other segments. Motion of objects is a superposition of rigid and deformable models. Rigid motion refers to the global motion of objects such as translation and rotation. In the case of tubular structures such as the uterine tubes and ligaments, these objects are subdivided into rigid volumetric segments. When an instrument touches/intersects a particular segment, all other segments belonging to the same object move according to a pre-defined physical/mathematical model. Therefore, movement is restricted to an object (e.g. left uterine tube, uterus, right ovary, etc.) rather than the entire anatomical structure (e.g. reproductive organs as a single mesh).

A model of deformable motion may be constructed such that the surface of an object is a group of points/particles. Surface deformation resulting from interactions with an instrument such as indentation and pulling is localised. This means that effects of a deformation propagate from the point of contact with an instrument to all neighbouring points lying within a pre-defined spherical volume determined by the force of contact. Hence, deformation may only affect a part of an object, rather than its entire mesh.

Preferably the virtual reality system described herein is used in the training of a surgical technique. Preferably the surgical technique is minimally invasive surgery. In a particularly preferred form of the invention the surgical technique is endoscopic surgery.

Preferably the first object is a model of an anatomical feature of the human body. More preferably the first object is an organ. Most preferably the organ is selected from the group including fallopian tube, uterus, ovary and ovarian ligament.

Preferably the second object is a model of a surgical instrument. The instrument may be any of the instruments known in the art of surgery and may be selected from the group including forceps, clamp, scissors, retractor, cauterisation device, endoscope laparoscopic probe, and haemostatic clip applicator.

The present invention also provides a method for training surgeons including a method and/or computer and/or virtual reality system described herein. The method for training may include other features well known in the art of teaching such as training manuals, lecture notes, practical demonstrations and the like.

wherein the algorithm does not consider a parameter in an equation of motion. Preferably, the algorithm does not consider a parameter selected from the group including velocity or acceleration. Applicants have found that an acceptable degree of realism is also provided by algorithms that do not consider a parameter of an equation of motion. This simplification of the calculations required to be performed by the central processing unit of a computer including a computer executable program embodying this method results in acceptable frame rates and acceptable realism.

The present invention will be further described with reference to the following non-limiting examples.

EXAMPLES

EXAMPLE 1 : DYNAMICS OF A FIXED-BASE SEGMENTED BODY

It is assumed that any fixed-base body can be represented as a hierarchy of segments, each with its own pivot point, forming a cantilever spring-mass system. Given that a model of the body to be represented is available in its resting state, the body can be segmented and spring stiffness for each segment found. A uterine tube model will be used throughout this document to illustrate the main concepts used. The methods for interactive motion of the tube to be described can be used on any fixed-base linked system.

System at equilibrium and its parameters For an equilibrium model of the tube shown in Figure 1 , the tube is firstly segmented into N parts. Each segment is assigned its own coordinate system, the origin of which is positioned on the top end of each segment, mid-way across.

A line connecting the top 2 pivot points forms an angle with the horizontal (αo), which is the initial angle for the cantilever mass-spring system (Figure 2). It is assumed that the total mass of the body, M is known. Thus, each segment has a mass m = M / N, provided the segments are equal. Otherwise, the mass of each link would be proportional to its size.

In a general cantilever spring-mass system, F = -k*d, where F is an applied force, k is the spring stiffness coefficient, and d is the distance by which the load end of the link is displaced (Figure 3).

F_S = F kd = mg kl sin θ = mg mg k = -

Isiaθ

In the tube system shown above, the spring stiffness for each segment (i) can be found as follows:

, ^m i § k; = j=i+l

L sin θ_;

The simulation begins after initialisation of the system, and is represented in Figure 4.

Interaction of a segmented model with an instrument: grasping

Each of the grasping surfaces of a laparoscopic instrument's jaws is represented by an intersection line. A segment is thus grasped when both of these lines intersect with the segment's geometry. The first time a segment is grasped, positions of its pivot point and axis are fixed with respect to the instrument until the segment is released. Once the grasped segment has been moved by an instrument, all other segments in the tube will change positions as well. Two algorithms are used to distribute the segments above and below the one intersected.

New positions of the segments above the one intersected are found with respect to their equilibrium position. These segments need to be distributed exactly between 2 fixed points: one is the fixed end of the segmented tube and the other is the coordinate system origin of the grasped segment. It is assumed that when moved by an external force, each segment would be attracted to its equilibrium position since this is the state in which the potential energy of the segmented system is at its minimum. The minimum energy algorithm is described in Figure 5.

After the first estimate, the pivot point of the first segment in the system is compared to its fixed point. If the error in position is > 0.001 , an iterative process commences and continues until the error reduces as specified. Firstly, the position of the first segment's pivot is translated to its fixed point (Figure 6).

The following segments' pivot points are found with respect to their calculated positions (Figure 7).

The same process is repeated from one fixed point to the next until a satisfactory solution is found (Figure 8).

Segments below the one grasped are distributed according to the cantilever mass-spring model described earlier. Their starting inclination is the axis of the grasped segment.

Interaction of a segmented model with an instrument: shaft manipulation

The segmented model can be moved by an instrument such as a long, pivoted laparoscopic instrument at interactive rates. Both the instrument and the segments have bounding volumes, cylinder and spheres respectively, encompassing their geometry to be used in intersection tests. The instrument also uses a line segment for intersection testing, which is directed down its longitudinal axis, from the instrument pivot point to the instrument tip (Figure 9).

When an instrument collides with the virtual segmented uterine tube, the intersection test returns the segment whose geometry was intersected by the line representing the instrument. If confirmed by the collision algorithm, this segment should change position in the next frame in response to movement by the instrument. The instrument is represented by an intersection line in Figure 9 for clarity. It is assumed that the segment will move with respect to the pivot point of the segment above it, and not its own pivot point. Outline of the segment's geometry can be of any shape, as shown in Figure 10. Based on the position and rotation of the instrument, the angle of rotation by which the segment needs to move is found in the following section. The angle and axis of rotation are then used to create a quaternion by which the intersected segment needs to rotate:

ff θ₂ - sin^"1 2--1 sm + γ a J

Therefore, for the segment intersected and moved by an instrument: Angle of rotation = θ₂ Axis of rotation is perpendicular to the plane of motion

Once the intersected segment moves in response to instrument motion, segments above and below it will need to move as well. The same algorithms as those described in the previous section are used to achieve this.

Effect of gravity Once an object such as the uterine tube is moved or picked up by an instrument, it must return to its resting position. The effect of gravity during free- fall of the tube is simulated via rotations of each segment. During each frame of free-fall, rotation of each segment's axis from its current to its initial orientation is found. This rotation, divided by a constant which sets the speed of free-fall, determines by how much each segment will fall in each frame until it reaches its resting state. The tube is back at equilibrium when every one of its segments returns to equilibrium.

Claims

CLAIMS:

1. A method for modelling the deformation of a first object in response to collision with a second object, the method including the steps of providing a segmented model of the first object, providing an algorithm governing the deformation of the first object in response to collision with the second object

wherein the algorithm does not include an equation of motion.

2. A method according to claim 1 wherein the equation of motion determines velocity.

3. A method according to claim 1 wherein the equation of motion determines acceleration.

4. A method for modelling the deformation of a first object in response to collision with a second object, the method including the steps of providing a segmented model of the first object, providing an algorithm governing the deformation of the first object in response to collision with the second object wherein the algorithm provides that the main parameter used to calculate the shape and/or position of the first object after collision is the spatial displacement of the second object.

5. A method according to any one of claims 1 to 4 wherein each segment of the first object has an independent system of co-ordinates.

6. A method according to any one of claims 1 to 5 wherein at least two segments of the first object are connected in a cantilevered arrangement.

7. A method according to claim 6 wherein the cantilevered arrangement is a cantilever mass-spring system,

8. A method according to claim 7 wherein the mass-spring system is substantially as described in Figure 1.

9. A method according to claim 8 wherein the cantilever mass-spring system behaves according to the equation F= -k*d, where F is an applied force, k is the spring stiffness coefficient, and d is the distance by which the load end of the link is displaced.

10. A method according to claim 9 wherein k is defined by the equation

mg k = -

Ismθ

11. A method according to claim 9 wherein the spring stiffness for each segment is defined by the equation

N

12. A method according to any one of claims 1 to 11 wherein the second object is a substantially rigid object.

13. A method according to claim 12 wherein the second object is substantially linear.

14. A method according to claim 12 wherein the spatial displacement of a segment of the second object is limited to rotation about a single point.

15. A method according to claim 14 wherein the angle of rotation by which the segment needs to move is found by trigonometry.

16. A method according to claim 15 wherein the angle of rotation by which the segment moves is defined by the equation

0₂ = sin -~1¹| | 22---- •1l | ssimr / + γ v. a

17. A method according to and one of claims 1 to 16 wherein the algorithm provides that deformation of the first object is effected by displacement of a segment adjacent to that which has been collided with by the second object.

18. A method according to any one of claims 1 to 17 wherein the first object is a model of an anatomical feature of the human body.

19. A method according to claim 18 wherein the anatomical feature is an organ.

20. A method according to claim 19 wherein the organ is selected from the group consisting of fallopian tube, uterus, ovary, and ovarian ligament.

21. A method according to any one of claims 1 to 20 wherein the second object is a model of a surgical instrument.

22. A method according to claim 21 wherein the surgical instrument is selected from the group consisting of forceps, clamp, scissors, retractor, cauterisation device, endoscope laparoscopic probe, and haemostatic clip applicator.

23. A computer executable program capable of implementing a method according to any one of claims 1 to 22.

24. A computer executable program according to claim 23 capable of running a simulation in about real-time.

25. A computer executable program according to claim 24 capable of providing a frame rate of at least about 24 frames per second.

26. A computer executable program according to claim 25 capable of providing a frame rate of at least about 30 frames per second.

27. A computer including a computer executable program according to any one of claim 23 to 26.

28. A computer according to claim 27 having a central processing unit with a clock speed higher than about 200 MHz.

29. A computer according to claim 28 wherein the clock speed is higher than about 100 MHz.

30. A computer according to any one of claims 27 to 29 wherein the computer has a central processing unit is selected from the group including Pentium 1 , Pentium 2, Pentium 3, Pentium 4, Celeron, MIPS RISC R10000 or better.

31. A computer according to claim 30 having a central processing unit is selected from the group including Pentium 4, Celeron, MIPS RISC R10000 or better.

32. A virtual reality system including a method according to any one of claims 1 to 22.

33. A virtual reality system including a computer executable program according to any one of claims 23 to 26.

34. A virtual reality system including a computer according to any one of claims 27 to 31.

35. A virtual reality system according to any one of claims 32 to 34 when used for training in surgical techniques.

36. A virtual reality system according to any one of claims 32 to 35 wherein the surgical techniques are used in a surgical speciality selected from the group consisting of gynaecological surgery, gall bladder surgery, neurosurgery, thoracic surgery, eye surgery, and orthopaedic surgery.

37. A virtual reality system according to claim 36 wherein the surgical techniques is used in gynaecological surgery.

38. A virtual reality system according to claim 37 capable of representing the female pelvic anatomical field approximating that viewed by an endoscopic camera during surgery.

39. A virtual reality system according to any one of claims 35 to 39 wherein the surgery is minimally invasive surgery.

40. A virtual reality system according to claim 39 wherein the minimally invasive surgery is endoscopic surgery.

41. A method for training surgeons including a method according to any one of claims 1 to 22.

42. A method for training surgeons including a computer executable program according to any one of claims 23 to 26.

43. A method for training surgeons including a computer according to any one of claims 27 to 31.

44. A method for training surgeons including a virtual reality system according to any one of claims 32 to 40.