A Soft Tissue Modeling Method Based on Constrained Energy Function and Finite Element Method

Abstract

This invention discloses a soft tissue modeling method based on constrained energy functions and the finite element method. First, a corresponding constrained energy function is constructed for the specific interaction effects between soft tissue and surgical instruments. Then, the constructed constrained energy function is integrated into the total energy function of the soft tissue. The continuous variational expression of the total energy is discretized into nodal displacement functions. Next, the displacement field is initialized, and the initial solution and iteration direction are calculated based on the gradient and Hessian matrix of the total energy function. By iteratively updating the displacement field and Lagrange multipliers, the optimal solution is gradually approximated. Finally, the convergence condition is checked; if the preset criteria are met, the iteration terminates; otherwise, the gradient and Hessian matrix are updated until the expected solution is reached. This method significantly improves computational efficiency while enhancing the accuracy of soft tissue deformation simulation, making it suitable for real-time simulation and interactive operation in complex surgical scenarios.

Description

Technical Field

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

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

[0003] Soft tissue modeling is a complex task, primarily due to the nonlinear, anisotropic, and viscoelastic mechanical properties of soft tissues. Traditional soft tissue modeling methods are often based on empirical formulas or simplifying assumptions, making it difficult to accurately reflect the true mechanical behavior of soft tissues. In practical applications, these methods typically fail to meet the requirements of precise deformation and real-time interaction during virtual surgery.

[0004] To address the aforementioned issues, the Finite Element Method (FEM) has been widely applied in soft tissue modeling. The FEM can simulate complex soft tissue mechanical behavior through discretization and numerical solution, exhibiting good accuracy and adaptability. However, standard FEM methods typically involve high computational costs, making them unsuitable for real-time simulations. Therefore, researchers have proposed many improved methods, such as multi-scale models, subspace methods, and acceleration algorithms, to enhance 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 the finite element method to further improve the accuracy and real-time performance of soft tissue deformation simulation remains a research challenge that urgently needs to be solved. Summary of the Invention

[0006] To address the shortcomings and challenges of existing technologies, this invention aims to provide a soft tissue modeling method that combines constrained energy functions and the finite element method. This method enables real-time interaction and deformation simulation between surgical instruments and diseased tissues during virtual surgery, thus providing efficient and reliable technical support for virtual surgical simulation.

[0007] This invention is achieved through the following technical solution:

[0008] In finite element method (FEM) simulations of soft tissue deformation, the interaction between surgical instruments and soft tissue is represented by applying a set of energy functions. The key to our proposed novel model lies in constructing additional energy functions and incorporating them into the total energy function. According to the principle of energy minimization, the equilibrium state of the system corresponds to the minimum of the total energy. Therefore, the deformation induced by the interaction between soft tissue and surgical instruments can be achieved by solving the problem of minimizing the total energy function.

[0009] The method includes the following steps:

[0010] S1: Construct corresponding constrained energy functions to describe the interaction characteristics of soft tissue and surgical instruments for specific interaction effects.

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

[0012] S3: Discretize the continuous variational expression of the total energy into nodal displacements u. h The function is used by computers to numerically solve for the deformation equilibrium state.

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

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

[0015] S6: Check the convergence condition. If the preset convergence criterion is met, terminate the iteration; otherwise, continue to update the gradient of the energy function and the Hessian matrix 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 interactive constraints into the solution process of the deformation model.

[0017] The interactive constraint energy function is represented by a general notation W. con In other words, 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 between surgical instruments and soft tissue.

[0020] Furthermore, in S3, the variational expression for the continuous total energy is discretized into nodal displacements u. h The function is used by computers to numerically solve for the deformation equilibrium state. Specifically, the equilibrium state of the system corresponds to the minimum value of the total energy function. The energy function W is found by calculating the first variation of the total energy function and setting it to zero. total The extreme points are shown in the following equation:

[0021] δW total =0

[0022] The variational expression for the material strain energy density function is:

[0023]

[0024] Where δC=δ(F) T F) is the variational expression of the deformation tensor C, where F is the deformation gradient matrix. The variational expression for the constraint terms is:

[0025]

[0026] The discrete representation of the nodal displacement u on the finite element mesh is u h Ω n Let n be the volume of the nth finite element. Discretize the total energy function as a function of nodal displacements:

[0027]

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

[0029]

[0030] In the formula, H is the Hessian matrix of the total energy function, ▽Wtotal h It is its gradient.

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

[0032]

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

[0034] Beneficial effects of the present invention

[0035] This invention innovatively combines constraint energy functions with the finite element method to achieve high accuracy and real-time performance in simulating soft tissue deformation. By constructing specific constraint energy functions to describe the interaction between surgical instruments and soft tissue, this method significantly enhances the model's ability to handle complex scenarios in virtual surgery. Compared to traditional finite element modeling techniques, this invention achieves a better balance between computational efficiency and interactivity, providing a solid theoretical and technical foundation for the accurate simulation of complex surgical scenarios. Attached Figure Description

[0036] Figure 1 This is a flowchart of the deformation calculation of the constrained energy finite element model described in this invention.

[0037] Figure 2 This is a cuboid soft tissue model diagram of an example of the present invention.

[0038] Figure 3 This is a diagram of friction-constrained plastic deformation in an example of the present invention. Detailed Implementation

[0039] To more clearly explain the technical details of the present invention, a specific embodiment of soft tissue modeling is now provided. This embodiment demonstrates how the constrained energy function model framework of the present invention can be applied to a surgical simulation system to describe the frictional interaction between surgical instruments and soft tissue, thereby causing plastic deformation of the tissue.

[0040] The cuboid soft tissue model in this example is as follows: Figure 2 As shown, the cuboid contains 1134 vertices and 3682 tetrahedrons. The mechanical parameters in this example 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 of which is as follows: Figure 1 As shown, it includes:

[0042] S1: Construct corresponding constrained energy functions to describe the interaction characteristics of soft tissue and surgical instruments for specific interaction effects.

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

[0044] S3: Discretize the continuous variational expression of the total energy into nodal displacements u. h The function is used by computers to numerically solve for the deformation equilibrium state.

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

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

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

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

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

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

[0051] Frictional force f acts on the contact surface, causing permanent deformation at points on the contact surface:

[0052] u p (e)=αf(e)

[0053] Here, α = 0.3 is a proportionality constant that describes the degree of influence of friction on permanent deformation. Friction can be expressed as:

[0054] f(e)=μN(e)

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

[0056] u p (e)=αμN(e)

[0057] The constraint energy function caused by friction at this time is:

[0058] Wcon =g(u t )=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 interactive constraints into the solution process of the deformation model. New frictional constraints are used to describe the permanent deformation of brain tissue under the frictional action of surgical instruments and are incorporated into the overall finite element simulation model.

[0060] The friction constraint energy function is represented by the symbol g(u). t The total energy function can be expressed 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 The constrained energy function is used to describe the frictional interaction between surgical instruments and soft tissue.

[0063] Furthermore, in S3, the variational expression for the continuous total energy is discretized into nodal displacements u. h The function is used by computers to numerically solve for the deformation equilibrium state. Specifically, the equilibrium state of the system corresponds to the minimum value of the total energy function. The energy function W is found by calculating the first variation of the total energy function and setting it to zero. total The extreme points are shown in the following equation:

[0064] δW total =0

[0065] The variational expression for the material strain energy density function is:

[0066]

[0067] Where δC=δ(F) T F) is the variational expression of the deformation tensor C, where F is the deformation gradient matrix. The variational expression for the constraint terms is:

[0068]

[0069] The discrete representation of the nodal displacement u on the finite element mesh is u h Ω n Let n be the volume of the nth finite element. Discretize the total energy function as a function of nodal displacements:

[0070]

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

[0072]

[0073] In the formula, H is the Hessian matrix of the total energy function, ▽Wtotal h It is its gradient.

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

[0075]

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

[0077] Finally, check the convergence condition. If the preset convergence criterion is met, terminate the iteration; otherwise, continue updating the gradient of the energy function and the Hessian matrix until the expected solution is reached. In this example, the cuboid soft tissue model comes into contact with the surgical instruments and undergoes friction-induced permanent deformation, as shown below. Figure 3 As shown.

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

[0079] The specific embodiments described herein are merely illustrative of the principles and spirit of the invention. Those skilled in the art can make various modifications, alterations, or equivalent substitutions to these embodiments without departing from the spirit and scope of the claims, and all such modifications or alterations should be considered to be included within the protection scope of the invention.

Claims

1. A soft tissue modeling method based on constrained energy function and finite element method, characterized in that... include: S1: Construct corresponding constrained energy functions to describe the interaction characteristics of soft tissue and surgical instruments for specific interaction effects; S2: The constructed constraint energy function is incorporated into the total energy function of the soft tissue, thereby introducing interactive constraints into the solution process of the deformation model; the expression for the total energy function of the soft tissue 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 constrained energy function used to describe the interaction between surgical instruments and soft tissue; S3: Discretize the continuous variational expression of the total energy into nodal displacements u. h The function is used by computers to numerically solve for the deformation equilibrium state; 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: Update the displacement field u and Lagrange multiplier λ based on the gradient of the total energy function and the Hessian matrix. * It gradually approaches the optimal solution; S6: Check the convergence condition. If the preset convergence criterion is met, terminate the iteration; otherwise, continue to update the gradient of the total energy function and the Hessian matrix until the expected solution is reached.

2. The method according to claim 1, characterized in that... Step S3 discretizes the variational expression for the continuous total energy into nodal displacements u. h The function is expressed as: Among them, Ω n C represents a discrete three-dimensional space. h It is the discretized right Cauchy-Green strain tensor, where n represents the discretization of the soft tissue into n elements, Wtotal h (u h W(C) represents the discretized total energy function. h ) h This represents the discretized strain energy function. This represents the constrained energy function for discretization.

3. The method according to claim 1, characterized in that... Step S4 obtains the initial solution and iteration direction using the following formula: Where H is the Hessian matrix of the total energy function. yes The gradient.

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