Resting twelve-lead electrocardiogram auxiliary diagnosis system based on correction model

Abstract

The application discloses a kind of resting twelve-lead electrocardiogram auxiliary diagnosis system based on correction model, belong to medical diagnosis technical field, including data acquisition module, label coding module, deep learning network construction module, conclusion sorting module and correction unit, for the output result of the verification and correction network group of correction model establishment;The application can accurately diagnose electrocardio signal by the combination of deep learning network and correction model, system utilizes the diagnostic process of automation and reliable correction model, provides fast and reliable heart disease diagnostic tool, at the same time, through flexible model fine-tuning and comprehensive sorting, accurate analysis of electrocardio signal is realized, such system provides the diagnostic result of high accuracy and strong explainability, effectively improves the auxiliary diagnosis effect of heart disease, doctor can rely on the accuracy and reliability of system, obtain more credible, fast and efficient diagnostic result.

Description

Technical Field

[0001] This invention relates to the field of medical diagnostic technology, and in particular to a resting 12-lead electrocardiogram auxiliary diagnostic system based on a calibration model. Background Technology

[0002] Electrocardiography (ECG) is one of the most commonly used methods for diagnosing heart disease in clinical practice. One of the most common diagnostic methods is a resting 12-lead ECG: the patient lies still in bed, and the doctor places electrodes in fixed positions on the patient's body, then takes an ECG measurement. A high-quality 10-second ECG is saved for diagnostic purposes. Generally, more than 200 types of diseases can be diagnosed from a 10-second resting ECG, and these disease terms have certain logical relationships and can be combined to form a composite conclusion. A good auxiliary diagnostic algorithm can improve the efficiency of doctors' diagnosis, provide prompts, and reduce misdiagnosis and missed diagnosis.

[0003] Currently, most high-accuracy auxiliary diagnostic systems are developed based on supervised deep learning algorithms. However, deep learning algorithms have poor interpretability, and the conclusions they produce are either A or non-A type conclusions. For some diseases that have not been trained, the reliability of conclusions obtained by such supervised deep learning is low. Therefore, to address this problem, this invention proposes a resting 12-lead electrocardiogram auxiliary diagnostic system based on a calibration model. Summary of the Invention

[0004] The purpose of this invention is to address the problems of insufficient interpretability and diagnostic accuracy of deep learning algorithms in existing electrocardiogram-assisted diagnosis, and to propose a resting 12-lead electrocardiogram-assisted diagnosis system based on a calibration model.

[0005] To achieve the above objectives, the present invention adopts the following technical solution:

[0006] A resting 12-lead electrocardiogram (ECG) auxiliary diagnostic system based on a calibration model includes a data acquisition module responsible for acquiring 12-lead ECG signals from patients at rest; a labeling and encoding module for labeling and encoding the acquired ECG signal data; a deep learning network construction module responsible for constructing M deep learning networks for ECG signal evaluation and judgment; a conclusion ranking module where experts rank completely correct results first based on their evaluation and judgment of the ECG signals; for other cases, the conclusion ranking module ranks the results according to missed diagnoses, misdiagnoses, and multiple diagnoses; and a calibration unit for establishing the validation of the calibration model and the output results of the calibration network group. The system fine-tunes the deep learning networks by acquiring sample data and ultimately diagnoses new ECG signals.

[0007] Preferably, the sampling frequency of the data acquisition module is 500Hz, and the length of each ECG signal is 10 seconds.

[0008] Preferably, the tag encoding module encodes the conclusions of the electrocardiogram signal and summarizes all disease types, using r = [r1, r2, ..., r N ,r N+1 As a label encoding for data, when the ECG signal conclusion contains the i-th conclusion, r i =1, otherwise r i =0.

[0009] Preferably, M deep learning networks are constructed, with a suggested ratio of 3 ≤ M ≤ 6. Each network contains at least two fully connected layers, and the networks are trained using labeled electrocardiogram signal data.

[0010] Preferably, when using a 6-layer CNN network structure, the recommended settings for each layer are as follows: the first layer is a convolutional layer containing 50 one-dimensional convolutional kernels of size 5, using the ReLU function as the activation function; the second layer is a convolutional layer containing 60 one-dimensional convolutional kernels of size 6, using the ReLU function as the activation function; the third layer is a max pooling layer with a kernel size and stride of 2; the fourth layer is a convolutional layer containing 70 one-dimensional convolutional kernels of size 3, using the ReLU function as the activation function; the fifth layer is a fully connected layer with 10 neurons as the output, using the ReLU function as the activation function; and the sixth layer is a fully connected layer with the number of output neurons equal to the number of separable dominant rhythm types, using the sigmoid function as the activation function.

[0011] Preferably, the calibration unit includes a calibration model construction module, a fine-tuning deep learning network module, and an electrocardiogram (ECG) signal diagnosis module. The calibration model construction module is used to verify and calibrate the output results of the network group, the fine-tuning deep learning network module is used to fine-tune the deep learning network using the collected sample data, and the ECG signal diagnosis module is used to diagnose new ECG signals.

[0012] Preferably, the output conclusions R of M deep learning networks are used. M*(N+1) As input, sort the result Y M ={y1, y2, ..., y M}, where y i =a, indicating that the conclusion output by the i-th deep learning network is ranked a-th and is used as the output to train the correction model.

[0013] Preferably, when the calibration model built by the calibration model construction module adopts a 3-layer LSTM network structure, the first layer is the encoding layer, and the second and third layers are the decoding layers. The first layer uses a unidirectional LSTM layer with 128 LSTM units as the encoding layer, the second layer uses a unidirectional LSTM layer with 64 LSTM units as the decoding layer, and the third layer is a fully connected layer with M neurons as the output. It is recommended to use the Softmax function as the activation function to output the ranking results corresponding to each neural network.

[0014] Preferably, when fine-tuning the deep learning network using the collected sample data, only the coefficients of the fully connected layers are adjusted, by minimizing the objective function Obj = net. J (net p m (x)) to optimize the network, where net J (.) indicates the output of the correction model when the input is a dot (.), net p m (x) represents the output of the m-th deep learning network constructed by the deep learning network building module when the input is x.

[0015] Preferably, for a new electrocardiogram signal, it is input into M deep learning networks to obtain M output vectors R of length N+1. M ={r1, r2, ..., r M}, where r i =[r1 i r2 i , ..., r N i r N+1 i ] represents the output of the i-th deep learning network. The RM is used as input to the correction model for correction, resulting in the ranking result Y. M ={y1, y2, ..., y M Assume Y M If the minimum value appears at the j-th position, then check r. j =[r1 j r2 j , ..., r n j r n+1 j If r n+1 j A value >0.5 indicates that the ECG signal cannot be used for diagnosis; otherwise, according to r... i j If r > 0.5, the presence or absence of a disease type is determined. i j If the value is greater than 0.5, it indicates that the disease type exists; otherwise, it indicates that the disease type is not included.

[0016] Compared with the prior art, the present invention provides a resting 12-lead electrocardiogram auxiliary diagnostic system based on a calibration model, which has the following beneficial effects:

[0017] This resting 12-lead electrocardiogram (ECG) auxiliary diagnostic system, based on a calibration model, combines deep learning networks and a calibration model to diagnose ECG signals with high accuracy. The system utilizes an automated diagnostic process and a reliable calibration model to provide a fast and reliable diagnostic tool for heart diseases. Furthermore, through flexible model fine-tuning and comprehensive sorting, it achieves precise analysis of ECG signals. This system provides highly accurate and interpretable diagnostic results, effectively improving the auxiliary diagnostic effect for heart diseases. Doctors can rely on the system's accuracy and reliability to obtain more credible, faster, and more efficient diagnostic results. Attached Figure Description

[0018] Figure 1 This is a connection block diagram of a resting 12-lead electrocardiogram auxiliary diagnostic system based on a calibration model proposed in this invention.

[0019] Figure 2 This is a functional schematic diagram of the correction unit of a resting 12-lead electrocardiogram auxiliary diagnostic system based on a correction model proposed in this invention.

[0020] Figure 3 This is a schematic diagram of the deep learning network structure of a resting 12-lead electrocardiogram auxiliary diagnostic system based on a calibration model proposed in this invention.

[0021] Figure 4 This is a schematic diagram of the model construction of a resting 12-lead electrocardiogram auxiliary diagnostic system based on a calibration model proposed in this invention. Detailed Implementation

[0022] The technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings of the embodiments of the present invention. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments.

[0023] In the description of this invention, it should be understood that the terms "upper", "lower", "front", "rear", "left", "right", "top", "bottom", "inner", "outer", etc., indicate the orientation or positional relationship based on the orientation or positional relationship shown in the accompanying drawings. They are only for the convenience of describing this invention and simplifying the description, and do not indicate or imply that the device or element referred to must have a specific orientation, or be constructed and operated in a specific orientation. Therefore, they should not be construed as limitations on this invention.

[0024] Reference Figures 1-4A resting 12-lead electrocardiogram (ECG) auxiliary diagnostic system based on a calibration model includes a data acquisition module responsible for acquiring 12-lead ECG signals from a patient in a resting state. These signals are collected from electrodes placed on the patient's body, and high-quality ECG data will be used for subsequent diagnostic processes. A labeling and encoding module is used to label and encode the acquired ECG signal data, assigning a correct disease label to each ECG signal for use in subsequent evaluation and judgment. A deep learning network construction module is responsible for constructing M deep learning networks for ECG signal evaluation and... These networks accept electrocardiogram (ECG) signals as input and evaluate and judge them through a learning process. Each network may contain structures such as convolutional layers, pooling layers, and fully connected layers for feature extraction and conclusion generation. A conclusion ranking module allows experts to rank the outputs of the deep learning networks based on accuracy, placing the most reliable conclusion first. A calibration unit is used to establish a calibration model for validation and to correct the outputs of the network group. The calibration unit fine-tunes the deep learning networks by collecting sample data to further improve their performance and accuracy. This helps optimize the network's output and further improve the reliability and accuracy of the assisted diagnostic system.

[0025] This technical solution can provide auxiliary diagnostic functions for evaluating and judging electrocardiogram signals, and improve the reliability and accuracy of diagnostic results through the verification and fine-tuning of the calibration model.

[0026] The data acquisition module can have a sampling frequency of 500Hz, and the length of each ECG signal is recommended to be 10 seconds. The purpose is to ensure high-quality sampling of the ECG signal. The sampling frequency refers to the number of times the signal is sampled per second.

[0027] For electrocardiogram (ECG) signals, high-frequency sampling can capture subtle changes in the signal more precisely. Generally, a sampling frequency of 500Hz has been widely accepted and applied to ECG data acquisition. Such a sampling frequency can provide sufficient detail, making subsequent analysis and diagnosis more accurate and reliable.

[0028] Each ECG signal is 10 seconds long to obtain a continuous ECG signal data that is commonly used in clinical practice. This duration is usually sufficient to capture the basic ECG waveform and characteristics, providing enough information for subsequent diagnostic analysis. At the same time, the longer signal length also helps to reduce the impact of noise and interference, and improve signal quality.

[0029] In summary, using a sampling frequency of 500Hz and a length of 10 seconds for each ECG signal can ensure high-quality sampling of ECG signals, providing accurate and reliable data for assisting in the evaluation and judgment of the diagnostic system.

[0030] The tag encoding module encodes the conclusions from the electrocardiogram (ECG) signal and summarizes all possible disease types. We use a binary encoding vector r = [r1, r2, ..., r N ,r N+1 ] is used as a label encoding for the data, where N represents the number of disease types, and r N+1 This indicates other or undiagnosed conditions.

[0031] In the encoding vector r, each element r i This indicates whether the i-th disease exists. If the conclusion of the electrocardiogram signal includes the i-th disease, that is, the disease is diagnosed on this signal, then r i The value of r is 1; otherwise, r i The value of is 0.

[0032] This encoding method can effectively represent the conclusion of each electrocardiogram signal and map it into a binary encoded vector. In this way, we can easily carry out subsequent data processing and analysis. For example, computer algorithms can be used to process the encoded vector to classify, predict and diagnose diseases.

[0033] It is important to note that the last element r N+1 Indicates other or undiagnosed conditions; when the conclusion of the electrocardiogram signal does not include any of the aforementioned diseases, nor are there any other known diseases, r N+1 The value of r is 1; otherwise, r N+1 The value of is 0.

[0034] To build deep learning networks, a suggested range of 3 ≤ M ≤ 6 networks can be considered, ensuring that each network contains at least 2 fully connected layers. In addition, the networks can be trained using labeled electrocardiogram signal data to improve the performance and accuracy of the model.

[0035] When using a 6-layer CNN network structure, the parameters of each layer can be set according to the following suggestions:

[0036] The first layer uses a convolutional layer with 50 one-dimensional kernels of size 5. This means that 50 filters of size 5 are used to extract features in the temporal dimension of the signal. The ReLU function can be used as the activation function to enhance the non-linear representation capability of the network. The second layer uses a convolutional layer with 60 one-dimensional kernels of size 6. This layer continues to extract higher-level features in the temporal dimension. Similarly, the ReLU function can be used as the activation function. The third layer is a max-pooling layer. Max-pooling operation with a kernel size and stride of 2 can be used to reduce the size of the feature map and retain the most... The fourth layer, which features a significant feature set, uses a convolutional layer with 70 one-dimensional kernels of size 3. This layer further extracts more abstract and higher-level features, also using the ReLU function as the activation function. The fifth layer is a fully connected layer with 10 neurons as the output. This layer transforms the feature mapping of the convolutional layer into a higher-dimensional feature representation, and uses the ReLU function as the activation function. The sixth layer is a fully connected layer with the number of neurons output as the number of separable dominant rhythms, using the sigmoid function as the activation function. This layer maps the final feature representation to the corresponding disease category.

[0037] The above suggestions are based on a 6-layer CNN network structure, which can be adjusted and optimized according to specific needs and the characteristics of the dataset. When training the network, labeled ECG signal data is used for supervised learning. The network performance is improved through iterative training, and an accurate ECG signal-assisted diagnostic model is finally obtained.

[0038] In a preferred embodiment, the calibration unit includes a calibration model construction module, a fine-tuning deep learning network module, and an electrocardiogram signal diagnosis module.

[0039] The calibration model construction module is used to build the calibration model. Its main function is to verify and calibrate the output results of the network group. In the preferred scheme, the calibration model adopts a 3-layer LSTM network structure. The first layer is the encoding layer, which adopts a unidirectional LSTM layer with 128 LSTM units. The second and third layers are the decoding layers, which adopt a unidirectional LSTM layer with 64 LSTM units and a fully connected layer, respectively. The output consists of M neurons. The Softmax function is used as the activation function to output the ranking results of each deep learning network.

[0040] The deep learning network fine-tuning module is used to fine-tune the deep learning network using collected sample data to further improve the network's performance and accuracy. In the preferred scheme, only the coefficients of the fully connected layers in the deep learning network are adjusted, by minimizing the objective function Obj = net. J (net p m (x) is used to optimize the network, where net J(.) indicates the output of the correction model when the input is a dot (.), net p m (x) represents the output of the m-th deep learning network constructed by the deep learning network building module when the input is x.

[0041] The ECG signal diagnostic module is used to diagnose new ECG signals. First, the ECG signal is input into M deep learning networks to obtain M output vectors R of length N+1. M ={r1, r2, ..., r M}, where r i =[r1 i r2 i , ..., r N i r N+1 i ] represents the output of the i-th deep learning network. The RM is used as input to the correction model for correction, resulting in the ranking result Y. M ={y1, y2, ..., y M Assume Y M If the minimum value appears at the j-th position, then check r. j =[r1 j r2 j , ..., r n j r n+1 j If r n+1 j A value >0.5 indicates that the ECG signal cannot be used for diagnosis; otherwise, according to r... i j If r > 0.5, the presence or absence of a disease type is determined. i j If the value is greater than 0.5, it indicates that the disease type exists; otherwise, it indicates that the disease type is not included.

[0042] Through the interaction of the above modules, the calibration unit can verify and correct the output results of the deep learning network, and ultimately make an accurate diagnosis of the new ECG signal. The construction of the calibration model, the fine-tuning of the deep learning network, and the ECG signal diagnosis module work together to improve the system's ability to analyze and judge ECG signals, and further enhance the accuracy and reliability of the resting 12-lead ECG auxiliary diagnosis system.

[0043] In summary, the resting 12-lead electrocardiogram (ECG) auxiliary diagnostic system proposed in this invention, by introducing a calibration model and fine-tuning methods, further improves the reliability and accuracy of auxiliary diagnosis, overcoming the limitations of traditional deep learning algorithms in terms of interpretability and reliability of conclusions. This system utilizes the calibration model to verify and calibrate the output results of the network group, ensuring the accuracy of diagnostic results. Simultaneously, through fine-tuning methods, the deep learning network is precisely adjusted based on the collected sample data, improving performance and robustness, thereby obtaining more reliable diagnostic conclusions. This approach not only helps improve doctors' diagnostic efficiency and reduce the possibility of misdiagnosis and missed diagnosis, but also better handles untrained disease conditions, providing more reliable conclusions. This represents a significant breakthrough in the development of auxiliary diagnostic systems in the field of ECG, providing a higher level of support for clinical diagnosis.

[0044] Furthermore, it should be noted that those skilled in the art will recognize that the units and algorithm steps of the various examples described in conjunction with the embodiments disclosed herein can be implemented in electronic hardware or a combination of computer software and electronic hardware. Whether these functions are implemented in hardware or software depends on the specific application and design constraints of the technical solution. Those skilled in the art can use different methods to implement the described functions for each specific application, but such implementation should not be considered to be beyond the scope of the present invention.

[0045] In the embodiments provided by this invention, it should be understood that the disclosed resting 12-lead electrocardiogram auxiliary diagnostic system based on the calibration model can be implemented in other ways. For example, the resting 12-lead electrocardiogram auxiliary diagnostic system based on the calibration model described above is merely illustrative. For example, the division of modules or units is merely a logical functional division, and there may be other division methods in actual implementation. For example, multiple units or components may be combined or integrated into another system, or some features may be ignored or not executed. Furthermore, the coupling or direct coupling or communication connection between the shown or discussed components may be indirect coupling or communication connection through some interfaces, devices, or units, and may be electrical, mechanical, or other forms.

[0046] Furthermore, the units described 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 can be selected to achieve the purpose of this embodiment according to actual needs.

[0047] In addition, the functional units in the various embodiments of the present invention can be integrated into one processing unit, or each unit can exist physically separately, or two or more units can be integrated into one unit. The integrated unit can be implemented in hardware or in the form of software functional units.

[0048] If the integrated modules / units are implemented as software functional units and sold or used as independent products, they can be stored in a computer-readable storage medium. Based on this understanding, the present invention can implement all or part of the processes in the methods of the above embodiments. Alternatively, a computer program that instructs related hardware can be stored in a computer-readable storage medium. When the computer program is executed by a processor, it can implement the steps of the various method embodiments described above. The computer program includes computer program code, which can be in the form of source code, object code, executable file, or some intermediate form. The computer-readable medium can include any entity or device capable of carrying computer program code, recording media, USB flash drives, portable hard drives, magnetic disks, optical disks, computer memory, read-only memory (ROM), random access memory (RAM), electrical carrier signals, telecommunication signals, and software distribution media, etc.

[0049] It should be noted that the content contained in computer-readable media may be appropriately added to or subtracted from the requirements of legislation and patent practice in a jurisdiction. For example, in some jurisdictions, according to legislation and patent practice, computer-readable media may not include electrical carrier signals and telecommunication signals.

[0050] The above description is only a preferred embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any equivalent substitutions or modifications made by those skilled in the art within the scope of the technology disclosed in the present invention, based on the technical solution and inventive concept of the present invention, should be covered within the scope of protection of the present invention.

Claims

1. A resting 12-lead electrocardiogram auxiliary diagnostic system based on a calibration model, characterized in that, include: The data acquisition module is responsible for acquiring 12-lead electrocardiogram signals from patients in a resting state; The tag encoding module is used to label and encode the acquired electrocardiogram signal data; The deep learning network construction module is responsible for building M deep learning networks for the evaluation and interpretation of electrocardiogram signals. In the conclusion ranking module, experts prioritize completely correct results based on their assessment and judgment of the electrocardiogram signals. For other cases, the module ranks the results according to the number of missed diagnoses, misdiagnoses, and multiple diagnoses. The correction unit uses the output conclusions R of M deep learning networks. M As input, sort the result Y M ={y1, y2, ..., y M As output, a calibration model is trained, used to establish the validation of the calibration model and the output results of the calibration network group. The deep learning network is fine-tuned by collecting sample data, and finally, new ECG signals are diagnosed. Where y i =a, indicating that the conclusion output by the i-th deep learning network is in the a-th position; For a new electrocardiogram (ECG) signal, it is input into M deep learning networks to obtain M output vectors R of length N+1. M ={r 1 r 2 , ..., r M }; Where r i =[r1 i r2 i , ..., r N i r N+1 i ] represents the output of the i-th deep learning network, R M The input correction model is used for correction to obtain the ranking result Y. M ={y1, y2, ..., y M Assume Y M If the minimum value appears at the j-th position, then check r. j =[r1 j r2 j , ..., r n j r n+1 j If r n+1 j A value >0.5 indicates that the ECG signal cannot be used for diagnosis; otherwise, according to r... i j A value greater than 0.5 indicates the presence or absence of a disease type; If r i j If the value is greater than 0.5, it indicates that the disease type exists; otherwise, it indicates that the disease type is not included.

2. The resting 12-lead electrocardiogram auxiliary diagnostic system based on a calibration model according to claim 1, characterized in that, The data acquisition module has a sampling frequency of 500Hz, and each ECG signal is 10 seconds long.

3. The resting 12-lead electrocardiogram auxiliary diagnostic system based on a calibration model according to claim 1, characterized in that, The label encoding module encodes the conclusions from the electrocardiogram signal and summarizes all disease types, using r=[r1,r2,…,r N ,r N+1 ] is used as a tag encoding for data; Where the conclusion of the electrocardiogram signal includes the i-th conclusion, r i =1, otherwise r i =0.

4. The resting 12-lead electrocardiogram auxiliary diagnostic system based on a calibration model according to claim 1, characterized in that, Construct M deep learning networks, where 3 ≤ M ≤ 6, each network containing at least 2 fully connected layers, and train the networks using labeled electrocardiogram signal data.

5. The resting 12-lead electrocardiogram auxiliary diagnostic system based on a calibration model according to claim 4, characterized in that, When using a 6-layer CNN network structure, the settings for each layer are as follows: The first layer is a convolutional layer containing 50 one-dimensional convolutional kernels of size 5, using the ReLU function as the activation function; The second layer is a convolutional layer containing 60 one-dimensional convolutional kernels of size 6, using the ReLU function as the activation function; The third layer is a max pooling layer, with a kernel size and step size of 2. The fourth layer is a convolutional layer containing 70 one-dimensional convolutional kernels of size 3, using the ReLU function as the activation function; The fifth layer is a fully connected layer with 10 neurons as the output, using the ReLU function as the activation function; The sixth layer is a fully connected layer with the number of output neurons equal to the number of separable dominant rhythm types, and uses the sigmoid function as the activation function.

6. The resting 12-lead electrocardiogram auxiliary diagnostic system based on a calibration model according to claim 1, characterized in that, The calibration unit includes a calibration model construction module, a fine-tuning deep learning network module, and an electrocardiogram signal diagnosis module; Among them, the calibration model construction module is used to verify and calibrate the output results of the network group, the fine-tuning deep learning network module is used to fine-tune the deep learning network using the collected sample data, and the electrocardiogram signal diagnosis module is used to diagnose new electrocardiogram signals.

7. A resting 12-lead electrocardiogram auxiliary diagnostic system based on a calibration model according to claim 6, characterized in that, When the calibration model built by the calibration model construction module adopts a 3-layer LSTM network structure, the first layer is the encoding layer, and the second and third layers are the decoding layers; The first layer uses a unidirectional LSTM layer with 128 LSTM units as the coding layer; The second layer uses a unidirectional LSTM layer with 64 LSTM units as the decoding layer; The third layer is a fully connected layer, with an output of M neurons. The activation function is the Softmax function, which outputs the ranking results corresponding to each neural network.

8. A resting 12-lead electrocardiogram auxiliary diagnostic system based on a calibration model according to claim 6, characterized in that, When fine-tuning a deep learning network using collected sample data, only the coefficients of the fully connected layers are adjusted, by minimizing the objective function. To optimize the network; Among them, net J (.) indicates the output of the correction model when the input is a dot (.). This represents the output of the m-th deep learning network constructed by the deep learning network building module when the input is x.

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