A bone tissue mechanical analysis method, system and device based on a physical information neural operator

Abstract

The application discloses a bone tissue mechanical analysis method based on a physical information neural operator, and the method comprises the following steps: acquiring CT image data of bone tissue and extracting a bone tissue mask; constructing a finite element model of the bone tissue and acquiring a finite element reference solution; constructing a first three-dimensional input tensor; inputting the first three-dimensional input tensor into a physical information neural operator network; constructing a structure weight field and superimposing a weighted high-frequency displacement residual to the overall displacement field to obtain a predicted displacement field; determining corresponding force balance equation residuals and boundary condition residuals based on the predicted displacement field, and training the physical information neural operator network; acquiring CT image data of bone tissue to be analyzed and constructing a second three-dimensional input tensor, and inputting the trained physical information neural operator network to obtain a predicted apparent displacement field; and the application effectively compensates for the problem that high-frequency displacement components of bone boundaries and material mutation regions are lost due to frequency domain truncation of a traditional FNO.

Description

Technical Field

[0001] This invention belongs to the field of medical image analysis technology, and in particular relates to a method, system and device for bone tissue biomechanical analysis based on physical information neural operators. Background Technology

[0002] With the development of medical imaging technology and digital healthcare, patient-specific bone biomechanical analysis based on CT images has extremely important clinical application value in areas such as bone strength evaluation, fracture risk assessment, and preoperative planning for orthopedic surgery. The mechanical response of bone (such as internal stress and strain distribution) is closely related to bone density and trabecular bone structure. Accurately obtaining these mechanical parameters is crucial for understanding the pathogenesis and development mechanisms of diseases such as osteoporosis and spinal degeneration.

[0003] Currently, conventional bone biomechanics analysis methods mainly rely on finite element analysis (FEA). The specific process involves: first, extracting the patient's CT images and performing grayscale segmentation; then, reconstructing a three-dimensional geometric model of the bone tissue, meshing it, and assigning heterogeneous material parameters to each element based on the mapping relationship between CT grayscale and bone density; finally, setting physiological loads and boundary conditions, and submitting the solution to software such as Abaqus for numerical iterative solving. In recent years, to overcome the timeliness bottleneck of finite element analysis, artificial intelligence technology has been introduced into this field, resulting in surrogate models based on deep neural networks, physical information neural networks (PINN), and neural operators (such as the Fourier neural operator FNO) to learn the mapping from image features to mechanical response.

[0004] However, the aforementioned existing technologies have significant drawbacks: traditional finite element analysis methods not only require cumbersome 3D modeling and mesh generation processes, resulting in long computation times, but also demand extremely high levels of modeling experience from professional operators, making it difficult to achieve large-scale, rapid, and universally applicable analysis in clinical scenarios. Furthermore, most existing pure data-driven AI models heavily rely on the distribution of training samples, and their prediction results often do not conform to fundamental physical laws such as force balance equations and boundary conditions, resulting in poor generalization ability. Existing PINN methods mostly employ fully connected networks, which severely limit their expressive power and computational efficiency when facing the complex 3D heterogeneous spatial characteristics of bone tissue. In addition, when solving globally, the neural operator (FNO) retains finite frequency modes, resulting in a "smoothing" effect at the bone cortex boundary and in regions of abrupt changes in material parameters, losing high-frequency displacement details, leading to inaccurate prediction of local stress peaks, and failing to effectively address local residual corrections related to load conditions. Summary of the Invention

[0005] In view of this, the present invention provides a method, system and device for bone tissue biomechanics analysis based on physical information neural operators, which solves the problems of complex online modeling, low computational efficiency and difficulty in taking into account both global physical laws and local structural mutation details in bone biomechanics analysis technology.

[0006] To achieve the above objectives, in a first aspect, the technical solution of this invention to solve the technical problem is to provide a bone tissue mechanics analysis method based on physical information neural operators, comprising: acquiring CT image data of bone tissue, performing anatomical orientation unification and resampling preprocessing, converting the processed CT grayscale values ​​into voxel-level material parameters and extracting a bone tissue mask; constructing a finite element model of the bone tissue based on the voxel-level material parameters, the bone tissue mask, and a set load field and boundary condition field, and obtaining a finite element reference solution; constructing a first three-dimensional input tensor based on the HU field of the CT image data, the voxel-level material parameters, the bone tissue mask, the load field, and the boundary condition field; inputting the first three-dimensional input tensor into a physical information neural operator network, and outputting the overall displacement field through parallel Fourier neural operator branches in the physical information neural operator network. The high-frequency displacement residuals are predicted through a three-dimensional convolutional neural network branch. A structural weight field is constructed based on the HU gradient normalized field and the bone tissue boundary indicator field to define the spatial action area of ​​the high-frequency displacement residuals in the material parameter abrupt change region and the bone tissue boundary region. The weighted high-frequency displacement residuals are then superimposed on the overall displacement field to obtain the predicted displacement field. Based on the predicted displacement field, the corresponding force balance equation residuals and boundary condition residuals are determined. The physical information neural operator network is trained by combining the finite element reference solution, the force balance equation residuals, and the boundary condition residuals. CT image data of the bone tissue to be analyzed is acquired and a second three-dimensional input tensor is constructed. This tensor is input into the trained physical information neural operator network to obtain the predicted apparent displacement field. The apparent strain field and apparent stress field are calculated based on the apparent displacement field and the voxel-level material parameters and then output.

[0007] In one possible embodiment, constructing a structural weight field based on the HU gradient normalized field and the bone tissue boundary indicator field includes: determining the structural gradient magnitude based on the HU field, and normalizing the structural gradient magnitude to obtain the HU gradient normalized field; determining the structural weight field based on the bone tissue boundary indicator field and the HU gradient normalized field, expressed as: ; in, This represents the HU gradient normalized field. Indicates the boundary indicator field of bone tissue. and The weights are non-negative; clip indicates that the calculation result is truncated and restricted to the interval between 0 and 1.

[0008] In one possible embodiment, converting the processed CT grayscale values ​​into voxel-level material parameters and extracting the bone tissue mask includes: according to Determine the apparent density, and according to Determine the elastic modulus; where, Apparent density is expressed in g / cm³. 3 E represents the elastic modulus, with units of MPa, and Poisson's ratio is set as a preset constant; the elastic modulus and the Poisson's ratio are used as the voxel-level material parameters; HU threshold segmentation is performed based on the preprocessed CT grayscale values, voxels with values ​​higher than the preset threshold are marked as bone tissue, and voxels with values ​​lower than the preset threshold are marked as background, thus obtaining the bone tissue mask.

[0009] In one possible embodiment, the physical information neural operator network is trained by combining the finite element reference solution, the force balance equation residuals, and the boundary condition residuals, including: constructing a supervised pre-training loss using the finite element reference solution to pre-train the physical information neural operator network; and, based on the pre-training, adding the force balance equation residuals, displacement boundary residuals, and traction boundary residuals for joint fine-tuning training.

[0010] In one possible embodiment, a finite element model of the bone tissue is constructed based on the voxel-level material parameters, the bone tissue mask, and the set load field and boundary condition field, and a finite element reference solution is obtained, including: converting each effective bone tissue voxel after uniform orientation and resampling into an eight-node hexahedral finite element; assigning the elastic modulus corresponding to each voxel to its corresponding eight-node hexahedral finite element according to the voxel-level material parameters; applying the boundary condition field and the boundary conditions and load conditions corresponding to the load field; and determining the output nodal displacement, element stress, and element strain through finite element solution; wherein the nodal displacement is determined according to the arithmetic mean of the eight nodal displacements of the corresponding element.

[0011] In one possible embodiment, the applied load conditions include: an axial compressive load sampled in the range of 500N to 3000N, and superimposed flexure, extension, or shear loads.

[0012] In one possible embodiment, CT image data of bone tissue is acquired and preprocessed for anatomical orientation unification and resampling, including: unifying the three-dimensional HU body data of the CT image data to a preset standard anatomical orientation; resampling the orientation-unified HU body data to an equidistant voxel spacing consistent with the training data; wherein, the HU body data uses continuous interpolation and the bone tissue mask uses nearest neighbor interpolation.

[0013] Secondly, this invention provides a bone tissue mechanics analysis system based on physical information neural operators, comprising: a CT image acquisition module for acquiring CT image data of bone tissue, performing anatomical orientation unification and resampling preprocessing, converting the processed CT grayscale values ​​into voxel-level material parameters and extracting the bone tissue mask; a finite element solution module for constructing a finite element model of the bone tissue based on the voxel-level material parameters, the bone tissue mask, and a set load field and boundary condition field, and obtaining a finite element reference solution; an input tensor construction module for constructing a first three-dimensional input tensor based on the HU field of the CT image data, the voxel-level material parameters, the bone tissue mask, the load field, and the boundary condition field; and a neural operator prediction module for inputting the first three-dimensional input tensor into a physical information neural operator network, outputting the overall displacement field through parallel Fourier neural operator branches in the physical information neural operator network, and predicting the overall displacement field through three... The system employs a three-dimensional convolutional neural network branch to predict high-frequency displacement residuals. A spatial fusion module constructs a structural weight field based on the HU gradient normalized field and the bone tissue boundary indicator field. This weight field is used to define the spatial action area of ​​the high-frequency displacement residuals in regions of abrupt changes in material parameters and bone tissue boundaries. The weighted high-frequency displacement residuals are then superimposed onto the overall displacement field to obtain the predicted displacement field. A physical constraint training module determines the corresponding force balance equation residuals and boundary condition residuals based on the predicted displacement field. It then trains the physical information neural operator network by combining the finite element reference solution, the force balance equation residuals, and the boundary condition residuals. An online prediction output module acquires CT image data of the bone tissue to be analyzed and constructs a second three-dimensional input tensor. This tensor is input into the trained physical information neural operator network to obtain the predicted apparent displacement field. The module then calculates the apparent strain field and apparent stress field based on the apparent displacement field and the voxel-level material parameters, and outputs the results.

[0014] Thirdly, the present invention provides an electronic device, including a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor executes the program to implement a bone tissue biomechanical analysis method based on physical information neural operators.

[0015] Fourthly, the present invention provides a computer-readable storage medium having a computer program stored thereon, characterized in that the computer program, when executed by a processor, implements a bone tissue biomechanical analysis method based on physical information neural operators.

[0016] Compared with the prior art, the beneficial effects of the present invention are as follows: By constructing a network architecture that parallels the global solution branch of the Fourier Neural Operator (FNO) and the local residual branch of the 3D Convolutional Neural Network (CNN), and by using a structural weight field constructed from the HU gradient normalization field and the bone tissue boundary indicator field to spatially weight and define the high-frequency displacement residuals output by the CNN, the problem of loss of high-frequency displacement components in bone boundaries and material abrupt regions caused by frequency domain truncation in traditional FNO is effectively compensated, thus improving the prediction accuracy of peak stress and displacement field. At the same time, by using the voxel-by-voxel hexahedral finite element reference solution corresponding one-to-one with regular voxels and combining it with the residuals of the force balance equation and boundary conditions to jointly train the network, the apparent mechanical response at the voxel scale can be output by inputting CT images for only one forward inference in the online prediction stage. This completely eliminates the cumbersome finite element modeling and Abaqus solution process in traditional methods, and the prediction results satisfy both the mechanical control equation and the patient-specific material distribution. Under the premise of ensuring mechanical consistency, it realizes the high efficiency, intelligence and individualization of bone mechanics analysis. Attached Figure Description

[0017] Figure 1 A flowchart illustrating the steps of a bone tissue biomechanical analysis method based on physical information neural operators according to an embodiment of the present invention; Figure 2 The structure diagram of the CNN-FNO mechanical solution network with structure-aware local residual correction; Figure 3 Flowchart for online prediction of input specification verification and system dimensional apparent mechanics output; Figure 4 This is a graph showing the change in loss during joint training. Figure 5 Figure showing the experimental results comparing the effectiveness of the structure-aware local residual correction module; Figure 6 This is a block diagram of a bone tissue biomechanical analysis system based on physical information neural operators according to an embodiment of the present invention. Detailed Implementation

[0018] The technical solutions of the embodiments of this application will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only a part of the embodiments of this application, and not all of the embodiments. Based on the embodiments of this application, all other embodiments obtained by those of ordinary skill in the art without creative effort are within the scope of protection of this application.

[0019] The first embodiment of this invention provides a bone tissue biomechanical analysis method based on physical information neural operators. This method generally includes an offline training phase and an online prediction phase. The offline training phase mainly completes image preprocessing, finite element reference solution generation, and model training. The online prediction phase outputs the apparent mechanical response at the voxel scale by calling the trained model.

[0020] like Figure 1 As shown, the bone tissue biomechanical analysis method based on physical information neural operators includes the following steps: S100: Acquire CT image data of bone tissue and its corresponding HU field, perform anatomical orientation unification and resampling preprocessing, convert voxel values ​​in the processed HU field into voxel-level material parameters and extract bone tissue mask. Specifically, during the offline training phase, the medical digital imaging and communications (DICOM) sequence file of the bone tissue to be analyzed can be read through the Picture Archiving and Communication System (PACS). The image matrix size and Hounsfield Unit (HU) grayscale spatial distribution in the file are analyzed. After spatial coordinate transformation and isometric resampling, the outer edge contour of the bone tissue is extracted using image segmentation to form a binary bone tissue mask. Simultaneously, the resampled HU values ​​are converted into voxel-level material parameters for spatial heterogeneity using an empirical mapping equation.

[0021] In an optional embodiment, the CT image data undergoes anatomical orientation unification and resampling preprocessing, including: The 3D body data of CT images is uniformly converted to a preset standard anatomical orientation; Specifically, the image orientation patient metadata in the DICOM image header file is parsed, and the three-dimensional HU body data with different scanning coordinate systems are uniformly converted to RAI (Right-Anterior-Inferior) anatomical orientation through an affine transformation matrix to ensure the consistency of spatial characteristics of the input data and eliminate structural interference introduced by differences in acquisition parameters of heterogeneous medical devices.

[0022] The HU body data after orientation unification is resampled to the same equidistant voxel spacing as the training data; Among them, the HU body data uses continuous interpolation, and the bone tissue mask uses nearest neighbor interpolation; Specifically, the Region of Interest (ROI) is cropped, and the image is scaled to a fixed isostatic voxel spacing (e.g., a 128×128×128 regular voxel matrix with side lengths of 0.6 to 1.0 mm). HU volume data uses a continuous interpolation algorithm to preserve smooth transitions in physical density gradient information, while the bone tissue mask is forced to use a nearest-neighbor interpolation algorithm to prevent the generation of geometrically meaningless floating-point artifacts at tissue boundaries.

[0023] By combining a unified baseline orientation transformation with a classification space difference interpolation scheme, the structural interference introduced by the differences in acquisition parameters of heterogeneous medical devices is eliminated, ensuring strict alignment of subsequent input tensor dimension features and robustness of network inference.

[0024] In an optional embodiment, converting the processed CT grayscale values ​​into voxel-level material parameters and extracting the bone tissue mask includes: according to Determine the apparent density, and according to Determine the elastic modulus; in, Apparent density is expressed in g / cm³. 3 E represents the elastic modulus, with units of MPa, and Poisson's ratio is set to a preset constant. The elastic modulus and the Poisson's ratio are used as the voxel-level material parameters; Based on the preprocessed CT grayscale values, HU threshold segmentation is performed, and voxels above the preset threshold are marked as bone tissue, while voxels below the preset threshold are marked as background, thus obtaining the bone tissue mask.

[0025] Specifically, bone mineral density is strongly correlated with the attenuation coefficient of CT images. The HU (Heat Hue) grayscale values ​​after resampling are extracted and substituted into this linear equation to directly derive the apparent density, reflecting the compactness of the internal microscopic bone structure. Subsequently, a nonlinear power function equation for bone biomechanics is used to exponentially amplify the apparent density, calculating the heterogeneous elastic modulus characterizing the medium's resistance to elastic deformation. Combined with a fixed empirical Poisson's ratio constant (e.g., 0.3) at the macroscopic level, this constitutes a complete material parameter field encompassing the patient's specific internal physical properties. Simultaneously, a uniform HU threshold is set (e.g., a HU value greater than 150 is considered a bone tissue region). This bone tissue mask is used to limit the effective calculation range, avoiding invalid mechanical calculations in non-bone tissue regions.

[0026] By deeply embedding the empirical equation of nonlinear biomechanical stiffness into the data preprocessing workflow and combining it with binarized mask extraction, a quantitative conversion from dimensionless image grayscale to high-fidelity engineering mechanical parameters was achieved. This process is based on voxel-based automatic execution, eliminating the tedious geometric reconstruction and mesh generation steps in traditional finite element modeling. At the same time, by directly mapping the generated material parameters and masks to subsequent input tensors, the spatial consistency between the model input features and physical and mechanical properties is ensured.

[0027] S200 constructs a finite element model of bone tissue based on voxel-level material parameters, bone tissue mask, and set load field and boundary condition field, and obtains the finite element reference solution. During the offline training data construction phase, voxel-level material parameters and the bone tissue mask are imported into the finite element solution environment. Based on the bone tissue mask, a regularized finite element mesh model perfectly aligned with the spatial coordinates of the medical image is directly generated, and the voxel-level material parameters are directly assigned to the corresponding mesh elements. According to the set load field and boundary condition field, corresponding external loads and fixed constraints are applied to the finite element model to establish a static analysis step, which is then submitted to the solver for finite element calculation. The nodal displacements, element stresses, and element strains output by the solver, after spatial alignment processing, serve as the finite element reference solution for supervising network convergence.

[0028] In an optional embodiment, a finite element model of the bone tissue is constructed based on voxel-level material parameters, a bone tissue mask, and defined load and boundary condition fields, and a finite element reference solution is obtained, including: Each effective bone tissue voxel after uniform orientation and resampling is converted into an eight-node hexahedral finite element. Abandoning the traditional tetrahedral mesh generation, a mesh is established using regular voxels. The voxels with a value of 1 inside the bone tissue mask are transformed in situ into C3D8 or C3D8R eight-node hexahedral elements with mechanical load-bearing capacity without damage. The element side length is strictly equal to the spatial resolution spacing of the image voxels. No load-bearing elements are established outside the bone tissue mask, thus avoiding distortion of anatomical geometric reconstruction.

[0029] The elastic modulus of each voxel is assigned to its corresponding eight-node hexahedral finite element according to the voxel-level material parameters. Based on a spatial topological one-to-one mapping mechanism, the heterogeneous elastic modulus values ​​calculated from each voxel are directly obtained and injected into spatially overlapping hexahedral elements as material stiffness properties. By automatically discretizing into multiple material gradients, all elements are set using an isotropic linear elastic material model, avoiding the attenuation of interpolation accuracy caused by cross-mesh mapping.

[0030] Apply the boundary conditions and load conditions corresponding to the boundary condition field and the load field, and determine the output nodal displacement, element stress and element strain through finite element solution; The node displacement is determined according to the arithmetic mean of the displacements of the eight nodes of the corresponding unit; Under a unified anatomical coordinate system, the upper and lower endplate regions of structures such as the lumbar vertebrae are determined. Translational fixed degree-of-freedom constraints are applied to the nodes of the lower endplate, and a reference point is established at the geometric center of the upper endplate to couple and transfer loads. After submitting the work and performing a reaction force balance quality check, the nodal displacements, element stresses, and element strains are extracted. Based on the corresponding hexahedral element shape functions, the arithmetic mean formula is used: ,in, For the center of the voxel, The displacement vectors of the eight nodes are represented as equivalent to the true values ​​of voxel displacements at the centroid of the unit. During loss calculation, solutions from the extra-bone region are excluded by a mask, achieving spatial alignment with the AI ​​input features. Using voxel-by-voxel lossless hexahedral modeling and central arithmetic mean displacement mapping, the data discontinuity between the pixel space of medical images and the node space of engineering simulation is bridged.

[0031] Furthermore, the applied load conditions include: axial compressive loads sampled in the range of 500N to 3000N, and superimposed flexure, extension, or shear loads.

[0032] Specifically, to fully cover the multidimensional stress states borne by bone tissue, axial compressive loads ranging from 500N to 3000N (such as the preset representative static compression condition of 2000N) are applied along the head-to-tail direction of the vertebral body at the reference point of the superior endplate. For the large-scale augmented dataset, transverse shear force, flexion moment, or extension moment are further superimposed using vector mechanics synthesis rules. Independent working conditions are established and solved again for different load directions and magnitudes to construct a composite stress sample space.

[0033] By combining orthogonal random combinations of multidimensional force directions and dynamic load amplitudes, the diversity of physical conditions in the true finite element method is enriched, enabling the agent network to have a strong generalization and deduction capability when facing unknown patient's actual force posture.

[0034] S300, based on HU field, voxel-level material parameters, bone tissue mask, load field and boundary condition field, constructs the first three-dimensional input tensor; Specifically, the extracted and generated geometric feature masks, heterogeneous material physics fields, boundary positions, and externally applied load scalars are extracted into a multi-channel array. These arrays are then stacked and stitched along the channel dimension to synthesize a multi-channel first 3D input tensor with a uniform size structure. ,in, Represents a multi-channel three-dimensional input tensor. Represents the set of real numbers. Indicates the number of input channels. , and These represent the dimensions of the regular voxels in the three spatial directions. Combined with the corresponding finite element reference solutions, a complete batch of training samples can be formed for offline model feeding and optimization.

[0035] S400, the first three-dimensional input tensor is input into the physical information neural operator network, the overall displacement field is output through the parallel Fourier neural operator branches in the physical information neural operator network, and the high-frequency displacement residual is predicted through the three-dimensional convolutional neural network branch; like Figure 2 As shown, let the input physical field function be a(x), which consists of the HU field, material field, bone tissue mask, load field, and boundary condition field. Let g be the true partial differential equation solver from the input physical field function a(x) to the true displacement function u(x). The network adopts a dual parallel branch topology: the FNO branch uses the standard Fourier layer to approximate the partial differential operator mapping in the frequency domain (with 64 hidden channels) to capture the non-local large-scale transmission of load within the entire skeleton and outputs the overall displacement field. , is represented as: The parallel CNN branches employ 3×3×3 three-dimensional convolutional kernels with four feature scales (32, 64, 96, and 128 channels respectively) for local spatial encoding, outputting high-frequency shift residuals. , is represented as: .

[0036] in, The solution operator learned for the FNO branch. For the solution operator learned by the CNN branch, For the trainable parameters of the FNO branch, For the trainable parameters of the CNN branch, For spatial location.

[0037] In this embodiment, the CNN branch is used to specifically target the prediction of drastic high-frequency displacement residual responses induced by complex boundaries or material mutations.

[0038] S500, based on the HU field and bone tissue mask, extracts the HU gradient normalized field and bone tissue boundary indicator field respectively, and constructs a structural weight field, superimposing the weighted high-frequency displacement residuals onto the overall displacement field. Specifically, to avoid meaningless oscillation artifacts generated by convolutional networks within homogeneous and smooth trabeculae, structural gating signals reflecting changes in anatomical material are extracted and used as attention constraint matrices. High-frequency displacement residuals are forced to activate only in areas of abrupt density changes and external geometric edges. Subsequently, element-wise addition is used to merge the truncated high-frequency residual terms into the macroscopic overall basic deformation, outputting the complete predicted displacement field.

[0039] In one possible embodiment, a structural weight field is constructed based on the HU gradient normalized field and the bone tissue boundary indicator field, including: The structural gradient magnitude is determined based on the HU field, and the normalized HU gradient magnitude is obtained by normalizing the structural gradient magnitude. Extract the orthogonal axial partial derivatives from the 3D image matrix to synthesize the structural gradient magnitude representing the severity of density abrupt changes. ,in Indicates spatial location The grayscale value of HU at that location Let represent the gradient differential operator with respect to spatial coordinates (x, y, z). The L2 norm (i.e., the square root of the sum of the squares of the components of a vector) is used to measure the magnitude of the gradient.

[0040] By introducing tiny positive constants to prevent division-by-zero overflow, a HU gradient normalized field constrained within a smooth probability interval is calculated. ,in This represents the maximum value of the structure gradient magnitude at all voxel locations in the 3D volume data. To prevent the use of tiny positive numbers with a denominator of 0, the value can be 10. -6 It is used to locate the boundary zone where the elastic modulus changes abruptly.

[0041] The structural weight field is determined based on the bone tissue boundary indicator field and the HU gradient normalized field, and is expressed as follows: ; in, For structural weight field, This represents the HU gradient normalized field. This indicates the boundary field of bone tissue (1 at the bone edge, 0 elsewhere). This means truncating the result to the range of 0 to 1. and These are non-negative weights used to adjust the contribution ratio of the two types of features in spatial attention.

[0042] It should be noted that, It can take the value 0.8. A value of 0.2 can be used to balance the contribution ratios of the HU gradient field and the bone boundary field, and + =1, used to ensure that the weight field is strictly limited to the interval [0,1].

[0043] In this embodiment, a discrete binary indicator field is obtained, with the bone edge location marked as 1 and the rest as 0. Normalized non-negative weighting factors are set to characterize the relative contributions of the normalized gradient field and the boundary field, ensuring their sum is 1 to avoid weight scale drift and guarantee comparability between different samples. A clipping function is used to restrict the calculation results to the interval between 0 and 1, constructing a gated structure weight field with unit boundaries. This weight matrix only performs the localization function of spatial position attention, does not independently predict mechanical response, and during fusion, it works with bone tissue masks and fixed boundary masks to cut off unreasonable repair predictions of the free space outside the bone and the rigid fixed area on the surface. The fusion equation is expressed as: ,in Indicates spatial location The bone tissue at the site is a mask. Indicates spatial location Displacement boundary mask for the boundary condition field. This indicates element-wise multiplication, and the effective masking of bone tissue prevents abnormal network speculation in the extra-bone free space.

[0044] By deeply embedding a gating suppression mechanism into the local residual pathway, the sluggish response of heterogeneous material structures caused by the low-pass characteristics of simple frequency domain neural operators is successfully compensated, the estimation error of peak equivalent stress is reduced, and the analytical upper limit of the model in handling local pathological lesions is enhanced.

[0045] S600 determines the corresponding force balance equation residuals and boundary condition residuals based on the predicted displacement field, and trains the physical information neural operator network by combining the finite element reference solution, force balance equation residuals and boundary condition residuals. Specifically, to overcome the problem that purely data-driven AI models often fail to conform to fundamental physical laws such as force balance equations and boundary conditions, a strategy of joint training using finite element supervised data and physical constraints is adopted. During training, the network not only fits the finite element reference solution but also needs to be constrained by the laws of mechanics during backpropagation, thereby ensuring the physical consistency of the prediction results.

[0046] In one possible embodiment, the physical information neural operator network is trained by combining the finite element reference solution, the force balance equation residuals, and the boundary condition residuals, including: A supervised pre-training loss is constructed using a finite element reference solution to pre-train the physical information neural operator network; Based on the pre-training, the residuals of the force balance equation and the residuals of the boundary conditions are added for joint fine-tuning training. The residuals of the boundary conditions include displacement boundary residuals and traction boundary residuals. During the offline supervision label formation stage, the three displacement components after voxelization are used as the primary supervision labels, and six independent stress components and six independent strain components can be used as auxiliary supervision labels. The coordinate directions in the Abaqus results are consistent with the standard RAI coordinate directions. The displacement supervision labels are represented as follows: ; in, Indicates spatial location Displacement monitoring label at the location, , , Representing spatial location The displacement components in the three coordinate directions at the location.

[0047] The stress monitoring label is represented as follows: ; in, Indicates spatial location Stress monitoring label at the location, Representing spatial location The three normal stress components at the location, Representing spatial location The three shear stress components at the location.

[0048] During the sample correspondence phase, each set of standardized HU fields, material parameter fields, load fields, and boundary condition fields is matched one-to-one with the voxelized finite element reference solution output by the same Abaqus working condition, forming a training sample: ; in, Indicates the first One data sample, Indicates spatial location The standardized HU grayscale value at that location Indicates spatial location The bone tissue at the site is a mask. and Representing spatial location The elastic modulus field and Poisson's ratio field at that location, Indicates spatial location The load field and boundary condition field at a given location are specifically characterized using masks, including displacement boundary masks and traction boundary masks. Indicates spatial location The displacement boundary mask is used to indicate the Dirichlet boundary (the region where the given displacement is zero or a fixed displacement). Indicates spatial location The traction boundary mask at the location is used to indicate the Neumann boundary (the area of ​​a given surface traction / external force). , and Representing spatial location The reference solution for finite element displacement, stress, and strain is provided.

[0049] During the data partitioning phase, training, validation, and test sets are divided according to patients. Different load conditions and all amplified samples of the same patient are assigned to the same data subset to avoid the same bone structure appearing in both the training and test sets.

[0050] Furthermore, combining the isotropic linear elastic constitutive relation, the predicted displacement field obtained from the final fusion described above... The strain tensor is predicted by calculating geometric equations. Then, the stress tensor is predicted by calculating the physical equations. ,in, Let represent the gradient differential operator with respect to spatial coordinates (x, y, z). This represents the transpose of the displacement gradient. Represents the trace of a tensor. This represents the second-order unit tensor. and Spatial location The first and second Lamé parameter fields (i.e., functions that vary with spatial location) are specifically calculated using the following formulas: ; ; It should be noted that here This refers only to the predicted mechanical response output by the network model, not the stress reference solution calculated by the finite element software in the offline stage. ( (Used only as a supervisory label for model training).

[0051] Force balance equation residuals Defined as ; Displacement boundary residuals Defined as ,in, Indicates a given displacement; Traction boundary residual Defined as ,in, Represents the outer normal vector of the boundary. Indicates the traction force on a given surface; To eliminate the impact of differences in physical dimensions and numerical magnitudes on the multi-objective joint optimization process of neural networks, this invention introduces a system feature physical scale to uniformly dimensionless the physical residuals corresponding to each loss term. The feature length of the system is defined as... The characteristic elastic modulus is The characteristic stress scale and surface traction force scale are Among them, the feature length The characteristic elastic modulus is the maximum value of the dimension of the computational domain in each coordinate direction. Take the arithmetic mean of the elastic modulus of each material element or voxel within the computational domain, the characteristic stress scale, and the surface traction force scale. The characteristic displacement is determined based on the load boundary conditions (assuming a real, non-negative load exists). It is defined according to the displacement, strain, and stress metric relationships in linear elasticity. for: ; Based on the physical dimensions of the stress divergence term in the equilibrium equations, the characteristic volume force density is defined. for: .

[0052] like Figure 4 As shown, the joint training loss is precisely expressed as ,in Indicates joint training losses; This represents the monitoring loss of the displacement reference solution; This indicates the monitoring loss of the stress reference solution; This represents the residual loss in the force balance equation; This represents the residual loss due to displacement boundary conditions; This represents the residual loss due to traction (stress) boundary conditions. ( ): The adjustment weight coefficients for the corresponding loss terms (preferred initial values ​​are 1, 1, 0.1, 0.1, 0.1).

[0053] All loss terms adopt a unified approach of "first making the residuals dimensionless, then calculating the mean square error". ; in: Indicates the first Physical constraints in the first The absolute residual vector at each sampling node; Indicates the first The number of sampling points corresponding to the class loss term; The system characteristic scale that has the same physical dimensions as the corresponding physical residual; To prevent the use of tiny positive numbers with a denominator of zero, the explicit mapping relationship between each loss term and the feature scale is as follows: Displacement-related residuals: ; Stress-related residuals: ; Equilibrium equation residuals: ; In addition, the supervised loss can optionally be calculated within the bone tissue mask, and the loss weights for material abrupt change regions and bone boundary neighborhoods can be increased based on the HU gradient field and the bone tissue boundary indicator field. The AdamW optimizer is used during training with an initial learning rate of 1×10⁻⁶. -4 Batch sizes range from 1 to 4.

[0054] S700 acquires CT image data of the bone tissue to be analyzed and constructs a second three-dimensional input tensor. The trained physical information neural operator network is input to obtain the predicted apparent displacement field. The apparent strain field and apparent stress field are determined and output based on the apparent displacement field and voxel-level material parameters. like Figure 3 As shown, the online prediction phase sequentially performs DICOM data reading, RAI orientation unification, Spacing unification, and ROI pruning, while verifying data integrity, input channel completeness, and tensor size consistency. No finite element model is built or Abaqus is run during the online phase. Subsequently, a patient-specific input physics field is constructed, and CNN-FNO forward inference is performed. Outputs include voxel-scale apparent displacement, strain, stress, and mechanical contour plots. It also supports cloud-based GPU batch analysis, lightweight edge quantization (model pruning, quantization) deployment, and standardized interface connections (DICOM input supported, 3D visualization connection supported for output). Furthermore, when the anatomical region, material parameters, load conditions, or boundary conditions of the data to be analyzed exceed the coverage of the training data, an out-of-range warning is output. This output represents the apparent mechanical response at the clinical CT voxel scale and under the assumption of an equivalent continuum material; it does not represent the actual microscopic stress within a single bone trabeculae, nor is it used as a standalone clinical diagnostic conclusion.

[0055] The following is a specific example to illustrate this application: (1) Offline training phase Using the lumbar vertebral body as the analysis object, an offline training dataset was constructed. The CT region was resampled into regular voxels of 128×128×128, with equidistant voxel side lengths set to 0.6–1.0 mm. The training dataset consisted of offline finite element numerical simulation data and case augmentation data (augmentation was generated through random geometric perturbations, material parameter perturbations, and random combinations of load conditions). The training, validation, and test sets were divided into three sets in a 7:1:2 ratio, with original samples and all perturbation samples from the same case grouped into the same subset. Axial compressive loads were sampled within the range of 500–3000 N, and flexion, extension, or shear loads could be superimposed.

[0056] A voxel-by-voxel C3D8 or C3D8R hexahedral model was created using Abaqus / Standard, and finite element supervision tags were generated. The tags include displacement components in three directions, six stress components, and six strain components; finite element generation and Abaqus solving were only used for offline supervision data generation.

[0057] The standardized HU field, material parameter field, load field, and boundary condition field are mapped one-to-one with the voxelized finite element reference solution output by the same Abaqus working condition to form a training sample.

[0058] The aforementioned multi-channel 3D voxel data are input in parallel into the FNO global solution branch, the CNN local residual branch, and the structural weight branch. The FNO branch outputs the global displacement solution, while the CNN branch simultaneously receives the HU field, elastic modulus field, Poisson's ratio field, load components, and boundary condition mask, outputting high-frequency displacement residuals corresponding to the current material, load, and boundary conditions. The structural weight branch calculates w(x) based on the HU gradient normalized field and the bone tissue boundary indicator field. This weight is only used to define the spatial region of residual correction and does not independently predict the mechanical response. The fixed boundary mask is used to suppress residual correction in a fixed region and superimposes the corrected residuals onto the global displacement field to obtain the final displacement field. Then, the voxel-scale apparent strain and apparent stress are calculated based on the displacement gradient and the E-ν isotropic linear elastic constitutive relation.

[0059] The model was trained using a GPU computing platform. The 3DCNN encoder was configured with four feature scales, with 32, 64, 96, and 128 channels at each scale. The FNO backbone consisted of four Fourier layers, each retaining no more than one-quarter of the input space size for low-frequency modes, with 64 hidden channels. The training batch size was set to 1–4, and the initial learning rate was 1 × 10⁻⁴. -4 The AdamW optimizer was used for 300 epochs of training, with learning rate decay and early stopping performed based on the validation set loss. Training employed a two-stage joint training approach: The first stage uses finite element displacement reference solution and stress reference solution to construct a supervised pre-training loss and perform pre-training. The second stage involves joint fine-tuning of the residuals of the force balance equation, displacement boundary, and traction boundary based on the pre-training.

[0060] The initial weights of the finite element supervision term, the equilibrium residual term, and the boundary residual term are set to 1, 0.1, and 0.1, respectively, and are adaptively adjusted according to the gradient magnitude of each loss term during training. Each loss term is recorded during training, and a surrogate model is obtained after training is completed.

[0061] (II) Online Prediction Phase During online inference, lumbar spine CT image data of the patient to be analyzed is acquired. DICOM data reading, RAI orientation unification, spacing unification, and ROI clipping are performed, and input standardization checks are conducted on data integrity, input channel completeness, and tensor size consistency. Subsequently, a patient-specific input physics field is constructed, composing the standardized HU field, material parameter field, load field, and boundary condition field into an input physics field function, and CNN-FNO forward inference is executed. No finite element model is built during the online phase, and Abaqus is not run.

[0062] The model outputs voxel-scale apparent displacement, apparent strain, apparent stress, mechanical contour plots, and mechanical risk assessment indicators. When the anatomical region, material parameters, load conditions, or boundary conditions of the data to be analyzed exceed the coverage of the training data, an out-of-range warning is output. This output represents the apparent mechanical response at the clinical CT voxel scale and under the assumption of an equivalent continuum material.

[0063] like Figure 5 As shown, to verify the effect of the structure-aware local residual correction mechanism proposed in this application on improving the accuracy of bone mechanical response prediction, an ablation experiment was conducted. While maintaining consistency in the training dataset, finite element reference solution, loss function, and training parameters, the following models were constructed: a basic model containing only Fourier neural operators (FNO) (Model A), an improved model with added 3D convolutional local residual branches but without structural weight constraints (Model B), and the complete model described in this invention, consisting of "FNO global solution branch + CNN high-frequency residual correction + structural weight constraint" (Model C).

[0064] The experimental results (mean) are as follows: Model A (pure FNO, without CNN residual correction): maximum equivalent stress error 8.72%, displacement field relative L2 error 4.58%, stress concentration location offset 2.35mm; Model B (FNO+CNN residual, unstructured weights): maximum equivalent stress error 5.63%, displacement field relative L2 error 3.08%, stress concentration location offset 1.42mm; Model C (complete model of this invention): maximum equivalent stress error 4.21%, displacement field relative L2 error 2.29%, stress concentration location offset 0.86mm.

[0065] Relative improvement (C relative to A): Maximum equivalent stress error decreased by 51.7%, displacement field error relative to L2 decreased by 50.0%, and stress concentration location offset decreased by 63.4%.

[0066] Conclusion: By introducing a structure-aware local residual correction module and combining it with structural weight constraints, significant improvements were achieved in peak stress, displacement field error, and the accuracy of stress concentration location positioning, effectively compensating for the insufficient response of FNO to local structural abrupt change regions.

[0067] like Figure 6 As shown, the second embodiment of the present invention also provides a bone tissue biomechanical analysis system 80 based on physical information neural operators, comprising: The CT image acquisition module 81 is used to acquire CT image data of bone tissue, perform anatomical orientation unification and resampling preprocessing, convert the processed CT grayscale values ​​into voxel-level material parameters and extract bone tissue masks. The finite element solution module 82 is used to construct a finite element model of the bone tissue based on the voxel-level material parameters, the bone tissue mask, and the set load field and boundary condition field, and to obtain a finite element reference solution. The input tensor construction module 83 is used to construct a first three-dimensional input tensor based on the HU field of the CT image data, the voxel-level material parameters, the bone tissue mask, the load field, and the boundary condition field. The neural operator prediction module 84 is used to input the first three-dimensional input tensor into the physical information neural operator network, output the overall displacement field through the parallel Fourier neural operator branches in the physical information neural operator network, and predict the high-frequency displacement residual through the three-dimensional convolutional neural network branch. Spatial fusion module 85 is used to construct a structural weight field based on the HU gradient normalized field and the bone tissue boundary indicator field, and to superimpose the weighted high-frequency displacement residuals onto the overall displacement field to obtain the predicted displacement field: ; in, Indicates spatial location The bone tissue at the site is a mask. Indicates spatial location Displacement boundary mask at the location, This indicates element-wise multiplication. Represents the overall displacement field. Indicates high-frequency displacement residuals. Represents the structural weight field. Indicates the predicted displacement field; The physical constraint training module 86 is used to determine the corresponding force balance equation residuals and boundary condition residuals based on the predicted displacement field, and to train the physical information neural operator network by combining the finite element reference solution, the force balance equation residuals and the boundary condition residuals. The online prediction output module 87 is used to acquire CT image data of the bone tissue to be analyzed and construct a second three-dimensional input tensor. The trained physical information neural operator network is input to obtain the predicted apparent displacement field, and the apparent strain field and apparent stress field are calculated and output based on the apparent displacement field and the voxel-level material parameters.

[0068] The third embodiment of the present invention provides an electronic device, including a memory, a processor, and a computer program stored in the memory and executable on the processor. When the processor executes the program, it implements a bone tissue biomechanical analysis method based on physical information neural operators.

[0069] The fourth embodiment of the present invention provides a computer-readable storage medium having a computer program stored thereon, which, when executed by a processor, implements a bone tissue biomechanical analysis method based on physical information neural operators.

[0070] Computer-readable storage media can be any type of media capable of non-volatilely storing digital program code, such as USB flash drives, portable hard drives, ROM (read-only memory), magnetic disks, or optically driven optical discs.

[0071] Those skilled in the art will understand that all or part of the processes in the methods of the above embodiments can be implemented by a computer program instructing related hardware. The computer program can be stored in a non-volatile computer-readable storage medium, and when executed, it can include the processes of the embodiments of the above methods. Any references to memory, databases, or other media used in the embodiments provided in this application can include at least one of non-volatile and volatile memory. Non-volatile memory can include read-only memory (ROM), magnetic tape, floppy disk, flash memory, optical memory, high-density embedded non-volatile memory, resistive random access memory (ReRAM), magnetic random access memory (MRAM), ferroelectric random access memory (FRAM), phase change memory (PCM), graphene memory, etc. Volatile memory can include random access memory (RAM) or external cache memory, etc. By way of illustration and not limitation, RAM can take many forms, such as Static Random Access Memory (SRAM) or Dynamic Random Access Memory (DRAM). The databases involved in the embodiments provided in this application may include at least one type of relational database and non-relational database. Non-relational databases may include, but are not limited to, blockchain-based distributed databases. The processors involved in the embodiments provided in this application may be general-purpose processors, central processing units, graphics processing units, digital signal processors, programmable logic devices, quantum computing-based data processing logic devices, etc., and are not limited to these. The technical features of the above embodiments can be combined in any way. For the sake of brevity, not all possible combinations of the technical features in the above embodiments are described; however, as long as the combination of these technical features does not contradict each other, it should be considered within the scope of this specification.

[0072] The specific embodiments of the present invention described above do not constitute a limitation on the scope of protection of the present invention. Any other corresponding changes and modifications made in accordance with the technical concept of the present invention should be included within the scope of protection of the claims of the present invention.

Claims

1. A method for analyzing bone tissue mechanics based on physical information neural operators, characterized in that, include: CT image data of bone tissue and its corresponding HU field are acquired, and anatomical orientation unification and resampling preprocessing are performed. The voxel values ​​in the processed HU field are converted into voxel-level material parameters and bone tissue mask is extracted. Based on the voxel-level material parameters, the bone tissue mask, and the set load field and boundary condition field, a finite element model of the bone tissue is constructed, and a finite element reference solution is obtained. Based on the HU field, the voxel-level material parameters, the bone tissue mask, the load field, and the boundary condition field, a first three-dimensional input tensor is constructed; The first three-dimensional input tensor is input into the physical information neural operator network, and the overall displacement field is output through the parallel Fourier neural operator branches in the physical information neural operator network. The high-frequency displacement residual is predicted through the three-dimensional convolutional neural network branch. Based on the HU field and the bone tissue mask, the HU gradient normalized field and the bone tissue boundary indicator field are extracted respectively, and a structural weight field is constructed. The weighted high-frequency displacement residuals are then superimposed onto the overall displacement field to obtain the predicted displacement field. ； in, Indicates spatial location The bone tissue at the site is a mask. Indicates spatial location Displacement boundary mask for the boundary condition field. This indicates element-wise multiplication. Represents the overall displacement field. Indicates high-frequency displacement residuals. Represents the structural weight field. Indicates the predicted displacement field; Based on the predicted displacement field, the corresponding force balance equation residuals and boundary condition residuals are determined. The physical information neural operator network is then trained by combining the finite element reference solution, the force balance equation residuals, and the boundary condition residuals. CT image data of the bone tissue to be analyzed is acquired and a second three-dimensional input tensor is constructed. The trained physical information neural operator network is input to obtain the predicted apparent displacement field. The apparent strain field and apparent stress field are calculated and output based on the apparent displacement field and the voxel-level material parameters.

2. The bone tissue biomechanical analysis method based on physical information neural operators as described in claim 1, characterized in that, A structural weight field is constructed based on the HU gradient normalized field and the bone tissue boundary indicator field, including: The structural gradient magnitude is determined based on the HU field, and the structural gradient magnitude is normalized to obtain the HU gradient normalized field. The structural weight field is determined based on the bone tissue boundary indicator field and the HU gradient normalized field, and is expressed as follows: ； in, This represents the HU gradient normalized field. Indicates the boundary indicator field of bone tissue. and The weights are non-negative; clip indicates that the calculation result is truncated and restricted to the interval between 0 and 1.

3. The bone tissue biomechanical analysis method based on physical information neural operators as described in claim 1, characterized in that, The processed CT grayscale values ​​were converted into voxel-level material parameters and bone tissue masks were extracted, including: according to Determine the apparent density, and according to Determine the elastic modulus; in, Apparent density is expressed in g / cm³. 3 E represents the elastic modulus, with units of MPa, and Poisson's ratio is set to a preset constant. The elastic modulus and the Poisson's ratio are used as the voxel-level material parameters; Based on the preprocessed CT grayscale values, HU threshold segmentation is performed, and voxels above the preset threshold are marked as bone tissue, while voxels below the preset threshold are marked as background, thus obtaining the bone tissue mask.

4. The bone tissue biomechanical analysis method based on physical information neural operators as described in claim 1, characterized in that, The physical information neural operator network is trained by combining the finite element reference solution, the force balance equation residuals, and the boundary condition residuals, including: The physical information neural operator network is pre-trained using the finite element reference solution to construct a supervised pre-training loss. Based on the pre-training, the residuals of the force balance equation and the residuals of the boundary conditions are added for joint fine-tuning training, wherein the residuals of the boundary conditions include displacement boundary residuals and traction boundary residuals.

5. The bone tissue biomechanical analysis method based on physical information neural operators as described in claim 1, characterized in that, Based on the voxel-level material parameters, the bone tissue mask, and the set load field and boundary condition field, a finite element model of the bone tissue is constructed, and a finite element reference solution is obtained, including: Each effective bone tissue voxel after uniform orientation and resampling is converted into an eight-node hexahedral finite element. According to the voxel-level material parameters, the elastic modulus of each voxel is assigned to its corresponding eight-node hexahedral finite element. Apply the boundary conditions and load conditions corresponding to the boundary condition field and the load field, and determine the output nodal displacement, element stress and element strain through finite element solution; The node displacement is determined according to the arithmetic mean of the displacements of the eight nodes of the corresponding unit.

6. The bone tissue biomechanical analysis method based on physical information neural operators as described in claim 5, characterized in that, The applied load conditions include: Axial compressive loads sampled in the range of 500N to 3000N, as well as superimposed flexure, extension, or shear loads.

7. The bone tissue biomechanical analysis method based on physical information neural operators as described in claim 1, characterized in that, Acquire CT images of bone tissue, perform anatomical orientation unification and resampling preprocessing, including: The 3D body data of CT images is uniformly converted to a preset standard anatomical orientation; The HU body data after orientation unification is resampled to the same equidistant voxel spacing as the training data; Among them, the HU body data uses continuous interpolation, and the bone tissue mask uses nearest neighbor interpolation.

8. A bone tissue biomechanical analysis system based on physical information neural operators, characterized in that, include: The CT image acquisition module is used to acquire CT image data of bone tissue and its corresponding HU field, perform anatomical orientation unification and resampling preprocessing, convert the voxel values ​​in the processed HU field into voxel-level material parameters and extract the bone tissue mask. The finite element solution module constructs a finite element model of the bone tissue based on the voxel-level material parameters, the bone tissue mask, and the set load field and boundary condition field, and obtains the finite element reference solution. The input tensor construction module constructs a first three-dimensional input tensor based on the HU field of the CT image data, the voxel-level material parameters, the bone tissue mask, the load field, and the boundary condition field. The neural operator prediction module inputs the first three-dimensional input tensor into the physical information neural operator network, outputs the overall displacement field through the parallel Fourier neural operator branches in the physical information neural operator network, and predicts the high-frequency displacement residuals through the three-dimensional convolutional neural network branches. The spatial fusion module is used to extract the HU gradient normalized field and the bone tissue boundary indicator field based on the HU field and the bone tissue mask, respectively, and construct a structural weight field. The weighted high-frequency displacement residuals are then superimposed onto the overall displacement field to obtain the predicted displacement field. ； in, Indicates spatial location The bone tissue at the site is a mask. Indicates spatial location Displacement boundary mask for the boundary condition field. This indicates element-wise multiplication. Represents the overall displacement field. Indicates high-frequency displacement residuals. Represents the structural weight field. Indicates the predicted displacement field; The physical constraint training module determines the corresponding force balance equation residuals and boundary condition residuals based on the predicted displacement field, and trains the physical information neural operator network by combining the finite element reference solution, force balance equation residuals and boundary condition residuals. The online prediction output module acquires CT image data of the bone tissue to be analyzed and constructs a second three-dimensional input tensor. It inputs the trained physical information neural operator network to obtain the predicted apparent displacement field, and calculates the apparent strain field and apparent stress field based on the apparent displacement field and the voxel-level material parameters for output.

9. An electronic device comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, characterized in that, When the processor executes the program, it implements the bone tissue biomechanical analysis method based on physical information neural operators as described in any one of claims 1 to 7.

10. A computer-readable storage medium having a computer program stored thereon, characterized in that, When executed by a processor, the program implements the bone tissue biomechanical analysis method based on physical information neural operators as described in any one of claims 1 to 7.

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