CN111488670B - Nonlinear mass point spring soft tissue deformation simulation method - Google Patents
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Abstract
The invention discloses a nonlinear mass spring soft tissue deformation simulation method, which comprises the following steps: acquiring medical image data of soft tissues, segmenting an organ image, and obtaining three-dimensional mesh data by utilizing Delaunay triangular patch mesh subdivision; on the basis of three-dimensional mesh generation, taking mesh nodes as mass points, wherein the sides connected between the nodes are spring components formed by connecting structural springs and damping springs in parallel to form a mass point spring soft tissue model, judging whether a surgical instrument and the mass point spring soft tissue model collide with each other, if so, calculating the stress of each mass point, updating the position and the speed of the mass point, and calculating the deformation value of the mass point on the surface of the mass point spring soft tissue model until no collision occurs between the surgical instrument and the mass point spring soft tissue model; and then outputting a simulation result, performing visual rendering, and ending the whole process. The invention can simulate the deformation of soft tissues more accurately in real time.
Description
Technical Field
The invention relates to the technical field of soft tissue deformation modeling, in particular to a nonlinear mass spring soft tissue deformation simulation method.
Background
Simulation of physical movement of organs and tissues is a core problem in the fields of anatomical teaching, surgical simulation, surgical training and the like. Human tissue and organs generally have soft tissue characteristics, which are usually expressed by material properties such as non-uniformity, anisotropy, quasi-incompressibility, nonlinearity, plasticity, viscoelasticity and the like, and the accurate and real-time simulation of the deformation process of the soft tissue is a very challenging research subject. The mass spring model is one of more models applied to the aspect of human soft tissue modeling, and most of the existing interactive soft tissue deformation simulation models based on the mass spring model simplify the constitutive relation of stress-strain of soft tissue into a linear relation. The method is low in computation complexity, easy to discretize time and further iterate computation, and low in computation accuracy.
Due to the requirement of real-time simulation calculation, the existing method mostly adopts a linear elasticity theory to describe the deformation of the soft tissue, namely, an online hysteresis relation between a stress component and infinitesimal strain is assumed. The linear elastic model has the advantages that the global stiffness matrix is constant in the whole simulation process, and can be pre-calculated, inverted and stored, so that the calculation efficiency is improved. In fact, biological soft tissue is a non-linear material that can be approximated only with small deformations. In addition, the large deformation, the large rotation and the large displacement in the soft tissue deformation can generate geometric nonlinearity, which is not solved by the classical linear elasticity theory.
In a word, the existing interactive simulation system based on the mass-spring model simplifies the soft tissue deformation into the linear elastic model in order to ensure that the model is solved in real time, so that the calculation efficiency is improved. The assumption of using linear elasticity as the basic model, while reducing the computational effort at run-time, limits the accuracy of modeling physical materials. More importantly, the linear elastic model can only approximate the soft tissue deformation under the conditions of small displacement and small deformation. However, biological soft tissue has material non-linear characteristics, and has geometric non-linear characteristics in the case where the amount of deformation is large.
Disclosure of Invention
The invention aims to provide a nonlinear mass spring soft tissue deformation simulation method aiming at the technical defects in the prior art, wherein Euler elasticity is used for describing spring deformation in a mass spring model so as to represent the nonlinearity of soft tissue, namely Euler elastic energy is used for describing the spring deformation in the mass spring model, and then the spring length is calculated so as to obtain a stress-strain relation meeting the nonlinearity, so that the constitutive relation of the nonlinearity of the soft tissue is represented.
The technical scheme adopted for realizing the purpose of the invention is as follows:
a nonlinear mass spring soft tissue deformation simulation method comprises the following steps:
acquiring medical image data of soft tissues, segmenting an organ image, and obtaining three-dimensional mesh data by utilizing Delaunay triangular patch mesh subdivision;
on the basis of three-dimensional mesh generation, mesh nodes are used as mass points, and edges connected between the nodes are spring components formed by connecting structural springs and damping springs in parallel to form a mass point spring soft tissue model;
detecting whether a collision occurs between the surgical instrument and the mass spring soft tissue model, if the collision occurs, calculating the stress of each mass, updating the position and the speed of the mass, and calculating the deformation value of the mass on the surface of the mass spring soft tissue model until the surgical instrument and the mass spring soft tissue model do not collide;
and outputting a simulation result, and finishing the whole process after performing visual rendering.
The invention considers the nonlinearity of biological soft tissue and can simulate the deformation of the soft tissue more accurately in real time. The nonlinear mass point spring soft tissue deformation simulation method provided by the invention is closer to the real soft tissue biomechanics characteristic, and can effectively ensure the simulation precision and the calculation complexity.
Drawings
FIG. 1 is a flow chart of a method for simulating non-linear mass spring soft tissue deformation;
FIG. 2 is a diagram of a liver triangulation visualization result;
FIG. 3 is a schematic view of a spring assembly between two mass points;
FIG. 4 is a schematic diagram of a neighborhood of a particle;
FIG. 5 is a schematic diagram of the parameter angles for calculating the total area of all triangular patches in a particle neighborhood;
FIG. 6 is a visualization result after visual rendering and a schematic diagram of motion simulation (the upper layer is in a static state, and the lower layer is in a motion state);
FIGS. 7-8 are graphs of the effect on stress-strain nonlinearity for the spheroid, liver parameter a, respectively:
fig. 9-10 are graphs of the effect on stress-strain nonlinearity for the sphere, liver parameter b, respectively.
Detailed Description
The invention is described in further detail below with reference to the figures and the specific embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention.
According to the invention, Euler elasticity is used for describing spring deformation in the mass spring model so as to represent the nonlinearity of soft tissue, namely Euler elasticity energy is used for describing the spring deformation in the mass spring model, and the calculated spring length can obtain a stress-strain relation meeting the nonlinearity so as to represent the constitutive relation of the nonlinearity of the soft tissue.
The mathematician Euler (Euler) develops the nonlinear constitutive relation of elastic energy and elastic lines according to the principle of minimum potential energy when studying the steady-state problem of the elastic soft rod under the action of external force. Thereafter, the mathematician Mumford introduced euler elastic energy into the field of computer vision. The core idea is that the deformation process of the cork stick is described by two geometric quantities of length and curvature, namely the following energy is considered:
E(l)＝∫l(a+bκ2(x))ds，
where a, b are two positive parameters, κ is the curvature of curve l at position x, and ds is the arc length.
As shown in FIG. 1, the nonlinear mass spring soft tissue deformation simulation method of the invention comprises the following steps:
three-dimensional mesh generation
And acquiring medical image data such as CT/MR of soft tissue, segmenting the organ to be researched, and obtaining three-dimensional mesh data by utilizing Delaunay triangular patch mesh subdivision. Taking a liver as an example, fig. 2 shows a liver triangulation visualization result, and a liver surface is formed by combining points, edges and triangular patches.
On the basis of three-dimensional mesh generation, mesh nodes are taken as mass points, edges connected between the nodes are spring assemblies formed by connecting structural springs and damping springs in parallel, and the initial model of soft tissue deformation is formed by referring to the graph shown in figure 3. FIG. 3 is a schematic view of a spring assembly, xi,xjAre two connected particles.
Second, collision detection
And detecting whether the surgical instrument collides with the soft tissue, if so, determining that the soft tissue is deformed under the action of force, and executing the next step, otherwise, outputting a simulation result and finishing the whole process.
Thirdly, all the particles are traversed, and the particle stress is calculated
Assume a total of n particles, for any particle xi1, …, n, with mass miE R and the resultant force f acting at that pointi∈R3. The particles on the soft tissue surface satisfy the following kinetic equation, newton's second law equation:
where M is a 3n x 3n diagonal quality matrix,
is the second derivative of displacement with respect to time, i.e. the acceleration of the particle.
1. External force applied to the calculated point
The external force applied to the mass point includes the gravity of the mass point and the artificially applied external force, i.e.
Fi ext＝mig+fu,
Wherein g is 9.8N/kg, f is standard gravitational accelerationuIs an artificially applied external force.
2. Computing damping force borne by particles
Since energy dissipation occurs during deformation, the viscous force is represented by the spring damping force, particle xiAnd xjDamping force therebetween
Comprises the following steps:
wherein,
damping constant, v, of the springiRepresenting the velocity of the particle i.
3. Calculating the spring force borne by the particles
Wherein,
spring rate of spring, /)0And l' represent the original length of the spring and the deformed spring length, respectively.
The invention uses Euler elastic energy to put forwardNew calculation of two particles xi,xjLength l of spring in between(i,j)The formula (c) is as follows:
wherein, κ(i,j)Using mass point xi,xjAverage curvature of
Is estimated from the mean value of (i.e. of)
Unlike the original Euler elastic energy, the curvature on the curved surface is used for replacing the curvature of the curved line for calculation, so that the calculation complexity is simplified. For a triangular mesh describing a smooth surface, there are many methods to estimate its vertex normal vector and curvature.
The invention calculates the particle normal vector and the average curvature using the following method.
Estimate the particle normal vector:
particle xiAnd the region of the triangular patch of adjacent particles is considered xiA neighborhood of (A), as in FIG. 4, x in the triangular mesh modeliIs a 'neighborhood' of the vertex.
Order to
Representing particles
The normal vector of (a) is,
is a particle xiThe set of surrounding triangular patches,
is a particle xiSet of surrounding triangular patches normal vectors, | xiIs xiThe total number of surrounding adjacent particles. Particle xiIs estimated as a weighted sum of the normal vectors of its surrounding triangular patches:
wherein,
is xiTo xjThe edge vector of (a) is calculated,
is the cross product.
Estimate the mean curvature of the particles:
in the present invention, the mean curvature manifold-based discretization method proposed by Matllieu-Desbrun is used to estimate the particle xiAverage curvature of
Wherein,
is any particle xiWith respect to the gradient operator of its coordinates (x, y, z), A is the particle xiTotal area of all triangular patches in a neighborhood, αj,βjIs defined as shown in FIG. 5, αj,βjTwo included angles are respectively formed, and the included angles are opposite angles of two adjacent triangular surface patches.
Fourthly, calculating the deformation value of the particles on the surface of the soft tissue
And (3) calculating the resultant force of the surrounding adjacent particles borne by the particles according to the above 1,2 and 3, and obtaining the deformation value of the particles on the surface of the soft tissue by using the kinetic equation of the particles on the surface of the soft tissue.
Wherein, the kinetic equation of the particle on the soft tissue surface is as follows:
wherein,
particle xiSubject to a resultant force fi＝fi ext-fi s-fi d.
fi dIs a particle xiResultant force of the applied damping, fi sIs a particle xiThe resultant force of the stressed springs;
the particle position and velocity are updated according to the following explicit framework Verlet integral to obtain the deformation state, i.e. the deformation value, of the particles on the soft tissue surface.
X is to bei(t + Δ t) is assigned to xi(t)，vi(t + Δ t) is assigned to vi(t) judging whether collision occurs.
In the formula, xi(t + Δ t) represents the position of the particle at time t + Δ t, vi(t + Δ t) represents the velocity of the particle at time t + Δ t, xi(t)，vi(t) represents the position and velocity of the particle at time t, respectively.
Visual rendering
The OpenGL is used to perform visual rendering on the real-time updated data, and a visualization result and a motion simulation diagram, taking a liver as an example, are given as shown in fig. 6.
Sixth, system stress-strain analysis
Taking a sphere and a liver as an example, figures 7-8 and 9-10 show the non-linear relationship of force and displacement for different parameters. For different a and b conditions, the deformation relationship between the spherical surface and the liver under external force is tested and compared. From the results, it can be seen that as a, b increases, the non-linearity of the system becomes stronger.
It should be noted that, for the non-linearity of the soft tissue, the spring coefficient in the deformation process may also be set according to the stress-strain relationship of the soft tissue, but the method has the premise that the real stress-strain relationship of the soft tissue is obtained through experimental tests, which is complex and costly, especially for the simulation of the human soft tissue.
Compared with the prior art, the invention has the following advantages:
1. the non-linear strain of the spring is realized by introducing Euler elasticity, the limitation of the existing mass point spring system for simulating soft tissue motion is improved, and the problem that the requirement of actual simulation of human tissue and organs cannot be met because the spring elasticity and the spring deformation quantity, namely stress-strain, in the traditional mass point spring model are in a linear relationship is solved.
2. Because of the elasticity of the spring in the mass spring system, the deformation is not only related to the length of the spring, but also to the degree of bending of the spring, and the curvature is a geometric quantity describing the degree of bending of the spring. Therefore, the method for describing the spring deformation by using the length and the curvature of the spring can simulate the soft tissue deformation process more accurately.
The invention uses the Euler elastic energy to depict the spring deformation, namely the spring deformation is controlled by the length and the curvature of the spring at the same time, and the curvature is a nonlinear function of the length, so that the elastic force obtained by calculation of the Euler elastic energy has a nonlinear relation with the length variation of the spring.
The method estimates the average curvature on the curved surface in a neighborhood of the surface grid, can effectively reduce the calculation cost, is suitable for parallel calculation, and can ensure the realization of real-time simulation in real simulation.
The mass point spring model with the nonlinear stress-strain relation can be used for simulating the deformation of soft tissues with the material properties of quasi-incompressibility, nonlinearity, viscoelasticity and the like, such as liver, kidney and the like.
The foregoing is only a preferred embodiment of the present invention, and it should be noted that, for those skilled in the art, various modifications and decorations can be made without departing from the principle of the present invention, and these modifications and decorations should also be regarded as the protection scope of the present invention.
Claims (1)
1. A nonlinear mass spring soft tissue deformation simulation method is characterized by comprising the following steps:
acquiring medical image data of soft tissues, segmenting an organ image, and obtaining three-dimensional mesh data by utilizing Delaunay triangular patch mesh subdivision;
on the basis of three-dimensional mesh generation, mesh nodes are used as mass points, and edges connected between the nodes are spring components formed by connecting structural springs and damping springs in parallel to form a mass point spring soft tissue model;
detecting whether a collision occurs between the surgical instrument and the mass spring soft tissue model, if the collision occurs, calculating the stress of each mass, updating the position and the speed of the mass, and calculating the deformation value of the mass on the surface of the mass spring soft tissue model until the surgical instrument and the mass spring soft tissue model do not collide;
outputting a simulation result, performing visual rendering, and ending the whole process;
and updating the position and the velocity of the particle according to the following explicit framework Verlet integral to obtain the particle x on the surface of the particle spring soft tissue modeliThe deformation value of (a);
fi＝fi ext-fi s-fi d，
fi ext＝mig+fu，
in the formula, miIs a particle xiMass of (f)iIs particle xiThe resultant force is exerted; f. ofi extIs particle xiThe external force includes mass point gravity mig and an artificially applied external force fu；fi dIs a particle xiResultant force of the applied damping, fi sIs a particle xiThe resultant force of the stressed springs;
is particle xiAnd xjThe damping force between the two springs is reduced,
damping coefficient of spring, viRepresents particle xiThe speed of (d);
is particle xiAnd xjThe force of the spring in between is,
spring rate of spring, /)0And l' represent the original length of the spring and the length of the deformed spring, respectively;
two particles xi,xjLength of spring in between l(i,j)The formula (c) is as follows:
wherein a and b are two positive parameters, curvature kappa(i,j)Using particle xi,xjAverage curvature of
Is estimated from the average of;
estimation of particle x using mean curvature manifold-based discretization methods proposed by Matllieu-DesbruniAverage curvature of
Wherein,
is any particle xiWith respect to the gradient operator of its coordinates (x, y, z), A is the particle xiTotal area of all triangular patches in a neighborhood, αj,βjAre respectively two included angles, which are mass points xiThe opposite angles of two adjacent triangular patches in a neighborhood;
wherein,
is xiTo xjThe edge vector of (a) is calculated,
is cross product, NiIs the estimated particle xiThe normal vector of (a) is calculated,
is particle xiThe surrounding triangular patch normal vector, | xiIs xiThe total number of surrounding adjacent particles.
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