CN113343513B - Method and device for simulating soft tissue deformation and path cutting - Google Patents
Method and device for simulating soft tissue deformation and path cutting Download PDFInfo
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Abstract
The invention discloses a method and a device for simulating soft tissue deformation and path cutting, wherein the method is used for simulating the examination and excision process of stomach lesion areas in a virtual operation, firstly, a finite element model is used for simulating the deformation process of stomach soft tissues, model reduction is realized by applying a model reduction method combining intrinsic orthogonal decomposition and Galerkin projection in the deformation process, then a cutting path is formed according to the intersection position formed by collision detection of surgical machinery and the soft tissues, and finally, bezier curves are used for drawing surface incisions generated after the soft tissues are cut. The invention not only improves the calculation speed and the real-time performance of soft tissue deformation simulation by using the finite element model, but also can create smoother and natural cutting paths and incision effects in the processing of the virtual operation cutting problem.
Description
Technical Field
The invention relates to a soft tissue force touch simulation method, in particular to a method and a device for simulating soft tissue deformation and path cutting.
Background
In recent years, the demand of remote medical consultation is increasingly prominent, and one key technical difficulty in the remote medical consultation is the accurate simulation of virtual lesion organs of patients, so that consultation specialists are prompted to propose a correct, scientific and proper treatment scheme, and finally accurate guidance is realized. In the existing virtual organ simulation method, a finite element model discretizes a soft tissue organ into finite unit bodies such as tetrahedrons and hexahedrons, the unit bodies are connected through grid nodes, displacement of any point on the unit body is expressed by using displacement of each unit node as a function of a variable, and deformation of the soft tissue organ is finally simulated by solving the node displacement to obtain a soft tissue deformation quantity. According to elastic mechanics, the complex solving domain is discretized to solve, so that the simulation method has the characteristic of high simulation precision. However, as the topological structure of the grid unit is continuously recombined in the deformation process of the model, the calculation amount is large, the calculation efficiency is low, the simulation instantaneity is greatly influenced, and the implementation in remote consultation is difficult.
Disclosure of Invention
The invention aims to: aiming at the problems of the finite element model, the invention provides a method and a device for simulating soft tissue deformation and path cutting, which can reduce deformation calculation amount and ensure simulation instantaneity, and when deformation reaches a limit, the deformation model is broken from the deformation model, so that the soft tissue deformation and path cutting are simulated.
The technical scheme is as follows: the invention provides a method for simulating soft tissue deformation and path cutting, which comprises the following steps:
(1) Performing deformation simulation on soft tissues to be simulated by using a finite element model, and realizing model reduction by using a model reduction method combining eigen orthogonal decomposition and Galerkin projection in the deformation simulation process;
(2) Forming a cutting path according to the position of an intersection point formed by collision detection of the surgical machine and the soft tissue;
(3) Bezier curves are used to map the surface incisions that are made after the soft tissue is cut.
Further, the process of using the finite element model to simulate the deformation of the soft tissue to be simulated in the step (1) is as follows:
wherein M represents a mass matrix, u represents a position vector, v represents a velocity vector, G represents an external force applied to the soft tissue, t represents an iteration time, F represents an internal force applied to the soft tissue, H T λ represents the constraint force generated by the surgical instrument when acting on the soft tissue surface, H represents the time interval matrix, λ represents the constraint parameter, and' represents the transpose;
discretizing the soft tissue into a series of tetrahedral grid units, and carrying out numerical calculation on the dynamic expression of the finite element model by using an implicit Euler method; consider the continuous iteration time to be discretized into intervals
At time interval t n ，t n+1 ]The specific numerical calculation formula is as follows: />
In the process, let
h denotes the time interval, i.e. h=t n+1 -t n ，/>
Represents a derivative symbol, d represents a derivative symbol, < ->
Represents F at t n Value of time of day->
Represents G at t n+1 A value of the time of day.
Further, the model reduction method in the step (1) is implemented as follows:
reducing the order of the finite element full order model by utilizing the intrinsic orthogonal decomposition, thereby obtaining a group of orthogonal basis functions phi= (phi) 1 ，φ 2 ，...，φ N ) The method can approximate the original sample to the greatest extent, and a specific calculation formula for obtaining the orthogonal basis function is as follows:
where J represents an error function between the solution vector samples of the full-order model and the basis function in the least squares sense,
for a discrete subset of parameter space Λ, -/->
Representation->
A certain component, t 0 And->
Two discrete moments, < +.>
Representation stored in the snapshot matrix->
Phi, data of (2) i Representing the ith component of the orthogonal basis functions;
the method comprises the steps of performing eigen orthogonal decomposition on a group of obtained basis functions, taking a space formed by the group of basis functions as a function space where a solution vector of a reduced order model is located, and then projecting a full order model to the reduced order space by using Galerkin projection; the specific calculation formula for solving the solution vector of the reduced-order model by using Galerkin projection is as follows:
where, alpha represents a coefficient vector of the orthogonal basis function phi,
further, the step (2) includes the steps of:
(21) The surgical instrument is simplified and abstracted into a line segment, and collision detection of the surgical instrument and soft tissues is that the intersection point of the line segment and a triangle unit is detected; assuming that two end points of a line segment are I and L respectively, and their position vectors are u respectively I (x I ，y I ，z I ) And u L (x L ，y L ，z L ) The spatial linear equation of the surgical instrument is specifically expressed as follows：
Wherein x, y and z respectively represent three unknowns of the space linear equation, and k represents a parameter of the space linear equation; any node coordinates on the line segment are:
(22) Determining a plane equation of a triangle unit intersecting the surgical instrument: assuming that three end points of the triangle unit of the collision are O, P, Q respectively, their position coordinates are u respectively O (x O ，y O ，z O )、u P (x P ，y P ，z P ) And u Q (x Q ，y Q ，z Q ) And the normal vector of the triangle unit is N (N) x ，n y ，n z ) The plane equation of the triangle unit is therefore:
Ux+Vy+Wz+T＝0
wherein U, V, W and T represent four parameters to be solved of the plane equation, and x, y and z represent unknowns of the plane equation;
according to the point French method for calculating the plane equation, the plane equation can be obtained through the coordinates of the point O and the normal vector N:
n x (x-x O )+n y (y-y O )+n z (z-z O )＝0
where u=n x ，V＝n y ，W＝n z ，T＝-n x x O -n y x O -n z z O ；
(23) The straight line equation and the plane equation are combined to obtain the position u of the intersection point S of the surgical instrument and the triangle unit plane at any discrete moment s (x S ，y S ，z S ) The specific calculation formula is as follows:
further, the Bezier curve in the step (3) is formed by three control points and adopting a Bemstein basis function of 2 times; the 2 nd order Bernstein basis function R i，2 The specific expression of (t) is:
further, the surface incision equation in the step (3) is:
wherein E (t) represents a position vector of a control point for drawing a cutting path at an iteration time t, R i，2 (t) represents the 2 nd order Bemstein basis function.
Based on the same inventive concept, the invention also provides a device for simulating soft tissue deformation and path cutting, which comprises a memory, a processor and a computer program stored on the memory and capable of running on the processor, and is characterized in that the computer program is loaded to the processor to realize the method for simulating soft tissue deformation and path cutting.
The beneficial effects are that: compared with the prior art, the invention has the beneficial effects that: (1) Constructing a soft tissue deformation model by using a finite element model based on a model reduced order method, obtaining a group of orthogonal basis functions by utilizing eigenvoice decomposition, and then projecting a solution vector of a finite element full-order model to a reduced order space formed by the group of orthogonal basis functions by utilizing Galerkin projection so as to obtain deformation displacement vectors of soft tissue nodes, thereby reducing deformation calculation amount, improving calculation efficiency and ensuring simulation instantaneity; (2) A linear model is adopted to simulate a cutting tool, and the surgical instrument is simplified into a straight line. The collision between the surgical instrument and the soft tissue is converted into the intersection detection of the straight line and the virtual soft tissue surface triangle unit, so that the simulated cutting process is realized, the real-time simulation is ensured, and the complexity of the cutting simulation is reduced; (3) The Bezier curve is adopted to draw the surface incision generated after the soft tissue is cut, the first end and the last end of the curve are controlled and the curve curvature is determined by regulating the curve end points and increasing the control points in the middle of the curve, the space complexity of the cutting procedure is greatly reduced, and the incision effect can be promoted to be smoother and more natural.
Drawings
FIG. 1 is a flow chart of the present invention;
FIG. 2 is a schematic diagram of a reduced order model according to the present invention;
FIG. 3 is a schematic view of the intersection of a surgical instrument and a triangle unit according to the present invention;
FIG. 4 is a schematic representation of a Bezier cut according to the present invention;
FIG. 5 is a graph of the effect of incisions based on different methods of cutting stomach soft tissue, wherein (a) is a graph of the effect of simulations based on the present invention; (B) is a simulated effect graph based on a B-spline curve; (c) Is a simulation effect graph based on a non-uniform rational B-spline curve.
Detailed Description
The invention is described in further detail below with reference to the accompanying drawings.
The invention provides a method for simulating soft tissue deformation and path cutting, which is shown in fig. 1 and specifically comprises the following steps:
step 1: and performing deformation simulation on the soft tissue to be simulated by using a finite element model, and realizing model reduction by applying a model reduction method combining eigen orthogonal decomposition and Galerkin projection in the deformation simulation process.
According to medical image data, a finite element model based on a model order reduction method is used for constructing a deformation model for human organs, a solving result of a group of finite element full-order models is taken as a sample, the full-order models are reduced to be order-reduction models through a model order reduction method combining eigenvoice orthogonal decomposition and Galerkin projection on the basis, and then the group of solution vectors are projected to an order-reduction space where the order-reduction models are located to calculate deformation displacement vectors of grid unit nodes. The three-dimensional geometric reconstruction of the gastric medical image data acquired by CT scan using OpenGL discretizes the stomach soft tissue into a series of tetrahedral mesh cells, and then deformation modeling using reduced order models, as shown in fig. 2.
The deformation displacement vector of the node is calculated by projecting the solution vector of the finite element full-order model into a reduced order space, and the method specifically comprises the following steps:
(a) Soft tissue deformation is simulated using a finite element model, whose dynamic expression is:
wherein M represents a mass matrix, u represents a position vector, v represents a velocity vector, G represents an external force applied to the soft tissue, t represents an iteration time, F represents an internal force applied to the soft tissue, H T λ represents the constraint force generated by the surgical instrument when acting on the soft tissue surface, H represents the time interval matrix, λ represents the constraint parameter, and' represents the transpose.
(b) Consider the discretization of successive iteration times t into intervals
At time interval t n ，t n+1 ]The finite element model dynamic expression is integrated:
where h denotes the time interval, i.e. h=t n+1 -t n ，
Represents a derivative symbol, d represents a derivative symbol, < ->
Represents F at t n Value of time of day->
Represents G at t n+1 A value of the time of day. />
(c) Discretizing the soft tissue into a series of tetrahedral grid units, and performing numerical calculation on the dynamic expression of the finite element model by using an implicit Euler method to obtain the following formula:
in the method, in the process of the invention,
representation->
A matrix formed by the method.
(d) Since the internal force F (u, t) experienced by soft tissue is a nonlinear function of position and velocity, it is iteratively calculated using a Newton-Raphson iteration:
in the method, in the process of the invention,
representing the internal force F at t of the soft tissue n A value of the time of day.
(f) Finally, a solving result of a group of finite element model full-order models is obtained, and a specific numerical calculation formula is as follows:
in the process, let
(g) Combining the set of solution vectors obtained by step (f)
Composing a snapshot matrix
And performing an eigen-orthogonal decomposition on u (t, lambda):
in which Φ= (Φ) 1 ，φ 2 ，...，φ N ) Representation of orthogonal basis functions based on eigen-orthogonal decomposition, alpha i Representing the basis function phi i Corresponding coefficients of (a) are provided.
(h) And cutting off the obtained orthogonal basis functions to approximate the original sample under the condition of reducing the dimension of the full-order model to the greatest extent. Defining an error function J to finally select an orthogonal basis function phi= (phi) representing the solution vector of the full-order model 1 ，φ 2 ，...，φ N )：
Where J represents an error function between the solution vector samples of the full-order model and the basis function in the least squares sense,
is a discrete subset of the parameter space lambda * Representation->
A certain component, t 0 And->
Two discrete moments, < +.>
Representation stored in the snapshot matrix->
Phi, data of (2) i Representing the i-th component of the orthogonal basis functions. />
(i) Obtaining a group of eigenvalue orthogonal decomposition base functions through the step (h), taking the space formed by the group of base functions as a function space where a solution vector of the reduced order model is located, and then projecting the full order model into the reduced order space by using Galerkin projection. The specific calculation formula for solving the solution vector of the reduced-order model by using Galerkin projection is as follows:
in the method, in the process of the invention,
step 2: the cutting path is formed according to the intersection point position formed by collision detection of the surgical machine and the soft tissue.
A cutting path is understood to mean a continuous length of fold line that the surgical instrument is stroked across the soft tissue surface in discrete times. Therefore, first, the surgical instrument is simplified and abstracted into a line segment, and collision detection of the surgical instrument and the soft tissue is that of the intersection point of the line segment and the triangle unit, as shown in fig. 3. First, assume that two end points of a line segment are I and L respectively, and their position vectors are u respectively I (x I ，y I ，z I ) And u L (x L ，y L ，z L ) The spatial linear equation of the surgical instrument is expressed as follows:
wherein x, y and z respectively represent three unknowns of the space linear equation, and k represents a parameter of the space linear equation; any node coordinates on the line segment are:
next, a plane equation of a triangle unit intersecting the surgical instrument is determined. Assuming that three end points of the triangle unit of the collision are O, P, Q respectively, their position coordinates are u respectively O (x O ，y O ，z O )、u P (x P ，y P ，z P ) And u Q (x Q ，y Q ，z Q ) And the normal vector of the triangle unit is N (N) x ，n y ，n z ) The plane equation of the triangle unit is therefore:
Ux+Vy+Wz+T＝0
where U, V, W and T represent four parameters to be solved for the plane equation and x, y and z represent unknowns for the plane equation.
Then, according to the point French method of calculating the plane equation, the plane equation is obtained by coordinates of the point O and the normal vector N:
n x (x-x O )+n y (y-y O )+n z (z-z O )＝0
where u=n x ，V＝n y ，W＝n z ，T＝-n x x O -n y x O -n z z O 。
Finally, the linear equation and the plane equation are combined, so that the position u of the intersection point S of the surgical instrument and the plane of the triangular unit at any discrete moment is obtained S (x S ，y S ，z S ) The specific calculation formula is as follows:
step 3: bezier curves are used to map the surface incisions that are made after the soft tissue is cut.
First, determining a start position A and an end position B of a cutting path, and acquiring all surface nodes of the cutting path, denoted as E 1 ＝{E 10 ，E 11 ，E 12 ，E 13 ，E 14 ，E 15 Nodes around the cut path are denoted as point set E 2 ＝{E 20 ，E 21 ，E 22 ，E 23 ，E 24 ，E 25 ，E 26 ，E 27 ，E 28 ，E 29 ，E 30 ，E 31 And distributed on both sides of the cutting path. Nodes C and D are selected as control points defining the curvature of the surface cut, as shown in fig. 4. Two quadratic bezier curves were then plotted according to A, B, C three vertices and A, B, D three vertices, respectively, to represent the incision created by the cutting operation.
Wherein, the positions of n+1 control points are assumed to be E i ＝(x i ，y i ，z i ) I=0, 1, …, n, and a position vector E (t) is generated by mixing the control points to describe E 0 And E is connected with n A path between the two approaches to the Bessel polynomial function:
where n represents the polynomial degree, typically determined by n+1 control points of the notch curve, R i，n (t) is a polynomial expression called the Bernstein basis function, defined as n times:
here, a quadratic bezier curve is chosen to achieve the reproduction of the notch, where the quadratic bezier curve is generated by 3 control points, n=2 is substituted into the Bernstein basis function, thus yielding a set of 2-degree Bernstein basis functions:
then, the equation for drawing the surface cut based on the quadratic bezier curve is:
finally, the invention, the B spline curve and the non-uniform rational B spline curve are adopted to carry out cutting simulation comparison on the virtual stomach soft tissue. Fig. 5 shows a graph of the effect of incision based on different methods of cutting stomach soft tissue. As can be seen from fig. 5, the fidelity of the simulated cut based on the present invention is higher, and the cut is more realistic, smooth and natural.
Based on the same inventive concept, the invention also provides a device for simulating soft tissue deformation and path cutting, which comprises a memory, a processor and a computer program stored on the memory and capable of running on the processor, wherein the computer program is loaded to the processor to realize the method for simulating soft tissue deformation and path cutting.
The invention can reduce the deformation calculation amount, simultaneously ensure the calculation efficiency and the calculation precision of simulating the deformation of the stomach soft tissue, improve the simulation instantaneity and enable an operator to feel the vivid cutting process of the soft tissue in the man-machine interaction process.
Claims (4)
1. A method for simulating soft tissue deformation and path cutting, comprising the steps of:
(1) Performing deformation simulation on soft tissues to be simulated by using a finite element model, and realizing model reduction by using a model reduction method combining eigen orthogonal decomposition and Galerkin projection in the deformation simulation process;
(2) Forming a cutting path according to the position of an intersection point formed by collision detection of the surgical machine and the soft tissue;
(3) Drawing a surface incision generated after the soft tissue is cut using a bezier curve;
the process of using the finite element model to simulate the deformation of the soft tissue to be simulated in the step (1) is as follows:
wherein M represents a mass matrix, u represents a position vector, v represents a velocity vector, G represents an external force applied to the soft tissue, t represents an iteration time, F represents an internal force applied to the soft tissue, H T λ represents the constraint force generated by the surgical instrument when acting on the soft tissue surface, H represents the time interval matrix, λ represents the constraint parameter, and' represents the transpose;
discretizing the soft tissue into a series of tetrahedral grid units, and carrying out numerical calculation on the dynamic expression of the finite element model by using an implicit Euler method; consider the continuous iteration time to be discretized into intervals
At time interval t n ,t n+1 ]The specific numerical calculation formula is as follows:
in the method, in the process of the invention,
h denotes the time interval, i.e. h=t n+1 -t n ，/>
Represents a derivative symbol, d represents a derivative symbol, < ->
Represents F at t n Value of time of day->
Represents G at t n+1 A value of time of day;
the model order reduction method in the step (1) is realized as follows:
reducing the order of the finite element full order model by utilizing the intrinsic orthogonal decomposition, thereby obtaining a group of orthogonal basis functions phi= (phi) 1 ,φ 2 ,…,φ N ) The method can approximate the original sample to the greatest extent, and a specific calculation formula for obtaining the orthogonal basis function is as follows:
where J represents an error function between the solution vector samples of the full-order model and the basis function in the least squares sense,
is a discrete subset of the parameter space lambda * Representation->
A certain component, t 0 And->
Two discrete moments, < +.>
Representation stored in the snapshot matrix->
Phi, data of (2) i Representing the ith component of the orthogonal basis functions;
the method comprises the steps of performing eigen orthogonal decomposition on a group of obtained basis functions, taking a space formed by the group of basis functions as a function space where a solution vector of a reduced order model is located, and then projecting a full order model to the reduced order space by using Galerkin projection; the specific calculation formula for solving the solution vector of the reduced-order model by using Galerkin projection is as follows:
where, alpha represents a coefficient vector of the orthogonal basis function phi,
the step (2) comprises the following steps:
(21) The surgical instrument is simplified and abstracted into a line segment, and collision detection of the surgical instrument and soft tissues is that the intersection point of the line segment and a triangle unit is detected; assuming that two end points of a line segment are I and L respectively, and their position vectors are u respectively I (x I ,y I ,z I ) And u L (x L ,y L ,z L ) The spatial linear equation of the surgical instrument is expressed as follows:
wherein x, y and z respectively represent three unknowns of the space linear equation, and k represents a parameter of the space linear equation; any node coordinates on the line segment are:
(22) Determining a plane equation of a triangle unit intersecting the surgical instrument: assuming that three end points of the triangle unit of the collision are O, P, Q respectively, their position coordinates are u respectively O (x O ,y O ,z O )、u P (x P ,y P ,z P ) And u Q (x Q ,y Q ,z Q ) And the normal vector of the triangle unit is N (N) x ,n y ,n z ) The plane equation of the triangle unit is therefore:
Ux+Vy+Wz+T＝0
wherein U, V, W and T represent four parameters to be solved of the plane equation, and x, y and z represent unknowns of the plane equation;
according to the point French method for calculating the plane equation, the plane equation can be obtained through the coordinates of the point O and the normal vector N:
n x (x-x O )+n y (y-y O )+n z (z-z O )＝0
where u=n x ，V＝n y ，W＝n z ，T＝-n x x O -n y x O -n z z O ；
(23) The straight line equation and the plane equation are combined to obtain the position u of the intersection point S of the surgical instrument and the triangle unit plane at any discrete moment S (x S ,y S ,z S ) The specific calculation formula is as follows:
2. the method for modeling soft tissue deformation and path cutting as defined in claim 1, wherein the bezier curve in step (3) is constructed by three control points and using 2-degree Bernstein basis functions; the 2 nd order Bernstein basis function R i,2 The specific expression of (t) is:
3. the method for modeling soft tissue deformation and path cutting as defined in claim 1, wherein the surface incision equation in step (3) is:
wherein E (t) represents a position vector of a control point for drawing a cutting path at an iteration time t, R i,2 (t) represents the Bernstein basis function 2 times.
4. An apparatus for simulating soft tissue deformation and path cutting comprising a memory, a processor and a computer program stored on the memory and executable on the processor, wherein the computer program when loaded into the processor implements the method for simulating soft tissue deformation and path cutting according to any of claims 1-3.
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