Soft tissue modeling method based on constraint energy function and finite element method

Abstract

The invention discloses a soft tissue modeling method based on a constraint energy function and a finite element method. The method comprises the following steps of: firstly, constructing a corresponding constraint energy function aiming at a specific interaction effect of soft tissue and a surgical instrument; and then, integrating the constructed constraint energy function into a total energy function of the soft tissue. Discretizing the continuous variation expression of the total energy into a nodal displacement function; next, a displacement field is initialized, and an initial solution and an iteration direction are calculated based on the gradient of the total energy function and a Hessian matrix; and the displacement field and the Lagrange multiplier are iteratively updated to gradually approach the optimal solution. And finally, checking a convergence condition, if a preset standard is met, terminating iteration, otherwise, continuing to update the gradient and the Hessian matrix until an expected solution is reached. According to the method, the calculation efficiency is remarkably improved while the soft tissue deformation simulation precision is improved, and the method is suitable for real-time simulation and interaction operation in a complex operation scene.

Description

Technical Field

[0001] The present invention belongs to the technical field of soft tissue deformation modeling, and specifically relates to a soft tissue modeling method based on a constrained energy function and a finite element method, which is suitable for real-time interaction and deformation simulation between surgical instruments and diseased tissues during virtual surgery. Background Art

[0002] In the field of modern medicine, virtual surgery simulation technology has developed rapidly and has become an important tool for surgical training, preoperative planning, and intraoperative navigation. Virtual surgery simulation can not only improve surgeons' operating skills, but also effectively reduce surgical risks and improve patients' postoperative recovery. However, the realism and accuracy of the virtual surgical environment largely rely on the precise modeling and real-time simulation of human soft tissues.

[0003] Soft tissue modeling is a complex task, mainly because soft tissue has nonlinear, anisotropic and viscoelastic mechanical properties. Traditional soft tissue modeling methods are mostly based on empirical formulas or simplified assumptions, which are difficult to accurately reflect the true mechanical behavior of soft tissue. In practical applications, these methods usually cannot meet the requirements of precise deformation and real-time interaction during virtual surgery.

[0004] In order to solve the above problems, the finite element method (FEM) is widely used in soft tissue modeling. The finite element method can simulate the complex mechanical behavior of soft tissue through discretization and numerical solution, and has good accuracy and adaptability. However, the standard finite element method usually has a large amount of calculation and is difficult to meet the needs of real-time simulation. Therefore, researchers have proposed many improved methods, such as multi-scale models, subspace methods, and acceleration algorithms, to improve computational efficiency.

[0005] On the other hand, constrained energy methods, as an effective energy minimization technique, have also been introduced into soft tissue modeling. By introducing physical or geometric constraints, the stability and realism of the model can be significantly improved. However, how to effectively combine constrained energy methods with finite element methods to further improve the accuracy and real-time performance of soft tissue deformation simulation is still a research problem that needs to be solved urgently. Summary of the invention

[0006] In view of the shortcomings and difficulties in the prior art, the present invention aims to provide a soft tissue modeling method combining a constrained energy function and a finite element method. The method can realize real-time interaction and deformation simulation between surgical instruments and diseased tissues during virtual surgery, thereby providing efficient and reliable technical support for virtual surgery simulation.

[0007] The present invention is achieved through the following technical solutions:

[0008] In the finite element deformation simulation of soft tissue, the interaction effect between surgical instruments and soft tissue is represented by applying a set of energy functions. The key to our proposed new model is to construct additional energy functions and incorporate them into the total energy function. According to the energy minimization principle, the equilibrium state of the system corresponds to the minimum value of the total energy. Therefore, the deformation caused by the interaction between soft tissue and surgical instruments can be realized by solving the minimization problem of the total energy function.

[0009] The method comprises the following steps:

[0010] S1: Aiming at the specific interaction effects between soft tissue and surgical instruments, the corresponding constraint energy function is constructed to describe its interaction characteristics.

[0011] S2: Incorporate the constructed constraint energy function into the total energy function of the soft tissue, thereby introducing the interaction constraints into the solution process of the deformation model.

[0012] S3: Discretize the continuous variational expression of total energy into node displacement u h The function is used for numerical solution of deformation equilibrium state by computer.

[0013] S4: Initialize the displacement field Secondly, the gradient and Hessian matrix of the total energy function are calculated to obtain the initial solution and iteration direction.

[0014] S5: Based on the energy function gradient and Hessian matrix, update the displacement field u and Lagrange multiplier λ * , gradually approaching the optimal solution.

[0015] S6: Check the convergence conditions. If the preset convergence criteria are met, terminate the iteration; otherwise, continue to update the gradient and Hessian matrix of the energy function until the expected solution is reached.

[0016] Furthermore, as described in S2, the constructed constraint energy function is incorporated into the total energy function of the soft tissue, thereby introducing the interaction constraints into the solution process of the deformation model.

[0017] The interaction constraint energy function is represented by a general symbol W con To express it, the total energy function can be written as:

[0018] W total =∫ Ω W(C)dV+W con

[0019] Where Ω is the volume of the deformed region, W(C) is the strain energy density function of the material, C is the right Cauchy-Green strain tensor, and W conIt is a constrained energy function used to describe the interaction effect between surgical instruments and soft tissues.

[0020] Furthermore, in S3, the continuous variational expression of the total energy is discretized into the node displacement u h The function is used by the computer to numerically solve the deformation equilibrium state. Specifically, the equilibrium state of the system corresponds to the minimum value of the total energy function. By calculating the first-order variation of the total energy function and making it zero, the energy function W is found total The extreme point of is as shown below:

[0021] δW total =0

[0022] For the material strain energy density function, its variational formula is:

[0023]

[0024] Where δC=δ(F T F) is the variation of the deformation tensor C, and F is the deformation gradient matrix. For the constraint term, its variational formula is:

[0025]

[0026] The node displacement u is discretely expressed on the finite element mesh as u h ,Ω n is the volume of the nth finite element unit, discretizing the total energy function as a function of the node displacement:

[0027]

[0028] Furthermore, the displacement field is initialized in S4 Secondly, the gradient and Hessian matrix of the total energy function are calculated to obtain the initial solution and iteration direction.

[0029]

[0030] Where H is the Hessian matrix of the total energy function, is its gradient.

[0031] Furthermore, in S5, based on the energy function gradient and the Hessian matrix, the displacement field u and the Lagrange multiplier λ are updated. * , gradually approaching the optimal solution.

[0032]

[0033] in is the updated Lagrange multiplier, u k is the updated displacement field.

[0034] Beneficial effects of the present invention

[0035] The soft tissue model of the present invention innovatively combines the constraint energy function with the finite element method to achieve high precision and real-time simulation of soft tissue deformation. By constructing a specific constraint energy function to describe the interaction between surgical instruments and soft tissues, this method significantly improves the model's ability to handle complex situations in virtual surgery. Compared with traditional finite element modeling technology, the present invention achieves a better balance between computational efficiency and interactivity, providing a solid theoretical and technical foundation for the accurate simulation of complex surgical scenes. BRIEF DESCRIPTION OF THE DRAWINGS

[0036] Figure 1 Flow chart of the constraint energy finite element model deformation calculation of the present invention

[0037] Figure 2 A rectangular soft tissue model diagram of the example described in the present invention

[0038] Figure 3 The friction-constrained plastic deformation diagram of the example described in the present invention is DETAILED DESCRIPTION

[0039] In order to explain the technical details of the present invention more clearly, a specific embodiment of soft tissue modeling is now described. This embodiment shows how to apply the constraint energy function model framework of the present invention in a surgical simulation system to describe the friction interaction when the surgical instrument contacts the soft tissue, thereby causing the plastic deformation of the tissue.

[0040] The cuboid soft tissue model in this example is as follows Figure 1 As shown, the cuboid contains 1134 vertices and 3682 tetrahedrons. In this example, the mechanical parameters are set as follows: Young's modulus E = 1.2 KPa, Poisson's ratio v = 0.5, shear modulus G = 0.3 KPa, and bulk modulus K = 15 KPa.

[0041] A soft tissue modeling method based on constrained energy function and finite element method, the process is as follows Figure 2 As shown, including:

[0042] S1: Aiming at the specific interaction effects between soft tissue and surgical instruments, the corresponding constraint energy function is constructed to describe its interaction characteristics.

[0043] S2: Incorporate the constructed constraint energy function into the total energy function of the soft tissue, thereby introducing the interaction constraints into the solution process of the deformation model.

[0044] S3: Discretize the continuous variational expression of total energy into node displacement u h The function is used for numerical solution of deformation equilibrium state by computer.

[0045] S4: Initialize the displacement field Secondly, the gradient and Hessian matrix of the total energy function are calculated to obtain the initial solution and iteration direction.

[0046] S5: Based on the energy function gradient and Hessian matrix, update the displacement field u and Lagrange multiplier λ * , gradually approaching the optimal solution.

[0047] S6: Check the convergence conditions. If the preset convergence criteria are met, terminate the iteration; otherwise, continue to update the gradient and Hessian matrix of the energy function until the expected solution is reached.

[0048] Furthermore, in S1, the corresponding friction constraint energy function is constructed for the plastic deformation effect caused by the contact between soft tissue and surgical instruments to describe the plastic deformation caused by the contact between soft tissue and surgical instruments. During the operation, the friction between the surgical instrument and the soft tissue will cause permanent deformation of the tissue. In order to accurately simulate this process, a new constraint can be introduced to describe the permanent deformation of the soft tissue under the friction of the surgical instrument and incorporated into the overall finite element simulation model.

[0049] In this example, the deformation field of brain tissue under the action of external force is u(e), where e is the spatial position vector. The permanent deformation field of brain tissue under the friction of surgical instruments is u p (e). The total deformation field of brain tissue is expressed as:

[0050] u t (e) = u(e) + u p (e)

[0051] The friction force f acts on the contact surface, causing permanent deformation of the points on the contact surface:

[0052] u p (e) = αf(e)

[0053] Among them, α = 0.3 is a proportionality coefficient, which describes the influence of friction on permanent deformation. Friction can be expressed as:

[0054] f(e)=μN(e)

[0055] Where μ is the friction coefficient, in this example μ = 0.3, and N(e) is the normal contact force. The permanent deformation field is expressed as:

[0056] u p (e) = αμN(e)

[0057] At this time, the constraint energy function caused by friction is:

[0058] W con =g(ut )=u t -u-αμN

[0059] Furthermore, as described in S2, the constructed constraint energy function is incorporated into the total energy function of the soft tissue, thereby introducing the interaction constraint into the solution process of the deformation model. A new friction constraint is used to describe the permanent deformation of the brain tissue under the friction of the surgical instrument, and is incorporated into the overall finite element simulation model.

[0060] The friction constraint energy function is represented by g(u t ), the total energy function can be written as:

[0061] W total =∫ Ω W(C)dV+W con =∫ Ω W(C)dV+g(u t )

[0062] where Ω is the volume of the deformed region, W(C) is the strain energy density function of the material, C is the right Cauchy-Green strain tensor, and g(u t ) is a constraint energy function used to describe the friction interaction effect between surgical instruments and soft tissues.

[0063] Furthermore, in S3, the continuous variational expression of the total energy is discretized into the node displacement u h The function is used by the computer to numerically solve the deformation equilibrium state. Specifically, the equilibrium state of the system corresponds to the minimum value of the total energy function. By calculating the first-order variation of the total energy function and making it zero, the energy function W is found total The extreme point of is as shown below:

[0064] δW total =0

[0065] For the material strain energy density function, its variational formula is:

[0066]

[0067] Where δC=δ(F T F) is the variation of the deformation tensor C, and F is the deformation gradient matrix. For the constraint term, its variational formula is:

[0068]

[0069] The node displacement u is discretely expressed on the finite element mesh as u h ,Ω n is the volume of the nth finite element unit, discretizing the total energy function as a function of the node displacement:

[0070]

[0071] Furthermore, the displacement field is initialized in S4 Secondly, the gradient and Hessian matrix of the total energy function are calculated to obtain the initial solution and iteration direction.

[0072]

[0073] Where H is the Hessian matrix of the total energy function, is its gradient.

[0074] Furthermore, in S5, based on the energy function gradient and the Hessian matrix, the displacement field u and the Lagrange multiplier λ are updated. * , gradually approaching the optimal solution.

[0075]

[0076] in is the updated Lagrange multiplier, u k is the updated displacement field.

[0077] Finally, the convergence condition is checked. If the preset convergence criteria are met, the iteration is terminated; otherwise, the gradient of the energy function and the Hessian matrix are continuously updated until the expected solution is reached. In this example, the cuboid soft tissue model contacts the surgical instrument and the friction-induced permanent deformation occurs. Figure 3 shown.

[0078] The present invention achieves high-precision and real-time simulation of soft tissue deformation by innovatively combining constraint energy functions with finite element methods. By using specific constraint energy functions to describe the interaction between surgical instruments and soft tissues, this method significantly enhances the model's ability to cope with complex situations in virtual surgery. Compared with traditional finite element modeling technology, the present invention achieves a better balance between computational efficiency and interactive experience, providing solid theoretical support and technical guarantee for the accurate simulation of complex surgical scenes.

[0079] The specific embodiments described herein are only used to illustrate the principles and spirit of the present invention. Those skilled in the art may make various modifications, changes or equivalent substitutions to these embodiments without departing from the spirit of the present invention and the scope of the claims, and these modifications or changes shall be deemed to be included in the protection scope of the present invention.

Claims

1. A soft tissue modeling method based on constrained energy function and finite element method, characterized in that include: S1: Aiming at the specific interaction effects between soft tissue and surgical instruments, the corresponding constraint energy function is constructed to describe its interaction characteristics; S2: Incorporate the constructed constraint energy function into the total energy function of the soft tissue, thereby introducing the interaction constraints into the solution process of the deformation model; S3: Discretize the continuous variational expression of total energy into node displacement u h The function is used for numerical solution of deformation equilibrium state by computer; S4: Initialize the displacement field Calculate the gradient and Hessian matrix of the total energy function to obtain the initial solution and iteration direction; S5: Based on the energy function gradient and Hessian matrix, update the displacement field u and Lagrange multiplier λ * , gradually approaching the optimal solution; S6: Check the convergence conditions. If the preset convergence criteria are met, terminate the iteration; otherwise, continue to update the gradient and Hessian matrix of the energy function until the expected solution is reached.

2. The method according to claim 1, characterized in that The total soft tissue energy function expression in step S2 is as follows: W total =∫ Ω W(C)dV+W con Where Ω is the volume of the deformed region, W(C) is the strain energy density function of the material, C is the right Cauchy-Green strain tensor, and W con It is a constraint energy function used to describe the interaction effect between surgical instruments and soft tissues.

3. The method according to claim 1, characterized in that Step S3 discretizes the continuous variational expression of total energy into node displacement u h The function is expressed as: Among them, Ω n Represents a discrete three-dimensional space, C h is the discretized right Cauchy-Green strain tensor, n represents the discretization of the soft group into n units, Wtotal h (u h ) represents the discretized total energy function, W(C h ) h represents the discretized strain energy function, Represents the discretized constraint energy function.

4. The method according to claim 1, characterized in that Step S4 obtains the initial solution and iteration direction by the following formula: Where H is the Hessian matrix of the total energy function, ▽Wtotal h is its gradient.

5. The method according to claim 1, characterized in that Step S5 updates the displacement field u and the Lagrange multiplier λ by the following equation: * : in, is the updated Lagrange multiplier, u k is the updated displacement field.

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