A physical information neural network-based post-corneal cross-linking prediction model construction method

Abstract

A kind of corneal cross-linking postoperative prediction model construction method based on physical information neural network, comprising: establishing hyperelastic constitutive model, its key mechanical parameters are characterized as spatial distribution function related to cross-linking treatment parameters, construct physical prior constraint;Obtain and pre-process preoperative corneal anterior surface height map and full layer thickness map, construct multi-modal input tensor and true value label by data enhancement, and divide into training set and test set;Build neural network with fusion coding-decoding structure and cyclic feedback mechanism, the network contains the branch of adaptive prediction individualized mechanical parameter, and the physical prior constraint is embedded in the training process in the form of differentiable loss function;The network is optimized by using training data, so that it learns the mapping relationship from preoperative morphology to postoperative morphology, for postoperative corneal morphology prediction of new data.The present application realizes the deep fusion of physical constraint and data-driven, significantly improves the prediction accuracy, efficiency and physical credibility.

Description

Technical Field

[0001] This invention relates to the interdisciplinary field of biomedical engineering and artificial intelligence, and in particular to a method for constructing a predictive model after corneal cross-linking based on a physical information neural network. Background Technology

[0002] Corneal cross-linking is a key method for treating corneal ectasias (such as keratoconus). It enhances the cross-linking between corneal collagen fibers through photochemical reactions, increasing corneal stiffness to slow or stop disease progression. However, there is a lack of precise theoretical predictive models for the impact of surgical parameters (such as UV irradiation intensity, duration, and spot shape) on postoperative corneal morphology. Current clinical treatment plans mainly rely on physician experience, and postoperative efficacy evaluation is largely limited to follow-up observation, making it difficult to achieve personalized, quantitative preoperative prediction and efficacy optimization.

[0003] In recent years, some studies have attempted to predict postoperative morphology based on preoperative corneal topography. One type of method relies entirely on data-driven deep learning models, such as directly fitting the image mapping relationship between preoperative and postoperative topography using convolutional neural networks. While this type of method can capture certain morphological statistical regularities, its prediction results lack physical interpretability, are easily affected by noise in the training data, and may generate morphological outputs that do not conform to the principles of corneal biomechanics, making it difficult to provide a reliable basis for clinical decision-making.

[0004] Another type of research focuses on computational mechanics methods such as finite element simulation, simulating the corneal mechanical response and morphological changes after surgery by establishing a constitutive model of the cornea and combining it with the physicochemical equations of the crosslinking process. As a multilayered collagen fiber structure, the cornea exhibits complex mechanical behavior. Existing constitutive models mainly include hyperelastic models, viscoelastic models, and super-viscoelastic models, as shown in Table 1. Among them, the HGO (Holzapfel–Gasser–Ogden) model, as a typical hyperelastic constitutive model, can effectively describe the anisotropic mechanical response of the cornea by separating the strain energy functions of the matrix and fibers and combining them with fiber dispersion parameters. It is suitable for simulating the directional enhancement effect of collagen fibers during the crosslinking process. In contrast, while viscoelastic models such as Maxwell and Kelvin can describe time-dependent stress relaxation or creep behavior, they are more suitable for slow loading or small deformation processes and are difficult to characterize the nonlinear large deformation characteristics of the cornea under intraocular pressure and crosslinking. While super-viscoelastic models such as Ogden can simultaneously describe large strain and time effects, they have numerous parameters, high computational costs, and weak physical interpretability. Therefore, HGO models are often chosen as the basis for corneal biomechanical modeling due to their clear structure, well-defined physical meaning of parameters, and ease of coupling with crosslinking parameters. Nevertheless, although this type of method has a clear physical mechanism, it is computationally expensive, the modeling process is complex, and it is highly dependent on the accurate acquisition of individualized corneal geometry and material parameters, making it difficult to achieve rapid and efficient real-time clinical prediction and treatment optimization.

[0005] Table 1: Classification and characteristics of common constitutive models used for the cornea

[0006]

[0007] Physical Information Neural Networks (PINNs), as an emerging modeling paradigm that integrates physical laws and data-driven approaches, can improve the generalization ability and physical consistency of prediction results under data-scarce conditions by embedding physical constraints such as governing equations and boundary conditions into the loss function of the neural network. Their successful applications in fluid mechanics and solid mechanics demonstrate PINN's potential to handle complex nonlinear and multi-physics coupled problems. However, effectively embedding constitutive models describing the anisotropic and hyperelastic mechanical behavior of the cornea into deep learning frameworks and achieving end-to-end, interpretable, and highly efficient prediction from preoperative multimodal imaging to postoperative morphology remains an unsolved technical challenge in the field of corneal cross-linking treatment planning.

[0008] It should be noted that the information disclosed in the background section above is only for understanding the background of this application, and therefore may include information that does not constitute prior art known to those skilled in the art. Summary of the Invention

[0009] The main objective of this invention is to overcome the deficiencies in the aforementioned background technology and provide a method for constructing a predictive model for corneal cross-linking based on a physical information neural network.

[0010] To achieve the above objectives, the present invention adopts the following technical solution:

[0011] A method for constructing a predictive model after corneal crosslinking based on a physical information neural network includes the following steps:

[0012] S1. Establish a hyperelastic constitutive model to describe the biomechanical behavior of the cornea, and characterize the key mechanical parameters therein as spatial distribution functions related to cross-linking treatment parameters, so as to construct physical prior constraints.

[0013] S2. Obtain clinical data including preoperative corneal anterior surface height map and full-thickness map, preprocess and augment the data, construct multimodal input tensors and corresponding postoperative morphological ground truth labels, and divide them into training set and test set;

[0014] S3. Build a recurrent physical information neural network prediction model. This model integrates an encoder-decoder structure and a recurrent feedback mechanism, and includes a branch for adaptively predicting individualized mechanical parameters from the input image. At the same time, the physical prior constraints are embedded into the network training process in the form of a differentiable loss function.

[0015] S4. The recurrent physical information neural network prediction model is trained using the training data. By jointly optimizing the image reconstruction loss, physical constraint loss, and clinical parameter loss, the model learns the mapping relationship from preoperative morphology and treatment parameters to postoperative morphology, which is used to predict postoperative corneal morphology based on new preoperative data.

[0016] Furthermore, step S1 specifically includes:

[0017] A hyperelastic constitutive model suitable for fibrous-reinforced biological tissues was selected as the mechanical description basis for the cornea;

[0018] The total strain energy function of the constitutive model is composed of the hyperelastic energy characterizing the nonlinear response of the matrix and anisotropic fibers, the time-dependent viscoelastic energy characterizing the time-dependent viscoelastic energy characterizing the rate-dependent dissipation.

[0019] A mapping relationship is established between the cross-linking treatment parameters and the key stiffness parameters in the constitutive model, so that the stromal stiffness and fiber stiffness of the cornea after surgery become functions of the treatment light intensity, time and spatial position.

[0020] The spatially varying stiffness function is simplified into an explicit stiffness field model distributed along the radial and tangential directions of the cornea, respectively. The scaling factor in this model is defined as a parameter that can be learned by the neural network, thereby transforming the physical laws of the constitutive model into a loss calculation basis that can be embedded in network training.

[0021] Furthermore, step S2 specifically includes:

[0022] The paired preoperative and postoperative corneal topography data collected clinically were cropped, denoised, and color-physical value decoded to obtain the corneal anterior surface height matrix and full-thickness matrix.

[0023] Based on the rotation invariance of the cornea, the height matrix and thickness matrix are rotated and resampled to achieve data augmentation, thereby expanding the training samples and improving the robustness of the model.

[0024] The preoperative corneal anterior surface elevation map and the preoperative corneal full-thickness thickness map are stacked along the channel dimension to form a dual-channel input tensor; the corresponding postoperative elevation map and thickness map are also stacked to form a dual-channel output ground truth label.

[0025] A case-based hold-out method was adopted to strictly separate data from different cases into training and testing sets, ensuring the clinical validity of the model's generalization ability assessment.

[0026] Furthermore, the specific method for color-physical value decoding of the corneal topography in step S2 includes:

[0027] Through interactive sampling, a mapping table between the color values ​​of the topographic legend bars and their corresponding physical height values ​​was established;

[0028] During decoding, for each pixel in the topographic map, the distance between its color feature vector and all sample color vectors in the mapping table is calculated;

[0029] The physical value corresponding to the nearest sample is selected as the decoding height value of the pixel.

[0030] Furthermore, the recurrent physical information neural network prediction model constructed in step S3 includes:

[0031] The dual-channel encoder consists of multiple cascaded convolutional modules, each containing a convolutional layer, a normalization layer, and an activation function. It also performs downsampling through pooling operations to extract multi-scale spatial features from the input dual-channel tensor.

[0032] The recurrent feedback decoder restores the feature map resolution through upsampling and makes skip connections with the features of the corresponding layer of the encoder. The decoder works in a multi-time step manner. At each time step, it receives the current corneal state and predicts a deformation increment. The increment is accumulated to the current state and fed back as the input for the next time step. Through iteration, it gradually approximates the postoperative morphology.

[0033] The adaptive physical parameter prediction branch, connected after the bottleneck features output by the encoder, automatically regresses the learnable scaling coefficients that characterize the individual corneal mechanical properties by globally compressing and nonlinearly mapping the high-dimensional features.

[0034] The physical constraint loss calculation module uses the scaling factor output by the adaptive prediction branch, combined with the stiffness distribution model in step S1, to generate a spatially varying stiffness field, and couples it with the deformation field predicted by the network to calculate the physical regularization loss following the principle of minimum potential energy.

[0035] Furthermore, the working mechanism of the adaptive physics parameter prediction branch includes:

[0036] Global average pooling is performed on the high-dimensional feature map output by the encoder to remove its spatial location information and obtain a global description vector representing the overall corneal morphological features.

[0037] The global description vector is input into a multilayer perceptron containing a fully connected layer for nonlinear transformation;

[0038] A monotonically increasing activation function that guarantees a positive output value is applied to the output layer of the multilayer perceptron. The final output corresponds to two learnable scaling coefficients for matrix hardness correction and fiber anisotropy correction, respectively, enabling the network to adaptively infer individualized mechanical properties based on the input image.

[0039] Furthermore, the specific process of the physical constraint loss calculation module includes:

[0040] A normalized spatial coordinate grid is constructed based on the input image size, and the radial distance and corneal depth corresponding to each pixel are calculated.

[0041] The proportional coefficient of the branch output is predicted using the adaptive physical parameters. Combined with the radial and tangential stiffness distribution functions defined in step S1, a radial stiffness field tensor and a tangential stiffness field tensor with the same size as the input image are generated through a broadcast mechanism.

[0042] The deformation displacement field predicted by the network at the current time step is multiplied with the stiffness field at the pixel level to calculate the local elastic strain energy of each pixel.

[0043] The local strain energy of all pixels is summed and averaged as a physical constraint loss term. This loss term constrains the deformation amplitude predicted by the model during training to match the spatial stiffness distribution, that is, it suppresses deformation in regions with high stiffness and allows greater deformation in regions with low stiffness.

[0044] Furthermore, the model training in step S4 employs a composite loss function, which is a weighted sum of the image reconstruction loss term, the physical constraint loss term, and the clinical parameter regression loss term.

[0045] The image reconstruction loss term incorporates dynamic weights based on the deformation amplitude during calculation to focus on areas of significant deformation caused by surgery.

[0046] The clinical parameter regression loss term extracts the maximum curvature clinical indicator from the predicted height map through a differentiable computation module and minimizes its difference from the true value.

[0047] Furthermore, in step S4, the recurrent physical information neural network prediction model is trained and predicted using a multi-time-step recursive residual mechanism:

[0048] The preoperative corneal morphology is used as the initial state input into the network;

[0049] In the first time step, the network predicts the first deformation increment based on the initial state and adds the increment to the initial state to obtain the state after the first update.

[0050] Using the updated state as input, we enter the second time step, where the network predicts the second deformation increment and then accumulates and updates the state again.

[0051] Repeat the above process until the preset number of time steps is reached, and output the final accumulated state as the predicted postoperative corneal morphology.

[0052] A computer program product includes a computer program that, when executed by a processor, implements the method for constructing a postoperative corneal cross-linking prediction model based on a physical information neural network.

[0053] The present invention has the following beneficial effects:

[0054] This invention proposes a method for constructing a predictive model for corneal cross-linking surgery based on a physical information neural network. By constructing a recurrent physical information neural network prediction model that integrates physical priors and data-driven approaches, it is possible to achieve high-precision and high-efficiency prediction of individualized corneal cross-linking postoperative morphology. This effectively overcomes the limitations of existing pure data-driven methods that lack physical interpretability and traditional mechanical simulation methods that have high computational costs.

[0055] This invention first establishes a hyperelastic constitutive model based on the properties of fiber-reinforced biological tissue and derives the spatial mapping relationship between its key mechanical parameters and cross-linking treatment parameters (such as ultraviolet light intensity, irradiation time, and treatment radius), transforming complex biomechanical behavior into explicit physical constraints that can be embedded in neural network training. Furthermore, by designing the scaling coefficients in the model as learnable parameters for the neural network and simplifying the complete strain energy density function into radial and tangential stiffness fields that vary with spatial location, the computational complexity of the physical loss function is significantly reduced while preserving the core biomechanical characteristics of the cornea (such as depth attenuation and anisotropy). Finally, by embedding the aforementioned biomechanical physical constraints into the network training process in the form of differentiable losses, a physical foundation is laid for achieving end-to-end real-time prediction.

[0056] In terms of model construction, this invention designs a neural network architecture that integrates an encoder-decoder structure and a recurrent feedback mechanism. This network simultaneously senses the anterior corneal surface height and full-thickness information through dual-channel input, and utilizes a multi-time-step recursive residual mechanism to simulate the gradual deformation process of the cornea during cross-linking, thereby progressively approximating a more physically realistic postoperative morphology. Crucially, the network's built-in adaptive physical parameter prediction branch can automatically regress learnable coefficients characterizing the mechanical properties of the individualized corneal images, achieving end-to-end inversion from image features to material properties, effectively solving the parameter uncertainty problem caused by individual differences. By coupling the aforementioned physical stiffness field with the deformation field predicted by the network at the pixel level, and constructing a physical constraint loss function based on the minimum potential energy principle, the network's prediction output is forced to strictly follow biomechanical laws, significantly suppressing non-physical noise and artifacts common in purely data-driven models.

[0057] Furthermore, this invention effectively expands scarce clinical samples through a rotation-invariant data augmentation strategy, improving the model's robustness to different scanning angles and pathological morphologies. During training, a composite optimization objective integrating image reconstruction loss, physical constraint loss, and clinical parameter regression loss is employed to ensure multidimensional consistency of prediction results in visual fidelity, mechanical rationality, and accuracy of clinical diagnostic indicators. Ultimately, the trained model can complete inference at millisecond speeds, achieving high-precision predictions while meeting the timeliness requirements of real-time clinical decision support, providing a reliable tool for personalized parameter planning and postoperative outcome prediction in corneal cross-linking surgery.

[0058] Other beneficial effects of the embodiments of the present invention will be further described below. Attached Figure Description

[0059] Figure 1 This is a flowchart illustrating the overall process of constructing a post-corneal cross-linking prediction model based on a physical information neural network, as described in this invention.

[0060] Figure 2 This is a corneal stiffness distribution diagram according to an embodiment of the present invention.

[0061] Figure 3 This is a diagram of a recurrent neural network structure based on physical information, as shown in an embodiment of the present invention.

[0062] Figure 4 This refers to the predicted deformation field displacement output by the network in this embodiment of the invention.

[0063] Figure 5 The physical parameter prediction branch learning curve is shown in the embodiment of the present invention.

[0064] Figure 6 This invention provides a comparison of the actual topographic maps and the no-pinn / pinn predicted topographic maps after surgery in Case 1 and Case 2.

[0065] Figure 7a Comparison of postoperative real topographic map and pinn predicted topographic map (detailed verification and surgical effect analysis) in Case 1 of this invention embodiment.

[0066] Figure 7b Comparison of postoperative real topographic map and pinn predicted topographic map (detailed verification and surgical effect analysis) in Case 2 of this invention embodiment. Detailed Implementation

[0067] The embodiments of the present invention will be described in detail below. It should be emphasized that the following description is merely exemplary and is not intended to limit the scope and application of the present invention.

[0068] Furthermore, the terms "first" and "second" are used for descriptive purposes only and should not be construed as indicating or implying relative importance or implicitly specifying the number of technical features indicated. Thus, a feature defined as "first" or "second" may explicitly or implicitly include one or more of that feature. In the description of embodiments of the present invention, "a plurality of" means two or more, unless otherwise explicitly specified.

[0069] This invention aims to solve the problem of accurate prediction of postoperative morphology after corneal cross-linking. It proposes a prediction method that integrates a hyperelastic constitutive model and a cyclic physical information neural network. By embedding biomechanical physical constraints into the network with differentiable loss and using adaptive parameter prediction branches to achieve individualized mechanical property inversion, the method significantly improves computational efficiency and clinical applicability while ensuring the physical interpretability of the prediction results.

[0070] See Figure 1 This invention provides a method for constructing a post-corneal crosslinking prediction model based on a physical information neural network, comprising the following steps:

[0071] Step S1: Construct physical constraints based on the hyperelastic constitutive model: Establish a hyperelastic constitutive model to describe the biomechanical behavior of the cornea, and characterize the key mechanical parameters in it as spatial distribution functions related to cross-linking treatment parameters to construct physical prior constraints.

[0072] In some embodiments, step S1 specifically includes: selecting a hyperelastic constitutive model suitable for fiber-reinforced biological tissue as the mechanical description basis of the cornea; the total strain energy function of the constitutive model is composed of hyperelastic energy characterizing the nonlinear response of the matrix and anisotropic fibers, viscoelastic energy characterizing time dependence, and viscoelastic energy characterizing rate-related dissipation; establishing a mapping relationship between crosslinking treatment parameters and key stiffness parameters in the constitutive model, so that the stromal stiffness and fiber stiffness of the cornea after surgery become functions of treatment light intensity, time, and spatial position; simplifying the spatially varying stiffness function into an explicit stiffness field model distributed along the radial and tangential directions of the cornea, respectively, and defining the scaling factor in the model as a parameter that can be learned by the neural network, thereby transforming the physical laws of the constitutive model into a loss calculation basis that can be embedded in network training.

[0073] Step S2, Multimodal Corneal Data Preparation and Enhancement: Acquire clinical data including preoperative corneal anterior surface height map and full-thickness map, preprocess and enhance the data, construct multimodal input tensors and corresponding postoperative morphological ground truth labels, and divide them into training set and test set.

[0074] In some embodiments, step S2 specifically includes: cropping, denoising, and color-physical value decoding of clinically collected paired preoperative and postoperative corneal topography data to obtain a corneal anterior surface height matrix and a full-thickness thickness matrix; based on the rotation invariance of the cornea, performing rotational resampling on the height matrix and thickness matrix to achieve data augmentation, thereby expanding the training samples and improving the robustness of the model; stacking the preoperative corneal anterior surface height map and the preoperative corneal full-thickness thickness map along the channel dimension to form a dual-channel input tensor; stacking the corresponding postoperative height map and thickness map in the same way to form a dual-channel output ground truth label; and using a case-based hold-out method to strictly separate the data of different cases into the training set and the test set to ensure the clinical validity of the model's generalization ability assessment.

[0075] In some embodiments, the specific method for color-physical value decoding of the corneal topography map in step S2 includes: establishing a mapping table between the color values ​​of the topographic map legend color bars and the corresponding physical height values ​​through interactive sampling; during decoding, for each pixel in the topographic map, calculating the distance between its color feature vector and all sample color vectors in the mapping table; and selecting the physical value corresponding to the nearest sample as the decoding height value of the pixel.

[0076] Step S3: Construct a recurrent physical information neural network prediction model: Build a recurrent physical neural network that integrates an encoder-decoder structure and a recurrent feedback mechanism. This network includes branches for adaptively predicting individualized mechanical parameters from the input image, and embeds the physical prior constraints from step S1 into the network training process in the form of a differentiable loss function.

[0077] In some embodiments, the recurrent physical information neural network prediction model constructed in step S3 includes: a dual-channel encoder, which consists of multiple cascaded convolutional modules, each module containing a convolutional layer, a normalization layer, and an activation function, and performs downsampling through pooling operations to extract multi-scale spatial features from the input dual-channel tensor; a recurrent feedback decoder, which restores the feature map resolution through upsampling operations and makes skip connections with the features of the corresponding layer of the encoder, the decoder operates in a multi-time-step manner, receiving the current corneal state and predicting a deformation increment at each time step, accumulating the increment to the current state and feeding it back as the input for the next time step, and iteratively approximating the postoperative morphology; an adaptive physical parameter prediction branch, connected after the bottleneck features output by the encoder, this branch automatically regresses a learnable scaling factor characterizing the individual corneal mechanical properties by performing global compression and nonlinear mapping on high-dimensional features; and a physical constraint loss calculation module, which uses the scaling factor output by the adaptive prediction branch, combined with the stiffness distribution model in step S1, to generate a spatially varying stiffness field, and couples it with the deformation field predicted by the network to calculate the physical regularization loss following the principle of minimum potential energy.

[0078] In some embodiments, the working mechanism of the adaptive physical parameter prediction branch includes: performing global average pooling on the high-dimensional feature map output by the encoder to remove its spatial location information and obtain a global description vector representing the overall corneal morphological features; inputting the global description vector into a multilayer perceptron containing a fully connected layer for nonlinear transformation; applying a monotonically increasing activation function that guarantees a positive output value to the output layer of the multilayer perceptron, and finally outputting two learnable scaling coefficients corresponding to the matrix stiffness correction and fiber anisotropy correction, respectively, so that the network can adaptively infer individualized mechanical properties based on the input image.

[0079] In some embodiments, the specific process of the physical constraint loss calculation module includes: constructing a normalized spatial coordinate grid based on the input image size, calculating the radial distance and corneal depth corresponding to each pixel; using the adaptive physical parameters to predict the scaling factor of the branch output, and combining the radial and tangential stiffness distribution functions defined in step S1, generating a radial stiffness field tensor and a tangential stiffness field tensor with the same size as the input image through a broadcast mechanism; multiplying the deformation displacement field predicted by the network at the current time step with the stiffness field at the pixel level, and calculating the local elastic strain energy of each pixel; summing and averaging the local strain energies of all pixels as the physical constraint loss term, which constrains the deformation amplitude predicted by the model during training to match the spatial stiffness distribution, that is, suppressing deformation in high stiffness regions and allowing greater deformation in low stiffness regions.

[0080] Step S4, Model Training: The recurrent physical information neural network prediction model constructed in Step S3 is trained using the training data prepared in Step S2. By jointly optimizing the image reconstruction loss, physical constraint loss, and clinical parameter loss, the model learns the mapping relationship from preoperative morphology and treatment parameters to postoperative morphology. The trained model can be used to predict postoperative corneal morphology based on new preoperative data.

[0081] In some embodiments, the model training in step S4 employs a composite loss function, which is a weighted sum of an image reconstruction loss term, a physical constraint loss term, and a clinical parameter regression loss term. The image reconstruction loss term incorporates dynamic weights based on deformation amplitude during calculation to focus on areas of significant deformation caused by surgery. The clinical parameter regression loss term extracts the maximum curvature clinical indicator from the predicted height map through a differentiable computation module and minimizes its difference from the true value.

[0082] In some embodiments, in step S4, the recurrent physical information neural network prediction model is trained and predicted through a multi-time-step recursive residual mechanism: the preoperative corneal morphology is input into the network as the initial state; in the first time step, the network predicts the first deformation increment based on the initial state, and adds the increment to the initial state to obtain the first updated state; the updated state is used as input to enter the second time step, the network predicts the second deformation increment, and the state is updated again by accumulating; the above process is repeated until a preset number of time steps is reached, and the finally accumulated state is output as the predicted postoperative corneal morphology.

[0083] This invention proposes a method for constructing a predictive model after corneal cross-linking surgery based on a physical information neural network. This method simplifies the hyperelastic constitutive model describing corneal biomechanical behavior and transforms its physical laws into a spatial stiffness field. This field is then embedded into a recurrent neural network architecture as a differentiable loss function, innovatively achieving a deep integration of physical constraints and data-driven approaches. This method not only endows the model with individualized mechanical inversion capabilities through an adaptive parameter prediction branch but also utilizes a recurrent residual mechanism to simulate the gradual process of corneal deformation. Ultimately, while ensuring that the prediction results strictly conform to biomechanical principles, it achieves high-precision and high-efficiency real-time clinical prediction, effectively resolving the core contradiction between interpretability, computational efficiency, and clinical application in traditional methods.

[0084] The following further describes specific embodiments of the present invention, algorithm examples, and experimental verification.

[0085] Physical loss based on HGO hyperelastic constitutive model

[0086] The HGO model is specifically designed for fiber-reinforced biological tissues, accurately capturing the directional mechanical contributions of collagen fibers in the cornea and supporting stability calculations under large deformations. Its strain energy function structure is clear, facilitating parameter physicalization and visualization, and it can be used in conjunction with factors such as UV irradiation and riboflavin distribution to establish predictive models. In this embodiment, the HGO model is selected for modeling and predicting the mechanical behavior of the cornea after cross-linking.

[0087]

[0088] The core idea of ​​the above four equations is to unify the hyperelastic response, viscoelastic memory effect, and rate-related viscous dissipation behavior of an organization into a single overall strain energy function.

[0089] First, equation (1) defines the total strain energy density function. It consists of three parts of energy: superelastic energy Viscoelastic energy and viscous dissipation energy Each section describes the mechanical response of the material under different mechanisms.

[0090] Equation (2) is the foundation of the entire model, primarily used to describe the isotropic elastic behavior of the corneal stroma and the nonlinear enhancement of the collagen fiber network in different directions. This part is responsible for simulating the structural deformation of the cornea under external force or intraocular pressure, and is suitable for scenarios with large deformations. Reflects the shear modulus of the corneal stroma. Used to describe directional tensile stiffness under fiber distribution.

[0091] Equation (3) is used to characterize the time-dependent response of tissue during loading, i.e., the viscoelastic effect. This part is simulated by the superposition of multiple viscoelastic units, reflecting the stress relaxation or creep characteristics of the cornea during or shortly after cross-linking surgery, especially the hysteresis behavior in the fiber direction.

[0092] Equation (4) represents the viscous dissipation of the material driven by the deformation rate, involving the strain rate tensor D. This part is often used to consider energy dissipation phenomena under high-speed loading or vibration, such as simulating the rapid tissue response caused by instantaneous irradiation during crosslinking.

[0093] The four equations above together construct a complete constitutive model of biological tissue, which retains the core structural response of hyperelasticity while allowing the introduction of time-dependent and rate-dependent dissipation mechanisms. This enables the model to comprehensively reflect the complex multi-scale mechanical behavior of the cornea before and after cross-linking surgery. The parameters to be fitted are shown in Table 2. The study by Ariza-Gracia et al. provides some reference values ​​for the parameters in this constitutive model.

[0094] Table 2: Parameters to be fitted in the corneal constitutive model

[0095]

[0096] After crosslinking, the constitutive model is affected by factors such as corneal surface riboflavin concentration, corneal surface UV irradiation intensity, crosslinking time, and treatment radius, resulting in varying degrees of change in the constitutive model. In this embodiment, the focus is on the impact of corneal surface UV irradiation intensity, crosslinking time, and treatment radius on corneal biomechanics; therefore, the derivation of these three parameters and their relationship with corneal biomechanics is discussed. The mapping relationship between them. To improve the simplicity and adjustability of the model, assume that the above parameters are related to... The impact is mainly reflected in and :

[0097]

[0098] The above three equations are based on the following assumptions:

[0099] 1. The crosslinking density [M] is determined by the photosensitizer concentration (C), ultraviolet light intensity (I), and time (T);

[0100] 2. Riboflavin diffusion decreases with corneal depth:

[0101]

[0102] 3. UVA radiation intensity decreases from the radiation center outwards, and also decreases with depth:

[0103] .

[0104] Based on this, during corneal crosslinking surgery, existing corneal models exhibit varying hyperelastic energies at different spatial locations, building upon their initial hyperelastic energy. The establishment of this constitutive model provides a solid mechanical foundation for subsequent prediction models. It not only serves as a physical prior in the PINN model but also possesses independent capabilities for finite element simulation or parameter inversion. This helps explain individual differences in crosslinking responses at the mechanistic level, laying the foundation for constructing a unified and generalizable prediction system.

[0105] Although the aforementioned HGO hyperelastic constitutive models (Equations 1 to 4) can theoretically and completely describe the biomechanical behavior of the cornea, in practical deep learning computing frameworks, directly performing automatic differentiation and backpropagation on the complete strain energy density function, which involves complex tensor operations, would incur huge computational overhead and make it difficult to guarantee gradient stability. To achieve a balance between physical realism and computational efficiency, the model undergoes reasonable dimensionality reduction and linearization.

[0106] The matrix shear modulus derived from the previous text (Formulas 6 and 7) With fiber tensile stiffness Based on the spatial evolution of crosslinking parameters, this embodiment further abstracts the corneal mechanical response into a stiffness distribution with two core dimensions: radial Young's modulus and tangential Young's modulus. These two moduli directly correspond to the contributions of the stroma and collagen fiber layers in the HGO model and are designed to vary with the corneal center distance. and depth Explicit functions that change.

[0107] Based on this, the originally complex strain energy density function is simplified into the following two governing equations, which are used to describe the local stiffness characteristics of the cornea at different spatial locations:

[0108]

[0109] in, Represents the distribution of stromal stiffness along the radial direction of the cornea. This represents the distribution of collagen fiber reinforcement stiffness along the tangential direction. The exponential term in the formula... This describes the characteristic that stiffness decreases exponentially with increasing corneal depth, which is consistent with the penetration loss law of ultraviolet light in the stroma; and The term describes the spatial distribution characteristics of stiffness along the radial direction of the cornea.

[0110] Most importantly, in the formula and Instead of being fixed physical constants, they are defined as learnable parameters in a neural network. During the training process of the physical information neural network, the network will automatically infer the appropriate parameters for the current case based on the input preoperative topographic map and thickness distribution. (Matrix hardness correction factor) and (Fiber anisotropy correction coefficient). This design cleverly transforms complex nonlinear constitutive relations into a dynamically adjustable "stiffness distribution map" (e.g., Figure 2 (As shown).

[0111] In this way, the abstract energy density concept in the HGO model is concretized into an observable and computable stiffness field distribution. This not only preserves the core characteristics of corneal biomechanics (i.e., depth dependence and anisotropy), but also greatly reduces the computational complexity of the physical loss function, making it possible to achieve end-to-end physical constraint prediction with limited computing power. Figure 2 The thermal map showing the spatial distribution of radial and tangential stiffness at different locations of the cornea, calculated based on this simplified model, intuitively reflects the non-uniform strengthening effect of cross-linking surgery on the local mechanical properties of the cornea.

[0112] Dataset preparation

[0113] To meet the training requirements of physical information neural networks for large data scale and multimodal inputs, while ensuring the model's generalization ability in clinical settings, this embodiment uses a pre-constructed standardized dataset containing high-precision corneal morphology data and clinical biomechanical parameters. Data preparation mainly covers three stages: data augmentation, multimodal tensor construction, and clinical-grade dataset partitioning.

[0114] 1. Data augmentation based on rotation invariance

[0115] Training deep learning models relies on large-scale, high-quality samples, while clinical corneal cross-linking surgery case data is relatively scarce. To address the overfitting problem that may result from insufficient samples, and considering the rotational invariance of corneal topography in polar coordinates, this embodiment employs a rotational data augmentation strategy.

[0116] Specifically, centered on the corneal apex, the original anterior and posterior surface topography and thickness distribution maps of each case were rotated and resampled in 10° increments, generating 36 derived samples at 0° to 350°. Through automated code processing, the 11 original clinical cases were expanded to include 396 groups (…). An augmented dataset of independent samples. This process not only greatly enriches the amount of training data, but also forces the network to learn the spatial isotropy of corneal morphological features, improving the model's robustness to different scanning angles.

[0117] 2. Construction of multimodal input-output pairs

[0118] Unlike traditional computer vision tasks that focus only on pixel-level regression of a single modality image, morphological reconstruction after corneal cross-linking surgery is essentially a biomechanical process involving three-dimensional geometric nonlinear transformations. Its final form is the result of the co-evolution of the geometric curvature of the anterior corneal surface and the full-thickness distribution of the stromal layer. To this end, this embodiment constructs a multimodal data input structure based on a dual-channel tensor, aiming to achieve implicit representation of the three-dimensional spatial entity of the cornea by stacking the channel dimensions of two-dimensional image data.

[0119] Specifically, the input tensor is formed by spatially registering and coupling the preoperative anterior corneal surface height map and the preoperative corneal full-thickness map. The anterior surface height map defines the upper surface geometric boundary of the cornea in a three-dimensional coordinate system, establishing the initial spatial curvature reference of the stroma; while the full-thickness map provides the amount of material accumulation in the stroma at different radial coordinates. This multimodal coupled data structure is a prerequisite for introducing physical mechanisms: when calculating strain energy density, the HGO hyperelastic constitutive model must calculate the exponential decay of collagen fiber stiffness based on the stroma depth. The introduction of thickness channels allows the network to accurately perceive geometric constraints along the optical axis, thereby ensuring that the subsequent calculation of the physical loss function has real anatomical significance.

[0120] Corresponding to the input, the network's output is also designed as a dual-channel structure containing postoperative height and thickness maps to ensure geometric consistency and reasonable volume distribution during deformation prediction. Furthermore, to enhance the model's sensitivity to key clinical diagnostic indicators, the maximum corneal curvature extracted from real postoperative images is introduced as an independent auxiliary supervision label during training. This design, by constructing a multi-task loss function, forces the network to accurately regress clinically relevant local steepness changes while optimizing the overall deformation field. Finally, all input and output data undergo a rigorous standardization preprocessing procedure, uniformly resampled into a 256×256 pixel matrix and normalized to a unit interval to eliminate dimensional differences and accelerate the convergence process of the neural network.

[0121] 3. Division of the Clinical Independent Test Set

[0122] To objectively evaluate the generalization performance of the prediction algorithm on unseen cases, this embodiment does not employ the traditional random shuffling method, but instead uses a more stringent subject-based hold-out approach. The dataset is strictly divided into training and test sets. The training set contains all rotated augmented samples from Cases 3 to 11, used for iterative updates of network weights and learning of physical parameters. The test set completely isolates all data from Cases 1 and 2. These two cases do not participate in any training process and are only used for final model inference and clinical accuracy validation. This partitioning strategy simulates a real-world clinical application scenario, namely, using a historical case database to train the model to predict the surgical outcomes of newly admitted patients, thereby maximizing the clinical reliability of the evaluation results.

[0123] Structure of Postoperative Topographic Mapping Prediction Algorithm Based on Physical Information Neural Network

[0124] To achieve effective coupling between the corneal hyperelastic constitutive model and the deep learning computing framework, this embodiment constructs a recurrent physical information neural network architecture. This algorithm aims to overcome the limitations of traditional convolutional neural networks that rely solely on statistical data for feature mapping. By introducing an adaptive prediction branch based on physical parameters and a recurrent feedback mechanism, it establishes a prediction paradigm that collaboratively optimizes both "data-driven" and "mechanism-driven" approaches.

[0125] 1. Cyclic Residual Backbone Network Architecture Design

[0126] like Figure 3 The backbone network architecture of this algorithm is based on the classic U-Net fully convolutional encoder-decoder model with significant improvements. To adapt to the high-dimensional nonlinear feature extraction requirements of corneal topography, the network adopts a symmetrical "U-shaped" structure, but the data flow direction has been reconstructed.

[0127] (1) Dual-channel feature encoder

[0128] The network input is designed as a two-channel tensor interface (Dimensions: The system receives preoperative corneal anterior surface height and full-thickness thickness maps, respectively. This design ensures that the network can simultaneously perceive the geometric curvature boundary conditions of the cornea and the thickness distribution constraints of the stromal layer.

[0129] The encoder consists of two consecutive convolutional modules. Each module contains two sets of cascaded structures: a 3×3 convolutional layer, a batch normalization layer, and a ReLU activation function. As the network depth increases, the number of feature channels increases sequentially from 2 to 32, then to 64, and finally reaches 128 at the bottleneck layer. 2×2 max pooling is used between layers for downsampling to expand the receptive field of neurons and extract multi-scale abstract semantic features.

[0130] (2) Loop Feedback Decoding Mechanism

[0131] Unlike conventional one-time inference networks, this embodiment takes into account the progressive deformation characteristics of corneal soft tissue during stress release and collagen cross-linking processes, and innovatively introduces a recurrent feedback mechanism in the network decoding stage. The decoder uses bilinear interpolation for upsampling and fuses high-resolution features from the same layer of the encoder through skip connections to recover lost spatial details.

[0132] To simulate the dynamic process of biological tissues tending towards a state of mechanical equilibrium, the network was configured as follows: One physical time step (set in this embodiment) At each time step The network outputs the incremental deformation field in the current state. (Includes height variation) With thickness variation This incremental field is accumulated to the corneal state of the previous time step through a residual connection. ,Right now This residual learning strategy not only extends the effective logical depth of the network, but also feeds it back to the network front end as prior input for the next time step. This also allows the model to gradually approximate a more physically realistic postoperative morphology, significantly improving its ability to capture details of minute deformations.

[0133] 2. Adaptive Physics Parameter Prediction Branch Construction

[0134] To address the parameter uncertainty caused by individual differences in traditional biomechanical modeling, this network embeds a lightweight physical parameter prediction branch in the deep feature bottleneck region.

[0135] (1) Global feature compression

[0136] like Figure 3 As shown in the flowchart of the physics-based recurrent residual neural network structure, the bottleneck layer b aggregates the highest-level features after four downsampling operations by the encoder, with a tensor dimension of 128 channels. Since the physical parameters (stromal stiffness coefficient and fiber anisotropy coefficient) are defined as global variables describing the overall mechanical properties of the cornea, rather than local variables that change drastically with pixel position, spatial decoupling of the feature map is first required. In the code implementation, an adaptive global average pooling layer is used to process the bottleneck layer features. This operation decouples the feature map into a 128-channel tensor. The three-dimensional feature volume is compressed into The process is mathematically equivalent to a weighted average of the geometric features of the entire cornea. It aims to strip away specific spatial location information (such as the specific coordinates of the cone's apex) and retain only a high-dimensional semantic descriptor containing the overall corneal curvature gradient, average thickness distribution, and potential texture features, providing the purest feature input for subsequent parametric regression.

[0137] (2) Nonlinear mapping and physical constraints

[0138] Next, the feature vector is fed into a multilayer perceptron (MLP). This sub-network contains two fully connected layers: the first layer reduces the dimension from 128 to 64 and is activated by ReLU; the second layer compresses the dimension to the final 2-dimensional output.

[0139] To ensure that the output parameters conform to the physical definition of non-negativity (stiffness coefficients cannot be negative), a Softplus activation function is used at the end of the output layer. This function guarantees the output matrix stiffness coefficient. With fiber anisotropy coefficient It is always greater than zero and has smooth gradient properties.

[0140] This design endows the neural network with the ability of "parameter self-evolution." As shown in Figure 4, these two parameters no longer depend on fixed human priors, but are automatically optimized during training using the gradient descent algorithm. The network can automatically infer corneal stiffness properties specific to the individualized corneal image input, achieving end-to-end inversion from image features to mechanical properties.

[0141] 3. Embedding mechanism of physical constraint loss function

[0142] To transform the HGO hyperelastic constitutive model into a differentiable regularized constraint, this embodiment constructs a composite loss function system that includes a physical energy term. The design of this physical loss term follows the principle of minimum potential energy, and its specific implementation logic is as follows:

[0143] (1) Tensor generation of dynamic stiffness field

[0144] When calculating the physical loss, the program first constructs a normalized mesh grid based on the size of the input image, generating a radial distance matrix for each pixel. and depth matrix Subsequently, the scalar parameters predicted by the PhysioHead branch of the network were used. and Combining formulas (10) and (11), two tensor fields with the same size as the image are generated through a broadcast mechanism: a radial matrix stiffness field. : Describes the isotropic elastic modulus distribution of the matrix layer, which decays with depth and radial position; tangential fiber stiffness field : Describes the anisotropic stiffness distribution of collagen fiber reinforcement as it varies with spatial location.

[0145] (2) The inference results on the pixel-level coupled test set based on strain energy density further verify the physical...

[0146] Physical loss function Defined as the average of the total strain energy across the entire field. In specific calculations, such as... Figure 4 As shown, the predicted deformation field output by the network is... (i.e., displacement) is coupled to the above stiffness field at the pixel level:

[0147]

[0148] This formula establishes a negative correlation penalty mechanism between stiffness and deformation: in areas with high stiffness (such as the peripheral cornea or high-strength cross-linked areas), (If the value is relatively large), in order to minimize the total energy, the network must predict smaller deformations. Conversely, in regions with lower stiffness (such as the apex of a cone), the network is allowed to undergo greater deformation to release stress.

[0149] (3) Composite loss optimization strategy

[0150] The final total loss function consists of three weighted components: a weighted image reconstruction loss (focusing on high-frequency textures), a physical strain energy loss (focusing on mechanical rationality), and a clinical parameter regression loss (focusing on metrics such as Kmax). Through this multi-objective joint optimization, the physical equations are no longer a post-processing step detached from the network, but rather participate directly in backpropagation as a core driving force. This forces the network to find a solution space that conforms to the laws of biomechanics while optimizing image similarity, thereby effectively eliminating non-physical noise and artifacts common in purely data-driven models.

[0151] Analysis of Training Results of Postoperative Topographic Mapping Prediction Algorithm Based on Physical Information Neural Network

[0152] To comprehensively evaluate the actual performance of this architecture after incorporating the HGO hyperelastic constitutive model, an in-depth analysis was conducted from three dimensions: the adaptive evolution characteristics of physical parameters, the quality of topological reconstruction in complex cases, and the differences compared with purely data-driven models. Experimental results strongly demonstrate the core role of physical constraint mechanisms in solving inverse corneal biomechanical problems, and verify that the model can output a reasonable deformation field that conforms to the laws of continuous medium mechanics while ensuring prediction accuracy.

[0153] First, the training dynamics of the physical parameter prediction branch reveal the network's ability to autonomously learn corneal material properties. For example... Figure 5 As shown, in the initial training stage, the matrix stiffness coefficient and fiber anisotropy coefficient exhibit significant oscillations, indicating that the network is in the exploratory phase of the solution space, attempting to find initial parameter values ​​that can balance image reconstruction error and physical potential energy. With the increase of iterations, the network gradually captures the nonlinear mapping relationship between corneal morphology and material properties, and the two key physical parameters quickly converge and stabilize within a specific numerical range. Notably, the converged parameters do not approach zero but stabilize within a positive range with clear physical meaning, and the fiber stiffness coefficient is significantly higher than the matrix stiffness coefficient. This is highly consistent with the biological fact of corneal collagen fiber reinforcement, proving that the model is not simply performing numerical fitting but has truly "learned" to use anisotropic material parameters to explain complex corneal deformation.

[0154] Figure 6 This study presents a comparison between the actual postoperative topographic maps and the predicted topographic maps (no-pinn / pinn) of cases 1-2. This is further demonstrated through a lateral comparison with models without physical constraints (e.g., ...). Figure 6 As shown in the diagram, the performance gain brought about by the physical mechanism can be clearly observed. Under the same network backbone and training settings, the introduction of the physical loss function significantly suppresses non-physical fluctuations in the prediction results, effectively eliminating local abnormal deformations caused by data scarcity or overfitting. The distribution of the error residual plot shows that the PINN model not only reduces the mean square error globally, but more importantly, corrects the underfitting bias of the pure data-driven model in stress concentration areas such as the corneal conus apex. In summary, the architecture of this invention, by internalizing biomechanical laws into the network's reasoning logic, successfully achieves a leap from "appearance similarity" to "mechanical similarity," providing a solution with both high accuracy and high interpretability for personalized morphological prediction after corneal cross-linking surgery.

[0155] The inference results on the test set further validated the robustness of the physical information neural network in handling complex geometric shapes. (For example, Case 1 and Case 2...) Figure 6 As shown in the figure, the PINN-based prediction algorithm exhibits stronger topology reconstruction capabilities. Visually, the smoothness and surface curvature changes are more continuous and natural compared to the PINN group. The postoperative real topographic maps and PINN-predicted topographic maps of Case 1 and Case 2 are compared as follows: Figure 7a and Figure 7bAs shown, the surgical effect is illustrated. The surgical effect map (i.e., the distribution of the difference between the postoperative image and the preoperative image) is a key indicator for evaluating whether the prediction algorithm has grasped the laws of corneal biomechanical response. The real surgical effect manifests as a centrally symmetrical and smoothly gradual morphological change, objectively reflecting the regular deformation process of the corneal stroma collagen fibers after cross-linking and hardening. The pinn algorithm in this invention produces a more uniform and smoother surgical effect map compared to pure graphics prediction algorithms.

[0156] In summary, this invention proposes a method for constructing a predictive model after corneal cross-linking surgery based on a physical information neural network. The significant innovative contribution of this invention lies in the first-ever construction of a "physics-data fusion-driven" prediction framework. This framework deeply embeds a constitutive model describing the anisotropic and hyperelastic mechanical behavior of the cornea into a recurrent neural network after reasonable simplification, using a differentiable physical loss. Furthermore, it innovatively designs branches that can adaptively infer individual mechanical parameters and a multi-step cyclic deformation simulation mechanism.

[0157] Compared with existing technologies, the significant technical advantages of this invention are: its prediction results have both physical interpretability and high accuracy, effectively overcoming the problems of lack of physical interpretability and physical distortion in pure data-driven models; at the same time, its computational efficiency is improved by orders of magnitude compared with traditional finite element simulation, realizing the leap from time-consuming simulation to millisecond-level clinical real-time prediction, providing a new solution with reliability, efficiency and practicality for personalized preoperative planning of corneal cross-linking surgery.

[0158] This invention also provides a storage medium for storing a computer program, which, when executed, performs at least the methods described above.

[0159] This invention also provides a control device, including a processor and a storage medium for storing a computer program; wherein the processor executes the computer program by performing at least the method described above.

[0160] This invention also provides a processor that executes a computer program, at least performing the methods described above.

[0161] The storage medium can be implemented by any type of non-volatile storage device, or a combination thereof. The non-volatile memory can be read-only memory (ROM), programmable read-only memory (PROM), erasable programmable read-only memory (EPROM), electrically erasable programmable read-only memory (EEPROM), magnetic random access memory (FRAM), flash memory, magnetic surface memory, optical disc or CD-ROM; magnetic surface memory can be disk storage or magnetic tape storage. The storage media described in the embodiments of this invention are intended to include, but are not limited to, these and any other suitable types of memory.

[0162] In the several embodiments provided by this invention, it should be understood that the disclosed systems and methods can be implemented in other ways. The device embodiments described above are merely illustrative. For example, the division of units is only a logical functional division, and in actual implementation, there may be other division methods, such as: multiple units or components can be combined, or integrated into another system, or some features can be ignored or not executed. In addition, the coupling or direct coupling or communication connection between the various components shown or discussed can be through some interfaces, and the indirect coupling or communication connection between devices or units can be electrical, mechanical, or other forms.

[0163] The units described above as separate components may or may not be physically separate. The components shown as units may or may not be physical units, that is, they may be located in one place or distributed across multiple network units. Some or all of the units may be selected to achieve the purpose of this embodiment according to actual needs.

[0164] In addition, in the various embodiments of the present invention, each functional unit can be integrated into one processing unit, or each unit can be a separate unit, or two or more units can be integrated into one unit; the integrated unit can be implemented in hardware or in the form of hardware plus software functional units.

[0165] Those skilled in the art will understand that all or part of the steps of the above method embodiments can be implemented by hardware related to program instructions. The aforementioned program can be stored in a computer-readable storage medium. When the program is executed, it performs the steps of the above method embodiments. The aforementioned storage medium includes various media capable of storing program code, such as mobile storage devices, read-only memory (ROM), random access memory (RAM), magnetic disks, or optical disks.

[0166] Alternatively, if the integrated units of this invention are implemented as software functional modules and sold or used as independent products, they can also be stored in a computer-readable storage medium. Based on this understanding, the technical solutions of the embodiments of this invention, or the parts that contribute to the prior art, can be embodied in the form of a software product. This computer software product is stored in a storage medium and includes several instructions to cause a computer device (which may be a personal computer, server, or network device, etc.) to execute all or part of the methods described in the various embodiments of this invention. The aforementioned storage medium includes various media capable of storing program code, such as mobile storage devices, ROM, RAM, magnetic disks, or optical disks.

[0167] The methods disclosed in the several method embodiments provided by this invention can be arbitrarily combined without conflict to obtain new method embodiments.

[0168] The features disclosed in the several product embodiments provided by this invention can be arbitrarily combined without conflict to obtain new product embodiments.

[0169] The features disclosed in the several method or device embodiments provided by the present invention can be arbitrarily combined without conflict to obtain new method or device embodiments.

[0170] The above description, in conjunction with specific preferred embodiments, provides a further detailed explanation of the present invention. It should not be construed that the specific implementation of the present invention is limited to these descriptions. For those skilled in the art, various equivalent substitutions or obvious modifications can be made without departing from the concept of the present invention, and all such modifications, achieving the same performance or application, should be considered within the scope of protection of the present invention.

Claims

1. A method for constructing a predictive model after corneal cross-linking surgery based on a physical information neural network, characterized in that, Includes the following steps: S1. Establish a hyperelastic constitutive model to describe the biomechanical behavior of the cornea, and characterize the key mechanical parameters therein as spatial distribution functions related to cross-linking treatment parameters, so as to construct physical prior constraints. S2. Obtain clinical data including preoperative corneal anterior surface height map and full-thickness map, preprocess and augment the data, construct multimodal input tensors and corresponding postoperative morphological ground truth labels, and divide them into training set and test set; S3. Build a recurrent physical information neural network prediction model. This model integrates an encoder-decoder structure and a recurrent feedback mechanism, and includes a branch for adaptively predicting individualized mechanical parameters from the input image. At the same time, the physical prior constraints are embedded into the network training process in the form of a differentiable loss function. The recurrent physical information neural network prediction model constructed in step S3 includes: A dual-channel encoder is used to extract multi-scale spatial features from an input dual-channel tensor. The recurrent feedback decoder works in a multi-time-step manner. At each time step, it receives the current corneal state and predicts a deformation increment. The increment is accumulated into the current state and fed back as the input for the next time step. Through iteration, it gradually approximates the postoperative morphology. The adaptive physical parameter prediction branch, connected after the bottleneck features output by the encoder, automatically regresses the learnable scaling coefficients that characterize the individual corneal mechanical properties by globally compressing and nonlinearly mapping the high-dimensional features. The physical constraint loss calculation module uses adaptive physical parameters to predict the proportional coefficient of the branch output, combines it with the stiffness distribution model in step S1 to generate a spatially varying stiffness field, and couples it with the deformation field predicted by the network to calculate the physical regularization loss that follows the principle of minimum potential energy. S4. The recurrent physical information neural network prediction model is trained using training data. By jointly optimizing the image reconstruction loss, physical constraint loss, and clinical parameter loss, the model learns the mapping relationship from preoperative morphology and treatment parameters to postoperative morphology, which is used to predict postoperative corneal morphology based on new preoperative data.

2. The method for constructing a post-corneal cross-linking prediction model based on a physical information neural network as described in claim 1, characterized in that, Step S1 specifically includes: A hyperelastic constitutive model suitable for fibrous-reinforced biological tissues was selected as the mechanical description basis for the cornea; The total strain energy function of the constitutive model is composed of the hyperelastic energy characterizing the nonlinear response of the matrix and anisotropic fibers, the time-dependent viscoelastic energy characterizing the time-dependent viscoelastic energy characterizing the rate-dependent dissipation. Establish a mapping relationship between cross-linking treatment parameters and key stiffness parameters in the constitutive model, so that the stromal stiffness and fiber stiffness of the cornea after surgery become stiffness functions that vary with treatment light intensity, time and spatial position; The stiffness function is simplified into an explicit stiffness field model distributed along the radial and tangential directions of the cornea, respectively, and the scaling factor in this model is defined as a parameter that can be learned by the neural network, thereby transforming the physical laws of the constitutive model into a loss calculation basis that can be embedded in network training.

3. The method for constructing a post-corneal cross-linking prediction model based on a physical information neural network as described in claim 1, characterized in that, Step S2 specifically includes: The paired preoperative and postoperative corneal topography data collected clinically were cropped, denoised, and color-physical value decoded to obtain the corneal anterior surface height matrix and full-thickness matrix. Based on the rotation invariance of the cornea, the height matrix and thickness matrix are rotated and resampled to achieve data augmentation, thereby expanding the training samples and improving the robustness of the model. The preoperative corneal anterior surface elevation map and the preoperative corneal full-thickness thickness map are stacked along the channel dimension to form a dual-channel input tensor; the corresponding postoperative elevation map and thickness map are also stacked to form a dual-channel output ground truth label. A case-based hold-out method was adopted to strictly separate data from different cases into training and testing sets, ensuring the clinical validity of the model's generalization ability assessment.

4. The method for constructing a post-corneal cross-linking prediction model based on a physical information neural network as described in claim 1, characterized in that, The specific methods for color-physical value decoding of corneal topography in step S2 include: Through interactive sampling, a mapping table between the color values ​​of the topographic legend bars and their corresponding physical height values ​​was established; During decoding, for each pixel in the topographic map, the distance between its color feature vector and all sample color vectors in the mapping table is calculated; The physical value corresponding to the nearest sample is selected as the decoding height value of the pixel.

5. The method for constructing a post-corneal cross-linking prediction model based on a physical information neural network as described in claim 1, characterized in that, The dual-channel encoder consists of multiple cascaded convolutional modules, each module containing a convolutional layer, a normalization layer and an activation function, and performs downsampling through pooling operations; The recurrent feedback decoder restores the feature map resolution through upsampling operations and makes skip connections with the features of the corresponding layer of the encoder.

6. The method for constructing a post-corneal cross-linking prediction model based on a physical information neural network as described in claim 1, characterized in that, The working mechanism of the adaptive physics parameter prediction branch includes: Global average pooling is performed on the high-dimensional feature map output by the encoder to remove its spatial location information and obtain a global description vector representing the overall corneal morphological features. The global description vector is input into a multilayer perceptron containing a fully connected layer for nonlinear transformation; A monotonically increasing activation function that guarantees a positive output value is applied to the output layer of the multilayer perceptron. The final output corresponds to two learnable scaling coefficients for matrix hardness correction and fiber anisotropy correction, respectively, enabling the network to adaptively infer individualized mechanical properties based on the input image.

7. The method for constructing a post-corneal cross-linking prediction model based on a physical information neural network as described in claim 1, characterized in that, The specific process of the physical constraint loss calculation module includes: A normalized spatial coordinate grid is constructed based on the input image size, and the radial distance and corneal depth corresponding to each pixel are calculated. The proportional coefficient of the branch output is predicted using the adaptive physical parameters. Combined with the radial and tangential stiffness distribution functions defined in step S1, a radial stiffness field tensor and a tangential stiffness field tensor with the same size as the input image are generated through a broadcast mechanism. The deformation displacement field predicted by the network at the current time step is multiplied with the stiffness field at the pixel level to calculate the local elastic strain energy of each pixel. The local strain energy of all pixels is summed and averaged as a physical constraint loss term. This loss term constrains the deformation amplitude predicted by the model during training to match the spatial stiffness distribution, that is, it suppresses deformation in regions with high stiffness and allows greater deformation in regions with low stiffness.

8. The method for constructing a predictive model after corneal cross-linking based on a physical information neural network as described in claim 1, characterized in that, The model training in step S4 uses a composite loss function, which is a weighted sum of the image reconstruction loss term, the physical constraint loss term, and the clinical parameter regression loss term. The image reconstruction loss term incorporates dynamic weights based on the deformation amplitude during calculation to focus on areas of significant deformation caused by surgery. The clinical parameter regression loss term extracts the maximum curvature clinical indicator from the predicted height map through a differentiable computation module and minimizes its difference from the true value.

9. The method for constructing a post-corneal cross-linking prediction model based on a physical information neural network as described in claim 1, characterized in that, In step S4, the recurrent physical information neural network prediction model is trained and predicted using a multi-time-step recursive residual mechanism: The preoperative corneal morphology is used as the initial state input into the network; In the first time step, the network predicts the first deformation increment based on the initial state and adds the increment to the initial state to obtain the state after the first update. Using the updated state as input, we enter the second time step, where the network predicts the second deformation increment and then accumulates and updates the state again. Repeat the above process until the preset number of time steps is reached, and output the final accumulated state as the predicted postoperative corneal morphology.

10. A computer program product, comprising a computer program, characterized in that, When the computer program is executed by the processor, it implements the method for constructing a post-corneal cross-linking prediction model based on a physical information neural network as described in any one of claims 1 to 9.

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