ANN-based Model Research Articles

Optimizing Energy-Efficient jet impingement cooling using an artificial neural network (ANN) surrogate model for high heat flux Semiconductors

As the demand for high-performance electronic devices and electric vehicles (EVs) increases, the importance of energy-efficient thermal management has become more important. This study presents the optimization of two high-heat flux (865.8 W/cm2) semiconductor thermal management systems targeting an EV inverter. The cooling system employs a jet impingement integrated with microposts. The thermal resistance (Rth.avg) and pumping power (Ppump) were evaluated through computational fluid dynamics (CFD) simulations by varying the jet nozzle parameters, micropost variables, and volume flow rate. To ensure the accuracy of the CFD model, experimental validation was also conducted, demonstrating less than a 5 % error. Then a surrogate model predicting Rth.avg and Ppump was developed using the artificial neural network (ANN) method. The sensitivity analysis subsequently identified the primary influencing factors of the system, and the Pareto optimal fronts were determined via the elitist non-dominated sorting genetic algorithm (NSGA-II). Through optimization, the approach provided multiple optimal designs for a broad spectrum of Ppump values, which was feasible within a brief period (less than 200 s) due to the rapid estimations by the ANN-based surrogate model. The experimentally validated design achieved a 65 % improvement in heat transfer coefficient (∼102.96 kW/(m2·K)) at similar pumping power levels (∼0.42 W) compared to previously reported jet impingement studies. Moreover, the design chosen through the optimization (Case B) projected a 140 % enhancement in heat transfer coefficient (∼150.2 kW/(m2·K)) with only a 63 % rise in Ppump (∼0.65 W). By exploring a more diverse range of multi-jet and micropost configurations, we were able to achieve higher performance. This study highlights the potential of combining jet impingement cooling with microposts as a highly attractive strategy for thermal management in high heat flux and multi-hot spot applications.

An efficient and invertible machine learning-driven multi-objective optimization architecture for light olefins separation system

The transience, cyclicity, and flexibility of process operation make the optimization of light olefins separation (LOS) system computationally intensive. An efficient and invertible architecture by incorporating machine learning techniques into optimization routines is proposed. Initially, a steady-state model of the LOS system was built allowing verification with industrial data and invocation through a COM interface. Then, artificial neuron networks (ANNs) were implemented to establish surrogate models for individual fitness calculation. Finally, to find the optimal operating parameters under the targets of energy and economy, an improved non-dominated sorting genetic algorithm was employed along with the ANN-based surrogate models. Optimization results demonstrates that the proposed architecture provides much better performance of computational efficiency with only 0.8 h, while the traditional surrogate models take more than 53 h. Furthermore, the reintroduction of optimized parameters into the LOS system model constructed by Aspen Plus V12.1 demonstrates the satisfactory fulfillment of the separation requirements and affirms the excellent invertibility of the proposed architecture. Essentially, it has been proved to be a powerful tool for multi-objective optimization that can be applied to large-scale chemical processes to derive insights for sustainable development and clean production.

Limitations and issues of conventional artificial neural network-based surrogate models for building energy retrofit

Artificial neural network (ANN) based surrogate models have been widely used in place of high-fidelity simulation tools. Error metrics such as mean absolute error, and root mean squared error (RMSE) have been widely used as de facto criteria. However, whether the ANN-based surrogate model can adequately reproduce the interwoven relationships and nonlinear causalities between design variables and simulated outputs are often overlooked. In this regard, the authors designed a case study regarding four ANN-based surrogate models. It was found that despite all of the models having low RMSEs, the models failed to adequately predict the causal relationships between input variables and energy use. In other words, the surrogate models were not always capable of providing accurate assessments of expected energy use reduction as a result of design changes. In this paper, we present a workflow for validating whether a surrogate model can reproduce the causal relationships between inputs and outputs.

ANN-based ground motion model for Turkey using stochastic simulation of earthquakes

SUMMARY Turkey is characterized by a high level of seismic activity attributed to its complex tectonic structure. The country has a dense network to record earthquake ground motions; however, to study previous earthquakes and to account for potential future ones, ground motion simulations are required. Ground motion simulation techniques offer an alternative means of generating region-specific time-series data for locations with limited seismic networks or regions with seismic data gaps, facilitating the study of potential catastrophic earthquakes. In this research, a local ground motion model (GMM) for Turkey is developed using region-specific simulated records, thus constructing a homogeneous data set. The simulations employ the stochastic finite-fault approach and utilize validated input-model parameters in distinct regions, namely Afyon, Erzincan, Duzce, Istanbul and Van. To overcome the limitations of linear regression-based models, artificial neural network is used to establish the form of equations and coefficients. The predictive input parameters encompass fault mechanism (FM), focal depth (FD), moment magnitude (Mw), Joyner and Boore distance (RJB) and average shear wave velocity in the top 30 m (Vs30). The data set comprises 7359 records with Mw ranging between 5.0 and 7.5 and RJB ranging from 0 to 272 km. The results are presented in terms of spectral ordinates within the period range of 0.03–2.0 s, as well as peak ground acceleration and peak ground velocity. The quantification of the GMM uncertainty is achieved through the analysis of residuals, enabling insights into inter- and intra-event uncertainties. The simulation results and the effectiveness of the model are verified by comparing the predicted values of ground motion parameters with the observed values recorded during previous events in the region. The results demonstrate the efficacy of the proposed model in simulating physical phenomena.

An Artificial Neural Network-based model that can predict inpatients’ personal thermal sensation in rehabilitation wards

Indoor thermal comfort is important to hospital designs as it affects healthcare outcomes. However, existing thermal sensation analysis models do not fully consider individual patient's preference and their effectiveness has not been validated in healthcare environments. The commonly used Predicted Mean Vote (PMV) model cannot process complex parameters such as individual differences, multiple patients' biosignals, medical activities, and spatial layout of wards, etc. To fill in the gaps, this research aims to develop an innovative model that can effectively predict in-patients' personal thermal sensation in rehabilitation wards. Based on previous research on machine learning (ML), a prototype of the Artificial Neural Network (ANN)-based model has been developed for this purpose. To test this model and assess its prediction accuracy in the real world, a case study was conducted in the rehabilitation department of a general hospital in Xuzhou, China. The results indicated that the ANN-based model effectively predicted patients' thermal sensation. Moreover, it was found from this study that the incorporation of spatial and health-related parameters into the ANN-based model could significantly improve the prediction accuracy. The best prediction accuracy of the ANN-based model was 8.10 % higher than that of the baseline model. It is therefore concluded that this model can be used to support architects and HVAC system engineers to make informed decisions in hospital designs and help medical staff allocate in-patients according to their preference.

Predicting the adsorption capacity of organic dye using zirconium-based metal–organic framework: a comparative analysis of RSM and ANN-based models

Predicting the adsorption capacity of organic dye using zirconium-based metal–organic framework: a comparative analysis of RSM and ANN-based models

Data-Driven Modeling of HVAC Systems for Operation of Virtual Power Plants Using a Digital Twin

Confronted with the climate crisis, the world is making tremendous efforts in energy transition, such as expanding renewable energy that does not emit carbon. The importance of virtual power plant (VPP) operation technology has emerged to secure grid flexibility in response to the expanding renewable energy implemented due to these efforts. Accordingly, VPPs, which include photovoltaics, wind turbines, heating, ventilation, and air conditioning (HVAC), load, and EV, have been constructed. HVAC, one of the component resources, is a system that controls and regulates temperature, humidity, and airflow. Since it responds sensitively to the building’s heat capacity and changes in the external environment, it requires continuous and stable control. In this paper, we used data-based modeling to implement the HVAC required for the optimal operation of VPP. Since accurately creating an equation-based HVAC model was difficult considering building information modeling and external environment variables, we used historical HVAC operation data to perform data-based modeling. The model was implemented using nonlinear regression and machine learning, such as a support vector machine and artificial neural network. Then, the data-based HVAC and the actual HVAC operation results were comparatively analyzed based on a case study, and the model’s goodness-of-fit was evaluated based on performance metrics. Model performance indicators confirmed that the ANN-based HVAC model was most similar to the actual HVAC system.

Bioinspired spiking spatiotemporal attention framework for lithium-ion batteries state-of-health estimation

State-of-health (SOH) estimation of batteries is crucial for ensuring the safety of energy storage systems. Prediction models based on external information (current, voltage, etc.) and artificial neural networks (ANN) are effective solutions. However, external information easily interferes, and the ANN-based model has data dependence, high energy consumption, and insufficient cognitive ability. This motivates us to utilize precise battery physical and chemical degradation information and brain-inspired spiking neural networks (SNNs) for accurate SOH estimation. Therefore, this study proposes a bioinspired spiking spatiotemporal attention neural network (SSA-Net) framework for battery health state monitoring by utilizing full-life-cycle electrochemical impedance spectroscopy (EIS). SSA-Net perfectly models brain neurons' information transmission mechanism and neuron dynamics, thereby endowing it with efficient spatiotemporal feature processing capabilities and low power consumption. Based on the designed spiking residual architecture, SSA-Net constructs a deep spiking information encoding framework achieving high gradient transfer efficiency. More importantly, this study proposes a novel SNN-based spiking spatiotemporal attention module, which realizes the enhancement of useful spiking features and discards worthless information through an adaptive spiking feature selection mechanism. Experimental results show that SSA-Net effectively extracts electrochemical features associated with battery degradation, facilitating precise modeling of the nonlinear relationship between EIS data and SOH and achieving competitive performance.

Predictive modeling on the surface tension and viscosity of ionic liquid-organic solvent mixtures via machine learning

Backgrounda comprehensive collection of reliable open-source data was compiled, encompassing 3454 data points for surface tension and 28,548 data points for viscosity of IL-organic solvent mixtures. Methodspredictive modeling was conducted for these two crucial properties using a combination of the GC method and three well-known machine learning algorithms: Artificial Neural Network (ANN), eXtreme Gradient Boosting (XGBoost), and Light Gradient Boosting Machine (LightGBM). The modeling results demonstrate that the GC models, employing all three machine learning algorithms, offer dependable predictions. Significant FindingsThe XGBoost-based model exhibits the most superior performance in surface tension predictions, achieving a square correlation coefficient (R2) of 0.9829 and a mean absolute error (MAE) of 0.0007 N/m for the randomly generated test dataset. In contrast, the ANN-based model with optimized neurons delivers the best predictive performance in viscosity, achieving an R2 of 0.9582 and an MAE of 0.0975 Pa·s. Furthermore, the Shapley Additive Explanations (SHAP) analysis is performed to evaluate the impact of each feature on the prediction of surface tension and viscosity in IL-organic solvent binary mixtures. The results reveal that the mole fraction of IL has the greatest positive effect on the prediction for both surface tension and viscosity.

Grey-box and ANN-based building models for multistep-ahead prediction of indoor temperature to implement model predictive control

Model-based predictive control (MPC) strategies for heating, ventilation, and air-conditioning (HVAC) systems present an opportunity to lower building energy consumption and operational costs. Such approaches rely on the development of a model to precisely forecast building thermal dynamics, such as room air temperature or heating/cooling rate, and make control-related decisions. The control-oriented modeling of building energy systems should be accurate in predicting indoor conditions and present low computational complexity. These features are the key challenge of implementing advanced control methods such as MPC. Extant studies on building modeling for MPC have focused on step-ahead forecasting techniques to forecast building thermal dynamics, while multistep-ahead forecasting is essential. Moreover, machine learning model suitable in case of the domain-based engineering expertise are also not available. To this aim, we perform a comparative analysis of the grey-box model based on a resistance-capacitance (RC) thermal network and a machine learning model composed of an artificial neural network (ANN) for multistep-ahead prediction of building thermal dynamics using current and historical data. Actual experimental data obtained from the Flexible Research Platform (FRP) in Oak Ridge National Laboratory (US) are used for estimation and validation purposes. The average root mean squared error (RMSE) of the grey-box and ANN models are 0.89 °C and 1.02 °C, respectively. The results indicate that the grey-box model outperforms the ANN model in the considered validation periods in terms of accuracy and prediction stability.

Machine learning models for efficient characterization of Schottky barrier photodiode internal parameters

We propose ANN-based models to analyze and extract the internal parameters of a Schottky photodiode (SPD) without presenting them with any knowledge of the highly nonlinear thermionic emission (TE) expression of the device current. We train, evaluate and demonstrate the ML models on thirty-six private datasets from three previously published devices, which denote current responses under illumination and ambient temperature of graphene oxide (GO) doped p-Si Schottky barrier diodes (SBDs). The GO doping levels are 0%, 1%, 3%, 5%, and 10%. The illumination ranged from dark (0 mW/cm2) to 30 mW/cm2. The predictions are then made completely at the intensity of 60 mW/cm2. For each diode, some values of the barrier height (phi ), ideality factor (n), and series resistance (R_s) independently calculated using the Cheung–Cheung method were included in the training dataset. The predictions are done at unspecified intensities on the model development data at 80 and 100 mW/cm2, and on external data at 5% and 20% GO doping which were not part of the development dataset. The ANN achieved a mean square error and mean absolute error score below 0.003 across all datasets. This demonstrates the effective learning capabilities of the ANN models in accurately capturing the photo responses of the photodiodes and accurately predicting the internal parameters of the Schottky Barrier Diodes (SBDs), all without relying on an inherent understanding of the thermionic emission (TE) equation for SBDs. The ANN models achieved high accuracy in this process. The proposed ML models can significantly reduce analysis time in device development cycles and can be applied to other datasets in various fields.

Neural-network-augmented projection-based model order reduction for mitigating the Kolmogorov barrier to reducibility

Inspired by our previous work on a quadratic approximation manifold [1], we propose in this paper a computationally tractable approach for combining a projection-based reduced-order model (PROM) and an artificial neural network (ANN) to mitigate the Kolmogorov barrier to reducibility of parametric and/or highly nonlinear, high-dimensional, physics-based models. The main objective of our PROM-ANN concept is to reduce the dimensionality of the online approximation of the solution beyond what is achievable using affine and quadratic approximation manifolds, while maintaining accuracy. In contrast to previous approaches that exploited one form or another of an ANN, the training of the ANN part of our PROM-ANN does not involve data whose dimension scales with that of the high-dimensional model; and the resulting PROM-ANN can be efficiently hyperreduced using any well-established hyperreduction method. Hence, unlike many other ANN-based model order reduction approaches, the PROM-ANN concept we propose in this paper should be practical for large-scale and industry-relevant computational problems. We demonstrate the computational tractability of its offline stage and the superior wall clock time performance of its online stage for a large-scale, parametric, two-dimensional, model problem that is representative of shock-dominated unsteady flow problems.

High-performance self-compacting concrete with recycled coarse aggregate: Soft-computing analysis of compressive strength

The growth of cities and industrialization has led to an increase in demand for concrete, resulting in resource depletion and environmental issues. Sustainable alternatives such as using recycled concrete aggregate (RCA) and industrial waste have been proposed to meet construction material demands while adhering to building codes and promoting sustainability. However, compressive strength (CS) is a crucial property of concrete, and the design parameters have different effects on CS for various grades. Recently, researchers have focused on partially replacing natural coarse aggregate (NCA) with RCA in concrete to achieve sustainability goals. This study aims to examine the influence of design parameters (w/c: water-cement ratio, w/b: water-binder ratio, A/c: total aggregate-cement ratio, FA/CA: fine-coarse aggregate ratio, SP: superplasticizer, w/s: water-solid ratio and RCA%) on concrete CS and address controversies in the insights gained from pairwise comparisons using Pearson's correlation coefficient (PCC) analysis. Additionally, five techniques (M5P, RF, SVM, LR, and ANNs) were used to predict the CS of high-performance self-compacting concrete (HP-SCC) with RCA, and the results were compared with an ANNs-based model as was the commonly used one in literature. The approaches were assessed based on their accuracy measured using correlation coefficient (CC), mean absolute error (MAE), Root Mean Square Error (RMSE), Mean absolute percentage error (MAPE), Scatter index (SI), and comprehensive measure (COM) indicators. Accordingly, the analysis indicated that SVM-PUK-based model is the most appropriate and effective technique to predict the CS of HP-SCC for the given datasets, with CC = 0.894, 0.900, MAE = 1.721, 3.813, RMSE = 5.137, 6.306, and MAPE = 4.5%, 7.6% for the training and testing stages, respectively. The uncertainty analysis results were 21%, 20.7%, 19%, 22%, and 19% for M5P, RF, SVM, LR, and ANN-based models, respectively, whereby all of them were under threshold of 35%. Moreover, according to sensitivity analysis, w/c, w/b, and w/s variables influences the most on CS prediction, while the RCA(%) variable has the least impact.

Early estimation of 28-day compressive strength of mortars using regression and neural network-based models

Concrete, the most commonly used construction material, presents a challenge in determining its compressive strength (CS) after 28 days of manufacture. This study aims to predict the compressive strength of conventionally cured (CC) mortar specimens using data from accelerated microwave-cured (MC) specimens. Two hundred and seventy-six specimens with different mixes and admixtures were divided into two groups: one half was subjected to MC and the other to CC. The CSs of the CC specimens were determined after 28 days, while the MC specimens were subjected to additional tests such as ultrasonics pulse velocity and three-point bending after microwave curing. Regression and artificial neural network models (ANN) were developed using the obtained data. While the data of the specimens subjected to MC were used as independent variables of the developed models, the compressive strength of the conventionally cured specimens (CC) was used as the dependent variable. In the models, 60% of the data were used for training, 20% for validation, and 20% for testing. The predictive capabilities of the developed models were compared using performance statistics such as root mean square error (RMSE), root mean absolute error (MAE), and Nash-Sutcliffe efficiency coefficient (NSE). As a result of the study, it was found that compared to all models, the highest performance values were obtained by the ANN-based models. Among the regression-based models, the multivariate adaptive regression splines method (MARS) obtained the highest performance values. Considering the calculated NSE values, it was determined that the ANN-based model gave similar results to the MARS model for the training and validation data sets but provided performance values approximately 25% higher than the MARS model for the test data set. In addition, the results obtained from the modeling studies showed that the CSs of concrete could be predicted very accurately and well in advance.

Improving wastewater treatment plant performance: an ANN-based predictive model for managing average daily overflow and resource allocation optimization using Tabu search

Improving wastewater treatment plant performance: an ANN-based predictive model for managing average daily overflow and resource allocation optimization using Tabu search

An RDL Modeling and Thermo-Mechanical Simulation Method of 2.5D/3D Advanced Package Considering the Layout Impact Based on Machine Learning.

The decreasing-width, increasing-aspect-ratio RDL presents significant challenges to the design for reliability (DFR) of an advanced package. Therefore, this paper proposes an ML-based RDL modeling and simulation method. In the method, RDL was divided into blocks and subdivided into pixels of metal percentage, and the RDL was digitalized as tensors. Then, an ANN-based surrogate model was built and trained using a subset of tensors to predict the equivalent material properties of each block. Lastly, all blocks were transformed into elements for simulations. For validation, line bending simulations were conducted on an RDL, with the reaction force as an accuracy indicator. The results show that neglecting layout impact caused critical errors as the substrate thinned. According to the method, the reaction force error was 2.81% and the layout impact could be accurately considered with 200 × 200 elements. For application, the TCT maximum temperature state simulation was conducted on a CPU chip. The simulation indicated that for an advanced package, the maximum stress was more likely to occur in RDL rather than in bumps; both RDL and bumps were critically impacted by layouts, and RDL stress was also impacted by vias/bumps. The proposed method precisely concerned layout impacts with few resources, presenting an opportunity for efficient improvement.

Application of multi-modality MRI-based radiomics in the pre-treatment prediction of RPS6K expression in hepatocellular carcinoma

In this study, we aim to develop and validate a radiomics model for pretreatment prediction of RPS6K expression in hepatocellular carcinoma (HCC) patients, thus helping clinical decision-making of mTOR-inhibitor (mTORi) therapy. We retrospectively enrolled 147 HCC patients, who underwent curative hepatic resection at First Affiliated Hospital Zhejiang University School of Medicine. RPS6K expression was determined with immunohistochemistry staining. Patients were randomly split into training or validation cohorts on a 7:3 ratio. Radiomics features were extracted from T2-weighted and diffusion-weighted images. Machine learning algorithms including multiple logistic regression (MLR), supporting vector machine (SVM), random forest (RF), and artificial neural network (ANN) were applied to construct the predictive model. A nomogram was further built to visualize the possibility of RPS6K expression. The area under the receiver operating characteristic (AUC) was used to evaluate the performance of diagnostic models. 174 radiomics features were confirmed correlated with RPS6K expression. Amongst all built models, the ANN-based hybrid model exhibited best predictive ability with AUC of 0.887 and 0.826 in training and validation cohorts. ALB was identified as the key clinical index, and the nomogram displayed further improved ability with AUC of 0.917 and 0.845. In this study, we proved MRI-based radiomics model and nomogram can accurately predict RPS6K expression non-invasively, thus providing help for clinical decision making for mTORi therapy.

ANN-based estimation model for the preconstruction cost of pavement rehabilitation projects

To combat prevalent roadway aging, the California Department of Transportation (Caltrans) implements routine rehabilitation projects, which require cost estimation models for better management. Several models have been developed to estimate the construction costs of new highway projects; however, few have targeted the cost of pavement rehabilitation, and, of these, most have focused primarily on the construction phase. This study presents an artificial neural network (ANN) estimation model that predicts the cost of the preconstruction phases (predesign, design, and bidding) of pavement rehabilitation projects. The model was developed using 139 projects maintained by Caltrans over the past five years. Pareto principle and sensitivity analysis were used to identify five project items that contributed significantly to the model’s high estimation performance, which was demonstrated by a coefficient of multiple determination value (R 2) of 76.9%. Thus, the proposed model will allow Caltrans to properly estimate its budget and manage its performance during the preconstruction phases of pavement rehabilitation projects. Moreover, it overcomes many limitations of conventional preconstruction cost estimation models and will allow Caltrans to better manage up to $80B that may be spent on its preconstruction activities. Finally, the model provides a benchmarking tool for tracking and controlling expenses.

Development of the estimation model for the maximum power point of building-applied photovoltaic systems based on machine learning

With increased photovoltaic (PV) applications in buildings, operational and maintenance (O&M) is crucial to preserving initial performance. Faults detection and diagnostics (FDD) methods, such as current and voltage measurement, are used to identify system errors by comparing measured values with calculated maximum power point (MPP) outputs. This method typically uses the One-Diode model to calculate MPP values and considers them as optimal performance outputs. However, the accuracy can inevitably vary based on PV installation conditions and types of solar cells. Data-driven models have been developed to address these complexities but require further exploration for practical building applications. This study developed a data-driven model using an artificial neural network (ANN) to estimate MPP in the context of FDD for various building-applied PV systems. Initial training was conducted with hourly-based simulated data, and the model was validated using real-world building-applied PV system data, including building-attached PV (BAPV) and building-integrated PV (BIPV) types. To improve the performance of the initial MPP estimation model, additional training and re-validation processes were conducted using the measured data of the BIPV system. Initial results showed acceptable performance for power, current, and voltage estimation with less than 30% CvRMSE for building-attached PV systems. Building-integrated PV systems, affected by external factors like shading, yielded less accuracy with over 30% CvRMSE. After additional training, the model showed improved performance, including around 11% CvRMSE for all parameters. This study indicates that the ANN-based MPP estimation model can be practically applied to FDD with acceptable accuracy.

Mathematical immunology: processes, models and data assimilation

The immune system is a complex multiscale multiphysical object. Understanding its functioning in the frame of systemic analysis implies the use of mathematical modelling, formulation of data consistency criterion, estimation of parameters, uncertainty analysis, and optimal model selection. In this work, we present some promising approaches to modelling the multi-physics immune processes, i.e., cell migration in lymph nodes (LN), lymph flow, homeostatic regulation of immune responses in chronic infections.
 To describe the spatial-temporal dynamics of immune responses in lymph LN, we propose a model of lymphocyte migration, based on the second Newtons law and considering three kinds of forces. The empirical distributions of three lymphocytes motility characteristics were used for model calibration using the KolmogorovSmirnov metric.
 Prediction of lymph flow in a lymph node requires costly computations, due to diversity of sizes, forms, inner structure of LNs and boundary conditions. We proposed an approach to lymph flow modelling based on replacing the full-fledged computational physics-based model with an artificial neural network (ANN), trained on the set of pre-formed results computed using an initial mechanistic model. The ANN-based model reduces the computational time for some lymph flow characteristics by four orders of magnitude.
 Calibration of MarchukPetrov model of antiviral immune response for SARS-CoV-2 infection was performed. To this end, we used previously published data on the viral load kinetics in nasopharynx of volunteers, and data on the observed ranges of interferon, antibodies and CTLs in the blood. The parameters, which have the most significant impact at different stages of infection process, were identified.
 Inhibition of immune mechanisms, e.g., T cell exhaustion, is a distinctive feature of chronic viral infections and malignant diseases. We propose a mathematical model for the studies of regulation parameters of four exhausted T cell subsets in order to examine the balance of their proliferation and differentiation determined by interaction with SIRPa+ PD-L1+ and XCR+1 dendritic cells. The model parameters are evaluated, in order to study the reinvigoration effect of aPD-L1 therapy on the homeostasis of exhausted cells.