CN113343513A - 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 PDF

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CN113343513A
CN113343513A CN202110514992.XA CN202110514992A CN113343513A CN 113343513 A CN113343513 A CN 113343513A CN 202110514992 A CN202110514992 A CN 202110514992A CN 113343513 A CN113343513 A CN 113343513A
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张小瑞
吴海伦
孙伟
宋爱国
刘佳
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Nanjing University of Information Science and Technology
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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 process of examining and removing a pathological change area of a stomach in a virtual operation. The invention not only improves the calculation speed and the real-time performance of soft tissue deformation simulation by using a finite element model, but also can create a smoother and natural cutting path and incision effect in the treatment of the virtual surgical cutting problem.

Description

Method and device for simulating soft tissue deformation and path cutting
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 remarkable, and a key technical difficulty in remote medical consultation is accurate simulation of a virtual pathological organ of a patient, so that consultation experts are prompted to put forward a correct, scientific and appropriate treatment scheme, and accurate guidance is finally realized. In the existing virtual organ simulation method, a finite element model disperses a soft tissue organ into finite unit bodies such as tetrahedrons, hexahedrons and the like, the unit bodies are connected through grid nodes, the displacement of any point on each unit body is represented by using the displacement of each unit node as a function of a variable, the displacement of the node is solved to obtain a soft tissue deformation quantity, and finally the deformation behavior of the soft tissue organ is simulated. The method is characterized in that a complex solution domain is discretized and solved according to elastic mechanics, so that the method has the characteristic of high simulation precision. However, the topological structure of the grid cells of the model can be continuously recombined in the deformation process, so that the calculation amount is large, the calculation efficiency is low, and the simulation real-time performance is greatly influenced, so that the implementation in the remote consultation is difficult.
Disclosure of Invention
The purpose of the invention is as follows: aiming at the problems of a finite element model, the invention provides a method and a device for simulating soft tissue deformation and path cutting, wherein the deformation calculation amount can be reduced, the simulation real-time performance can be ensured, and when the deformation reaches the limit, the deformation model is broken from the position.
The technical scheme is as follows: the invention provides a method for simulating soft tissue deformation and path cutting, which specifically comprises the following steps:
(1) carrying out deformation simulation on soft tissues to be simulated by using a finite element model, and realizing model reduction by applying a model reduction method combining intrinsic orthogonal decomposition and Galerkin projection in the deformation simulation process;
(2) forming a cutting path according to an intersection point position formed by collision detection of the surgical machine and the soft tissue;
(3) bezier curves are used to map the surface cuts that are made after the soft tissue has been cut.
Further, the process of simulating deformation of the soft tissue to be simulated by using the finite element model in the step (1) is as follows:
Figure BDA0003059900920000011
in the formula, 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 iteration time, F represents an internal force applied to the soft tissue, and HTλ represents a constraint force, generated by the surgical instrument acting on the soft tissue surface, H represents a time interval matrix, λ represents a constraint parameter,' represents a transpose;
dispersing the soft tissue into a series of tetrahedral mesh units, and performing numerical calculation on a dynamic expression of the finite element model by using an implicit Euler method; consider the discretization of successive iteration times into intervals
Figure BDA00030599009200000214
At time intervals tn，tn+1]The specific numerical calculation formula is as follows:
Figure BDA0003059900920000021
in the formula
Figure BDA0003059900920000022
h denotes a time interval, i.e. h-tn+1-tn，
Figure BDA0003059900920000023
Representing the derivative sign, d the differential sign,
Figure BDA0003059900920000024
denotes that F is at tnThe value of the time of day is,
Figure BDA0003059900920000025
denotes G at tn+1The value of the time of day.
Further, the model order reduction method in step (1) is implemented as follows:
reducing the finite element full-order model by intrinsic orthogonal decomposition to obtain a set of orthogonal basis functions phi (phi)1，φ2，...，φN) So that the original sample can be approximated to the maximum extent, and the specific calculation formula for obtaining the orthogonal basis function is as follows:
Figure BDA0003059900920000026
in the formula, J represents an error function between a solution vector sample of the full-order model and a basis function in the least square sense,
Figure BDA0003059900920000027
for one discrete subset of the parameter space a,
Figure BDA0003059900920000028
to represent
Figure BDA0003059900920000029
A certain component of, t0And
Figure BDA00030599009200000210
respectively representing two discrete instants of the iteration time t,
Figure BDA00030599009200000211
representation storage in snapshot matrix
Figure BDA00030599009200000212
Data of (a), phiiRepresents the ith component of the orthogonal basis function;
decomposing a basis function through the obtained group of intrinsic orthogonal decomposition basis functions, taking a space formed by the group of basis functions as a function space where a reduced order model solution vector is located, and then projecting the full order model to the reduced order space by using Galerkin projection; the specific calculation formula for solving the reduced order model solution vector by using Galerkin projection is as follows:
Figure BDA00030599009200000213
wherein alpha represents a coefficient vector of the orthogonal basis function phi,
Figure BDA0003059900920000031
further, the step (2) comprises the steps of:
(21) simplifying and abstracting the surgical instruments into a line segment, wherein the collision detection of the surgical instruments and the soft tissue is the intersection point detection of the line segment and the triangular unit; suppose that the two end points of the line segment are I and L, respectively, and their position vectors are u, respectivelyI(xI，yI，zI) And uL(xL，yL，zL) Then, the spatial linear equation of the surgical instrument is specifically expressed as follows:
Figure BDA0003059900920000032
in the formula, x, y and z respectively represent three unknowns of a space linear equation, and k represents a parameter of the space linear equation; the coordinates of any node on the line segment are:
Figure BDA0003059900920000033
(22) determining a plane equation for a trigonometric cell intersecting the surgical instrument: let O, P, Q be the three endpoints of the triangle unit of the collision, and u be the position coordinates of the triangle unit of the collisionO(xO，yO，zO)、uP(xP，yP，zP) And uQ(xQ，yQ，zQ) And the normal vector of the triangle unit is N (N)x，ny，nz) Therefore, the plane equation of the triangle unit is:
Ux+Vy+Wz+T＝0
in the formula, U, V, W and T represent four parameters to be solved of a plane equation, and x, y and z represent unknowns of the plane equation;
according to the point normal method for calculating the plane equation, the plane equation can be obtained through the coordinates of the point O and the normal vector N:
nx(x-xO)+ny(y-yO)+nz(z-zO)＝0
wherein U is nx，V＝ny，W＝nz，T＝-nxxO-nyxO-nzzO；
(23) Simultaneous linear equation and plane equation to obtain the position u of the intersection S of the surgical instrument and the plane of the triangle unit at any discrete moments(xS，yS，zS) The specific calculation formula is as follows:
Figure BDA0003059900920000034
further, the Bezier curve in the step (3) is formed by three control points and 2-time Bemstein basis functions; the 2-degree Bernstein basis function Ri，2The specific expression of (t) is as follows:
Figure BDA0003059900920000041
further, the surface notching equation in step (3) is as follows:
Figure BDA0003059900920000042
wherein E (t) represents a position vector of a control point for drawing a cutting path at an iteration time t, Ri，2(t) represents the 2-fold Bemstein basis function.
Based on the same inventive concept, the present invention also provides 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 as described above.
Has the advantages 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 order reduction method, obtaining a group of orthogonal basis functions by using intrinsic orthogonal decomposition, and then projecting a solution vector of the finite element full-order model to a reduced order space formed by the group of orthogonal basis functions through Galerkin projection so as to obtain a deformation displacement vector of a soft tissue node, thereby reducing deformation calculation amount, improving calculation efficiency and ensuring simulation real-time property; (2) a linear model is adopted to simulate a cutting tool, and the surgical instrument is simplified into a straight line. The collision of the surgical instrument and the soft tissue is converted into the intersection detection of the straight line and the triangular unit on the surface of the virtual soft tissue, so that the simulated cutting process is realized, the real-time simulation is ensured, and the complexity of cutting simulation is reduced; (3) the Bezier curve is adopted to draw the surface incision generated after the soft tissue is cut, the end points of the curve are regulated, the control points in the middle of the curve are added to control the beginning and the end of the curve and determine the curvature of the curve, the space complexity of a cutting procedure is greatly reduced, and the incision effect can be made to be smoother and more natural.
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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 according to the present invention with a triangle unit;
FIG. 4 is a schematic representation of a Bezier curve cut according to the present invention;
FIG. 5 is a graph of the effect of incisions on the soft tissues of the stomach based on various methods, wherein (a) is a graph of the simulated effect based on the present invention; (b) a simulation effect graph based on a B spline curve; (c) is a simulation effect graph based on the non-uniform rational B-spline curve.
Detailed Description
The present invention will be described in further detail with reference to the accompanying drawings.
The invention provides a method for simulating soft tissue deformation and path cutting, which specifically comprises the following steps as shown in figure 1:
step 1: and (3) carrying out 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 intrinsic orthogonal decomposition and Galerkin projection in the deformation simulation process.
According to medical image data, a deformation model is built for a human body organ by using a finite element model based on a model order reduction method, a group of solving results of a finite element full-order model are used as samples, on the basis, the full-order model is reduced into the order-reduced model by a model order reduction method combining intrinsic orthogonal decomposition and Galerkin projection, and then the group of solving vectors are projected to a order-reduced space where the order-reduced model is located to calculate deformation displacement vectors of grid unit nodes. The method comprises the steps of performing three-dimensional geometric reconstruction on stomach medical image data acquired by CT scanning by using OpenGL, discretizing stomach soft tissues into a series of tetrahedral mesh units, and then performing deformation modeling by using a reduced order model, as shown in figure 2.
Calculating the deformation displacement vector of the node by projecting the solution vector of the finite element full-order model into a reduced order space, and specifically comprising the following steps:
(a) simulating soft tissue deformation by using a finite element model, wherein the dynamic expression is as follows:
Figure BDA0003059900920000051
in the formula, 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 iteration time, F represents an internal force applied to the soft tissue, and HTλ represents the constraint force generated by the surgical instrument 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
Figure BDA0003059900920000052
At time intervals tn，tn+1]Integrating the finite element model dynamic expression:
Figure BDA0003059900920000053
Figure BDA0003059900920000054
wherein h represents a time interval, i.e. h ═ tn+1-tn，
Figure BDA0003059900920000061
Representing the derivative sign, d the differential sign,
Figure BDA0003059900920000062
denotes that F is at tnThe value of the time of day is,
Figure BDA0003059900920000063
denotes G at tn+1The value of the time of day.
(c) The soft tissue is discretized into a series of tetrahedral mesh units, and the dynamic expression of the finite element model is numerically calculated by using an implicit Euler method, so that the following formula can be obtained:
Figure BDA0003059900920000064
Figure BDA0003059900920000065
in the formula (I), the compound is shown in the specification,
Figure BDA0003059900920000066
to represent
Figure BDA0003059900920000067
The formed matrix.
(d) Since the internal force F (u, t) experienced by the soft tissue is a non-linear function with respect to position and velocity, it is iteratively calculated using a Newton-Raphson iteration:
Figure BDA0003059900920000068
in the formula (I), the compound is shown in the specification,
Figure BDA0003059900920000069
Figure BDA00030599009200000610
indicates that the soft tissue is subjected to an internal force F at tnThe value of the time of day.
(f) And finally, obtaining a solving result of a group of finite element model full-order models, wherein the specific numerical calculation formula is as follows:
Figure BDA00030599009200000611
in the formula
Figure BDA00030599009200000612
(g) Solving the set of solution vectors obtained in the step (f)
Figure BDA00030599009200000613
Composing a snapshot matrix
Figure BDA00030599009200000614
And carrying out eigen-orthogonal decomposition on u (t, lambda):
Figure BDA00030599009200000615
wherein phi is (phi)1，φ2，...，φN) Representation based on eigen-orthogonal decompositionOrthogonal basis function, αiExpressing the basis function phiiThe corresponding coefficient of (a).
(h) And truncating the obtained orthogonal basis functions to approximate the original sample under the condition of reducing the dimension of the full-order model to the maximum extent. Defining an error function J to finally select (phi) an orthogonal basis function phi representing a solution vector of the full-order model1，φ2，...，φN)：
Figure BDA0003059900920000071
In the formula, J represents an error function between a solution vector sample of the full-order model and a basis function in the least square sense,
Figure BDA0003059900920000072
as a discrete subset of the parameter space Λ, λ*To represent
Figure BDA0003059900920000073
A certain component of, t0And
Figure BDA0003059900920000074
respectively representing two discrete instants of the iteration time t,
Figure BDA0003059900920000075
representation storage in snapshot matrix
Figure BDA0003059900920000076
Data of (a), phiiRepresenting the ith component of the orthogonal basis function.
(i) And (h) obtaining a group of eigen-orthogonal decomposition basis functions through the step (h), using a space formed by the group of basis functions as a function space where a reduced order model solution vector is located, and then projecting the full order model to the reduced order space by using Galerkin projection. The specific calculation formula for solving the reduced order model solution vector by using Galerkin projection is as follows:
Figure BDA0003059900920000077
in the formula (I), the compound is shown in the specification,
Figure BDA0003059900920000078
step 2: and forming a cutting path according to the intersection point position formed by the collision detection of the surgical machine and the soft tissue.
A cutting path may be understood as a continuous line of weakness traversed by the surgical instrument at a soft tissue surface in discrete times. Therefore, the surgical instrument is simplified and abstracted into a line segment, and the collision detection of the surgical instrument and the soft tissue is the intersection detection of the line segment and the triangular unit, as shown in fig. 3. First, assume that the two end points of a line segment are I and L, respectively, and their position vectors are u, respectivelyI(xI，yI，zI) And uL(xL，yL，zL) Then, the spatial linear equation of the surgical instrument is specifically expressed as follows:
Figure BDA0003059900920000079
in the formula, x, y and z respectively represent three unknowns of a space linear equation, and k represents a parameter of the space linear equation; the coordinates of any node on the line segment are:
Figure BDA00030599009200000710
second, a plane equation for a triangle cell intersecting the surgical instrument is determined. Let O, P, Q be the three endpoints of the triangle unit of the collision, and u be the position coordinates of the triangle unit of the collisionO(xO，yO，zO)、uP(xP，yP，zP) And uQ(xQ，yQ，zQ) And the normal vector of the triangle unit is N (N)x，ny，nz) Therefore, the plane equation of the triangle unit is:
Ux+Vy+Wz+T＝0
in the formula, U, V, W and T represent four parameters to be solved for the plane equation, and x, y and z represent unknowns of the plane equation.
Then, according to a point normal method for calculating a plane equation, a plane equation can be obtained from the coordinates of the point O and the normal vector N:
nx(x-xO)+ny(y-yO)+nz(z-zO)＝0
wherein U is nx，V＝ny，W＝nz，T＝-nxxO-nyxO-nzzO。
Finally, a linear equation and a 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 obtainedS(xS，yS，zS) The specific calculation formula is as follows:
Figure BDA0003059900920000081
and step 3: bezier curves are used to map the surface cuts that are made after the soft tissue has been cut.
Firstly, determining a starting position A and an end position B of a cutting path, and acquiring all surface nodes passed by the cutting path, and recording the surface nodes as E1＝{E10，E11，E12，E13，E14，E15Recording the peripheral nodes of the cutting path as a point set E2＝{E20，E21，E22，E23，E24，E25，E26，E27，E28，E29，E30，E31And the cutting edges are distributed on two sides of the cutting path. Nodes C and D are selected as control points that specify the degree of surface cut curvature, as shown in fig. 4. Two quadratic bezier curves were then plotted from A, B, C and A, B, D vertices to represent the cuts resulting from the cutting operation.
Wherein, assume n +1Position of control point Ei＝(xi，yi，zi) I is 0, 1, …, n, and a position vector E (t) is generated by combining these control points to describe E0And EnThe path between which the bezier polynomial function is approximated:
Figure BDA0003059900920000082
where n represents the polynomial degree, typically determined by n +1 control points of the notch curve, Ri，n(t) is a polynomial expression called Bernstein basis function of degree n, defined as:
Figure BDA0003059900920000091
here, a quadratic bezier curve is selected to realize the reconstruction of the notch, where the quadratic bezier curve is generated by 3 control points, and n-2 is substituted into Bernstein basis functions, thereby obtaining 2 sets of Bernstein basis functions:
Figure BDA0003059900920000092
then, an equation for drawing the surface cut is obtained based on the quadratic bezier curve as follows:
Figure BDA0003059900920000093
and finally, performing cutting simulation comparison on the virtual stomach soft tissue by respectively adopting the invention, the B-spline curve and the non-uniform rational B-spline curve. FIG. 5 shows a graph of the effect of incisions on the soft tissues of the stomach based on different methods. 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 real, smooth and natural.
Based on the same inventive concept, the present invention also provides 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, which when loaded into the processor implements the above-described method for simulating soft tissue deformation and path cutting.
The invention can reduce deformation calculation amount, ensure calculation efficiency and calculation precision of simulated stomach soft tissue deformation, improve simulation real-time performance, and enable an operator to feel the vivid cutting process of the soft tissue in the human-computer interaction process.

Claims (7)

1. A method for simulating soft tissue deformation and path cutting, comprising the steps of:
(1) carrying out deformation simulation on soft tissues to be simulated by using a finite element model, and realizing model reduction by applying a model reduction method combining intrinsic orthogonal decomposition and Galerkin projection in the deformation simulation process;
(2) forming a cutting path according to an intersection point position formed by collision detection of the surgical machine and the soft tissue;
(3) bezier curves are used to map the surface cuts that are made after the soft tissue has been cut.
2. A method for simulating soft tissue deformation and path cutting as claimed in claim 1, wherein the process of simulating the deformation of the soft tissue to be simulated by using the finite element model in step (1) is as follows:
Figure FDA0003059900910000011
in the formula, 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 iteration time, F represents an internal force applied to the soft tissue, and HTλ represents a constraint force, generated by the surgical instrument acting on the soft tissue surface, H represents a time interval matrix, λ represents a constraint parameter,' represents a transpose;
discretizing soft tissue into a series of tetrahedral meshesA unit, which uses an implicit Euler method to carry out numerical calculation on the dynamic expression of the finite element model; consider the discretization of successive iteration times into intervals
Figure FDA0003059900910000012
At time intervals tn，tn+1]The specific numerical calculation formula is as follows:
Figure FDA0003059900910000013
in the formula (I), the compound is shown in the specification,
Figure FDA0003059900910000014
h denotes a time interval, i.e. h-tn+1-tn，
Figure FDA0003059900910000015
Representing the derivative sign, d the differential sign,
Figure FDA0003059900910000016
denotes that F is at tnThe value of the time of day is,
Figure FDA0003059900910000017
denotes G at tn+1The value of the time of day.
3. The method for simulating soft tissue deformation and path cutting as claimed in claim 1, wherein the model reduction method of step (1) is implemented as follows:
reducing the finite element full-order model by intrinsic orthogonal decomposition to obtain a set of orthogonal basis functions phi (phi)1，φ2，...，φN) So that the original sample can be approximated to the maximum extent, and the specific calculation formula for obtaining the orthogonal basis function is as follows:
Figure FDA0003059900910000021
in the formula, J represents an error function between a solution vector sample of the full-order model and a basis function in the least square sense,
Figure FDA0003059900910000022
as a discrete subset of the parameter space Λ, λ*To represent
Figure FDA0003059900910000023
A certain component of, t0And
Figure FDA0003059900910000024
respectively representing two discrete instants of the iteration time t,
Figure FDA0003059900910000025
representation storage in snapshot matrix
Figure FDA0003059900910000026
Data of (a), phiiRepresents the ith component of the orthogonal basis function;
decomposing a basis function through the obtained group of intrinsic orthogonal decomposition basis functions, taking a space formed by the group of basis functions as a function space where a reduced order model solution vector is located, and then projecting the full order model to the reduced order space by using Galerkin projection; the specific calculation formula for solving the reduced order model solution vector by using Galerkin projection is as follows:
Figure FDA0003059900910000027
wherein alpha represents a coefficient vector of the orthogonal basis function phi,
Figure FDA0003059900910000028
4. a method for simulating soft tissue deformation and path cutting as claimed in claim 1, wherein the step (2) comprises the steps of:
(21) simplifying and abstracting the surgical instruments into a line segment, wherein the collision detection of the surgical instruments and the soft tissue is the intersection point detection of the line segment and the triangular unit; suppose that the two end points of the line segment are I and L, respectively, and their position vectors are u, respectivelyI(xI，yI，zI) And uL(xL，yL，zL) Then, the spatial linear equation of the surgical instrument is specifically expressed as follows:
Figure FDA0003059900910000029
in the formula, x, y and z respectively represent three unknowns of a space linear equation, and k represents a parameter of the space linear equation; the coordinates of any node on the line segment are:
Figure FDA00030599009100000210
(22) determining a plane equation for a trigonometric cell intersecting the surgical instrument: let O, P, Q be the three endpoints of the triangle unit of the collision, and u be the position coordinates of the triangle unit of the collisionO(xO，yO，zO)、uP(xP，yP，zP) And uQ(xQ，yQ，zQ) And the normal vector of the triangle unit is N (N)x，ny，nz) Therefore, the plane equation of the triangle unit is:
Ux+Vy+Wz+T＝0
in the formula, U, V, W and T represent four parameters to be solved of a plane equation, and x, y and z represent unknowns of the plane equation;
according to the point normal method for calculating the plane equation, the plane equation can be obtained through the coordinates of the point O and the normal vector N:
nx(x-xO)+ny(y-yO)+nz(z-zO)＝0
wherein U is nx，V＝ny，W＝nz，T＝-nxxO-nyxO-nzzO；
(23) Simultaneous linear equation and plane equation to obtain the position u of the intersection S of the surgical instrument and the plane of the triangle unit at any discrete momentS(xS，yS，zS) The specific calculation formula is as follows:
Figure FDA0003059900910000031
5. the method for simulating soft tissue deformation and path cutting as claimed in claim 1, wherein the Bezier curve in step (3) is formed by three control points and using 2 Bernstein basis functions; the 2-degree Bernstein basis function Ri，2The specific expression of (t) is as follows:
Figure FDA0003059900910000032
6. a method for simulating soft tissue deformation and path cutting as claimed in claim 1, wherein the surface cut equation of step (3) is:
Figure FDA0003059900910000033
wherein E (t) represents a position vector of a control point for drawing a cutting path at an iteration time t, Ri，2(t) represents the 2-degree Bernstein basis function.
7. 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, characterized in that the computer program, when loaded into the processor, implements the method for simulating soft tissue deformation and path cutting according to any of claims 1-6.
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