Cardiac Purkinje networks are a fundamental part of the conduction system and are known to initiate a variety of cardiac arrhythmias. However, patient-specific modeling of Purkinje networks remains a challenge due to their high morphological complexity. This work presents a novel method based on optimization principles for the generation of Purkinje networks that combines geometric and activation accuracy in branch size, bifurcation angles, and Purkinje-ventricular-junction activation times. Three biventricular meshes with increasing levels of complexity are used to evaluate the performance of our approach. Purkinje-tissue coupled monodomain simulations are executed to evaluate the generated networks in a realistic scenario using the most recent Purkinje/ventricular human cellular models and physiological values for the Purkinje-ventricular-junction characteristic delay. The results demonstrate that the new method can generate patient-specific Purkinje networks with controlled morphological metrics and specified local activation times at the Purkinje-ventricular junctions. The VCS comprises the His-bundle, left and right bundles branches, and the Purkinje fiber network (PN). The system provides ventricular contraction from apex to base, together with synchronization of the left and right ventricles, which allows the correct coordination of electrical activity in mammalian species5. Furthermore, it has been shown that Purkinje cells and ventricular cells have different electrophysiological properties7. For instance, Purkinje cells’ action potentials (APs) have a faster depolarization and a more negative plateau phase. This characteristic results in longer action potential duration (APD) when compared to ventricular cells8, which makes Purkinje cells more susceptible to developing pro-arrhythmic abnormalities, like early-after-depolarizations and delayed after-depolarization. This contrast in the APD form is likely apparent at the Purkinje-ventricular-junction (PVJ) sites. Due to the distinct shape of the action potential of Purkinje and ventricular cells, electrotonic effects and APD dispersion occur between the cells from the Purkinje fiber and those from the ventricular tissue6. In addition, at the PVJ sites, a characteristic delay in conduction, 3–5 ms, is observed from the Purkinje to the ventricular tissue. Mainly, this phenomenon arises when a small volume  tries to depolarize a large volume  and is also referred to as source-sink mismatch11. Therefore, to properly study cardiac electrophysiology via computational modeling, reliable models of the PN are needed and must take into account both morphological and electrical properties.In the current clinical and modeling literature, several studies have implicated the PN as both trigger and maintainer of deadly cardiac arrhythmias, such as ventricular fibrillation (VF)12. An essential step towards CDTs relies on the ability of models to replicate the ventricular activation sequence under a broad range of conditions. However, despite recent progress on cardiac functional and imaging techniques, there is not a valid technique to extract a PN from clinical data, which makes the development of a coupled Purkinje-ventricular patient-specific computational model challenging14. Therefore, as a result of their complex geometry, manual generation of PNs can become a highly demanding and time-consuming task15, making the development of methods that can automatically generate PNs a requirement to advance studies in this field.Personalized models of cardiac electrophysiology that match clinical observation with high fidelity, referred to as cardiac digital twins (CDTs), are promising tools for tailoring cardiac precision therapies19. A wide variety of methods have been used to address the generation of realistic PNs. In particular, PNs can be generated using image processing techniques by extracting the structure from images of dissected ventricles and them projecting these flat networks onto realistic endocardial surfaces17. Such models gather both the geometry and the local activation time (LAT) from the processed network. On the other hand, the generated PN may not be acceptable for other subjects due to biological variability. Another set of widely utilized methods for PN generation is fractal trees. In such models, the L-System is commonly described as an alternative to automatically produce PNs using a pre-defined set of rules that not only can prevent collisions20 but also enhance its geometry21. An alternative to the L-System is the fractal method proposed by Costabal et al.22 which allows the automatic generation of PNs using controllable curvature of the branches, enhancing the geometry of the tree, especially in irregular surfaces. Moreover, fractal trees and image processing techniques can be combined to construct the VCS. For instance, in the work of Bordas et al.23, from magnetic resonance images of a rabbit, it was possible to identify the Purkinje network partially. After manually constructing the His-Bundle, a fractal method was applied to extend the PN to the myocardium. Furthermore, there are models which are LAT constrained. Activation times data is acquired from patients and utilized to grow the networks, like in the work of Vergara et al.13 and in the recent study of Barber et al.24.Purkinje networks are known to have a complex morphology with several branches and bifurcations, as can be seen in physiological images25 and was inspired by the Constructive Constrained Optimization (CCO) method that can generate detailed and realistic vascular trees26. A major advantage of this method is the flexibility of using any cost function in the method’s optimization process and including as many topological and electrical metrics as needed .Another class of methods is based on optimization principlesWe develop a novel method using optimization principles to generate patient-specific PNs with geometrical and activation accuracy. The new method, named Constructive Optimization (CO), is a modified version of the classical CCO method that is traditionally utilized for the construction of vascular networks. The CO is focused on generating patient-specific PN models and as such it considers a cost functions that relies on electrical and geometrical metrics. Both methods rely on optimization principles to generate a topology. However, their main difference is that the CCO considers vascular concepts in its equations and has a set of constraints that all segments in the tree must fulfill during the optimization process. These restrictions are associated with blood flow and hydrodynamic resistance of the segments from the growing tree, which are not present in the CO method for Purkinje networks.In this work, we have evaluated the generated PN models by coupling them to biventricular meshes and comparing the LATs obtained in the simulations to different references. The simulations are based on modern cellular electrophysiology models for human Purkinje and ventricular cells and a fast parallel cardiac simulator. Our results show that the generated patient-specific PNs can accurately reproduce important geometrical and electrical features. In addition, the new PN models also correctly reproduced the anterograde physiological delay at the Purkinje-ventricular junctions. Therefore, we believe the results presented in this work are an essential step towards a better understanding of Purkinje networks and provide a valuable tool to study the role of patient-specific models and their impact, for instance, in the simulations of cardiac arrhythmias.25 in tri-dimensional domains following a cost function that minimizes the total length of the network, is the foundation to develop the PN models of this work. In this work, the new method uses as input a given cloud of points representing the surface to be covered, an initial root position, and a prescribed cost function to be minimized. The method aims to generate a tree with branches that satisfy a set of restrictions and, at the same time, minimize the user-specified cost function.Constructive Optimization (CO), a method developed to generate Purkinje networksS with distal locations for the terminal branches and a proximal location of the root branch To generate a Purkinje network the framework requires an endocardium surface Two different cost functions are used in the optimization procedure. One for placing new intermediate branches, cost function PreProcessing subroutine is responsible for dividing the set S into two subsets: intermediate points, A The main loop consists of branch generation using geodesic pathways and the minimization of the two cost functions until a stop criterion is reached.PostProcessing subroutine connects any remaining active PVJs, via pruning and other techniques.Finally, a To emphasize, set S contains the locations of the active PVJs, The novel patient-specific Purkinje generation method (Algorithm 1) can be summarized as follows: In addition, the PN is represented by a graph data structure, where the nodes represent points over the surface, and the edges are segments that link two points. Based on this data structure, each segment has access to its parent and offspring. A terminal segment is defined as a segment with no offspring, and a branch is a set of segments between bifurcations. For more details related to the geometrical concepts and data structures used by the method, please refer to Supplementary Material section In the following subsections each step of the method is explained in more detail.PreProcessing step is executed on the provided endocardium surface S, and divide them into two sets: intermediate and active PVJ points. An intermediate point is mainly used to increase the endocardium coverage of the growing PN with additional branches. On the other hand, an active PVJ point has a corresponding LAT, which is used as a target in the optimization process.Before growing the PN, a PreProcessing subroutine reorders the indices of the intermediate points in S, as illustrated in Fig. S so that the numbering of points increases with the distance to the initial root position. The user can also provide an input set of active PVJ points inside the set S. In this case, during the PreProcessing subroutine, we pre-process the PVJs by sorting them according to their LATs and return these points as set First, the After this initial step, the method constructs the root branch, which can be done in two different forms: using only the given initial root position In the case an initial PN is provided, this structure is used as the initial topology of the tree. This input offers flexibility and advantages, in particular in two scenarios. Firstly, well-known structures of the VCS, like the Left-Bundle-Branch (LBB) and Right-Bundle-Branch (RBB), can be constructed using different techniques and supplied as the initial network. The second advantage of this feature is extending an already created PN by adding other branches to its topology.Therefore, if the initial PN is not provided, given the initial root position 25 was applied in the present work. Before adding a new terminal branch, the threshold distance The distance criterion consists of checking whether the minimum distance mentclass2pt{minimB is generated by calling the GenerateTerminal subroutine. When selecting the new branch B, we try to minimize the total length of the PN. To do so, we select the shortest branch that connects s, subject to the following restrictions, s of the PN by solving minimization problem  across the cable is given byG is the internal conductivity, d is the diameter of the cable. To evaluate a new branch in terms of LAT, we temporarily connect that branch to the PN via a segment, s. Next, we sum to the current LAT of the segment s the times of conduction velocity obtained via the cable equation, of every segment in the pathway between s. To speed up the minimization process for selecting branch s of the PN using the cable equation applied to a straight line that connects s and s of the PN by solving minimization problem  is a function that returns the LAT of a given segment s using the cable equation, T(PVJ) is the known value for the LAT of the PVJ, and the subscript a refers to active PVJ points. Finally, if at least one PVJ is connected to the PN, the procedure is repeated until there are no updates to the tree. The whole procedure of connecting an active PVJ, To simulate the LATs we use the cable equation, which is commonly used to describe the electrical flow in neurons and cardiac cells problem  alongsidmentclass2pt{minimPostProcessing step is applied and illustrated in Fig. After we reach the stopping criterion of the main loop, a However, after this procedure, some PVJs may still be unconnected. This scenario could happen if  problem  by increminimum network. More information and details regarding all the steps and parameters of the method are supplied in the Supplementary Material section After this step, all the active PVJs that the user specified are connected, and the geometric and electric metrics are computed in this topology, which is referred to as the Three different meshes were used to evaluate the robustness of our method. The first considers a simplified mesh that captures the main structures of the biventricular system. Secondly, a canine mesh is the target of the study. A human patient-specific mesh is utilized as the final experiment to evaluate our model. These meshes are shown in Fig. 29. In addition, a simplified PN was generated using a fractal method22. This simplified PN was considered as our reference for comparisons. The branches of the reference PN activate the endocardium at the early region sites of the LV and RV as reported in30. For instance, these regions are close to the apex in the left and right ventricles and the posterior-basal region of the left ventricle. The locations of the PVJs and the His-bundle were manually selected and constructed to provide an activation pattern similar to what is observed physiologically30. All the terminals of the reference PN are considered active PVJs, representing a total of 43 and 20 PVJs in the LV and RV, respectively. Detailed information about the simplified mesh construction, as well as the parameters set used to generate the reference PN with the fractal method are described in the Supplementary Material, Section The simplified biventricular synthetic mesh was constructed via ellipsoids, for the left ventricle (LV) and right ventricle (RV) endocardium and epicardium. The main idea consists of subtracting the ellipsoids and cutting their surfaces along the base plane, as shown in Fig. 17 and later processed into a graph representation25 was considered as our reference for comparison and can be visualized in Fig. 25 only the LV was considered in the experiments. In the present work, both ventricles are utilized to better evaluate the proposed method. Similar to the simplified mesh case, fiber orientation and transmural distance of the mesh were calculated using the LDRB algorithm29. Furthermore, since only the reference PNs of the LV and RV were available, we manually built the His-bundle structure. The provided data has 130 and 98 PVJs over the LV and RV, respectively.A biventricular canine mesh whose PN was reconstructed from histological images28 was used to further evaluate our novel method in a more realistic scenario, as illustrated in Fig. TM, Webster BioSense Inc.)32, with 31 and 16 active PVJ points on the LV and RV surfaces, respectively. To allow time for the Purkinje to grow branches and activate the estimated PVJs, a A biventricular human patient-specific mesh utilized previouslyS, which is the main input parameter of the method (see the previous section), by sub-sampling the set of points on the endocardial surfaces of the meshes.For each of these meshes, we randomly generated the sets 33, which is given byV is the transmembrane potential, Cardiac myocytes are electrically coupled through gap junctions, a specialized set of proteins that allow the spreading of the electrical wave of excitation from one cell to another in the tissue. This phenomenon can be described by a reaction-diffusion equation, called the monodomain model7 for human Purkinje cells and the recent version of the ToRORd model35 for human ventricular cells were applied to provide a sinus rhythm activation.In the present work, when the monodomain model was applied, the Purkinje networks were solved using the one-dimensional form of the equation. On the other hand, for ventricular tissue modeling, the three-dimensional form of Eq.  was usedMonoAlg3D, which is an efficient parallel cardiac solver36. This particular solver can handle non-uniform, non-conforming adaptive meshes and, most importantly, use the graphics processing unit to speed up the solution of both the diffusion and reaction terms associated with the model.To numerically solve the monodomain equations and simulate the action potential propagation in both the Purkinje networks and the ventricular tissue, we used the 30. For the Purkinje domain the conductivity was set to Regarding the parameters to enable the solution of the monodomain model, the surface-to-volume ratio was set to ToRORd model for human ventricular cells35 and the Trovato et al. model for Purkinje cells7 were used mainly because these two cellular models are now considered the latest models in terms human cellular model available in the literature, presenting important advances over previous models like the classical TT3 model37 and the O’Hara-Rudy model38 in several aspects. Similarly, the Trovato et al., Purkinje model provides several features that the Stewart et al., model39 was not able to capture. Moreover, to the best of our knowledge, there was no published work that coupled both models in a Purkinje-biventricular simulation.In addition, for the cellular dynamics, the 25. This approach was used to capture essential features of propagation from Purkinje and ventricular cells accurately including at the PVJ sites, such as the characteristic delay that happens at these locations10, giving rise to a more realistic simulation in terms of cardiac electrophysiology. For a detailed description of how the monodomain model is solved for the Purkinje-coupled simulations check Supplementary Material section To properly evaluate the activation time generated by the PNs, the monodomain equation was solved both for the Purkinje and ventricular domain, instead of the coupled Eikonal model seen in the previous work from7. Constraining the values to those observed in experiments40, we arrived at The parameters of the cable equation used in the generation of the PNs were chosen by comparing the conduction velocity of Eq.  and thati attached to the Purkinje cell, To model the Purkinje-Ventricular-Junctions sites, an additional passive current, Similarly, for the Purkinje cells, a current, To properly evaluate the geometrical features of our PN models, the mean and standard deviation of the branch size and angle of the bifurcations are computed together with the total number of branches and bifurcations.N is the number of samples. Finally, the percentage of the active PVJs within a specific range of error , given by the values To assess the activation accuracy of our model, the minimum/maximum LAT from the active PVJ sites were computed along with the maximum absolute error observed in these points in the generated PN. In addition, the root mean square error (RMSE) and relative root mean square error (RRMSE) between the LAT from the active PVJ sites of the reference points and the generated networks were calculated as follows:To evaluate the robustness of our method, we performed a sensitivity analysis over the input parameters. Our baseline simulation considered the generation of 10 PN trees for the simplified mesh with the following parameter set: Based on the results of Table Analyzing the other parameters, we could verify that the Activation accuracy and execution time are most sensitive to the number mentclass2pt{minimAs a result of this initial analysis, we performed a new set of simulations varying entclass1pt{minimaWhen S, which is the main input parameter of the method, by sub-sampling the set of points on the endocardial surfaces of the meshes. A total of 200 PNs were generated for each mesh, including 100 PNs for the LV and the remainder for the RV. RMSE was calculated for each PN by comparing the LATs at the reference active PVJ sites. The PN samples were sorted by their RMSE values, and the ten best PNs for each ventricle region were used to compute the metrics and statistics. Based on the results of the sensitivity analysis, the parameters of the performed simulations were set to PostProcessing procedure). Consequently, all terminals of the resultant PNs were connected to active PVJs. More details regarding the geometrical and activation results from the ten best PNs for each mesh are presented in the tables within the Supplementary Material, Sections The method proposed in this work was evaluated using three different biventricular meshes of increasing complexity. For each mesh, we randomly generated 100 sets able Eq.  with PurS, the method generated different PN morphologies with similar LATs at the active PVJ sites. For this reason, even for the same set of PVJs, it is possible to have multiple PN solutions that share LATs close to the target values. When the best and worst PNs were compared, as shown in Fig. Figure Analyzing the geometrical results given in Fig. Furthermore, the bifurcation angle and branch size values of the patient-specific mesh varied more than the simplified and canine meshes. For instance, the mean value of the bifurcation angle is approximately As can be seen in Fig. From the RMSE results in Fig. Finally, the fraction of PVJs that were connected within a LAT error tolerance of 2 ms .In addition to the geometrical and activation results, we measured important features related to the computational performance of the method. The execution time spent on each section of the program, the percentage of time required for the geodesic pathway calculus and the percentage of PVJs connected in each step were calculated and are presented in Tables PostProcessing subroutine. A further reason is related to PVJs being already too close to the tree, which means that these points could be connected only by removing the distance criterion constraint, as confirmed in Table PostProcessing function for the simplified mesh.The results in Table Table Table 36 with parameters as previously described in the Methods section.To assess the generated PNs in 3D cardiac simulations, we simulate the activation of the ten best PNs for each mesh using a high-performance monodomain solverFor the patient-specific mesh, as depicted in Fig. To further investigate the differences in LATs presented in Fig. From Supplementary Material Fig. Therefore, this analysis provides support for the notion that the differing PNs are the primary cause of the observed disparities in LATs, particularly regarding the locations of the potential PVJ sites.Figure 41 or parallel simulation using modern GPUs36, that for the monodomain simulations done in this work took on average The reported total execution times for constructing the patient-specific geometries demonstrate the method’s effectiveness in generating the PN in less than 20 min. This achievement is particularly impressive when compared to the time a specialist would typically spend on the initial manual steps involved in defining the PN, such as data importation and seed selection. Moreover, when considering the time required for generating ventricular geometry or simulating propagation, it becomes evident that the method remains highly efficient. It is worth noting that these results hold true even when considering state-of-the-art implementations for automatic segmentation, which can take in general As a final validation experiment, a subset of the original EAM points used to estimate the active PVJs in the patient-specific mesh was selected to compare the errors of the best and worst generated networks. Only EAM points within healthy regions of the endocardial surface in the LV and RV were considered. The points inside or near the damaged tissue that could be affected by fibrosis were eliminated, so that 100 and 47 EAM points were selected in the LV and RV, respectively.3D monodomain simulations were performed considering our method’s best and worst generated PNs. Tissue conductivity was manually calibrated to reproduce a tissue LAT close to the ones provided by the EAM points. As shown in Fig. 43 which decreases the conduction velocity. We have also considered the diameter of the Purkinje branches to be constant. However, physiological imaging studies have observed variations in the diameter of the structure depending on the location19. To address these limitations and improve the generation method, we propose, in the near future, incorporating a dynamic adjustment of the diameter of the Purkinje branches. This adjustment would account for the observed variations in diameter and allow for different conduction velocities along the network. By introducing this dynamic feature, we can enhance the physiological relevance and accuracy of our simulations.Our simulations consider that the conduction velocity across the PN is constant to avoid an extra layer of complexity in the generation method. However, it is known that at bifurcations, there is a “push and pull” effectIn addition, based on the results it was noticed that the PVJ characteristic delay influences the activation of the ventricles. In the near future, we plan to take this delay into account in the active cost function by combining a data-based model, trained by monodomain simulations, and the cable equation.44. It is worth noting that several factors can influence the activity of a specific PVJ, including source-sink mismatch, electrotonic effects, and cellular dynamics45. Furthermore, emerging evidence suggests that non-functional PVJs can exhibit both pro-arrhythmic and anti-arrhythmic properties44, as they may serve as sites for re-entrant circuits or only become activated during network failures.In the present study, we have solely focused on functional PVJs, also known as active PVJs. This choice stems from the limited knowledge regarding both the density and activity of the junctionsTo advance our model and extend its applicability, it would be valuable to explore other techniques that incorporate non-functional PVJs as additional input parameters. This expansion would augment the range of applications for our novel method, providing a more comprehensive understanding of the complex behavior of the cardiac system.47.In this study, our monodomain simulations are focused on the activation of PVJs originating from a single pulse initiated at the His-bundle. The primary objective was to match the LATs of the PVJ points and validate our Purkinje generation method. However, we acknowledge that the density and coverage of PVJs within a specific PN can influence its propensity to generate reentry pathways, potentially leading to ventricular fibrillationTo address this aspect comprehensively in future work, uncertainty quantification studies can be employed. By generating multiple PNs capable of reproducing similar LATs, these studies can assess the probability of PNs with reentry pathways for individual patients. Such investigations would enhance our understanding of the relationship between PVJ characteristics, PN topology, and the risk of ventricular arrhythmia, contributing to personalized assessments and interventions in clinical settings.48 and its presence may influence behavior of these states50.Currently, implementing models for patient-specific PNs which combine geometric and activation precision are essential for more accurate development of cardiac electrical simulations. In addition, the recent increase in the development of cardiac digital twin (CDT) models makes the usage of Purkinje generation methods an essential step towards more realistic simulations. Furthermore, to study cardiac pathologies such as arrhythmias, models that consider PNs have substantial importance since this structure is known to be a source of ventricular fibrillationIn this work, we presented a new method for the automatic construction of the PN to be used in cardiac electrical simulations. The method is an extension of our previous work, which is based on optimization principles. The method uses a new cost function that relies on electrical principles using the classic cable equation and targets LATs of a given set of active PVJs. We demonstrated the capabilities of the proposed method with the generation of different PNs for three biventricular meshes of increasing complexity. In addition, the electrical activity of the best PNs generated by our method was evaluated on a biventricular Purkinje-coupled monodomain simulation using the latest human ventricular/Purkinje cellular models and a high-performance GPU-accelerated solver.Our results, both in terms of activation times and geometric features, indicate that the novel method can capture key aspects of the PNs  while preserving the structure within the endocardial surfaces, even near complex regions such as the papillary muscles. Furthermore, the method is flexible and scalable by allowing PNs generated by other methods to be used as initial roots. As a result, our method can further extend these initial networks to satisfy desired objectives, such as increasing the PN endocardial coverage or linking additional PVJs.In the future, the applicability and performance of the method could be further improved and investigated. For instance, the use of additional electrical information, such as ECG readings, could be used to further constrain PN generation. The execution time of the method could be reduced by using parallel programming. Moreover, uncertainty quantification studies may provide new insights into improving our models and methods.Finally, the novel method provides a new alternative for generating Purkinje networks for patient-specific heart models with controlled morphological metrics and specified local activation times at the Purkinje-ventricular junctions.Supplementary Information.


Noncontact heart rate monitoring techniques based on millimeter-wave radar have advantages in unique medical scenarios. However, the accuracy of the existing noncontact heart rate estimation methods is still limited by interference, such as DC offsets, respiratory harmonics, and environmental noise. Additionally, these methods still require longer observation times. Most deep learning methods related to heart rate estimation still need to collect more heart rate marker data for training. To address the above problems, this paper introduces a radar signal-based heart rate estimation network named the “masked phase autoencoders with a vision transformer network” (MVN). This network is grounded on masked autoencoders (MAEs) for self-supervised pretraining and a vision transformer (ViT) for transfer learning. During the phase preprocessing stage, phase differencing and interpolation smoothing are performed on the input phase signal. In the self-supervised pretraining step, masked self-supervised training is performed on the phase signal using the MAE network. In the transfer learning stage, the encoder segment of the MAE network is integrated with the ViT network to enable transfer learning using labeled heart rate data. The innovative MVN offers a dual advantage—it not only reduces the cost associated with heart rate data acquisition but also adeptly addresses the issue of respiratory harmonic interference, which is an improvement over conventional signal processing methods. The experimental results show that the process in this paper improves the accuracy of heart rate estimation while reducing the requisite observation time. Heart rate is a vital sign of the presence and quality of life activity ; it is aA radar system, a noncontact vital sign monitoring technology, is one of the frontier technologies in this field . The maiAs an emerging technology, millimeter-wave radar technology still faces some difficulties and challenges when monitoring human heart rates. Due to the low frequencies and small amplitudes of heartbeat signals, they are susceptible to interference from DC offsets, respiratory harmonics , and envTo address the above issues, many approaches have employed machine learning and deep learning techniques to process and optimize raw radar data, rather than relying solely on traditional signal processing techniques. For example, one study employed an artificial neural network (ANN) to extract heart rate signals , achieviTo address these problems found in existing research, this paper proposes a heart rate estimation algorithm for radar signals based on self-supervised MAE pretraining, i.e., a heart rate estimation network called the “masked phase autoencoders with a vision transformer network” (MVN) using FMCW radar. The MVN not only solves the problem of respiratory harmonic interference, which is difficult to solve with some of the existing signal processing methods , to furtOverall, the method is divided into four stages. In the subject localization stage, DC and noise reduction are performed using mean reduction to more accurately extract the positions of subjects. During the phase preprocessing stage, the phase signals corresponding to the subject positions are phase-differenced and interpolated inward to smooth out phases with excessive jump amplitudes. In the network training stage, masked self-supervised learning is performed on unlabeled radar phase data via self-supervised pretraining  using maTraditional heart rate monitoring methods typically use electrocardiogram (ECG) technology. Nevertheless, it is challenging to satisfy the demand for prolonged monitoring due to limitations, such as the need to touch electrodes and the inconvenience of portability. In addition, some people using ECG electrodes may suffer from discomfort, such as dermatitis, which further limits the application of ECG technology. Therefore, medical institutions must adopt noncontact heart rate monitoring systems to meet the demand for long-term, efficient, and safe monitoring. Noncontact vital sign monitoring technology based on radar sensors is a research area of great interest. Compared to traditional heart rate monitoring methods, radar has many benefits. First, it is convenient for patients to use without undressing or wearing electrodes, and it avoids concerns about patient privacy. Second, the technology is suitable for both in-bed and out-of-bed situations, enabling 24 h hospital and daily monitoring. However, some aspects could be improved regarding the application of this technology. Its signal power and signal-to-noise ratio are low because electromagnetic waves suffer path losses as they pass through the body, and heart rate signals have low energy and are susceptible to interference from respiratory signals. In addition, self-motion and movement with the body can also pose challenges during radar-based vital sign monitoring. Therefore, obtaining accurate heart rate signals is a challenging problem.To cope with these problems, the current solution strategies focus on three directions: signal processing-based methods, machine learning-based methods, and deep learning-based methods.Among the signal processing-based approaches, Sharpe et al.  used FMCAmong the available machine learning-based methods, Malešević et al.  proposedAmong the available deep learning-based methods, Shih-Hsuan Lai et al.  performeOverall, the existing heart rate estimation algorithms are weak in terms of noise immunity, and their robustness still needs improvement. In addition, the heart rate signal estimation methods using neural networks also face the problems of high data acquisition costs and the fact that different data characteristics are possessed by different subjects.Among the existing algorithms for heart rate estimation using neural networks for FMCW radar, CNNs  are wideCombining the above reasons, this paper introduces the ideas of self-supervised pretraining and transfer learning in natural language processing  and propThe ViT model is attributable to the study of Dosovitskiy et al. . CombiniThe MAE model is derived from the work of Kaiming He et al. , which wCompared to traditional convolutional neural networks (CNNs) and recurrent neural networks (RNNs), MVN heart rate estimation networks use a self-attention mechanism to process the input data and acquire the following advantages:Better generalization capability: The architectures of CNNs and RNNs are typically fixed. For instance, the filter size in CNNs remains constant, and RNNs sequentially process sequences of a set length. This suggests that they might not be particularly adept at handling data with varying sizes and shapes. In contrast, the self-attention mechanism can dynamically and concurrently assign weights to each element in the input data, allowing it to handle data of different sizes and structures more naturally. Moreover, networks based on the transformer architecture can often stack more layers without encountering vanishing or exploding gradients. Consequently, the MVN, which utilizes a self-attention mechanism, can more effectively manage data with diverse sizes and scales, enhancing the model’s generalization capabilities and making it more adaptable to larger or more complex tasks.Better efficiency: Compared to CNNs and RNNs, the self-attentive mechanism can parallelize computations better when processing input data. Therefore, it performs faster in terms of its training and inference speeds.Better cross-domain performance: The heart rate estimation network (MVN) is a transformer-based model, with its core built around a self-attention mechanism. This mechanism allows the network to process all elements in a sequence simultaneously, allocating distinct weights to each element. Such a feature enables a transformer to account for the long-range dependencies within a sequence, unlike CNNs, which are constrained by the sizes of their convolutional kernels, or RNNs, which rely on recursive sequence processing. In recent years, due to these capabilities, transformers have excelled in various domain-specific tasks, including speech recognition , signal Combining the above advantages, the MAE network can learn a large amount of scale-free radar phase data during self-supervised learning, which results in a better ability to extract features from radar data. Subsequently, transfer learning is performed on the feature vector output by the MAE network using the ViT network to complete the mapping of the radar phase to the heart rate using less labeled data. This reduces the acquisition cost while improving the robustness and accuracy of heart rate recovery.The original heart rate signal acquisition process derived from FMCW radar theory  is shownTXC(t) and transmits it through the antenna TX, whose frequency increases linearly with time, as shown in Equation (1):First, the FMCW radar signal generator periodically generates linear FM pulses  as a , as shown in Equation (2):The antenna c is the speed of light, where il and ir are the distance from the object to the radar in the i-th distance cell and its reflection coefficient, respectively. After obtaining the echo signal, the mixer mixes the transmitted signal TX(tC) and the received signal RX(tC) and outputs the IF signal IF(tY), as shown in Equation (3):TXA is the amplitude of the IF signal, and if and iφ are the frequency and phase of the reflected signal at the i-th distance unit, respectively. From part if of the above equation, we can see that the objects reflected at different distances produce different frequencies in the IF signal, so the FFT transform of the IF signal can distinguish the different reflected objects. The minimum distance at which the FMCW radar can distinguish different targets, i.e., the distance resolution resl, is shown in Equation (4):i(tl) = i + hi(tl), where il is the distance from the human body to the radar and i(th) is the small-amplitude regular vibration of the thorax over time. At this point, iφ in Equation (3) is equivalent to:After confining the reflected object to the human body, the vital signals of the human body cause the thorax to undergo small regular vibrations. In turn, the vibration of the human thorax can be described as t, where cλ is the wavelength of the FMCW radar signal. Based on this function, i(th), the regular vibration of the human chest, can be obtained. Thus, by extracting the phase information of the IF signal and processing it, the heartbeat signal corresponding to the human body can be obtained.At this point, the phase in the IF signal becomes a function of time The structure of the millimeter-wave radar heart rate estimation algorithm based on self-supervised transfer learning proposed in this paper is shown in It mainly consists of four parts: person localization, phase preprocessing, MVN training, and heart rate estimation. In the person localization stage, the person location is obtained using the FFT with the Hanning window function with mean decimation. During the phase preprocessing stage, the initial phase signal is received using the inverse tangent function. Subsequently, phase differencing, phase unwrapping, and interpolation smoothing are applied to obtain a cleaner phase signal. In the MVN training stage, masked self-supervised training is performed on the radar phase data using the MAE network. Subsequently, a small amount of data with heart rate markers are used for transfer learning training during the transfer learning process of heart rate estimation. The final heart rate estimation step is performed using the MVN to obtain a heart rate estimate for the window corresponding to the input phase data.This step uses the FFT with a Hanning window  and meanBecause the phase information of the target is contained in the IF signal, this step mainly calculates the distance unit at the location where the FFT is located in the human chest cavity. However, due to the presence of DC and static clutter in an IF signal, it is easily confused with the echo signal generated by the human thorax, thus affecting the accuracy of thorax localization. Therefore, the FFT with the Hanning window function and the mean reduction method is used here to reduce its influence and improve the accuracy of human localization as follows:if part of Equation (3), the frequency of the object in the IF signal is proportional to its distance. Hence, the position of the peak in the frequency directly corresponds to the distance cell where the current target is located. Subsequently, the complex data in each row of the distance slow-time matrix are modulated, and the modal value of this row represents the reflected power at a fixed distance cell. The distance slow-time maps without DC, clutter removal, static objects, human thorax, and distance cell where the zero frequency is located all have high reflection values. Because the vital signs have periodicity, the distance unit data where the chest cavity is located also have periodicity. In contrast, the multipath reflections from static objects and the environment have little or irregular motion. DC signals have no phases, and the data distribution of the distance units where they are located is not periodic. Therefore, this paper proposes the averaging-based reduction method to eliminate the effects of static clutter and DC by subtracting the average value at each distance, i.e., each row in the matrix. The calculation process is shown in Equation (6):ijdu denotes the data in row i and column j of the matrix and ijdu′ represents the data in row i and column j of the matrix after executing the mean reduction method.The Hanning window function is first added to each frame of the IF signal, and subsequently, the FFT is performed on each frame to obtain a long time–frequency matrix. From the After applying mean reduction and de-DC to the long time–frequency matrix, the human chest signal with its periodicity is preserved, and the effects of static clutter and DC are diminished. Subsequently, a peak frequency localization operation is performed to obtain the distance unit where the human chest cavity is located.This step works by extracting phase information at this distance after obtaining the distance unit where the human chest cavity is located by localizing the long time–frequency matrix, restoring the human chest displacement, and further optimizing the phase using phase unwrapping, phase differencing, and interpolation smoothing operations.wrapφ(p) of the target over time is obtained by extracting the phase of the distance unit where the human body is located using the inverse tangent function, as shown in Equation (7):Q(p) and I(p) are the imaginary and real parts of the p-th frame of the complex variable signal in the distance cell where the chest cavity is located, respectively. Due to the nature of the inverse tangent function, the derived phases are both wrapped in the interval . However, the physical displacement of the human chest cavity is greater than cλ/4, so the phase change in Equation (5) exceeds the  interval. In this case, a correction must be made by adding or subtracting 2π in the phase-unwrapping operation. This operation is computed as shown in Equation (8) and is performed only for phases that vary beyond the interval .For the complex data contained in the long time–frequency matrix, the phase changes continuously with time, which is expressed in the matrix as the phase changing constantly with each frame of the IF signal. In contrast, each row in the matrix corresponds to one frame of the IF signal. The phase change φ(p) exhibited by the human thorax after the unwrapping operation is thus obtained. According to the functional form of Equation (5), the complex signal produced after unwrapping can be expressed as Equation (9). In this equation, chestl is the distance from the human thorax to the radar, while chesth(p) describes the regular vibration of the human thorax. Specifically, chesth(p) encompasses the thoracic cavity vibrations induced by both breathing and heartbeats. Breathing results primarily from the expansion and contraction of the lungs, while heartbeats arise from the rhythmic pulsation of the heart. Each of these vibrations possesses distinct frequencies and amplitudes, yet they exhibit pronounced nonlinearity.The phase variation To eliminate the baseline phase drift caused by body movements, it is necessary to perform a differential operation on the phase after the untwisting process, i.e., to determine the difference between two consecutive phase values to obtain a phase differential signal, which can be performed to eliminate the phase drift while preserving the heart rate signal. The phase difference is calculated as shown in Equation (10):insertφ(p) obtained after interpolation is shown in Equation (11):Because thoracic micromotion makes the phase change according to a nonlinear law, the irregular noise in space can quickly destroy the continuity of the phase signal change. When the phase difference exceeds a certain threshold, it is considered that the phase here jumps due to a disturbance, and the inward interpolation method can be used to smooth the phase with an excessive jump amplitude. The phase The above work enables the heart rate prediction network (MVN) to always receive cleaner heart rate phase signals, further enhancing the robustness of the heart rate estimation process and reducing the influences of the surrounding environment and subject anomalies on the phase while preserving the heart rate component.To solve the problems of the existing FMCW radar method in heart rate monitoring, a heart rate estimation network (MVN) based on FMCW radar is proposed in this paper. The network is based on an MAE network with a ViT transfer learning network as a post-network. The MAE encoder has a better feature extraction ability for the phase data encountered during the learning process, and the self-supervised pretraining method enables the deep network to learn autonomously on unlabeled data. Based on this, the ViT network is used as the post-transfer learning module to improve the accuracy and robustness of heart rate estimation while reducing the cost of labeled data acquisition and preserving the feature extraction capability of the MAE network.The training process of the network is divided into two phases. In the self-supervised pretraining stage, the network uses the MAE network for the masked self-supervised learning of unlabeled radar phase data; during the transfer learning phase, the masked self-encoder part of the MAE is used as the MVN encoder, and the ViT network is used as the MVN. In the transfer learning stage, the masked self-encoder part of the MAE is used as the MVN encoder, and the ViT network is used as the MVN decoder to perform transfer learning on a small amount of phase data with heart rate markers.The MVN uses an asymmetric encoder–decoder architecture in both stages, where the number of encoder layers is larger than the number of decoder layers. Such an asymmetric coder–decoder architecture can reduce the time consumption of transfer learning while retaining more parameters of the self-supervised pretrained model. The architecture of one of the deep MVN subnetworks is shown in The unlabeled phase data are converted into a radar phase signal matrix in the self-supervised pretraining stage. Some phase signals in this matrix are subsequently masked and labeled, identifying which signals need to be predicted or ignored. Next, the unmarked signal matrix is fed to the MAE encoder for forward propagation. After the signals pass through the encoder, marker vectors are added to the output semantic vectors in their original order and then sent to the MAE decoder to recover the actual radar phase signal matrix. This is a process of recovering from the encoded semantic space to the original signal space and is an essential step in self-supervised learning. After training and convergence are completed, the parameters of the MAE encoder are extracted and saved for the subsequent transfer learning phase.During the transfer learning phase, a small amount of phase data with heart rate markers are used for training. The signal matrix is first fed to the MAE decoder to obtain an output semantic vector, which is subsequently provided to the ViT network. In the ViT network, the output head vector is processed by an MLP head to output the corresponding window heart rate value. In the above transfer learning process, the method in this paper locks the gradient of the MAE encoder network. That is, the method in this paper does not train the encoder but only the ViT network. This operation can keep the parameters of the encoder unchanged to utilize the feature representations learned during the self-supervised pretraining stage. It also focuses on tuning the ViT network to better adapt it to the target task.In addition, as seen in the phase data shown in The asymmetric encoder and decoder of the MAE contain multiple transformer encoder layers, where each of these layers uses layer normalization . A schemLayer normalization (LN) first calculates the mean and variance of each feature for each sample, unlike batch normalization (BN) , which nCompared to BN, LN has the advantage of avoiding performance problems on small batches while not requiring the same level of performance as instance normalization (IN). Normalization is performed for each channel. In addition, LN can be used in networks, such as RNNs and transformers, to improve their training speeds and accuracies because it does not depend on the size and order of the given batches.The MAE network model is shown in An MAE is an autoencoding network model that reconstructs the original signal from partial observations. Like all autoencoders, the method in this paper has an encoder that maps observations to semantic vectors and a decoder that reconstructs the original signal from the semantic vectors. Unlike traditional autoencoders, we adopted an asymmetric design such that the larger encoder operates only on a portion of the observed signal (without the mask tokens), while the lighter decoder reconstructs the complete signal from the semantic vectors and mask tokens. In the actual computation process, this paper used an MAE encoder containing eight transformer encoder layers and an MAE decoder containing five transformer encoder layers.The MAE encoder includes an embedding layer, class tokens, a position embedding layer, a mask layer, a multilayer transformer encoder layer, and LN. The decoder consists of an embedding layer, a mask token layer, a position embedding layer, a transformer encoder layer, LN, and a linear layer.The input feature dimensionality of the transformer encoder layer of the MAE encoder is 128, while the input feature dimensionality of the MAE decoder is 64. The number of multihead attention mechanisms used by the transformer encoder layers of both coders is 16, and the dimensionality of the feedforward network model is 2048. Finally, the output vector dimensionality of the linear layer is the same as that of the phase data, which is used to predict the phase signal being masked for processing.First, the MAE encoder performs embedding coding on the radar phase data, and after adding a learnable embedding header vector and a positional embedding, it performs the masking process and inputs the data into the transformer encoder layer. In the masking layer, the authors of this paper chose a masking ratio of 40%, i.e., 40% of the unlabeled phase data are randomly marked by the masking operation, while the added learnable embedding header vectors are not masked in this operation. The rest of the data that are not masked and labeled are output as semantic vectors via LN after the multilayer transformer encoder layer computation.Subsequently, the MAE decoder complements the input semantic vectors with corresponding learnable mask vectors (mask tokens) and positional embeddings in the order in which they are masked. Then, the recovered raw phase data are output through the multilayer transformer encoder layer, LN, and a linear fully connected layer. The training process uses the adaptive moment estimation (Adam) optimizer  and(MSE)After completing the self-supervised pretraining process, this paper added the ViT decoder after the MAE encoder part to output the corresponding heart rate estimates, and the architecture diagram is shown in In the ViT transfer learning network, the data output by the MAE encoder are first processed with embedding encoding and positional embedding, followed by LN of the multilayer transformer encoder layer. For a better transfer learning effect in heart rate prediction, the ViT transfer learning network only performs MLP head calculation on the extracted embedding header vectors, which reduces the computational effort while obtaining heart rate values estimated in the current window. Among them, the ViT network uses a five-layer transformer encoder layer, a single transformer encoder layer with an input feature dimensionality of 64, 16 multihead attention mechanisms, 2048 feedforward network model dimensions, and LN for internal normalization.During the transfer learning training process, the model parameters of the MAE encoder are fixed and not trained. For the ViT network, the optimizer still uses the Adam method, the error statistics are MSE values, and a small amount of radar phase data with heart rate markers are used to conduct supervised training on the ViT network. After several rounds of training, the optimal model parameters are selected. The radar acquisition data are subsequently input into this network according to the processing flow, and the heart rate estimates corresponding to the current phase data are calculated.The hardware system used for the experiments consisted of two modules: Module I was a millimeter-wave radar platform consisting of an IWR1642BOOST radar sensor evaluation board and a DCA1000EVM radar data acquisition board from Texas Instruments. Module II was an Neulog Pulse Sensor NUL-208 sensor from Neulog, which was used to collect the actual heart rates of the subjects. The equipment used is shown in The experiment was conducted in a study setting with the subject sitting quietly at approximately 0.5–1.0 m in front of the radar, maintaining a natural breathing state. During the test, the heart position on the chest was kept in a substantially vertical relationship with the transmission direction of the radar antenna. During heart rate monitoring, the FMCW radar and the body-worn heart rate sensor simultaneously measured the subject. The FMCW radar parameters were set as shown in In this paper, a total of 17 volunteers were measured, including 8 females and 9 males, and each volunteer measured 20 sets of data, each set lasting for 60 s with a total of 3000 frames. The heart rate estimation window length was divided into 2.56 s and 5.12 s. Finally, 30,060 and 28,724 heart rate estimation windows were generated, which were divided into training and test sets at a ratio of 8:2. During the self-supervised pretraining stage, the entire training set was used for self-supervised training; during the transfer learning stage, a small portion of the training set was labeled with heart rate information, and the rest was unlabeled; i.e., a small portion of the training set was used for transfer learning training with labeled heart rate data. After completing the above training process, the MVN was finally evaluated on the test set.Five comparison experiments were designed on the same test set with the same window length to test the performance of the method in this paper.The first set of experiments verified the effectiveness of the method used in this paper in the subject location stage and compared the effects of the long time–frequency matrix before and after utilizing the mean reduction method.The second set of experiments verified the effectiveness of the method used in this paper in the phase preprocessing stage and compared the effects of phase preprocessing before and after this operation.The third set of experiments verified the effectiveness of masked self-supervised pretraining, and the network model with the pretraining module removed was chosen for comparison with the method in this paper. The same test sets were determined at 2.56 s and 5.12 s observation times, and the network’s convergence speeds and heart rate estimation accuracies achieved with and without the pretraining module were compared.The fourth set of experiments was a validation experiment used to verify the effectiveness of ViT transfer learning, and the network model without ViT transfer learning was chosen for comparison with the method in this paper. The head vector was extracted at the end of the network model without ViT migration learning, and a linear layer was added for the heart rate output. The same test sets were determined at 2.56 s and 5.12 s observation times, and the convergence speeds and heart rate estimation accuracies achieved by the network with and without ViT transfer learning were compared.(1)The classic FMCW radar life signal estimation algorithm  was used(2)Koyanaka et al.  Operatio(3)Toda et al. , i.e., h(4)Lai et al. , i.e., h(5)The MVN using only labeled data. No unlabeled data were used in the self-supervised pretraining stage, and only the part of the data with heart rate labels was used in both learning stages. This method was used only in the training stage of the MVN, and the rest of the operations were consistent with those in this paper.The fifth set of experiments compared the heart rate estimation accuracies of multiple algorithms on test sets with different labeled data densities under 2.56 s and 5.12 s observation times. The person localization and phase extraction stages remained consistently processed, and the comparison methods only varied in the heart rate estimation stage. Data densities of 32%, 24%, and 16% were selected for different markers. The comparison algorithms are as follows:m is the number of data points and iy is the true value of the i-th data point. i-th data point. The MSE measures the average squared difference between the predicted and true values, with smaller values indicating that better predictive performance is achieved by the model.The error functions utilized during network training were all measured according to the MSE; given a set of true values and a set of predicted values, the MSE was calculated as shown in Equation (12):N is the total number of division windows, f(n) denotes the heart rate estimate for the nth window, and F(n) denotes the heart rate reference value of the contact device for the n-th window.The performance evaluation metric for the heart rate estimation task used the average absolute error percentage (AAEP) statistical metric to measure the magnitude of the error between the measured and reference values. The AAEP is defined as shown in Equation (13):(1)The first group of experimentsIn this experiment, a single set of 60 s data was used for comparison purposes, the FFT obtained the long time–frequency matrix for each row, and its spectrum was plotted as shown in From (2)The second group of experimentsThe first part of this experiment used a single set of 60 s data for comparison purposes. First, phase extraction was executed by computing the data from the columns corresponding to the positions of the individuals in the long time–frequency matrix. This was achieved by using the inverse tangent function as described in Equation (7), and the resulting phase signal is depicted in π, +π] interval return to the normal range, but the baseline drift caused by respiration and noise remains. The before-and-after comparison in The before-and-after comparison shown in In the second part of this experiment, 32% of the phase data with heart rate markers were selected from the whole dataset for training, and the error was verified using the same test set. As seen in In addition, the MAE encoder, which has phase differencing and interpolation smoothing processes, has a minor mean squared error (MSE) in terms of recovering phase data compared to that induced on the original data. This indicates that the phase preprocessing operation makes the deep neural network receive a purer radar signal, which enhances the ability of the MAE encoder to precisely extract the radar phase signal, thus further enhancing the accuracy and robustness of heart rate estimation.(3)The third group of experiments−5, and the training process was stopped after 2000 epochs. The MSE convergence curve of the MVN with the self-supervised pretrained MAE decoder removed is shown as the orange line in This experiment used 32% of the phase data with heart rate markers from the whole dataset for training, and the same test set was used to verify the error. Comparisons were made only between model versions with and without the MAE decoder. The learning rate was set to 5 × 10As shown in (4)The fourth group of experiments−5, and the training process was stopped after 2000 training epochs. The MSE convergence curve of the MVN with ViT transfer learning removed is shown as the orange line in This experiment used 32% of the phase data with heart rate markers from the whole dataset for training, and the same test set was used to verify the error and to make a comparison only between model versions with and without ViT transfer learning. The learning rate was set to 5 × 10As shown in As shown in (5)The fifth group of experimentsIn addition, it can also be seen from This study introduces a heart rate estimation algorithm for radar signals based on self-supervised MAE pretraining, utilizing a 77-GHz FMCW radar. The experiments were conducted in an environment with subjects seated approximately 0.5–1.0 m from the radar, with minimal movement. While this method may not match the precision of contact-based heart rate monitors, it demonstrates the feasibility of noncontact heart rate monitoring using radar. It also addresses some challenges associated with noncontact vital sign estimation using FMCW radar, such as interference from DC and harmonics and the high cost of data collection. Experimental validations show that our method achieves an average absolute error percentage of 8.10% with a sample density of 32% in a 2.56 s time window. This is a significant improvement over the results of other studies, such as the 18.57% value of Koyanaka et al.  and the Regarding the construction of the deep network in the proposed method, we drew inspiration from effective natural language processing network models, adopting an innovative heart rate estimation approach. We began by using easily accessible, low-cost radar signals derived from seated individuals as a dataset for self-supervised pretraining. This strategy endowed the model with enhanced feature extraction capabilities, setting the stage for a subsequent transfer learning step with rich semantic vectors. We then employed a limited set of high-cost data with heart rate labels for transfer learning. This step ensured the model’s ability to accurately estimate heart rates even with limited labeled samples. By integrating self-supervised pretraining with transfer learning, we successfully improved the accuracy of heart rate estimation while reducing the data collection costs and complexity of the network. Even with limited labeled samples, this method guarantees accurate heart rate estimates. Additionally, by combining certain signal processing techniques with deeper network models, we achieved good reductions in the interference caused by respiratory harmonics and noise, shortened the required radar observation time, and enhanced the overall efficiency and practicality of the network.While this solution offers several innovative advantages, it has some limitations. The experimental environment is restricted to subjects in a seated position. Given the high distance resolution of the FMCW radar system, it is sensitive to human movement. Minor movements can easily mask heart rate signals ; hence, This paper proposes a noncontact heart rate estimation method based on FMCW radar. In the subject localization stage, this paper uses the FFT with a Hanning window and mean decimation to increase the accuracy of chest position extraction; during the phase preprocessing stage, phase unwrapping, phase differencing, and interpolation smoothing are performed on the phase signal. These operations eliminate the baseline drift and amplitude jumps and ensure the purity and stability of the obtained heart rate signal. In the training stage of the MVN, masked self-supervised phase signal training is performed using the MAE network. Heart rate marker data transfer learning is then performed using the encoder part of the MAE network with a ViT network. The MVN model combining the MAE encoder and the ViT network can take full advantage of the feature performance achieving in the self-supervised pretraining stage without altering the encoder parameters. In this way, the ViT network is trained to better adapt to specific tasks. This approach enables the MVN model to reduce its training time consumption while enhancing the accuracy and robustness of the final heart rate estimation results. The network also solves the respiratory harmonic interference problem to a large extent. From the experimental results, the method in this paper exhibits high accuracy even within a short observation window. Moreover, it maintains excellent performance even in cases with sparsely labeled data. The method developed in this paper improves the accuracy of heart rate estimation while reducing the cost of radar data labeling, and it demonstrates good robustness in complex environments. These features make it more suitable for practical applications and provide a new noncontact vital sign monitoring solution.To address the future focus of this work, this paper will further compare the proposed method with other contactless solutions and combine the developed algorithmic solution with different environments to adapt to different detected person states, such as people located outdoors and people exercising. Another interesting future work topic is the reconstruction of the ECG and PPG of a detected person, as many studies have attempted to determine the presence or absence of cardiovascular diseases based on ECG signals. Therefore, the potential ability of this algorithm to detect whether a user has cardiovascular disease is worth investigating. In addition, due to the episodic nature of cardiovascular disease, wearing an ECG device for a limited period of time may not be sufficient for monitoring abnormal activity in the cardiovascular system, whereas the FMCW radar has the ability to continuously monitor a patient, and cardiovascular disease detection techniques based on this device are also of great value.


We studied two problems called the Traveling Repairman Problem (TRPTW) and Traveling Salesman Problem (TSPTW) with time windows. The TRPTW wants to minimize the sum of travel durations between a depot and customer locations, while the TSPTW aims to minimize the total time to visit all customers. In these two problems, the deliveries are made during a specific time window given by the customers. The difference between the TRPTW and TSPTW is that the TRPTW takes a customer-oriented view, whereas the TSPTW is server-oriented. Existing algorithms have been developed for solving two problems independently in the literature. However, the literature does not have an algorithm that simultaneously solves two problems. Multifactorial Evolutionary Algorithm (MFEA) is a variant of the Evolutionary Algorithm (EA), aiming to solve multiple factorial tasks simultaneously. The main advantage of the approach is to allow transferrable knowledge between tasks. Therefore, it can improve the solution quality for multitasks. This article presents an efficient algorithm that combines the MFEA framework and Randomized Variable Neighborhood Search (RVNS) to solve two problems simultaneously. The proposed algorithm has transferrable knowledge between tasks from the MFEA and the ability to exploit good solution space from RVNS. The proposed algorithm is compared directly to the state-of-the-art MFEA on numerous datasets. Experimental results show the proposed algorithm outperforms the state-of-the-art MFEA in many cases. In addition, it finds several new best-known solutions. The Traveling Salesman Problem with Time Windows (TSPTW) , and TraThe Travelling Salesman Problem with Time Windows (TSPTW) is a popular NP-hard combinatorial optimization problem studied much in the literature . The algIn the Travelling Repairman Problem with Time Windows (TRPTW), there is an exact algorithm and three metaheuristics algorithms in the literature: (1) These above algorithms are the state-of-the-art metaheuristics in current. However, they are designed to solve each problem independently. That means they cannot solve both two problems well simultaneously. In To date, the MFEA framework  has beenvia genetic information transfer. Their dMFEA-II controls the knowledge transfer by adjusting the crossover probability value. The technique allows good knowledge to transfer between tasks. Although the results are promising, the drawback of the above two algorithms  are used. It opens up the chance for knowledge transfer through crossover-based exchange between tasks.The combination between the MFEA and the RVNS is to have good transferrable knowledge between tasks from the MFEA and the ability to exploit good solution spaces from the RVNS. However, focusing only on reducing cost function maybe lead the search to infeasible solution spaces like the algorithms . TherefoNumerical experiments show that the proposed algorithm reaches nearly optimal solutions simultaneously in a short time for two problems. Moreover, it obtains better solutions than the previous MFEA algorithms in many cases.This article introduces the first algorithm combining the MFEA framework and RVNS to solve two tasks simultaneously. The combination is to have positive transferrable knowledge between tasks from the MFEA and the ability to exploit good solution spaces from RVNS. The major contributions of this work are as follows:The rest of this article is organized as follows. Sections 1 and 2 present the literature and preliminary, respectively. Section 3 describes the proposed algorithm. Computational evaluations are reported in Section 4. Section 5 presents the conclusions and future work.https://lopez-ibanez.eu/tsptw-instances), the objective function cost (using the function cost of the TRPTW) of this solution for the TRPTW is 7,519, while the known-best cost for this instance for the TRPTW is 6,351 . Thus, the difference between the two objective function costs is 15.5%. It implies that a good metaheuristic algorithm for the TSPTW does not produce a good solution for the TRPTW and vice versa. The above algorithms are the best algorithms for two problems. However, they only solve each problem independently but cannot simultaneously produce good solutions for two problems.We consider an example that describes the difference between two problems in a specific instance. If we use the optimal solution of n40w160.002 instance for the TSPTW  algorithms perform well in practice and validate their performance through experiments. This approach is suitable for a problem with large sizes.For NP-hard problems, we have three approaches to solve the problem, specifically, (1) exact algorithms, (2) approximation algorithms, and (3) heuristic (or metaheuristic) algorithms:Previously, several metaheuristics have been proposed to solve the TSPTW  and the This article proposes an MFEA approach to solve two problems simultaneously. Our MFEA solves two tasks simultaneously: the first task is the TRPTW, and the second is the TSPTW. Experiment results indicate its efficiency: (1) for small instances, the proposed algorithm obtains the optimal solutions in both two problems; (2) for large ones, our solutions are close to the optimal ones, even much better than those of the previous MFEA approaches.k optimization problems are needed to be performed simultaneously. Without loss of generality, tasks are assumed to be minimization problems. The P has a set of properties as follows: Factorial Cost, Factorial Rank, Scalar Fitness, and Skill Factor. These properties allow us to sort and select individuals in the population.Factorial Cost Factorial rank Scalar-fitness i.e., Skill-factor The overview of multifactorial optimization is introduced in SP individuals (SP is the size of population) in the unified search space and then evaluate it by calculating the skill-factor of each individual. After the initialization, the iteration begins to produce the offsprings and assign skill-factors to them. Selective evaluation guarantees that the skill-factor of each new offspring is selected randomly among those of the parents. The offspring and the parent are merged in a new population with SP solutions with the best skill-factors for the next generation.The pseudo-code of the basic MFEA is described in rmp). Specifically, the child is created using crossover from parents with the same skill-factor. Otherwise, the child is generated by a crossover with a rmp value or by a mutation when parents own different skill-factors. A large rmp value generates more information exchanging between tasks. Also, in the traditional GA, the fitness of child is evaluated directly, while the skill-factor is assigned to it in the MFEA.The MFEA is also unlike multiobjective optimization. In multiobjective optimization, we have one problem with many objective functions. On the other hand, the MFEA solves many tasks at the same time. In addition, multiobjective optimization generally uses a single representative space, while the MFEA unifies multiple representative spaces for many tasks.Running two algorithms for two tasks independently is not the idea of the MFEA approach. When two algorithms run independently, each task is represented by its own search space. There is no transferrable knowledge between tasks. Otherwise, in the MFEA, two tasks use the unified search space, and transferrable knowledge between tasks is done. It can increase convergence and improve the quality of solutions for multitasks. SP individuals is then generated in the second step. All solutions for the population must be feasible. After that, the iteration begins until the termination criterion is satisfied. Parents are selected to produce offsprings using crossover or mutation and then assign skill-factors to them. The offsprings are then added to the current population. The individuals of the population are evaluated to update their scalar-fitness and skill-factor. We select the best solutions regarding skill-factor from the current population and convert them from the unified representation to each task’s one. It is then fed into the RVNS step to find the best solution for each task. The output of the RVNS is then converted to the unified search space. Finally, it is added to the population. The Elitist strategy keeps the SP solutions with the best skill-factors for the next generation.This section introduces the pseudocode of the proposed MFEA+RNVS. The TSPTW task corresponds to a particular task in the MFEA, while another is the TRPTW task. The flow of the proposed algorithm is described in In the literature, various representations are proposed for two problems. Among these representations, the permutation representation shows efficiency compared to the others. In the permutation, each individual is coded as a set of Sp individuals in the initial population for the genetic step.Each feasible solution is created from the RVNS to take a role as an individual in the population. Therefore, we have RCL) is created by ordering all non-selected vertices based on a greedy function that evaluates the benefit of including them in the solution. One vertex is then chosen from RCL randomly. Since all vertices are visited, we receive a solution. If this solution is a feasible one, it is an initial solution, and this step stops. Conversely, a repair procedure based on the RVNS with many neighborhoods  minus the number of vertices with the same position on T and T in the population. The larger R is, the better solution T is.The metric distance between two solutions is the minimum number of transformations from one to another. We define the distance R values to mate. We choose the tournament selection operator  or if the parents have the same skill-factor. When parents have the same skill-factor, we have inter-crossover. Otherwise, the intra-crossover is applied to parents with different skill-factor. It opens up the chance for knowledge transfer by using crossover-based exchange between tasks. In The first type is related to the position of certain genes in parents .The second selects genes alternately from both parents, without genes’ repetition .The third is an order-based crossover .The crossover is implemented with the predefined probability (NO), another crossover operator is replaced randomly. Using multiple crossovers helps the population be more diverse than one crossover. Therefore, these operators prevent the algorithm from premature convergence. If the offsprings are infeasible, the fix procedure is invoked to convert them to feasible ones. The offsprings’ skill-factors are randomly set to the one of the father or mother. The detail in this step is described in Initially, we select a crossover randomly. If any improvement of the best solution is found, the current crossover operator is continued to use. Otherwise, if the improvement of the best solution is not found after the number of generations , population size (SP), group size (NG) in the selection, and crossover rates (rmp) influence the algorithm’s results. Many efforts in the literature studied the algorithm sensitivity to parameter changes. We found that no work shows which values are the best for all cases. However, the following suggestions help us in choosing parameter values:A large generation number does not improve performance. Besides, it consumes much time to run. A small value makes the algorithm fail to reach the best solution .A higher crossover value obtains new individuals more quickly while a low crossover rate may cause stagnation .A large population size can increase the population diversity. However, it can be unhelpful in the algorithm and increase the running time of it .Increased selection pressure can be provided by simply increasing the group size. When the selection pressure is too low, the convergence rate is slower, while if it is too high, the chance of the algorithm prematurely converges .The The experiments are conducted on a personal computer equipped with a Xeon E-2234 CPU and 16 GB bytes of RAM. The program was coded in CBased on the suggestions, we determine a suitable range for each parameter in The fourth sets in the majority are the instances proposed by http://comopt.ifi.uni-heidelberg.de/software/TSPLIB95/) includes fourteen instances from 50 to 100 instances.The TSPLIB  to (1) the optimal solution (notation: OPT); and (2) the known best solution (notation: KBS) of the previous metaheuristics  as follows:The efficiency of the metaheuristic algorithm can be evaluated by comparing the best solution found by the proposed algorithm  . The optimal or best solutions in the TRP are obtained from In Tables, p value)  was implp value) . In the In this experiment, a new selection operator for the MFEA+RVNS algorithm effectively balances knowledge transfer and diversity. Due to being too expensive in computation, we choose some instances to evaluate the efficiency of this operator. In In Generally speaking, algorithms get stuck into local optimum because there is a lack of balance between exploration and exploitation. Exploration helps the search to explore extension spaces on a global scale, while exploitation helps the search to focus on local space by exploiting the information that a current solution is reached in this space. In this experiment, the balance between exploration and exploitation is considered. We run the MFEA with or without the RVNS on the same selected instances. In vs its average metric distance to all other local minima (the distance metric and its average is defined in “Selection operator”). The results are illustrated in In gap is −0.28%  algorithms is developed to solve the TRPTW only). The obtained results are impressive since it can be observed that the proposed algorithm finds not only near-optimal solutions but also the new best-known ones for two problems simultaneously. It also indicates the efficiency of positive transferrable knowledge control techniques between optimization tasks in improving the solution quality.In It is impossible to expect that the MFEA+RVNS always outperform in comparison with the state-of-the-art metaheuristic algorithms for the TSPTW and TRPTW in all cases because their algorithms are designed to solve each problem independently. We adopt the proposed algorithm in the experiment to solve the TSP and TRP problems. Otherwise, we also use three algorithms  to solveStatistical tests are used to check whether the difference between the proposed algorithm and the remaining ones is significant or not. A non-parametric test (Friedman and Quad test) is carried out in the group of the algorithms to check if a significant difference between them is found. The output of the Friedman test in vs its average metric distance to all other local minima of two algorithms (YA and OA). The black and red points are still the results of the MFEA and RVNS, respectively. Their algorithm explores widely in the solution space, implying a good capacity for exploration. However, exploitation capacity is not good enough. Moreover, The normalized objective cost (see formulation 4) can be used to analyze the convergence trends of our MFEA+RVNS algorithm. The convergence trend of the two strategies is described in K is the number of tasks). Therefore, we consider the worst-case situation when the number of generations for multitasking is K times the one for single-tasking. If multitasking obtains better solutions than single-tasking in this case, we can say that multitasking saves computational efforts. The experimental results are described in When multitasking is run with the same number of generations as single-tasking, on average, it only consumes In short, the efficiency of multitasking is better in comparison with single-tasking because of the process of transferring knowledge during multitasking. It is an impressive advantage of the evolutionary multitasking paradigm.In this article, our contribution is threefold. Firstly, we propose a new selection operator that balances skill-factor and population diversity. The skill-factor effectively transfers elite genes between tasks, while diversity in the population is important when it meets a bottleneck against the information transfer. Secondly, multiple crossover schemes are applied in the proposed MFEA+RVNS. They help the algorithm have good enough diversity. In addition, two types of crossover (intra- and inter-) are used. It opens up the chance for knowledge transfer through crossover-based exchange between tasks. Lastly, the combination between the MFEA and the RVNS is to have good transferrable knowledge between tasks from the MFEA and the ability to exploit good solution spaces from the RVNS. However, focusing only on reducing cost function maybe lead the search to infeasible solution spaces like the algorithm . TherefoExtensive experiments on the benchmark dataset indicate that the proposed algorithm simultaneously obtains good solutions for both problems. In addition, it obtains better solutions than the other MFEA algorithms in many cases. More interestingly, it finds the new best-known solutions compared to the state-of-the-art metaheuristics only for the TRPTW in many cases.In future work, we will study how to apply multiple population ideas for multitasking. Many researchers is interested in the approach . The app