The popularity of video services such as video call or video on-demand has made it impossible for people to live without them in their daily lives. It can be anticipated that the explosive growth of vehicular communication owing to the widespread use of in-vehicle video infotainment applications in the future will result in increasing fragmentation and congestion of the wireless transmission spectrum. Accordingly, effective bandwidth management algorithms are demanded to achieve efficient communication and stable scalability in next-generation vehicular networks. To the best of our current knowledge, a noticeable gap remains in the existing literature regarding the application of the latest advancements in network communication technologies. Specifically, this gap is evident in the lack of exploration regarding how cutting-edge technologies can be effectively employed to optimize bandwidth allocation, especially in the realm of video service applications within the forthcoming vehicular networks. In light of this void, this paper presents a seamless integration of cutting-edge 6G communication technologies, such as terahertz (THz) and visible light communication (VLC), with the existing 5G millimeter-wave and sub-6 GHz base stations. This integration facilitates the creation of a network environment characterized by high transmission rates and extensive coverage. Our primary aim is to ensure the uninterrupted playback of real-time video applications for vehicle users. These video applications encompass video conferencing, live video, and on-demand video services. The outcomes of our simulations convincingly indicate that the proposed strategy adeptly addresses the challenge of bandwidth competition among vehicle users. Moreover, it notably boosts the efficient utilization of bandwidth from less crowded base stations, optimizes the fulfillment of bandwidth prerequisites for various video applications, and elevates the overall video quality experienced by users. Consequently, our findings serve as a successful validation of the practicality and effectiveness of the proposed methodology. As the greenhouse effect worsens, environmental regulations for vehicles are now becoming more stringent in numerous countries. In order to meet the increasingly stringent regulations on greenhouse gas emissions and to solve the problems associated with traditional engine vehicles, such as air pollution, climate change, and high fuel prices, electric vehicles (EV), which use green energy to reduce carbon emissions, have been recognized as the future alternative to traditional automobiles.Affected by the COVID-19 epidemic that started in late 2019, academics and organizations have been turning to video conferencing to match the COVID-19 precautionary measures, as well as to improve efficiency and convenience. Video conferencing applications account for an increasing share of Internet applications and bandwidth usage , and theEncoding plays a vital role in modern digital media, ensuring efficient and reliable transmission and playback of videos in various situations. Currently, a significant portion of videos use Advanced Video Coding (AVC)  as theirCommencing in 2019, the official deployment of 5G witnessed the incorporation of sub-6 GHz and millimeter wave (mmWave) frequencies . In respBeyond tackling the issues of the next-generation network by enhancing network architecture and optimizing algorithms for resource allocation, the forthcoming generation of networks will predominantly revolve around 6G technology. Thus, 6G promises to usher in an era characterized by continuous connectivity, immersive experiences, robust support for a multitude of simultaneous users, ultra-low latency, universal accessibility, unmatched data capacity, unwavering reliability, and stringent security measures . In receScholars have already conducted research on the applications of THz in vehicle communications. Li et al.  analyzedTo address these limitations, the dense deployment of base stations is required to extend coverage. Moreover, the expenses associated with THz equipment are substantial . To exteAlongside the THz frequency range, another set of frequencies currently unutilized in existing communication standards is VLC. This alternative spectrum brings forth specific advantages over other communication technologies. Firstly, VLC is relatively safe for human eyes  and can ectively . The aboBeyond the realm of 6G technology, the significance of edge computing has risen substantially within contemporary communication networks. This strategy, involving the proximity of computation and resources to users, harmonizes seamlessly with the high-speed and low-latency benefits inherent in 6G. It is foreseeable that within the landscape of vehicular networks in the years to come, the fusion of 6G technology and edge computing will engender heightened efficacy in data processing and real-time applications. The advancement of mobile edge computing has also inspired the development of edge transcoding technology, where service providers can distribute transcoding tasks from central servers to edge servers closer to users for video transcoding. Edge transcoders can convert high-bitrate video into low-bitrate versions for user selection, improving the performance of video services by reducing transmission delay . FurtherAs the demand for future multimedia traffic continues to rise, bandwidth shortages become unavoidable during peak periods. At present, multiple research endeavors have introduced techniques to allocate bandwidth and improve mobile network throughput, aiming to tackle the challenge of allocating network resources efficiently among users. Yuan et al.  introducThe advent of 6G technology holds the promise of significantly enhancing vehicular mobility. Vehicles operating at high speeds can leverage the capabilities of 6G connectivity to establish cooperative communication, thereby ensuring safe following distances and smooth traffic flow . ConsequBased on the literature cited above, it is evident that most recent studies have primarily focused on RSUs and base stations for video downloading and transmission, while overlooking the potential of vehicular communication technology to enhance these processes and neglecting the prioritization of real-time applications. With the expanding user base of video applications, it is apparent that perceived video quality could be adversely affected by bandwidth constraints during peak traffic hours. In view of different characteristics and requirements of multimedia applications, future bandwidth allocation schemes should be planned according to the multimedia requirements so that the real-time multimedia applications can be prioritized to receive sufficient bandwidth.To the best of our knowledge, there is a noticeable gap in the existing literature when it comes to exploring the utilization of the most recent 6G network communication technologies for distributing bandwidth in multimedia applications within upcoming vehicular networks. In light of this, our paper employs THz base stations, as well as mmWave and sub-6 GHz band base stations, to fulfill the bandwidth requirements for video applications targeting EV users. Due to the continuous growth of video traffic, service providers need to offer high-quality streaming approaches to meet user expectations. In addition, this paper adopts EVs as edge devices, enabling them to provide edge computing, storage, and communication capabilities. VLC technology is being utilized in V2V communications to provide high data rate and high security for inter-vehicle video transmissions. Considering the different characteristics and requirements of multimedia applications, this paper employs the most widely used AVC with high encoding speed for real-time video conferencing sessions and live videos, and adopts the VVC with high compression rate for non-real-time VoDs, so that the users can watch high-resolution videos at a lower bit rate.In view of the significant time delay of traditional centralized computing for multimedia applications during peak hours and the increasing computational power of edge nodes, a decentralized computing architecture is adopted in this paper.Through the utilization of the fundamental technologies of 6G, such as THz frequencies, in conjunction with the existing 5G mmWave and sub-6 GHz base stations, this paper facilitates the creation of a network environment characterized by elevated transmission rates and extensive coverage.In order to ensure uninterrupted real-time video playback for EV users, this paper chooses the low-latency AVC method as the preferred approach. This decision is especially crucial for applications such as real-time video conferencing and live video. Moreover, the introduced algorithm places high priority on efficiently allocating bandwidth for real-time video content. All of these efforts are directed towards achieving the best possible performance standards for users of EVs.This paper demonstrates the capability to employ VLC for V2V communication, thereby enabling the redistribution of bandwidth from less-congested base stations located in alternate road sections. Furthermore, this paper capitalizes on base stations equipped with ample bandwidth and fosters collaboration among other EVs to pre-download video segments. This strategic approach leads to noteworthy improvements in bandwidth availability for various video applications, while also markedly enhancing the efficient utilization of bandwidth resources from less-congested base stations. Additionally, this paper successfully mitigates prolonged delays in downloading video content from distant servers and alleviates congestion in vehicular networks arising from a substantial influx of video applications by EV users during peak hours. Consequently, this paper effectively heightens users’ perceived video quality across all genres of video applications.The main contributions of this paper can be outlined as follows:This paper employs a decentralized computing architecture, as illustrated in In the realm of advancing chip technology, there is a prevailing anticipation that forthcoming next-generation vehicular networks will prominently integrate chips for a diverse range of computational functions. Addressing the burgeoning demand for decentralized computing, as introduced in this paper, dedicated servers are strategically deployed at both RSUs and base stations. Their primary purpose is to facilitate multimedia processing and computational transmissions.Within the context of RSUs, these servers assume a pivotal role in several key aspects. They diligently record the bandwidth requirements of electric vehicle users within their respective managed segments. Concurrently, these servers store crucial information regarding the available bandwidth originating from base stations. Moreover, these servers harness their computational prowess to efficiently execute various RSU modules, optimizing their operational performance.On the flip side, base stations also reap the benefits of these servers. Once an EV user initiates a video service application, the server situated at the base station undertakes the task of intelligently prioritizing the storage of specific video files. This strategic maneuver effectively mitigates potential download latency concerns, enhancing the overall user experience. Additionally, leveraging the robust computational capabilities of these servers, video transcoding emerges as a feasible capability. This translates to offering users an array of video resolutions to choose from, tailored to their preferences and requirements.If the coverage area of the base station of the road section that the EV traverses cannot meet the bandwidth requirement for the EV user’s video application, the managing RSU of the road section will activate the “Real-time Bandwidth Allocation” module to assist in allocating the bandwidth to meet the minimum requirement for the video application according to the characteristics of the video application and the pre-set video quality requirement of the EV user. In the case of video conferencing or live video, the RSU starts with the EV, adds other EVs arriving at the road section at the same time to form a fleet, and continues to expand the fleet until the last fleet member is on a road section where the base station’s coverage can provide bandwidth to the requesting EV. For the scenario involving VoD, this paper proposes a strategy wherein a separate EV traversing the same road segment pre-downloads the necessary video segment when in a bandwidth-accessible area. Subsequently, when the two EVs cross paths, the pre-downloaded video segment is transmitted from the previously prepared EV to the requesting EV user via V2V communication. Notably, the EV using the video application will also track the EV users’ satisfaction with the video playback quality at regular intervals during the video playback process, and if there is a need to adjust the video playback quality, it will notify the “Real-time Bandwidth Allocation” module of the managing RSU(s) for subsequent bandwidth adjustments of the video usage.The modules shown in Before the start of the EV, the EV user starts the OBU to set the departure point, departure time, and destination, and then starts to run this module. The EV first downloads the global real-time road traffic information from the cloud, and uses Dijkstra’s algorithm to estimate the shortest route from the departure point to the destination based on the average traveling time of each road section. Since the arrival time of EVs at each road segment is affected by the traffic conditions at the time of arrival at the road section, this module calculates the arrival time at each road section based on the latest traffic condition information of each road segment of the shortest driving route, and notifies the RSU managing the road section of the driving route and the estimated time of arrival at each road section.To avoid discrepancies between the latest estimated arrival times of EVs and their expected arrival times at road sections due to ad hoc changes in itineraries by EV users or road congestion during peak hours, this module recalculates the arrival time at each road section at regular intervals based on the latest traffic condition information sent from the RSU that manages each road section the EV travels through, and uses a machine learning technique to recalculate the arrival time at each road section. If the recalculated arrival times at the road sections are too different from the original predicted times, the module sends the updated times to the RSUs that manage the road sections of the traveled route. Given the effectiveness of machine learning in predicting road travel times in the literature ,59, the The steps of this module are described below.Step 1: Before the EV starts its journey, it first downloads the global real-time traffic condition information from the cloud, and after setting the starting location and time as well as the destination, it estimates the shortest path from the starting location to the destination using Dijkstra’s algorithm based on the cost of the average traveling time of each road section.Step 2: The estimated time of arrival at each road section of the traveled route is given by:Step 3: The estimated time of arrival at each intersection is transmitted to the RSU that manages each road section, along with the EV’s route.Step 4: This module performs the background execution mode here. After the preset time interval, the latest traffic condition information is obtained from the RSU managing the road section, and the arrival time at each road section of the traveled route is recalculated using Equation (1).Step 5: If the updated EV arrives at the traveled section in a time that exceeds the system’s predefined thresholds due to a temporary trip change or peak hour congestion, the RSU governing the traveled section will be notified of the revised arrival time.Step 6: Before the EV reaches its destination, go back to step 4 to continue.After an EV user activates the video application while the EV is in motion, this module obtains the requirements and specifications of the application from the server of the video application software provider, and adjusts the quality and bandwidth requirements of the video application in a timely manner according to the user video quality requirements pre-set by the EV user. Then, this module transmits the video bandwidth demand to the managing RSU of each road section that the application travels through during the video usage period, and each RSU carries out the bandwidth allocation according to the demand specification of the video application while the EV traverses on the managing road section.The steps of this module are described below.Step 1: After an EV user activates a video application, the requirements and specifications for the video application is retrieved from the video application software vendor’s server.Step 2: The RSU along the route informs the minimum bandwidth that can be provided by the base station coverage of the road section it manages.Step 3: Using the driving durations of EVs across individual road sections and considering the resolution prerequisites for video applications by EV users, the calculation of the essential minimum bandwidth required for video application within each time slot or video segment throughout the driving duration takes place. This calculation is followed by an examination of the bandwidth provisioned by the base station within the coverage area of the corresponding road section to ascertain if it aligns with the stipulated requirements of the video application.As shown in Equations (2) and (3), if the quantities of Equation (6) clarifies that the pre-download of a video segment must be completed before the playback can begin. Meanwhile, Equation (7) guarantees that the buffer space available in the EV is sufficient to accommodate a video segment prior to its playback.Step 4: If all of the base station coverage areas of the EV routes can meet the minimum bandwidth requirement, this module notifies each RSU of the road section of the bandwidth used in the road section and proceed to Step 6. On the other hand, this module notifies the RSUs along the road sections with insufficient bandwidth to assist in adjusting the bandwidth. The optimization objectives concerning the video quality requirements of EV users can be formulated by:Equation (8) allocates the required bandwidth for the video application in terms of the EV user’s perceived video quality. Equations (9) and (10) delineate the upload and download durations for video conferencing/live video during each time slot. Equations (11) and (12) present the distinct quality metrics utilized to assess the effectiveness of video conferencing and live video, following the guidelines outlined in the cited reference . FurtherEquations (14)–(16) ensure that the video resolution remains equal to or exceeds the minimum resolution defined by the video service provider. Simultaneously, Equation (17) guarantees the complete pre-download of individual video segments before initiating their playback. Moreover, Equations (18) and (19) delineate the rebuffering time for each video segment.Equations (20) and (21) establish the prerequisite that the base stations covering the road segments traversed by the electric vehicle (EV) must possess the necessary upload and download bandwidth to support video conferencing and live video, as required. Similarly, Equation (22) imposes an analogous requirement for VOD. Additionally, Equation (23) ensures that the buffer space within the EV is appropriately dimensioned to accommodate a video segment before its playback.To be specific, the optimization problem outlined in Equation (8) is being approximated in this scenario by lowering the requested video resolution from the EV user to align with the minimum resolution established by the video service provider.Step 5: Should the bandwidth obtained in the previous step continue to be inadequate for supporting the minimum resolution specified by the video service provider, the EV user will receive a notification.Step 6: This module operates in the background execution mode. Following the time interval established by the system, the process returns to Step 2 to readjust the quality and bandwidth prerequisites of the video application, provided that the video application remains active.If the video bandwidth requirement of the EV user cannot be met when the EV arrives at the road section managed by the RSU, this module assists in allocating the bandwidth according to the characteristics of the video application. If the user’s video application is video conferencing or live video, the bandwidth originally allocated for other VoD downloads in the same time slot will be reallocated to the required video conferencing or live video. Should the video bandwidth requirement remain unfulfilled, the EV will assume the role of the ultimate member within a fleet. Subsequently, the fleet will undergo expansion until the leading EV member enters a road segment where the base station’s coverage can sufficiently accommodate the demanded video bandwidth. The bandwidth is then transferred from the leading EV member to the requesting EV through V2V communication.In the case of VoD, if a video segment cannot be downloaded in time before the video segment is played, the RSU checks the routes of other EVs that arrive at the same road section at the same time as the EV playing the VoD to find out whether there is a base station that can provide bandwidth for pre-downloading the required video segment. Once a base station that can support bandwidth is found, the EV traveling through the road section covered by the base station will download the required video segment to the on-board storage of the EV. When the EV storing the downloaded video segment arrives at the same road section as the requesting EV, the video segment can be transmitted to the VoD user via V2V communication.The steps of this module are described below.Step 1: For VoD, proceed to Step 8.Step 2: The RSU allocates the bandwidth of other VoD(s) in the same time slot to the demanding video conferencing/live video.Step 3: If the bandwidth requirement for video conferencing/live video is satisfied, go to Step 6. Otherwise, proceed to the next step.Step 4: In cases where the bandwidth necessary for video conferencing or live video is inadequate, the RSU initiates the process by including the EV as the final member of a fleet for bandwidth transfer and incorporating other EVs into a fleet. This fleet expansion persists until the foremost fleet member reaches a road segment where the base station’s coverage can sufficiently supply the required bandwidth for the requesting EV. The establishment of the fleet must fulfill the subsequent requirement:Equation (24) serves as the foundation for configuring the fleet established during the transmission relay, optimized for the minimum fleet length. Building upon this, Equation (25) outlines the specific composition of the fleet, where Equation (26) identifies the designated EV as the terminal component of this fleet. To guarantee a harmonized sequence, Equation (27) specifies the scheduling of the initiation time for video conferencing or live video subsequent to the EV’s departure, while also ensuring the timely conclusion of the application before the EV reaches its destination.Equations (28) and (29) guarantee that the leading fleet member reaches a road segment where the base station’s coverage can adequately provide the necessary bandwidth for the requesting EV. Additionally, Equations (30) and (31) establish that the V2V communication bandwidth between two successive fleet members must not fall below the relayed bandwidth acquired by the requesting EV.To provide specific details, the process of forming the fleet as defined in Equation (24) entails a step-by-step assessment of all EVs that reach the same road segment as the requesting EV. Starting from the selected EV, the expansion of the fleet persists until the farthest member of the fleet enters a road segment where the coverage from the base station is proficient in supplying the required bandwidth for the EV that initiated the request.Step 5: In instances where a certain VoD allocates bandwidth to video conferencing/live video, resulting in an inability to meet the minimum bandwidth requirement of the VoD due to reduced bandwidth availability, this module advances to the subsequent step in order to acquire the necessary bandwidth for the VoD. Conversely, the module concludes.Step 6: To examine whether there are base stations in the coverage area of the road sections that can provide bandwidth for EVs to pre-download the required video segment(s):Step 7: If bandwidth is available in the base station coverage area of another road section, pre-download the video segment from the base station to the on-board storage of the EV while passing through the base station coverage area. Notify the managing RSU of the base station and this module ends. If not, proceed to the next step.Step 8: Check if other EVs arrive at the same road section as EV Step 9: If it is not possible to find an EV that arrives at the same time on the same road section as EV Step 10: View the routes of the EVs arriving at the same section of the road at the same time as Step 11: If EV In this paper, the proposed algorithm is simulated using a personal computer with an Intel Core i7 2.9 GHz CPU and 64 GB RAM. To assess the effectiveness of the proposed algorithm, two comparison targets are employed. The first target is the conventional first-come, first-served (FCFS) method. The second reference benchmark is a contemporary state-of-the-art algorithm that combines the DQN and AQCS [The historical traffic data are obtained from a website of traffic volume counts for New York City . Video aThree types of base stations, including THz , mmWave The red curve represents the bandwidth requirements of video applications.The yellow curve depicts the bandwidth allocated to video applications using the conventional FCFS method.The green curve showcases the bandwidth allocated to video applications when employing the DQN+AQCS algorithm.Finally, the purple curve demonstrates the bandwidth allocation achieved by video applications when utilizing the proposed algorithm.The yellow curve in As observed in the purple curve in However, as depicted by the yellow curve in Conversely, as shown on the purple curve in Although some studies in the literature have proposed bandwidth allocation algorithms to provide video services to EV users, there are no bandwidth allocation algorithms for video services that utilize emerging communication technologies for next-generation vehicular networks. In view of this, this paper integrates the THz base stations of 6G wireless networks with the existing mmWave and sub-6 GHz band base stations to provide the required bandwidth for EV users. A bandwidth management mechanism that suits the video quality requirements of different EV users is proposed to allocate the bandwidth from the three types of base stations mentioned above to the in-vehicle video services. Meanwhile, VLC technology is utilized in V2V communications to provide a high data rate and high security for inter-vehicle video transmissions.A series of simulations is conducted to comprehensively compare the performance of the proposed algorithm against both the conventional FCFS method and a contemporary state-of-the-art approach known as DQN+AQCS. This rigorous evaluation process allowed for a thorough assessment of how the proposed algorithm performed in comparison to these two benchmark methods. The simulation results clearly demonstrated the efficacy of the proposed mechanisms, including video pre-downloading for EV users and bandwidth relay, in conjunction with the utilization of VLC for V2V communication. These mechanisms collectively enhanced the adaptability of base station bandwidth allocation, particularly during peak hours. This enhancement ensures that video conferencing, live video, and VoD applications for EV users can reliably access the necessary bandwidth support even in scenarios of bandwidth scarcity on congested road segments during peak times. Ultimately, the perceived video quality for EV users can be maintained at satisfactory levels.In summary, the algorithm presented in this paper offers a dual advantage. Specifically, it effectively addresses the challenge of bandwidth contention among EV users, which stems from the constraints of base station bandwidth. Additionally, it represents a notable advancement over both the FCFS and DQN+AQCS approaches. To be more precise, it ensures access to sufficient bandwidth for real-time live video applications and guarantees uninterrupted playback of the vast majority of video applications even during periods of heightened demand. This enhancement significantly elevates the overall quality of the video viewing experience for EV users. Remarkably, even when the available bandwidth from congested base stations reaches its peak capacity, the proposed algorithm demonstrates a substantial improvement of 30% and 17% in average video quality compared to the FCFS approach and the DQN+AQCS method, respectively. This further solidifies the effectiveness of our proposed algorithm in providing satisfactory video quality even within the constraints of bandwidth limitations.


The BDS multipath delay error is highly related to the surrounding monitoring environment, which cannot be eliminated or mitigated by applying the double difference observation model. In the actual monitoring environment, due to the complexity of the BDS constellation, it is difficult for existing algorithms to consider GEO, IGSO, MEO and other different orbital types of satellites for real-time and efficient multipath error reduction. Therefore, we propose a novel BDS dual-frequency multipath error reduction method for real deformation monitoring for BDS considering various satellite orbit types. This method extracts the single error residual of each satellite based on the assumption of “zero mean” and divides the appropriate grid density of GEO and IGSO/MEO, respectively, to construct a dual-frequency multipath hemispherical map model suitable for BDS satellites with different orbital types. This method can realize the multipath error elimination of the observed values of different orbits and different frequencies. The results of simulation experiments and real deformation monitoring data demonstrate that this method can effectively eliminate low-frequency multipath delay errors in the observation domain and coordinate domain. After multipath correction, the precision of the horizontal coordinates and height coordinates are 1.7 mm and 4.6 mm. The precision of the horizontal coordinate and height coordinate is increased by 50% and 60%, respectively. The fixed rate of ambiguity increased by 5–7%. Beidou Navigation Satellite System (BDS) has completed the “three step” strategy ,2. Now, Recently, the application of Global Navigation Satellite System (GNSS) technology in deformation monitoring has become an important means to monitor the structural health of buildings due to its advantages in automation, its real-time and large-scale applications and its ability to operate in all weather conditions . HoweverBDS has three different orbit types: the geostationary orbit (GEO), the inclined geostationary orbit (IGSO) and the medium Earth orbit (MEO) . Using tOver nearly 30 years, many GNSS multipath mitigation methods have been developed which can be classified into three main branches . The secThe last and most common method is data post-processing approaches, including weight adjustment based on data signal to noise ratio (SNR) techniques ,19; ray-Careful site selection can reduce the impact of multipath error but is limited in its usefulness . To dateFor BDS multipath error, Ye et al. studied the multipath repeat cycle of BDS satellites, and proposed a SF carrier phase multipath elimination approach of BDS system . A multiThe SF method faced a computational challenge to precisely calculate the orbit repeat time of each satellite at the end of multipath model for the BDS system with a heterogeneous constellation. Fuhrmann proposed a multipath stacking method based on the repeatability of multipath space. This method divides the grid into equal area, which can also effectively mitigate the multipath effect . HoweverHowever, the GNSS deformation monitoring environment is complicated, and GNSS signals are reflected by surrounding water bodies and vegetation, resulting in serious interference on direct GNSS signals, which greatly affects the accuracy of GNSS deformation monitoring. It is not universal to verify the multipath mitigating method only through simulation experiments. The current research of the MHM model mostly uses simulation experiments to verify the performance, and there are few experiments based on the real deformation monitoring situation. This study attempts to fill this research gap using two groups of experiments, including a simulation deformation monitoring scene and a real river culvert deformation monitoring scene. These experiments are designed to verify the multipath error mitigating performance of the BDS B1/B2 dual frequency MHM model, both in the observation domain and the coordinate domain. The basic principle of the multipath SD method and BDS dual-frequency positioning model are described in the section on the BDS dual-frequency precise positioning model. The spatial repetition characteristics of carrier-phase residuals and the BDS dual-frequency MHM model are built in the section on the MHM model for BDS dual-frequency carrier phase multipath mitigation. The experiments and analysis of these results is presented in the section on experiment analysis. A discussion and conclusions are in the section on conclusions.In the process of relative positioning, ionospheric errors, tropospheric errors and receiver clock errors are eliminated by means of a double difference combination model. Generally, the remaining DD residuals are regarded as multipath errors, but considering that the reference satellite will change with time, the DD residuals are converted to single-error residuals. In this section, the double frequency observation equation of BDS is listed and the theoretical method of converting the DD residue to the SD residue is derived.f, When BDS is used for deformation monitoring, the double difference (DD) observation equation is generally used to eliminate the receiver and satellite error. The multipath error is highly related to the monitoring environment, and it cannot be eliminated or mitigated by constructing the double difference observation equation. Assuming In relative positioning, the satellite with the highest elevation angle is generally selected as the reference satellite, then the DD observation equations are formed between other satellites and the reference satellite. Given that the satellite orbit of BDS is heterogeneous and the reference satellite changes frequently, it is difficult to correctly reflect the multipath error in the DD carrier phase residual. It is necessary to convert the DD residual into the SD residual .Assume that there are The conversion relationship between dual-frequency SD residual and dual-frequency double difference residual can be expressed as:If we can reconstruct SD residuals from DD residuals using the transition matrix which contains the “zero mean” assumption essed as ,37:(4)0dAfter the above derivation, DD residuals can be converted to SD residuals. At this stage, the SD residue contains multipath error and measurement noise, and it is generally considered that the low-frequency component of the SD residual is the multipath error.The multipath effect refers to that the signal transmitted by the satellite being reflected by the surrounding environment of the receiver, causing the actual observed value received by the receiver to deviate from the actual observed value transmitted by the satellite. In the process of signal transmission, the GNSS positioning accuracy can be greatly improved by mitigating the multipath effect as much as possible.D represents the horizontal distance between the receiver antenna and the reflector around the antenna, λ and θ are respectively the wavelength and the incidence angle of the reflected signal, α is the reflection coefficient of the reflector, and D), incident angle (θ) and reflection coefficient (α). The reflective surface, antenna and their properties remain unchanged in most cases of deformation monitoring, which means that the horizontal distance and reflection coefficient are constant. Hence it is only the incidence angle of reflected signals that could change multipath errors. Meanwhile, the incidence angle of reflected signals is determined by the position of satellites in the sky which periodically repeats in specific orbits. This indicates that we can mitigate multipath effects for current data by subtracting the multipath error models extracted from the last period data.The path deviation of satellite signals caused by multipath effects can be expressed as: When the environment around the monitoring station is static, the multipath error is highly correlated with the incidence angle of the satellite signal, and the variation characteristics of the satellite signal incidence angle are consistent in the spatial domain. However, the variation characteristics of different satellites orbit types or different observation frequencies vary. Taking the carrier phase observations of different frequencies and different satellites as the focus, the relationship between DD residual and SD of carrier phase observation is constructed, and the carrier phase SD residual is obtained ,28. The To verify the repetition characteristics of BDS multipath error in the spatial domain, a set of BDS data were collected from 27 September to 10 October 2017 at a frequency of 1 Hz and using 5° as the cutoff elevation angle at the roof of the SESI building, China University of Mining and Technology, as shown in By extracting the SD residuals of the B1 frequency carrier observations of BDS MEO C12 satellites for many days, the relationship between the SD residuals and elevation/azimuth angle was constructed respectively. Based on the above characteristics in the spatial domain of the observations, taking the carrier phase SD residuals of different frequencies and various satellites as the research focus, an MHM model is constructed with the receiver as the coordinate origin and the azimuth and elevation angles as the longitude and latitude. The model is mainly divided into two parts: the hemispherical geometric model and the hemispherical grid parameter fitting.The SD residual variation in the BDS carrier phase observations is related to the satellites’ azimuth, elevation angles, and BDS frequency. The Multipath Hemispherical grid is constructed with the receiver as the coordinate origin and the azimuth and elevation angles as the longitude and latitude. Assuming that the min elevation angle is Due to the BDS GEO satellites operating on a small range orbit, the common hemispherical grid dividing method is too large for GEO satellites. It is necessary to construct a special and refined grid model for GEO satellites. We can obtain the minimum elevation angle We refer to the data used in modeling as first-phase data and the corrected data as second-phase data. Since the repetition cycle of a BDS MEO satellite is 7 days, the data of 7 consecutive days are used as the first-phase data, and the data of the eighth day, that is, the data to be corrected by multipath, are used as the second-phase data.Firstly, the wavelet analysis method is used to denoise the SD residuals between the stations obtained from the first-phase data solution, and the multipath errors of each satellite operating in each grid are extracted, which are distinguished by azimuth angle and elevation angle. Among them, only the SD residuals between satellite stations at the successful time of fixed ambiguity are extracted. The mesh parameters of the MHM geometric model were fitted with the multipath errors extracted from the first-phase data. The parameters in each grid are distinguished according to the satellites PRN and satellite frequency. Assuming that this is satellite In summary, the construction flow chart of the BDS dual frequency MHM is given in It is worth mentioning that this method can also be applied to medium-to-long baselines and Network RTK . For exaIn order to verify the effectiveness of the proposed method, the multipath simulation experiment of CUMT building roof  and actuThe data set described in Due to the limitation of the length of this paper, the performance of the MHM method of BDS GEO/IGSO/MEO satellites is discussed and analyzed by taking C01 (GEO), C08 (IGSO) and C12 (MEO) satellites as examples. Four schemes were used to compare the performances of single-frequency/dual-frequency BDS multipath mitigation. These schemes are listed in The single-frequency comparative experiment shows that the MHM method has a significant correction effect on the baseline sequence, the average improvement accuracy is 71%, and the average ambiguity fixed rate is increased by 4%. The dual-frequency comparative experiment shows that the MHM method also results in an obvious improvement for the baseline residuals, and the average improvement accuracy is 50%. The ambiguity success rate is 100%. By constructing the multipath error model, the baseline accuracy can be effectively improved, and the baseline sequence after multipath mitigation presents the characteristics of white noise.A GNSS monitoring receiver was set on the river culvert roof to monitor the building deformation caused by river confluence. In The data on DOY 355–364 were collected to build the MHM, improve it, and extract the carrier phase residual sequences of all the satellites before and after multipath correction. The MHM models were used to mitigate the data on DOY 365 and obtain the refined baseline series to evaluate the MHM method performance.To analyze the performance of the MHM method in a realistic deformation monitoring scenario, the C01, C12, and C16 satellites were used as an example of BDS GEO/IGSO/MEO satellites. The same schemes were used to compare the performances of single-frequency/dual-frequency BDS multipath mitigation for real deformation monitoring scenarios, which can be found in The single-frequency comparative experiment shows that the MHM method has a significant correction effect on the baseline sequence, the average improvement accuracy is 83.3%, and the average ambiguity fixed rate is increased by 7%. The dual-frequency comparative experiment shows that the MHM method also results in obvious improvement for the baseline residuals, improving the accuracy by 57.2%. The ambiguity success rate is almost 99.6~99.8%. By constructing the multipath error model, the baseline accuracy can be effectively improved, and the baseline sequence after multipath mitigation presents the characteristics of white noise. The dual-frequency error EMS of the E, N, and U components are improved to 1.4 mm, 1.5 mm and 3.7 mm, respectively.We proposed an SD residual-based multipath hemispherical map for the BDS dual-frequency carrier phase for dynamically mitigating the multipath errors in the dual-frequency carrier phase data for a real deformation monitoring scenario. In the proposed method, DD residuals are converted to SD residuals, and the multipath errors of each satellite are corrected respectively considering the differences between BDS satellites of different orbital types, which solves the problem that DD residuals change with the change in reference satellites. The developed method takes the elevation and the azimuth angles of the satellites as independent variables to construct the MHM model. The real scenario experiment results indicate that the MHM method has a significant correction effect on the baseline sequence: the average improvement accuracy is 83.3% and 57.2% for single-frequency and dual-frequency, respectively; the average ambiguity fixed rate is increased by 7% for single-frequency data; and the dual-frequency error EMS of E, N, and U components is improved to 1.4 mm, 1.5 mm and 3.7 mm, respectively.


To analyze breathing modes with infrared thermography.Cross-sectional observational exploratory study conducted in 20 female participants with a mean age of 26.0-years. The thermograms were made following the principles of the American Academy of Thermology and the Brazilian Thermology Society. The camera FLIR A315  was used for the tests. The recordings consisted of the participants breathing normally through the nose for 2 min and simulating oral/oronasal breathing for another 2 min. The thermograms were analyzed with the FLIR Tools software. An ellipse was placed between the nostrils and the lip commissures to obtain the mean temperatures. The collection was made by two independent researchers, and the normalized non-dimensional temperature was calculated.The temperature in nasal breathing is higher than in oral/oronasal breathing both for inhaling and exhaling when measured in the region of the mouth. The exhaling temperatures were higher than the inhaling ones in oral/oronasal breathing (through the nose and the mouth) and nasal breathing (only through the nose). The temperature difference between exhaling and inhaling (ΔT) was greater in oral/oronasal breathing when measured in the region of the mouth.The thermographic assessment of breathing modes may be made by comparing the mean temperatures of the mouth, using an ellipse.Study without consistently applied reference standards. The breathing mode encompasses the path traveled by the airflow in the upper airways during breathing.Within the area there is still no objective “gold standard” technique to evaluate the breathing mode, many times done in a subjective way and depending on the report of the signs and symptoms by the patient/caregiver.Infrared thermography is known to be a non-contact and non-invasive method.Studies have employed thermography in the evaluation of breathing, mainly using it as a technique to quantify the respiratory flow or rate.Based on the aforementioned assumptions, this research has as its main objective to analyze the breathing mode by means of infrared thermography. In that way, it will be possible to develop a protocol of collection and analysis of the cutaneous temperatures of the nose and mouth during nasal breathing and during oral/oronasal breathing, helping the speech-language clinical practice through an objective and efficient evaluation of the breathing mode.This research is a cross-sectional observational study of exploratory type. It was developed with self-declared nasal breathing participants at the Functional Health Observatory in Speech-language Therapy at the UFMG  School of Medicine. All the research participants were speech-language therapists or speech-language therapy students in the clinical cycle, and, therefore, had knowledge about the physiology of breathing and the clinical signs involving mouth breathing. Data collection was only started after approval by the Institution’s Research Ethics Committee by the following opinion: CAAE 32465518.0000.5149 and all signed the informed consent form.Among the inclusion criteria for the study were: being 18-years of age or older; being a nasal breather; not presenting any physical, neurological, and/or cognitive alteration that would hinder or make it difficult to collaborate during the exam.Regarding the exclusion criteria for the participants, the following were established: the use of bandages on the skin and/or other factors that would prevent exposure to a temperature-balanced environment; physical exercise, acupuncture, massages, or having undergone electrical stimulation, saunas, or prolonged exposure to the sun up to 24 h before the measurements were taken; the use of bronchodilators and vasoactive medications; the presence of fever, allergic rhinitis, flu and/or cold symptoms on the day of the tests; the presence of orofacial pain; men with facial beards, and women in the fertile or menstrual period.The following three steps were observed for the selection of participants and classification of the breathing mode, in addition to self-reporting of nasal breathing: evaluation of the signs and symptoms of mouth breathing, lip seal evaluation, and evaluation of orofacial muscle tone.To evaluate the signs and symptoms a questionnaire containing major and minor signs for mouth breathing was applied among the participants.After the selection of participants by means of the described selection criteria, the final sample was composed of twenty nasal breathing women, totaling 240 thermograms.The thermography records were performed following all topics recommended by the American Academy of Thermology and the Brazilian Association of Thermology.A FLIR A315 camera  and a FLIR 18 mm lens were used for the tests. The equipment was positioned on a tripod at a distance of approximately one meter from the face. The volunteer was seated on a fixed chair with a back resting against the wall. The thermal camera was frontally positioned on a tripod with an angle between 80° and 90° for better viewing of the nose and mouth simultaneously. A grid panel was also inserted behind the participant to allow head alignment during image recording. A sequence of 5 frames per second was used to take the videos.The tests consisted of three different moments, initially the nasal breathing was requested for two minutes of recording. Subsequently, the patient was asked to simulate oral/oronasal breathing as a form of training, therefore without filming. Only in the third stage, after training, the participant was recorded for two minutes during oral/oronasal breathing.After data collection, the nose and mouth temperatures were obtained and analyzed. To obtain this information the Flir Tools software was used. An ellipse was positioned between the lip commissures, the cupid’s bow and the lower limit of the lower lip and chin to obtain the thermal values of the mouth. To obtain nasal temperature, another ellipse was positioned around the nostrils, the nose apex, and the nasolabial angle . The choFor the selection of the video images the following procedures were performed: all the two-minutes videos were divided into three moments of 40 s and in each of these moments, two distinct images were selected, that is, one image for inspiration and one image for expiration. Therefore, six images (three of inspiration and three of expiration) were selected from the nasal breathing videos and six images from the oral/oronasal breathing videos (three of inspiration and three of expiration), totaling twelve images per participant. Therefore, the final data analysis was composed of 240 thermal images.The criterion used to select the best inspiration image was based on the physiology of breathing, and the selected image was the one that presented the lowest minimum temperature. Similarly, we chose the one with the highest maximum temperature for the selection of the expiration image. The collection of temperatures was performed by two independent evaluators, as well as replicated by the same evaluator for agreement analysis.As the infrared thermography technique measures the absolute temperature, it is influenced by the individual and local temperatures. To correct for these factors, the normalized non-dimensional temperature was calculated, which consists of quantitative numerical values ranging from 0 to 1.T the temperature of the assessed surface (nose and mouth); T∞ the ambient room temperature at the time of the examination; and Tb the participant’s tympanic temperature.Where θ is the corrected non-dimensional temperature; Data were analyzed using the data analysis program IBM SPSS version 25. To test the distribution of the continuous variables, the Shapiro–Wilk test was applied, indicating a non-symmetric distribution (<0.05). Comparisons between the corrected non-dimensional nose and mouth temperatures in nasal breathing and oral/oronasal breathing during inspiration and expiration and the ΔT difference (expiration‒inspiration) of nasal and oral/oronasal breathing were done using Wilcoxon’s test. Three temperature measurements were collected for each variable studied, so for statistical analysis the median of these three measurements was adopted as the parameter analyzed throughout the study. The significance level adopted was 5%.The intra evaluator and inter evaluator agreement analysis was done by means of the Intraclass Correlation Coefficient (ICC), since these are continuous measurements. In this analysis we used only the temperature data obtained from the mouth due to the statistically significant values obtained in this study. All analyses showed desirable agreement, i.e., above 0.800 .Table 1IWhen comparing nose temperatures between Nasal Breathing (NB) and Oral/Oronasal Breathing (OB) both at inspiration and expiration, no significant differences were observed . StatistComparing the temperature difference between expiration and inspiration (ΔT) in nasal and oral/oronasal breathing allows to notice differences with statistical relevance in the comparison with the temperatures in the mouth region .Table 4CThe results obtained illustrate the applicability of infrared thermography for the evaluation and classification of the breathing mode. It is important to emphasize that the analysis used the non-dimensional normalized temperature, which allows the interpretation of thermal values without the metabolic influence of each individual, as well as the environmental temperature of the room at the time of testing, thus making the analyzed results more reliable.According to the findings we can consider that the mouth can be used as a Region of Interest (ROI) for thermographic analysis of the breathing mode.When comparing the temperatures during nasal breathing and oral/oronasal breathing, we noticed a statistically significant difference when comparing the temperatures of the mouth, both during inspiration and expiration, being higher during nasal breathing. This result may be associated with the fact that in nasal breathing there is no contact with atmospheric air, making it possible to keep the body temperature stable in the mouth. In oral/oronasal breathing, on the other hand, air intake and outlet seem to be able to decrease the body temperature.Results with statistical relevance for the mouth were also observed in the analysis of the temperature difference between expiration and inspiration (ΔT). In this case the ΔT of oral/oronasal breathing was higher than that of nasal breathing, revealing greater airflow through the mouth during oral/oronasal breathing.The comparison between inspiration and expiration temperatures showed statistically significant differences in the temperatures collected in the mouth and nose during oral/oronasal breathing, and in the temperatures collected in the nose during nasal breathing. In all those three situations the values in expiration were higher than in inspiration. This fact is concurrent with the literature, since during expiration the warm air from the lungs is exhaled, thus warming the nose and mouth, while during inspiration there is cooling of the nasal vestibule and mouth.This research has great importance for Orofacial Motricity practice, since it presents an objective, simple, non-invasive and visual technique for defining the breathing mode, facilitating its evaluation, which is often subjective and lacking a “gold standard” procedure. Moreover, it is known that mouth breathing is a frequent symptom in childhood,However, despite the satisfactory results, this research has some limitations. The sample size of the study is relatively small, i.e., it was composed of twenty participants, corresponding to a total of 240 thermograms; therefore, further investigations with a larger sample size are suggested. Moreover, to evaluate oral/oronasal breathing, data were collected from nasal breathing subjects simulating this breathing mode; therefore, data from patients with a multidisciplinary diagnosis of oral/oronasal breathing were not used.This research is innovative and no other studies with similar objective were found, only one recent studyIt was possible to perform the thermographic evaluation of the breathing mode by comparing the mean temperatures of the mouth, using an ellipse to define the region of interest to be analyzed. The calculation of the corrected non-dimensional temperature was relevant in the analyses since it allowed correcting the influence of the environmental temperature and the core temperature of the individual during thermographic evaluations.YCT was responsible for preparing the research design, collecting, and analyzing data, and writing the manuscript. RMMM, MPP, RAMF and ARM performed the overall guidance of the work, supervising the preparation of the research project, data analysis and writing of the article.None.The authors declare no conflicts of interest.


In this study, we propose a novel micromechanical model for the brain white matter, which is described as a heterogeneous material with a complex network of axon fibers embedded in a soft ground matrix. We developed this model in the framework of RVE-based multiscale theories in combination with the finite element method and the embedded element technique for embedding the fibers. Microstructural features such as axon diameter, orientation and tortuosity are incorporated into the model through distributions derived from histological data. The constitutive law of both the fibers and the matrix is described by isotropic one-term Ogden functions. The hyperelastic response of the tissue is derived by homogenizing the microscopic stress fields with multiscale boundary conditions to ensure kinematic compatibility. The macroscale homogenized stress is employed in an inverse parameter identification procedure to determine the hyperelastic constants of axons and ground matrix, based on experiments on human corpus callosum. Our results demonstrate the fundamental effect of axon tortuosity on the mechanical behavior of the brain’s white matter. By combining histological information with the multiscale theory, the proposed framework can substantially contribute to the understanding of mechanotransduction phenomena, shed light on the biomechanics of a healthy brain, and potentially provide insights into neurodegenerative processes. Gray matter consists primarily of neuronal cell bodies, whereas white matter is characterized by a more complex microstructure, including oligodendrocytes, fibrous astrocytes, microglia, capillaries, and a dense network of myelinated axon fibers embedded in a soft extracellular matrix3. There is evidence that pathological and physiological changes in the brain microstructure, such as those resulting from traumatic brain injury, growing tumors, neurodegenerative disorders including Alzheimer’s disease, or even healthy aging, directly affect the mechanical properties of brain tissue9. In addition, recent studies have shown that growth and development of axons are not only related to chemical but also to mechanical stimuli and can be affected by changes in tissue stiffness12. Therefore, there is a need to improve our understanding of the interactions between the brain microstructure and its mechanical behavior through the development of new computational models. In this way, we could better investigate mechanotransduction phenomena that contribute to normal brain function and shed light on the underlying mechanisms of brain neurodegeneration.In the human brain, white and gray matter form a highly inhomogeneous structure, which leads to region-dependent mechanical behavior15; (ii) micromechanical multiscale models, where the microstructure occurs explicitly at the element level and the macroscopic response is obtained through the homogenization of the microscopic fields. With regard to the latter, multiscale models based on representative volume elements (RVEs) were combined with numerical homogenization for fiber-reinforced soft tissues such as tendons16, arteries19, myocardium20 and brain tissue. These multiscale models adopt a physical representation of the microstructure to simulate the homogenized response of the material at the macroscale22. Multiscale models of brain tissue have been developed mainly for the white matter, with particular emphasis on the brain stem and the corpus callosum (CC). Both regions are characterized by bundles of highly oriented axon fibers33. For the brain RVE, most studies proposed a micrometer-scale cubic geometry comprising a limited number of axons modeled as straight or wavy cylindrical inclusions uniformly distributed within a ground matrix33. Diffusion tensor imaging34 and, more recently, three-dimensional (3D) imaging with polarized light35 have enabled the reconstruction of the orientation of axon fibers. A more realistic fiber architecture, including random location and orientation of the axons, was included in the model presented by Hoursan et al. 31.Microstructure-informed computational models are developed as powerful tools that connect the microstructure of the tissue with its macroscopic mechanical response. These models can be roughly divided into two groups: (i) structure-based continuum models, in which the heterogeneous microstructure occurs indirectly in the formulation of the constitutive law36. Given the difficulty of directly quantifying the material properties of axons, this is a valuable consideration. From a mechanical point of view, brain tissue exhibits a soft, time-dependent material behavior40. Neuroimaging techniques highlight a distinct structural anisotropy42, although there is disagreement whether this is also reflected in the macroscopic mechanical anisotropy44. In addition, current predictions of the hyperelastic and viscoelastic material properties of white matter are inconsistent. According to reports, axons are three to thirteen times stiffer than the ground matrix46. Histological images also indicate that the axons follow wavy streams49, an aspect that can have relevant implications for the performance of the micromechanical model27. Overall, a micromechanical model based on data obtained from histological information combined with parameter identification performed at multiple loading modes simultaneously could lead to a better understanding of the relationship between microstructure and macroscopic mechanical behavior.Multiscale models provide a convenient tool to characterize the viscoelastic and hyperelastic mechanical properties of brain tissue microstructure. The macroscale response derived from the stress homogenization can be fitted to experimental data to determine the mechanical parameters of the constituents by an inverse procedure50 and more recently to collagen networks by Dalbosco et al.19, is optimal for simulating complex architectures of one-dimensional (1D) fibers. The fiber tortuosity is indirectly included in the constitutive law via the concept of the recruitment stretch51. The second objective is to investigate how the microstructure correlates with the macroscopic, possibly anisotropic, mechanical response of the white matter. For this purpose, the homogenized response is used to determine the mechanical parameters of axon fibers and the ground matrix, through an inverse parameter identification scheme based on existing experiments2. The model is then validated against load cases other then those used to identify the material parameters. Finally, the influence of fiber tortuosity and orientation on the homogenized response of the RVE is assessed.The main objective of this study is to propose a new microscale RVE of the brain white matter. The micromechanical model includes several microstructural parameters, such as axon diameter, orientation and tortuosity, based on histological information. For each of these parameters, appropriate probability density functions (PDFs) are calibrated to the experimental distribution observed in human CC. Furthermore, we propose an efficient implementation in a finite element framework based on the embedded element technique. This technique, which was applied to the modeling of axon fibers19, axons are distributed in the matrix at random locations with different diameter, orientation, and tortuosity. We considered the outer diameter of axons, including the myelin sheath. Fiber tortuosity refers to the waviness of the axon fibers in the unloaded state, here implemented through the concept of recruitment stretch (see Methods). The volume of axon fibers is defined by the fiber volume fraction 2. An illustration of the RVE is shown in Fig. We designed a microscopic RVE based on a simplified two-component representation of the brain white matter. A continuous 3D medium describes the ground matrix, including the extracellular matrix and the non-neuronal cells, reinforced by 1D myelinated axon fibers. Following the work from our group on arterial tissueWe included microstructural features of axon fibers in the RVE from existing histological information of brain CC, in the form of distribution functions. Namely, we considered three parameters: fiber diameter, fiber tortuosity and fiber orientation.52. We fitted the experimental data of the genu (the bending of the anterior CC) to a generalized extreme value distribution defined by the following PDFd, 53. The experimental distribution and fitting are shown in Fig. Scanning electron micrographs show that the diameter of myelinated axons in the human CC follows a certain distribution55. In the absence of information from human brain, we employed data derived from scanning electron micrographs of sheep CC, which suggested a log-normal distribution of the axon tortuosity 48. More precisely, we defined a straightness parameter 51. In summary, the stretch of a tortuous fiber consists of a first stage of unbending or fiber recruitment, with no mechanical contribution, and a subsequent phase in which the fully stretched fiber starts to bear loads. This concept is illustrated in Fig. From histological images of the white matter, we could observe wave-like streams, suggesting that axons can be characterized by a certain tortuosityentclass1pt{minimaentclass1pt{minima54, we assumed a bivariate von Mises PDF, which requires a pair of concentration parameters 56. In vivo diffusion tensor imaging studies can provide the orientation of the fibers with millimeter-scale resolution57, which is not applicable to the microscopic range of this work. Furthermore, increasing evidence confirms that axons in the human CC are highly unidirectional47. Therefore, we assumed perfect alignment of the axons . The macroscale response is provided by the homogenized stress, which is calculated based on the rule of mixture and the assumption of continuity in the displacement fields of matrix and fibers (no-slip condition)19. The homogenized first Piola–Kirchhoff  stress To ensure kinematic compatibility between the micro and macroscales, periodic boundary conditions are applied to the RVE Results of the RVE size convergence analyses are illustrated in Fig. 2 for the human brain CC, from compression FF and tension FF, TT  and employed the experimental data from Budday et al. TT Fig. .4. We postulated an analogous material model for the axon fibers, with a modified formulation of the strain-energy function of the single fiber We assumed that the ground matrix behaves like a nearly incompressible isotropic material, whose strain-energy function 50. Therefore, the material parameters In total, four material parameters must be determined: shear moduli of ground matrix and fibers, entclass1pt{minimaWe performed analyses on RVEs with edge length entclass1pt{minima2. In particular, the homogenized stress-stretch curve is able to replicate the nonlinearity shown by the experimental data in tension  or compressed orthogonally (compression TT), suggesting that recruited axon fibers provide an important contribution in these cases. In particular, tension FF shows a characteristic stiffening in the macroscopic stress-stretch curve, a feature that previous micromechanical models could not predict32.Experimental data suggest that compressing the CC along the fiber axis (compression FF) or stretching it orthogonally (tension TT) results in a softer mechanical response32. We believe the difference is due to the simplifying assumption that axon tortuosity was neglected in previous micromechanical models. As argued by Meaney45, the wavy structure of axons and their distribution in tissue are critical factors that significantly affect the homogenized response of the RVE and consequently the inverse parameter identification. In addition, the fact that axons appear as soft fibers is consistent with their role being distinct from the structural function of collagen fibers2. We emphasize that the identified material parameters correspond to the material properties of the embedded fibers based on the modified strain-energy function. In micromechanical models that are not based on fiber embedding, the material properties of the actual axons are defined from the sum of the strain-energy functions of the ground matrix and the axon fibers.Regarding the material parameters identified, our findings show that the axons are only slightly stiffer than the ground matrix, whereas previous studies reported that the shear modulus of the axons is several times larger2, which showed a very similar mechanical response for the CC in tension FF and tension TT at low stretches  using an inverse parameter identification procedure. Second, we investigated the relationship between brain microstructure and macroscopic mechanical response of the tissue. To provide a realistic and repeatable representative microstructure, a random fiber network of axons was developed based on statistical distribution functions fitted to imaging data. These include the dispersion of the axon diameter, axon fiber tortuosity and orientation. To increase the computational efficiency of the simulation, straight one-dimensional axon fibers were embedded in the volume of the ground matrix and tortuosity was included into the constitutive law through the concept of recruitment stretch.The representative size of the volume element was determined with a convergence study performed on forty randomly generated RVEs with different sizes. By simultaneously fitting the homogenized response of the RVE to macroscale experimental data for compression and tension, the parameters of one-term Ogden models were inversely identified for both the ground matrix and the axons. To the best of the authors’ knowledge, this is the first study in which a micromechanical model of the brain is adopted to characterize the material parameters of the microstructure, including the influence of orientation and tortuosity. Incorporating the concept of fiber tortuosity and recruitment stretch, we found that axons are only slightly stiffer than the ground matrix, which is in contrast to existing similar models but in good agreement with experimental observations. Based on the identified material parameters, the simulated homogenized response showed strong agreement with the experimental data across different deformation modes. Although existing phenomenological models can fit the tension-compression response of the human brain , recent microstructural models available in the literature have failed to reach this goal. In contrast, our model was able to accurately capture the response under tension based on the observed distribution of axon tortuosity without the need to introduce additional fitting parameters.Finally, we would like to mention some limitations of the proposed model. Although it is widely accepted that brain tissue exhibits time-dependent behavior, the current study did not consider viscoelasticity or stress relaxation originating from the interaction with cerebrospinal fluid. In addition, the current study focused on a single region of the human brain white matter, and was based on the assumption of perfect fiber alignment. Existing imaging data from diffusion tensor imaging were not applicable to this study due to poor resolution at the microscopic scale of the RVE. In the future, 3D polarized light imaging could be applied to human brain tissue and potentially provide more accurate data to refine the model.54, whereThe orientation of a fiber with unit reference vector entclass1pt{minima58. In the previous equation, l is the deformed length of the straight fiber  using the embedded element technique. In the embedded element technique, a guest domain (axon fibers) is superposed on a host domain (ground matrix). The formulation implemented in Abaqus FEA constrains the translational degrees of freedom of the embedded nodes to the elements of the surrounding matrix. In our model, the ground matrix is meshed by first-order hexahedral mixed-formulation elements (C3D8H) and the fibers are meshed by first-order mixed-formulation truss elements (T3D2H). To ensure the condition of no-slip between the matrix and fibers, the fibers are carefully meshed so that the embedded nodes lie precisely on the edge or face of the host elements. A mesh convergence analysis for the RVE with 32. Based on the rule of mixtures and the no-slip condition, we can write the total energy W stored in the continuum medium asi-th embedded fiber and The embedded element technique creates a stiffness redundancy for the matrix volume occupied by the fibers. Various approaches are discussed in the literature to address the stiffness redundancy59. The constraints are implemented using a custom Python script and are provided in detail in the Supplementary Material available online.In the multiscale theory based on RVEs and stress homogenization, periodic boundary conditions are applied to ensure that opposite faces at the boundary of the RVE deform identically. Mathematically, we require that each pair of points 2. First, the ground matrix parameters We devised a two-step optimization algorithm to characterize the mechanical parameters of the ground matrix and the embedded axon fibers based on experimental data60. In order to improve the computational efficiency of the finite element analyses, the simulations and post-processing were parallelized. Sensitivity analyses were carried out to verify that the optimal values are independent of the initial guesses and the boundary values. The algorithm is summarized in the Supplementary Material available online.The process of identifying the material parameters was performed using a custom Python script integrated into the Abaqus FEA software. A trust–region reflective algorithm was employedSupplementary Information 1.


To improve the assessment of carotid plaque vulnerability, a comprehensive characterization of their composition is paramount. Multispectral photoacoustic imaging (MSPAI) can provide plaque composition based on their absorption spectra. However, although various spectral unmixing methods have been developed to characterize different tissue constituents, plaque analysis remains a challenge since its composition is highly complex and diverse. In this study, we employed an adapted piecewise convex multiple-model endmember detection method to identify carotid plaque constituents. Additionally, we explore the selection of the imaging wavelengths in linear models by conditioning the coefficient matrix and its synergy with our unmixing approach. We verified our method using plaque mimicking phantoms and performed ex-vivo MSPAI on carotid endarterectomy samples in a spectral range from 500 to 1300 nm to identify the main spectral features of plaque materials for vulnerability assessment. After imaging, the samples were processed for histological analysis to validate the photoacoustic decomposition. Results show that our approach can perform spectral unmixing and classification of highly heterogeneous biological samples without requiring an extensive fluence correction, enabling the identification of relevant components to assess plaque vulnerability. The rupture of vulnerable carotid plaques is considered one of the main causes of ischemic cerebrovascular disease and a leading source of mortality and morbidity worldwide6. Previous studies have explored the formation and evolution of plaques and how they acquire the connotation of ‘vulnerable’ or ‘prone to rupture’10. It has been found that the development of lipid-rich regions inside the plaque triggers an immune response from the body, resulting in tissue degradation and the formation of a thin fibrous cap that separates the lumen of the plaque material11. Accordingly, the pathological assessment of vulnerable plaque relies on the identification of lipid-rich necrotic areas and the thickness of the thin fibrous cap6. Further studies have shown that intra-plaque hemorrhage is another relevant indicator of vulnerability in plaques14. The presence of intra-plaque hemorrhage is associated with either blood infiltration through the fibrous cap or the formation of new micro-vascularization13, both equally concerning. In the case of lumen infiltration, there is a high possibility of rupture or material infiltration that may lead to clogging formation15. On the other hand, neo-vascularization implies inflammation and the delivery of more macrophages and other biological constituents that promote plaque progression13.Atherosclerosis is a systemic disease characterized by the accumulation of highly modified heterogeneous biomaterials inside the artery walls, forming plaque lesions that narrow the luminal size16. However, several studies indicate that a large number of patients that undergo surgery didn’t actually have a vulnerable plaque, and treatment through medication would have been possible17. Overtreatment is partially due to the current diagnosis based on the grade of stenosis, a geometrical relationship between the transversal size of the lumen and the size of the carotid18. As an alternative, some studies illustrate the benefits of using plaque composition as an additional criterion for plaque vulnerability assessment15. Plaque composition provides valuable information on the possible mechanical characteristics of the plaques and their progression; specifically, lipid pools and hemorrhage promote macrophage infiltration, causing plaque growth and plaque instability15.One of the principal treatments for ischemic cerebrovascular events is carotid endarterectomy surgery, in which the plaque is removed from the carotid to prevent its rupture and clot formation19, phase-contrast computed tomography21, and optical-based spectroscopy techniques23 to determine the composition of carotid plaque. However, these techniques have disadvantages like the use of ionizing radiation, the impossibility of use in patients with implants, and limited penetration, respectively11. As an alternative, another promising image technique that provides composition information is multispectral photoacoustic imaging (MSPAI)26. In MSPAI, pulse light is used to illuminate the tissue, which will selectively absorb part of the optical energy that leads to the generation of an acoustic pressure wave due to thermoelastic expansion, detectable with conventional US probes27. The most outstanding feature of MSPAI is its optical contrast, allowing material determination from the characteristic absorption spectra of tissue while reaching imaging depths up to several centimeters, which clearly surpasses that of most purely optic techniques. Nonetheless, the accurate analysis of PA spectral modulation in patients remains a challenge due to tissue heterogeneity and inter-subject variability29.Ultrasound (US) and Computed Tomography (CT) imaging are the core imaging techniques used in the medical diagnosis of vulnerable plaques. Although they provide an accurate estimation of the plaque geometry, the usual approach does not provide information on the plaque composition. Some studies exemplify the use of magnetic resonance imaging (MRI)30. To identify imaged components, MSPAI relies on the response of tissue chromophores, the optical absorbing molecules, to different wavelengths. However, the modulation of the PA signals in volumetric samples does not match with literature absorption coefficients due to the effect of local fluence variations, also called spectral coloring. It has been demonstrated that without an accurate correction of light fluence, linear unmixing approaches are not feasible for complex heterogeneous samples, usually leading to erroneous estimations31. Besides, the exact composition of the plaque and surrounding tissue is unknown, increasing the complexity of fluence estimation; thus, we need new methods that can work without prior knowledge of the absorption spectra of the sample. The most common solution is the use of blind unmixing techniques, where linear models are employed to fit the acquired data and estimate the constituent spectra  and their proportion distribution33. The endmembers correspond to the representative PA spectra of the different absorbers in the sample, and the proportion maps determine the percentage of each endmember per pixel. Considering the high heterogeneity of biological tissue, the problem at hand requires an unmixing method that does not work under the assumption of pure pixels consisting of a single component. In this work, we adapted a Piecewise Convex Multiple-Model Endmember Detection (PCOMMEND)34 for the unmixing of MSPAI that deals with the nonhomogeneous illumination conditions of photoacoustic imaging (PAI). PCOMMEND is a pixel-based method without pure pixel assumption that extends the linear mixing model to multiple sets of endmembers, allowing the solution of non-convex problems. This property supports the unmixing of datasets with limited bandwidth and fluence variations, problems that greatly hinder the use of linear unmixing techniques. By splitting the data into multiple regions with similar fluence based on the amplitude of the signal, it is possible to obtain a better fitting of the endmembers in the model and find an average representation of the spectra in the sample. In contrast to other promising blind unmixing approaches like non-negative matrix factorization (NNMF), the multiple-model approach obtains a non-convex solution by optimizing individual convex problems, making the solution more stable34.Photoacoustic (PA) signals are proportional to the product of the local fluence and the absorption coefficient of the tissue of interest36, while others optimize the model fitting40. In this work, we aim to select the most suitable set of wavelengths for an accurate unmixing of the data; therefore, we introduce an additional technique designed to minimize the number of variables required for solving linear models. For this purpose, one of the most widely used methods for imaging wavelength selection is model conditioning, which can be achieved by optimizing the model’s coefficient matrix39.One of the main challenges for the spectral analysis of multiple materials in biological samples is that spectral markers can be distributed over a wide spectral range, making in-vivo implementation impractical due to the low repetition rate of tunable lasers. Acquisition time optimization is essential for MSPAI to be viable in medical applications; hence, it is necessary to optimize the number of imaging wavelengths to identify as many components as possible. Different approaches for wavelength selection have been proposed, with some focusing on optimizing in-vivo imaging conditions43.In this study, we demonstrate unmixing of multi-wavelength PA signals of carotid plaques based on a multiple-model approach (PCOMMEND), demonstrating the first implementation of such an approach in the unmixing of spectral signals from highly heterogeneous biological tissue. The method performance is first evaluated quantitatively on a plaque mimicking phantom. Next, MSPAI is performed on ex-vivo carotid endarterectomy samples, and the unmixing results are validated by comparison with histology. Moreover, we optimize (and limit) the number of imaging wavelengths through the conditioning of the model to investigate how our approach can work for in-vivo scenarios. Our results show that it is possible to identify multiple plaque constituents with a significant correlation with histology staining. Besides, the capability of splitting the initial dataset into multiple models has proven to support the blind unmixing of low amplitude signals. Four plaque materials were classified as repeatable over the different samples. Two of them, hemorrhages and lipids, have a significant diagnostic value. The other two, collagen and smooth muscle cells (MSC), may become relevant in light of recent studiesThe imaging setup is presented in Fig. 45. Additional endmembers were found for the background and PVA, but since they do not provide additional biological information and for the sake of clarity, they are not displayed. In Fig. The US and PA images of the phantom are shown in Fig. The mean percentage error of the regions misidentified after unmixing is presented in Table Considering that the same approach can be used in specific spectral windows, we evaluate the performance of our wavelength selection in the phantom data out of a broad spectral range from 500 to 1300 nm, including as many spectral markers as possible. Given that the maximum number of endmembers required to unmix the phantom data was six, the data was optimized for the same number of wavelengths. In Fig. Since the blind unmixing problem is solved using a least square minimization, unmixing with a limited number of wavelengths is more sensitive to outliers and noise. Table We present three selected cases of plaque morphologies that exemplify the diagnostic relevance of MSPAI: One stable plaque with low lipid content, and two vulnerable plaques with a lipid-rich necrotic core in one sample and an intraplaque hemorrhage in the other.Figure Figure Figure In this study, we presented a blind unmixing approach for MSPAI, based on endmember detection, that can identify multiple components in human plaques and may improve the assessment of vulnerability in plaques. After analyzing the endmembers detected from PA data acquired in the designed plaque phantom and nine human plaques, we found that four endmembers had a high correlation over different acquisitions. By comparing the mean representation of the spectral modulation observed in the unmixed endmembers and the reference spectra in the literature, we can interpret the detected endmembers to the tissue composition of collagen, lipid, hemorrhages, and SMC, which are highly relevant to the assessment of plaque vulnerability. In general, all the unmixed results agree with the corresponding histological staining.17. Similarly, lipids signals were mainly observed as speckle-like regions and usually overlapped with other materials. As expected, the typical absorption peak of lipids is found at around 1190 nm, while the spectral features of other chromophores were usually found in the visible range. It indicates that although lipids usually are mixed with other degraded plaque tissues, they can still be independently classified. The additional material was classified as smooth muscle cells based on the pathophysiology of plaques since it was present as a cellular matrix in most plaques11. Lastly, the photoacoustic signature of collagen could be detected throughout the whole plaques. However, in those plaques with a large lipid pool, collagen would be detected predominantly in the outer part of the plaque. Recent studies indicate that collagen remodeling could play a fundamental role in the mechanical stability of plaques, suggesting that its identification could provide valuable information for vulnerability assessment43.Two of the classifications, hemorrhage and lipids are indicators of plaque vulnerability and potential markers to assess therapeutic efficacy and disease progression. The spectrum classified as hemorrhage mainly presents the spectral features of methemoglobin and generates high-intensity signals, making them easily recognized. Besides, hemorrhage regions display a strong diffuse signal in contrast with the images expected from luminal bloodUnlike traditional linear unmixing methods, the method proposed does not require prior knowledge of the sample chromophores, and the number of endmembers can be easily determined by analyzing their normalized cross-correlation. Another major advantage of the method proposed is its capability to differentiate complex mixtures instead of pure signals, making it a valuable tool for studying highly heterogeneous samples, such as biological tissue. Moreover, the principal characteristic of our approach compared to other blind unmixing techniques in MSPAI is the solution of multiple convex models that are initialized using K-means clustering, generating an initial differentiation in the data based on intensity. Given the broad dynamic range and intrinsic bandwidth limitation of MSPAI, defining multiple models (starting with intensity) prevents low-contrast regions and photoacoustic speckles from being misclassified as background noise.Additionally, we evaluated the method with a limited number of selected wavelengths, proving the stability and feasibility of such an approach for in-vivo applications where acquisition time, imaging wavelengths, and system complexity are usually restricted. Although there is a trade-off between the number of wavelengths used and the accuracy of material decomposition, we observed an overall error in the prediction on phantom below 10%. Note, however, that the prediction error in the phantom experiments is also due to poor sampling of the transition between filled materials due to its discrete distribution, which is not the case in human plaque tissue. Therefore, such error would be less significant in the human plaques, which was confirmed by the comparison between the unmixing results in human plaques and histology.While promising results have been acquired, some challenges remain, especially aiming for in-vivo applications. Bearing in mind that surrounding tissue will be present for in-vivo applications, we will perform further studies with more complex surrounding tissue around the plaque to understand and compensate for the effect of muscle, fat, and skin on the spectral coloring of PA signals. Besides, strongly scattering media will decrease the signal amplitude reducing the SNR and the penetration depth. This problem requires the development of better illumination schemes and intensity compensation algorithms, fields that develop in parallel to unmixing research. Nevertheless, we believe the multiple-model approach has proved to have good performance even under the effect of spectral coloring and considerable variation in the signal intensity between components. Please note that the wavelengths used in this paper were selected for an in-vitro setting and may not be suitable for in vivo applications. To be able to apply the method in-vivo, further modifications of the proposed method, such as constraining the wavelengths, repeating the wavelength selection procedure in a relevant in-vivo setting, and the improvement of the PA acquisition system (such as optimizing the probe design) are likely required. Finally, we must mention that the scope of our unmixing approach on MSPAI is not limited to the materials presented in this study. Other components may be identified by improving the system resolution, spectral sampling, and SNR, which may lead to the identification of subtypes of collagen and lipids.The phantom was prepared using a solution of 15wt% PVA  and 85wt% water and three freezing-thaw cycles to achieve a consistent stiffness. The channels were filled with three biological materials: oleate cholesterol ; plaque tissue was obtained from the outer part of an endarterectomy plaque; porcine blood from a local butcher house prepared with a solution of 10 wt% sodium citrate tribasic dihydrate and 90 wt% blood.We scanned nine endarterectomy plaques obtained at the Catharina Hospital in Eindhoven. The study was approved by the local Ethical Committee  and conducted according to relevant guidelines and regulations. All patients gave their informed consent to participate in this study.34. PCOMMEND is an iterative unmixing approach that employs the least square method to estimate the set of endmembers that best represent a hyperspectral image composed of convex domains. It solves the standard linear mixture model and imposes non-negative constraints for the endmembers and proportions to ensure physical meaning. PCOMMEND employs an initial clustering to define multiple convex regions, allowing the solution of complex hyperspectral sets. We adapted this property to reduce the estimation error due to bandwidth limitation and fluence artifacts that can lead to a non-convex solution space. After convergence, the endmembers obtained for the bandwidth-limited region are projected into the endmembers of the other model, and then the proportions are recalculated.For the blind unmixing of the MSPAI data, we adapted a Piecewise Convex Multiple-Model Endmember method (PCOMMEND) by Zare et al.45, different types of cholesterol22, methemoglobin for hemorrhages46, and previous acquisitions contrasted with histology for SMC.We use two clusters during initialization, one for the bandwidth-limited signal and another for the rest of the modulated signal. We use a regularization parameter a = 0.1 to constrain the size of each cluster. The number of endmembers of each sample was determined by considering six endmembers, the number of selected wavelengths, and then reducing the number while monitoring the normalized cross-correlation between endmembers to identify the number of independent components. Finally, the endmembers are classified based on their modulation by comparing them with the spectra of common plaque constituents, such as collagen39; however, instead of using the smallest principal value, we employ the condition number as a criterion for the model as suggested by Mazhar et al.38.Our wavelength selection is based on the iterative approach presented by Luke et al.We define the conditioning matrix by taking the endmembers from the blind unmixing for 162 wavelengths. Then, the condition number of the matrix is iteratively optimized to determine the set of wavelengths that gives a well-conditioned problem. We found that the optimal number of endmembers for the phantom data was six; therefore, the same number of wavelengths was selected. For the ex-vivo samples, a set of representative spectra acquired from the endarterectomy samples was used to create the observation matrix to be optimized. We identified four materials of interest, and the background accounts for five wavelengths. Through experimental observations, we found that a sixth wavelength significantly increases the performance of the blind unmixing by reducing uncertainty due to laser power variation. The addition of one more wavelength didn't yield any improvement in the plaque unmixing.We validate the method in nine endarterectomy plaques scanned under the same imaging conditions as the phantom. Although all samples were imaged for 162 wavelengths, we split the data into two sets to exemplify the method's performance with a limited number of wavelengths. Four plaques are used for the wavelength selection algorithm, and the other five are analyzed with the wavelengths selected. Because plaque composition is distinct from the materials used in the phantom, different selected wavelengths were obtained for both experiments. The selected wavelengths for plaque imaging are: .For the co-registration of the US-PA images and the histology images, we used a grid placed under the plaque to localize the imaging plane. Then, the same grid is employed to cut the sample into sections of 4 mm thickness with the imaging plane located in the center. Afterward, we compared the shape of the plaque in the US image with the hematoxylin and eosin (H&E) histology of sections.Plaque measurements were compared with the histology sections. Three of the principal plaque constituents were validated using Masson's Trichrome (MT) for collagen (blue staining), Oil Red O (ORO) for lipids (red staining), and Martius Scarlet Blue (MSB) for blood and fibrin (orange-red staining). Even though neither technique provides quantitative information, it is possible to make a fair assessment of the similarity between the unmixing and the histology by investigating the correspondence of the identified materials and their approximate location.In conclusion, we have demonstrated the feasibility and advantages of using MSPAI to assess multiple components in carotid plaques with a limited number of wavelengths. The identified materials: hemorrhage, lipid, and collagen, have an immediate application in the assessment of plaque vulnerability and progression. To this end, we introduce a modified piecewise convex multiple-model endmember approach as an effective blind unmixing technique for highly heterogeneous biological samples, despite the presence of spectral coloring and bandwidth-limitation effects. Furthermore, we exemplify the possibility of wavelength selection based on the conditioning of the coefficient matrix of a linear model and its functionality on blind unmixing. In future studies, the performance of MSPAI will be studied in the presence of luminal blood and surrounding tissue as the next step towards in-vivo applications. Finally, we believe the methods presented in this work can be implemented in other spectroscopy modalities for biological samples, where the exact spectral absorption of the components is unknown, making it a universal approach for in vivo morphology assessment using MSPAI.