Eleonore Dufresne

Denis Fortun
email: dfortun@unistra.fr

Stéphane Kremer

Vincent Noblet

Benoit Combès

Stefano Cerri

Renaud Hedouin

Eléonore Dufresne

A unified framework for focal intensity change detection and deformable image registration. Application to the monitoring of multiple sclerosis lesions in longitudinal 3D brain MRI

Keywords: deformable D registration, change detection, longitudinal analysis, multiple sclerosis, joint minimization, alternating direction method of multipliers (ADMM)

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

. Introduction

Multiple sclerosis (MS) is an auto-immune neurodegenerative disease characterized by the inflammation of the myelin coating that surrounds the nerves. As a consequence, the transmission of nervous impulses is impaired, causing motor, cognitive and sensorial disabilities. The evolution of MS is characterized by the apparition of focal lesions in the brain and in the spinal cord, and by a progressive atrophy of brain tissues. Both phenomena can be monitored thanks to Magnetic Resonance Imaging (MRI) [START_REF] Kaunzner | MRI in the assessment and monitoring of multiple sclerosis: an update on best practice[END_REF]. In the clinical routine, the evolution of the lesion load and of the brain atrophy is generally assessed qualitatively. However, the precise quantification of lesion changes over time may be of great interest to finely characterize the course of the pathology and to evaluate at the early stage the effect of a therapeutic strategy [START_REF] Mcnamara | Current and emerging therapies in multiple sclerosis: implications for the radiologist, part 1-mechanisms, efficacy, and safety[END_REF]. Since the manual delineation of lesion changes in MRI is a tedious and time consuming task, which is prone to both intra-and inter-observer variability, there is a great need for efficient and reliable automated tools [START_REF] Altay | Reliability of classifying multiple sclerosis disease activity using magnetic resonance imaging in a multiple sclerosis clinic[END_REF].

Most change detection methods dedicated to lesion monitoring rely on a sequential scheme that first consists in removing all changes that are not of interest in order to detect in a second step only the evolution of lesions [START_REF] Radke | Image change detection algorithms: a systematic survey[END_REF]. To correct for global intensity changes induced by the difference of MRI acquisition setups, algorithms for bias field inhomogeneity correction [START_REF] Song | A review of methods for bias correction in medical images[END_REF] and histogrambased intensity normalization procedure [START_REF] Shinohara | Statistical normalization techniques for magnetic resonance imaging[END_REF] are generally considered. Then, geometrical discrepancies due to variation in patient positioning, acquisition-related geometrical distortion and brain atrophy are corrected thanks to registration algorithms involving either rigid, affine or deformable transformations. Finally, the remaining changes corresponding to the evolution of lesions are detected. This final step generally consists in thresholding an intensity-based [START_REF] Bosc | Automatic change detection in multimodal serial MRI: application to multiple sclerosis lesion evolution[END_REF] or deformation-based [START_REF] Rey | Automatic detection and segmentation of evolving processes in 3D medical images: application to multiple sclerosis[END_REF] feature map. The threshold can be chosen according to some statistical modeling in order to control the expected number of false positive detections [START_REF] Rousseau | An a contrario approach for change detection in 3D multimodal images: application to multiple sclerosis in MRI[END_REF].

The main flaw of the sequential procedure is that it implicitly assumes that each correction step can be performed while not being influenced by the changes that remain to be corrected in the next steps. As a consequence, the sequence order of the correction procedures should be carefully chosen. Moreover, for each correction step, a trade-off should be found between its performance (i.e., the ability of the method to accurately and specifically correct a given kind of change) and its robustness (i.e., the ability not to be biased by another kind of remaining changes). This observation advocates for a unified formulation of the change detection problem allowing to estimate all the different kinds of changes jointly.

In this paper, we address more specifically the interplay between deformable image registration and focal intensity change detection. When deformable registration is performed in the presence of appearing lesions, the estimated transformation tends to make these new lesions disappear in order to minimize the dissimilarity between the two images. This is the reason why the most common practice is to consider only rigid or affine registration in order not to alter lesion shape. However, such linear transforms can only compensate for difference in patient positioning but are not able to capture the complex deformations induced by brain atrophy, which typically occurs in MS. These remaining deformations may yield to spurious detections in atrophied areas, especially in the cortex and around the ventricles.

We propose to account for the intertwining of deformable registration and focal intensity change detection by estimating them jointly. To this end, we show that these two separate tasks can be formulated as a single optimization problem involving a unique energy that models their coupling. Basically, areas corresponding to detected changes are ignored in the registration similarity criterion, which prevents the lesion elimination effect described above. Solving this issue allows us to use of deformable registration, which in turn prevents from detecting spurious changes in atrophied areas. We propose an efficient alternating optimization scheme to solve this unified optimization problem. We focus on demonstrating the benefits of this joint formulation in the particular case of a standard intensity-based data similarity criterion. Nevertheless, the proposed approach is versatile and could easily be extended to more elaborated modeling choices. Experimental analysis is performed on the BrainWeb synthetic dataset and on two annotated real datasets. We first demonstrate in each case the benefits of considering a deformable registration as compared to an affine registration only in order to reduce the number of false detections. Then, we highlight the benefit of the proposed joint formulation as compared to the standard sequential scheme in terms of change detection accuracy. A preliminary version of this work has been published as a conference paper in [START_REF] Dufresne | Joint registration and change detection in longitudinal brain MRI[END_REF]. In this paper, we provide a more extensive experimental analysis, which helps to better characterize the behavior of the proposed method and better understand why it outperforms the sequential approach.

The paper is organized as follows. In Section 2, the sequential approach, which will be considered as the reference baseline method of this work, is described and its limitations are discussed. In Section 3, we introduce the proposed joint model and the optmization strategies that has been set up estimate both change detection map and deformation field. In Section 4, we give implementation details. Finally, we present and discuss experimental results in Section 5.2.

. The sequential approach

The conventional sequential approach consists of three main steps. First, the images are corrected for global intensity variations, then they are spatially registered, and finally the focal intensity changes due to the evolution of lesions are detected. In this section, we give a brief overview of the common practices in the registration and change detection steps. Our goal is not to cover a comprehensive scope of the field but to formulate the general principles underlying existing methods. In the remainder of this article, we will denote I 1 , I 2 : → R the . /fnimg. .

baseline and follow-up MRI images, where ⊂ R 3 is the image domain.

. . Registration

The registration problem can be formulated as:

w = argmin w x∈ ρ(I 1 , I 2 , w, x) + λ 1 (w), (1)
where w : → R 3 represents the transformation, ρ(•) is a data similarity term, and (•) is a regularizer weighted by a scalar λ 1 > 0. An overview of deformable registration methods in medical imaging can be found in [START_REF] Sotiras | Deformable medical image registration: a survey[END_REF].

Several transformation models can be considered relying either on a parametric representation (e.g., rigid, affine, polynomial, or Bspline-based) or on a non-parametric deformable mapping (i.e., a displacement vector is estimated for each voxel).

The role of the data term is to penalize dissimilarity between I 1 and I 2 warped with the estimated transformation. For monomodal registration, it is common to use intensity-based measures such as the sum of squared intensity differences or the cross correlation. In the multimodal case, mutual information is one of the most widely used similarity metric [START_REF] Kaunzner | MRI in the assessment and monitoring of multiple sclerosis: an update on best practice[END_REF].

In the context of deformable registration, considering the data term only can lead to an ill-posed problem. To overcome this issue, the data term has to be balanced with an additional regularization term (w) that enforces some constraints on the deformation field. For instance, penalizing the ℓ 2 or ℓ 1 norm of the gradient of w helps to promote smooth solutions.

. . Change detection

The change detection step generally consists in thresholding a map of feature differences between registered baseline and follow-up images (see the survey Lladó et al., 2012). These maps can be calculated directly from either intensity-based [START_REF] Sweeney | Automatic lesion incidence estimation and detection in multiple sclerosis using multisequence longitudinal MRI[END_REF][START_REF] Ganiler | A subtraction pipeline for automatic detection of new appearing multiple sclerosis lesions in longitudinal studies[END_REF][START_REF] Cabezas | Improved automatic detection of new T2 lesions in multiple sclerosis using deformation fields[END_REF], or deformation-based [START_REF] Rey | Automatic detection and segmentation of evolving processes in 3D medical images: application to multiple sclerosis[END_REF][START_REF] Cabezas | Improved automatic detection of new T2 lesions in multiple sclerosis using deformation fields[END_REF][START_REF] Salem | A supervised framework with intensity subtraction and deformation field features for the detection of new T2-w lesions in multiple sclerosis[END_REF] features, and sometimes integrate other kind of information [START_REF] Elliott | Temporally consistent probabilistic detection of new multiple sclerosis lesions in brain MRI[END_REF][START_REF] Sweeney | Automatic lesion incidence estimation and detection in multiple sclerosis using multisequence longitudinal MRI[END_REF] .

In the perspective of integrating both registration and change detection in a single joint optimization problem, we advocate that they should both rely on the same data similarity. Consequently, we model the binary change map c : → {0, 1} defined at each voxel x as follow:

c(x) =    0 if ρ(I 1 , I 2 , w, x) ≤ λ 2 1 otherwise, (2)
λ 2 ∈ R + * being the detection threshold. The thresholding scheme (Equation 2) can be reformulated as the following optimization problem:

c(x) = argmin c : →{0,1} x∈ (1 -c(x)) ρ(I 1 , I 2 , w, x) + λ 2 c(x). (3)
Since simple thresholding can yield noisy results, most MS lesion change detection methods also integrate a denoising step in post-processing to obtain the final change map. The denoising can be realized jointly with the change detection by integrating a regularization term (•) in Equation (3):

c(x) = argmin c : →{0,1} x∈ (1 -c(x)) ρ(I 1 , I 2 , w, x) + λ 2 c(x) + λ 3 (c), (4)
Where λ 3 weights the regularization term.

. . Limitation of the sequential approach

The main limitation of the sequential approach is illustrated in Figure 1 involving the baseline (Figure 1) and follow-up (Figure 1) MRI acquisitions of a patient suffering from MS. One can observe in the follow-up scan the apparition of a new lesion and a slight enlargement of the ventricle reflecting the brain atrophy process. In the case of affine registration (Figures 1), we can see on the subtraction image that the lesion is well detected, but that spurious detection occur around the ventricles and in the cortical regions due to brain tissue atrophy. Using a deformable registration (Figures 1) helps to remove these spurious detection by compensating the ventricles enlargement and the cortical atrophy. However, it also tends to make the new lesion disappear (Figure 1), thus altering the shape of the corresponding detection (Figure 1). The goal of the proposed joint approach (Figures 1) is to perform an accurate atrophy correction while preserving the shape of appearing and evolving lesions, even when the lesion-to-tissue contrast is quite low.

. Joint approach . . General formulation

To overcome the limitations of the sequential approach, we advocate a joint modeling of registration and change detection. The two steps are fundamentally intertwined, since registration aims at finding correspondences between images, while change detection determines regions that does not admit correspondences. Therefore, both tasks should be defined with the same objective function to work in synergy. We formulate FIGURE Warped follow-up images and subtraction images obtained with the sequential a ne and deformable registration pipelines and the proposed joint deformable approach, on an example of the dataset described in COM (

).

the following joint minimization problem that achieves this goal by unifying the principles described previously:

w, c = argmin w,c x∈ (1 -c(x)) ρ(I 1 , I 2 , w, x) + λ 2 c(x) +λ 1 (w) + λ 3 (c). (5)
With this model, the data term is cancelled in change regions (where c(x) = 1), so that the estimation of the transformation is only driven by the regularization term, thus producing smoothed deformation field in these areas. By this way, it prevents the lesion disappearing effect observed in Figure 1.

. . Modeling choices

The formulation (Equation 5) is versatile and could be instantiated with a variety of data and regularization terms. In this paper, the goal is to demonstrate the superiority of the joint formulation over the sequential approach under standard modeling choices, which are detailed in the sequel.

First, we assume that, thanks to the intensity normalization step done as a preprocessing, intensities of both images are comparable, thus allowing us to consider a data term based on intensity difference. Consequently, we consider the following standard similarity measure:

ρ(I 1 , I 2 , w, x) = 1 σ 2 I 2 (x -w(x)) -I 1 (x) 2 2 , ( 6
)
Where σ is a normalization constant defined by the median absolute deviation of the intensity differences between I 1 and I 2 . This data term has been used for motion estimation [START_REF] Bruhn | Lucas/kanade meets horn/schunck: combining local and global optic flow methods[END_REF] and is representative of intensity-based features commonly used in change detection methods [START_REF] Sweeney | Automatic lesion incidence estimation and detection in multiple sclerosis using multisequence longitudinal MRI[END_REF][START_REF] Ganiler | A subtraction pipeline for automatic detection of new appearing multiple sclerosis lesions in longitudinal studies[END_REF].

Secondly, we assume that the deformations induced by brain tissue atrophy are complex but still locally smooth. This is why we consider a non-parametric representation of the transformation field w while introducing a first order Tikhonov regularization term:

(w) = x∈ ∇w(x) 2 2 , ( 7
)
Where ∇• is the gradient operator.

Finally, we assume that the detected changes should be spatially coherent. Consequently, the change map c is regularized with a standard binary Potts model:

(c) = x∈ y∈N (x) (1 -δ c(x), c(y) ), ( 8
)
Where δ is the Kronecker function equal to 1 if its argument is true and N (x) is the 6-neighborhood of x.

. . Optimization

To solve the optimization problem (Equation 5), we rely on an alternating minimization strategy: at each iteration, we successively minimize with respect to (w.r.t.) each variable, while keeping the other fixed. We detail in this section the optimization strategies dedicated to each of the two subproblems.

Notice that, since the problem is nonconvex, convergence toward the global minimum cannot unfortunately be guaranteed. However, the results presented in Section 5.2 on several datasets suggest that the optimization process converges in practice toward a satisfying solution.

Minimization w.r.t. w

To make the problem tractable, we consider the linearized version of the data term (Equation 6) obtained by replacing I 2 (yw(x)) with its Taylor expansion around y:

ρ l (I 1 , I 2 , w, x) = 1 σ 2 ∇ ⊤ I 2 (x) w(x) + I t (x) 2 2 , (9)
Where I t (y) = I 2 (y) -I 1 (y) is the temporal derivative. Since the Taylor expansion is valid only for small deformations, we embed the estimation in a coarse-to-fine scheme, which is a common practice in registration and motion estimation [START_REF] Hill | Medical image registration[END_REF].

After Taylor development of I 2 (yw(x)), the new data term ρ l in Equation ( 9) is the composition of a quadratic function with a linear function of w, which yields a convex term. Since the regularization term (Equation 7) is also convex, the whole optimization problem is convex. By substituting ρ by ρ l in Equation ( 5), the optimisation problem can be addressed with a variety of efficient optimization methods. We chose to consider the alternated direction method of multipliers (ADMM) framework [START_REF] Boyd | Distributed optimization and statistical learning via the alternating direction method of multipliers[END_REF]. To this end, we introduce a splitting variable z that decouples the two terms of Equation ( 5) that depend on w, and we formulate the problem in the constrained form:

min w x∈ (1 -c(x)) ρ l (I 1 , I 2 , w, x) + λ 1 (z) s.t w = z (10)
The ADMM algorithm is based on the minimization of the augmented Lagrangian associated with Equation (10) w.r.t. w and z, and a gradient ascent on the dual variable [START_REF] Boyd | Distributed optimization and statistical learning via the alternating direction method of multipliers[END_REF]. It leads to the following iterative updates of w and z (see [START_REF] Fortun | Fast piecewise-affine motion estimation without segmentation[END_REF] for a similar derivation with different data and regularization terms):

w k+1 = prox x (1-c(x))ρ l (I 1 ,I 2 ,•,x) z k - α k µ (11) z k+1 = prox λ 1 w k+1 + α k µ ( 12
)
α k+1 = α k + µ(w k+1 -z k+1 ) ( 13
)
Where prox f (x) = argmin y 1 2 xy 2 2 + f (y) denotes the proximity operator of a function f . The subproblem (Equation 11) is voxel-wise and quadratic, and it admits a simple closed form solution. The subproblem (Equation 12) is equivalent to a denoising operation with the regularizer (•), and it also has a closed form linear solution that can be computed efficiently in the Fourier domain. µ is the parameter associated with the quadratic penalty in the Augmented Lagrangian associated with Equation ( 10). The ADMM algorithm is derived from this Augmented Lagrangian and the update (Equations 11, 12) are its minimization w.r.t. w and z. Intuitively, µ controls how fast the constraint w = z is imposed through the optimization process. Thus, even if it is not strictly speaking a step size, it has a similar impact on the convergence speed. Notice that the ADMM framework is flexible enough to cope with different data and regularization terms with low computational cost. The requirement is to be able to design a splitting of the cost function such that the proximity operators of each iteration have computationally efficient solutions. Examples of admissible models comprise data terms based on the ℓ 1 penalty function or cross-correlation [START_REF] Vogel | An evaluation of data costs for optical flow[END_REF], and regularizations by total variation or Nuclear norm of the Jacobian [START_REF] Bostan | Improved variational denoising of flow fields with application to phase-contrast MRI data[END_REF].

Minimization w.r.t. c When w is fixed, the estimation of c amounts to a binary segmentation problem with Potts regularization:

c = argmin c x∈ λ 2 -ρ(I 1 , I 2 , w, x) c(x) + λ 3 (c). ( 14
)
We solve it with a graph-cut method [START_REF] Boykov | Fast approximate energy minimization via graph cuts[END_REF], which is able to find an exact solution with very low computational cost.

. Implementation details . . Pre-processing

Before applying the change detection framework, the input images require to be pre-processed as follows. First, images are corrected for bias field inhomogeneity using the N4 algorithm [START_REF] Tustison | N4ITK: improved N3 bias correction with robust B-spline approximation[END_REF]. Then, a global scaling of the intensity is performed in order to enforce the median value of the intensities inside the brainmask to be equal to 100. Images are then resampled to 1 mm isotropic resolution. These two steps aim at harmonizing all input images in terms of intensity range and spatial resolution. The follow-up scan is then rigidly registered on the baseline image using ANTs library [START_REF] Avants | A reproducible evaluation of ANTs similarity metric performance in brain image registration[END_REF] with default parameters and mutual information metric. Differential bias field inhomogeneity is corrected thanks to the method described in [START_REF] Lewis | Correction of differential intensity inhomogeneity in longitudinal MR images[END_REF], while considering a 21 × 21 × 21 median filter size.

. . Post-processing

The detection maps are post-processed by discarding the connected components smaller than 3mm 3 and the detections https://github.com/ANTsX/ANTs Frontiers in Neuroimaging frontiersin.org outside brain parenchyma (i.e., the union of gray and white matter). Notice that the binary change detection map can also be computed to reflect either positive or negative intensity changes only. Brain parenchyma masks are computed from T1-weighted images using the FAST method provided in the FSL library [START_REF] Zhang | Segmentation of brain mr images through a hidden markov random field model and the expectationmaximization algorithm[END_REF] or alternatively from FLAIR images using SAMSEG brain parcellation tool [START_REF] Cerri | A contrast-adaptive method for simultaneous wholebrain and lesion segmentation in multiple sclerosis[END_REF], in the case where no T1-weighted image is available.

. . Hyperparameter setting

Three hyperparameters have to be set in the proposed joint formulation (Equation 5): λ 1 , controlling the spatial regularization of the deformation field, λ 2 acting as a threshold for the map of intensity differences, and λ 3 , controlling the spatial regularization of the change map. Here, we suggest strategies to find out relevant parameters settings.

The value of λ 1 should be chosen to optimally estimate longitudinal brain atrophy, since it is the main source of brain deformation in MS. Thus, we consider a subset of 21 images from the dataset OASIS-3 (LaMontagne et al., 2019) that contains longitudinal Alzheimer and normal aging MRI data that exhibit various pattern of longitudinal brain atrophy. We determine the optimal value of λ 1 by selecting the one that leads to the best registration performance on this subset of OASIS-3. To this end, we derive a registration quality metric from the provided segmentation maps of brain structures obtained with Freesurfer. Concretely, the Dice score is computed for each structure between the segmentation maps of the baseline image and of the registered follow-up image. The global registration quality metric is then computed as the sum over all the regions of the median Dice score observed for each region. This procedure leads us to find λ 1 = 70 as an optimal value.

The values of λ 2 and λ 3 have to be set to find the best compromise regarding: (i) The expected intensity difference, (ii) the noise level that corrupts the images, and (iii) the spatial extent of the changes. Here, we suggest an approach to find out optimized setting for each of the two considered databases (see Sections 5.1.2, 3.2). In practice, λ 2 and λ 3 have been fixed to maximize the overall performance of the affine sequential approach (see Section 5.1.5) in terms of local Dice Similarity Coefficient (local DSC, see Section 5.1.4) for each dataset. Considering the local DSC ensures to focus on the ability of the detection scheme (Equation 14) to recover the detected changes while not being influenced by false positive detections that can occur in other parts of the brain. Considering the affine transformation model ensures that the registration step does not to alter the geometry of evolving regions. This procedure leads https://surfer.nmr.mgh.harvard.edu/fswiki/Samseg us to find λ 2 = 16 and λ 3 = 5 as optimal setting for LesjakDB dataset and λ 2 = 25 and λ 3 = 3 for MSSEG-2 dataset.

. . Convergence and stopping criteria

The iterations of the alternated minimization of Equation ( 5) and of the ADMM algorithm (Equations 11-13) are stopped when a stopping criterion is verified or when a maximum number of iterations is reached. The stopping criterion is a threshold on the norm of the relative changes between two consecutive iterations, and is set to 10 -3 for the alternated minimization and 2.10 -3 for ADMM. The maximum number of iterations is set to 5 for the alternated minimization and 300 for ADMM.

. Experimental evaluation . . Evaluation framework

In this section, we report results obtained on one synthetic dataset and two publicly available real patients datasets. The synthetic dataset offer the advantage to have an unambiguously defined ground truth change detection map, while controlling the amount of noise, bias field inhomogeneity and brain atrophy that corrupt the images. The real datasets are used to evaluate the proposed approach in conditions that are closer to the clinical routine, while considering different acquisition conditions and various pathological evolution. The first real patients dataset, denoted in the sequel as LesjakDB [START_REF] Lesjak | Validation of whitematter lesion change detection methods on a novel publicly available MRI image database[END_REF], is dedicated to assess the ability of methods to detect every kinds of MS lesion evolutions (shrinkage, growth, new and disappearing), whereas the second dataset, denoted in the sequel as MSSEG-2 (COM, 2021), only focus on the ability to detect new appearing lesions.

. . . Synthetic dataset

We evaluate the proposed method on T2-weighted synthetic volumes generated with the Brainweb simulator [START_REF] Cocosco | Brainweb: online interface to a 3D mri simulated brain database[END_REF], while considering the normal anatomical model (i.e., without lesion) and two multiple sclerosis anatomical models with moderate and severe lesion load. The images are simulated at a 1 mm 3 isotropic resolution (image size: 181 x 217 x 181) with bias field inhomogeneity (20%). To simulate realistic brain atrophy for the follow-up image, we applied a deformation field that has been estimated using a deformable registration [START_REF] Avants | Symmetric diffeomorphic image registration with cross-correlation: evaluating automated labeling of elderly and neurodegenerative brain[END_REF] from two T1-weighted MRI scans acquired 4 years apart of a patient suffering from MS that exhibits a significant brain atrophy evolution (in-house dataset). It should be noted that these real data images were first affinely registered onto the brainweb image to ensure the estimated deformable registration to be consistent with the underlying anatomy.

The visual inspection confirms that the simulated image exhibits a realistic atrophy pattern. Gaussian additive noise was added with a standard deviation fixed at 5% of the mean intensity in the brightest tissue (cerebrospinal fluid in the T2-weighted simulation). We consider several scenarios of simulated longitudinal acquisition that are summarized in Table 1.

.

. . Real dataset LesjakDB: All kinds of lesion evolution

LesjakDB dataset [START_REF] Lesjak | Validation of whitematter lesion change detection methods on a novel publicly available MRI image database[END_REF] is composed of 20 longitudinal MRI acquisitions of MS patients with two timepoints. The median time between the baseline and followup studies was 311 days, ranging from 81 to 723 days. Each MRI acquisition consists in a 2D T1-weighted, a 2D T2weighted and 2D-FLAIR sequences. Change detection was conducted on the FLAIR images only. The FLAIR image size is 256 × 256 × 49 with an anisotropic spatial resolution of 0.9 × 0.9 × 3 mm. Ground truth change detection maps are also provided, which were obtained from manual annotations done by two expert raters. We adjusted some of the ground truth annotations that did not match the real lesion changes. The annotated changes include appearing, growing, shrinking and disappearing lesions. Ground-truth detection maps are compared to binary detection maps that include both positive and negative intensity changes.

. . . Real dataset MSSEG-: Only appearing lesions

MSSEG-2 dataset (COM, 2021) is composed of 100 pairs of FLAIR MRI scans from MS patients acquired on various MR scanners. The provided ground-truth is limited to new appearing lesions, and was build from the consensus of manual annotations delineated by four experts. The dataset is separated into training (40 patients) and testing (60 patients) sets. Since the proposed approach does not require any training step, we consider the whole dataset for testing. However, we distinguish two subgroups of data, namely MSSEG-2-Change corresponding the 61 subjects that exhibit at least one new appearing lesion and MSSEG-2-NoChange corresponding the 39 subjects that do not exhibit any new appearing lesion. Since the provided groundtruth is limited to new appearing lesions, they are compared only to the positive binary change detection maps obtained with the different methods. Notice, that the proposed method framework does not discriminate appearing from evolving lesion. Consequently, lesion evolutions, which are not labeled in the ground truth detection maps, are erroneously considered as false positive detection, thus introducing a bias in some of the evaluation metrics.

. . . Metrics

We report four metrics to evaluate the performance of the methods to detect changes, namely the Dice Similarity Coefficient (DSC), the Positive Predictive Value (PPV), the True Positive Ratio (TPR) and the local DSC. Let TP, TN, FP, and FN be the number of voxels from estimated change detection map that correspond to True Positive, True Negative, False Positive and False Negative, respectively. The DSC is defined as:

DSC = 2TP/(2TP + FP + FN)
and reflects the overall good overlap between the detection map and the ground truth.

The PPV is defined as:

PPV = TP/(TP + FP)
and reflects the proportion of relevant detections among all the detected changes.

The TPR is defined as:

TPR = TP/(TP + FN)
and reflects the proportion of the ground-truth changes that have been detected. The local DSC correspond the DSC computed on a restricted area defined as the dilation with a 4-voxel radius spherical structuring element to the ground truth. This metric enables us to focus the evaluation on the local spatial accuracy of the detection method

In addition to the voxel-wise metrics, we also report lesion-wise metrics, namely the Lesion True Positive Ratio (L-TPR) and the Lesion Positive Predictive Value (L-PPV). These metrics have been evaluated thanks to the animaSegPerfAnalyzer validation tool while considering the same hyperparameters as in [START_REF] Commowick | Objective evaluation of multiple sclerosis lesion segmentation using a data management and processing infrastructure[END_REF].

Since all these metrics are not relevant for data that do not exhibit any changes, we consider in that specific case FIGURE Evolution of DSC across iterations of the alternating optimization scheme of the joint approach on the synthetic dataset (blue: Lesion appearance, simulated atrophy, orange: Lesion growth, simulated atrophy).

the number of detected connected components as well as the volume of detected changes to characterize the false positive detections.

. . . Variants used for comparison

To demonstrate the benefits of the proposed joint modeling, we consider three variants of the change detection framework:

• joint: the proposed joint change detection and registration method described in Section 3.

• sequential: The sequential counterpart of the proposed method, which successively performs deformable The median ± the median absolute deviation (MAD) computed over all subjects are reported for each metric. The MSSEG-2-Change inverse experiment consist in swapping the baseline and follow-up images to evaluate the ability of the methods to detect disappearing lesions. The bold values indicate the highest scores among the three methods.

registration and change detection. For the two steps, we use the same model and optimization algorithms as in substeps of the joint approach described in Section 3.3.

• affine: The sequential approach where the deformable registration has been replaced by affine registration, which corresponds to the most common case. The affine registration was estimated using ANTs library [START_REF] Avants | A reproducible evaluation of ANTs similarity metric performance in brain image registration[END_REF] with default parameters and mutual information metric. Then, the thresholding and https://github.com/ANTsX/ANTs smoothing of the change map routine follows model (Equation 14).

. . Results

. . . Synthetic dataset

First, a qualitative visual comparison of the three methods is provided in Figure 2 for the lesion appearance with atrophy scenario. The affine method succeeds to detect almost all the lesion areas, but it suffers from false positive detection around the ventricles due to brain atrophy. Both the sequential and the joint methods compensated for brain atrophy deformation since none of them exhibit false detections around the ventricles. However, the sequential method failed to detect the whole lesion areas due to the overcompensation of lesion changes. This limitation is overcome by the joint approach that succeeds to detect the entire lesion areas.

A quantitative comparison of the three methods under four scenarios is provided in Table 2. First, we consider the appearance of lesion without atrophy. Unsurprisingly, this scenario is the most favorable for the affine method since there is no geometric difference to compensate. The sequential approach

FIGURE

Correlation between the performance of the three methods and the ground-truth evolving lesion load (cm ) of the subjects of LesjakDB.

yields to significantly lower values of DSC and local DSC. This is due to the lesion over-compensation effect, as supported by the observed low TPR value (i.e., lack of sensitivity) and high PPV value (i.e., high specificity). Finally, the joint approach overcomes the shortcoming of the sequential approach and have performances similar to the affine method, with a slight tendency to underestimate the detected area. Similar observations can be made about the second scenario involving lesion growth without atrophy.

The conclusions are drastically different for the two scenarios involving simulated atrophy. The performance of the affine method is significantly hampered by the numerous false detections due to the atrophy. This is illustrated by the significant decrease of the DSC and PPV values compared to the cases without atrophy, while the TPR and local DSC values are less modified. The sequential approach succeeds to compensate for the simulated brain atrophy, as highlighted by the high PPV value, but still underestimates the changes to detect, as indicated by the low TPR value. The joint approach clearly outperforms the two previous approaches in terms of detection accuracy, as objectified by the significantly higher DSC value.

The behavior of the joint approach can be monitored through the iterations of the alternating optimization scheme (see Figure 3). We can see that the DSC increases across the iterations, and the convergence is reached in a few iterations. Concerning the computational burden of the joint approach, it is about 24min on one single core (Intel(R) Xeon(R) Gold 6130 CPU @ 2.10GHz) for an experiment on the synthetic dataset (image size: 181 x 217 x 181).

FIGURE

Qualitative comparison of the binary change detection maps obtained with the sequential (purple) and joint (green) approaches as compared to the ground truth (underlying transparent red) on one selected subject from the MSSEG-dataset.

. . . LesjakDB

First, a qualitative visual comparison of the three methods is provided in Figure 4. We can draw similar conclusions as for the synthetic dataset (see Figure 2). The affine method demonstrates a high sensitivity (i.e., the lesion evolution is well detected) but a lack of specificity (i.e., numerous false positive detections are detected around the ventricles and in the posterior part of the cortex). Conversely, the sequential method has high specificity but lacks sensibility. The joint approach provides the best visual results, thus illustrating its ability to achieve both high sensitivity and high specificity. The quantitative evaluation shown in Table 3 (first row) and in Figure 5 confirms the conclusions of the visual analysis. The high sensitivity of the affine method is objectified at the voxel level by a statistically significantly higher TPR than the two other methods. At the lesion level, all the three methods exhibit similar L-TPR values, thus emphasizing their ability to detect the same amount of changing areas. Both the sequential and joint methods yield significantly higher PPV and L-PPV compared to the affine method, which illustrates their ability to reduce the number of false detections induced by brain atrophy at both voxel and lesion levels. This result highlights the benefit of using deformable registration in the context of MS lesion monitoring. The significantly lower TPR achieved by the sequential method compared to the joint method is the consequence of the lesion over-compensation effect. Finally, the joint approach significantly outperforms the two other approaches in term of voxel-wise global accuracy (see DSC).

Figure 6 highlights the variability of the performance of the methods across the subjects. It is interesting to notice that, although the performance of the detection methods greatly varies from one subject to the other, the ranking among the three methods appears to be highly consistent across the subjects. When investigating for the factors that may explain the observed variability, it appears that the volume of the ground-truth seems to play a prominent role: the larger is the volume to detect, the better is the performance of the change detection algorithm (see Figure 7).

. . . MSSEG-

Similarly as for the synthetic and LesjakDB datasets, the qualitative visual comparison of the two approaches based on deformable registration highlights the lack of sensitivity of the sequential method due the lesion over-compensation effect (see Figure 8). The resulting change detection map (purple) is too small compared to the ground-truth (underlying transparent red) due to the deformable registration that significantly shrinks the lesion. With the joint approach, the shape of the lesion is almost preserved in the warped follow-up image and the change detection map (green) matches almost perfectly the ground-truth.

The quantitative evaluation on the subset MSSEG-2-Change is reported in the second row of Table 3 and in the upper part of Figure 9.

The fact that both the sequential and joint approaches lead to significantly higher PPV values as compared to the affine approach advocates the use of deformable registration to reduce the number of false detections. The benefit of considering the joint over the sequential approach to overcome the lesion overcompensation effect is clearly demonstrated by the significantly higher TPR and local DSC values obtained with joint method.

It is also interesting to notice that the lesion overcompensation effect does not affect the special case of disappearing lesion. Indeed, when registering an image without lesion on a image with a lesion, the dissimilarity in the area of the disappearing lesion cannot be corrected by the transport of intensity of the registration (this is in fact only the case for non symmetric image registration method, see [START_REF] Noblet | A topology preserving non-rigid registration method using a symmetric similarity function -application to 3-D brain images[END_REF] for further explanations). To illustrate this phenomenon, we consider the MSSEG-2-Change inverse experiment (see the third row of both Table 3 and the bottom part of Figure 9) that consist in swapping the baseline and the follow-up image, so that the ground-truth now correspond to disappearing lesions. The same conclusion can be drawn from the DSC, PPV, and TPR as compared to the MSSEG-2-Change experiment. The most interesting point concern the local DSC that focuses the evaluation on the disappearing lesion. In that case, there is no significant difference any more between sequential and joint approaches contrary to the MSSEG-2-Change experiment, showing the absence of lesion over-compensation effect in the specific scenario of detecting disappearing lesions.

Note that all the results presented above in this section are evaluated on the 61 subjects of MSSEG-2-Change (i.e., subject presenting at least one new appearing lesion). Indeed, the presented metrics cannot be computed anymore for the 39 subjects of MSSEG-2-NoChange since the ground-truth change detection map is empty. This is why we only report the volume of detected changes for this subset of MSSEG-2 (see Figure 10). We can notice that both sequential and joint approaches lead to significantly lower volume of detected changes as compared to the affine, which appears in line with previous findings that support the use of deformable registration to reduce the number of false detections. Also note that the joint method yields consistently to slightly higher volume of detected changes as compared to the sequential method. This is also the consequence of the lesion over-compensation effect that affects the sequential appproach.

. Conclusion and perspectives

We have presented a method that unifies registration and change detection for the analysis of longitudinal brain MRI. It is based on the joint modeling of these two tasks as the minimization of a single objective function, for which we have developed an efficient alternating optimization method. The proposed approach has been evaluated in the context of the follow-up of multiple sclerosis lesion, which requires deformable registration to capture characteristic brain atrophy, but also with the potential caveat to shrink appearing lesions. In this context, the conventional sequential detection pipeline leads to large detection inaccuracies around new appearing lesions. We have demonstrated on simulated and real data that the proposed joint approach is able to combine the ability of deformable registration to correct brain atrophy, and a good preservation of the lesions shape to ensure accurate change detection.

The implementation presented in this paper of the proposed joint model relies in fact on quite simple modeling assumptions. The versatility of the optimization approach opens the way for more sophisticated models that could be handled in the same framework. Alternative data fidelity terms such as cross correlation or mutual information, and regularizers such as total variation could be considered to potentially improve the performance of the method. Another perspective is to improve the convergence of the alternating optimization strategy to ensure a better robustness to local minima. To this end we could consider a fuzzy change detection map to turn its estimation into a continuous optimization problem. This would allow us to use more robust optimization approaches such as Proximal Alternating Linearized Minimization [START_REF] Bolte | Proximal alternating linearized minimization for nonconvex and nonsmooth problems[END_REF]. Finally, while we have addressed in this paper the registration problem, the unification principle could be extended to other steps of the change detection pipeline. In particular, the intensity normalization and the bias field inhomogeneity correction of the MRI acquisitions are crucial pre-processing tasks that are impacted by the presence of evolving lesions. Therefore, the integration of these two tasks in a single unified model would be a natural extension of the proposed framework. organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

FIGURE

  FIGUREQualitative comparison of the binary change detection maps obtained with the three methods on the synthetic dataset (lesion appearance with atrophy scenario).

FIGURE

  FIGUREQualitative comparison of the binary change detection maps obtained with the three methods and of the jacobian of the deformation field estimated with the sequential and joint approaches on one selected subject from the MSSEG-dataset. Hyperparameters: λ = (for sequential and joint methods), λ = , λ = . I : Baseline, I : Follow-up, sequential, joint, Ground-truth, and a ne.

FIGURE

  FIGUREBoxplots corresponding to the results summarized in Table for LesjakDB (N Subject = ). Statistical significancy is evaluated thanks to the Wilcoxon signed-rank test between each pair of methods while applying Benjamini/Hochberg FDR correction.

FIGURE

  FIGUREMetrics reporting the performance of the three methods for each subject of LesjakDB.

FIGURE

  FIGUREBoxplots corresponding to the results summarized in Table for both MSSEG--Change and MSSEG--Change inverse (N Subject = ). Statistical significancy is evaluated thanks to the Wilcoxon signed-rank test between each pair of methods while applying Benjamini/Hochberg FDR correction.

FIGURE

  FIGURENumber of connected components (left) and volume in cm (right) of the detected changes computed for the subjects of MSSEG--NoChange datasets (i.e., subjects without new appearing lesion). Statistical significancy is evaluated thanks to the Wilcoxon signed-rank test between each pair of methods while appling Benjamini/Hochberg FDR correction.

Figures 4

 4 Figures 4 show the Jacobian of the deformation fields obtained by the sequential and joint methods, respectively. The specific pattern characterized by the alternance of both high and low values of the jacobian (see areas highlighted by the red squares in Figure4) reflects the high local contraction and dilation induced by the deformation field to make the lesion disappear, thus explaining the lack of sensitivity of the detection results.The quantitative evaluation shown in Table3(first row) and in Figure5confirms the conclusions of the visual analysis. The high sensitivity of the affine method is objectified at the voxel level by a statistically significantly higher TPR than the two other methods. At the lesion level, all the three methods exhibit similar L-TPR values, thus emphasizing their ability to detect the same amount of changing areas. Both the sequential and joint methods yield significantly higher PPV and L-PPV compared to the affine method, which illustrates their ability to reduce the number of false detections induced by brain atrophy at both voxel and lesion levels. This result highlights the benefit of using deformable registration in the context of MS lesion monitoring. The significantly lower TPR achieved by the sequential method compared to the joint method is the consequence of the lesion over-compensation effect. Finally, the joint approach significantly outperforms the two other approaches in term of voxel-wise global accuracy (see DSC).Figure6highlights the variability of the performance of the methods across the subjects. It is interesting to notice that, although the performance of the detection methods greatly varies from one subject to the other, the ranking among the three methods appears to be highly consistent across the subjects. When investigating for the factors that may explain the observed variability, it appears that the volume of the ground-truth seems to play a prominent role: the larger is the volume to detect, the

  TABLE Simulated longitudinal acquisitions.

	Scenario	Baseline image	Follow-up image	Simulated atrophy
	Lesion appearance without atrophy	Normal	Moderate	No
	Lesion growth without atrophy	Moderate	Severe	No
	Lesion appearance with atrophy	Normal	Moderate	Yes
	Lesion growth with atrophy	Moderate	Severe	Yes

  TABLE Results computed on the synthetic dataset.

					.	/fnimg.	.
	Scenario	Method	DSC	PPV	TPR	Local DSC
		affine	0.830	0.782	0.885	0.830
	Lesion appearance no atrophy	sequential	0.684	0.921	0.544	0.684
		joint	0.814	0.887	0.751	0.814
		affine	0.766	0.635	0.964	0.770
	Lesion growth no atrophy	sequential	0.685	0.734	0.641	0.688
		joint	0.806	0.726	0.902	0.808
		affine	0.460	0.329	0.767	0.810
	Lesion appearance simulated atrophy	sequential	0.626	0.960	0.465	0.627
		joint	0.743	0.925	0.621	0.744
		affine	0.652	0.505	0.919	0.827
	Lesion growth simulated atrophy	sequential	0.753	0.869	0.664	0.754
		joint	0.847	0.833	0.861	0.848

The bold values indicate the highest scores among the four methods.

  TABLE Results computed on LesjakDB and MSSEG--Change datasets.

	DataSet	Method	Local DSC	DSC	PPV	TPR	L-PPV	L-TPR
	LesjakDB	sequential	0.424 ± 0.139	0.317 ± 0.117	0.293 ± 0.152	0.323 ± 0.125	0.088 ± 0.051	0.577 ± 0.197
		joint	0.501 ± 0.179	0.353 ± 0.144	0.323 ± 0.148	0.447 ± 0.207	0.103 ± 0.061	0.574 ± 0.208
		affine	0.626 ± 0.224	0.142 ± 0.165	0.081 ± 0.139	0.633 ± 0.269	0.015 ± 0.042	0.840 ± 0.271
	MSSEG-2-Change	sequential	0.520 ± 0.196	0.310 ± 0.178	0.298 ± 0.264	0.379 ± 0.176	0.095 ± 0.150	0.872 ± 0.307
		joint	0.579 ± 0.219	0.356 ± 0.208	0.336 ± 0.254	0.474 ± 0.222	0.111 ± 0.155	0.872 ± 0.304
		affine	0.626 ± 0.224	0.142 ± 0.165	0.081 ± 0.139	0.633 ± 0.269	0.015 ± 0.042	0.840 ± 0.271
	MSSEG-2-Change inverse	sequential	0.625 ± 0.217	0.348 ± 0.216	0.290 ± 0.244	0.550 ± 0.243	0.094 ± 0.151	0.977 ± 0.292
		joint	0.655 ± 0.237	0.378 ± 0.233	0.312 ± 0.250	0.619 ± 0.266	0.091 ± 0.164	0.947 ± 0.304

affine 0.539 ± 0.174 0.152 ± 0.087 0.086 ± 0.060 0.603 ± 0.223 0.022 ± 0.022 0.576 ± 0.195

Frontiers in Neuroimaging frontiersin.org

Funding

This work was funded by the Région Grand Est and Philips Healthcare.

Data availability statement

Publicly available datasets were analyzed in this study. This data can be found at: https://brainweb.bic.mni.mcgill. ca/selection_ms.html; http://lit.fe.uni-lj.si/tools.php?lang=eng; https://portal.fli-iam.irisa.fr/msseg-2/.

Author contributions

The proposed method was coded by ED. ED, DF, and VN equally contributed to the writing of the article and the interpretation of the results. SK provided a clinical expertise for the analysis of the results and the real annotated databases. All authors contributed to the article and approved the submitted version.

Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher's note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated


Sonja Kallio
email: sonja.kallio@insa-strasbourg.fr

Monica Siroux

A Review -Renewable energy based micro-cogeneration and hybrid energy systems

To reduce carbon and greenhouse gas emissions, the more efficient and environmentally friendly energy production in the building sector is required. The deployment of renewable energy based microcogeneration units in the decentralized hybrid energy systems is a part of the solution. The micro combined heat and power (micro-CHP), or co-generation, units produce simultaneously heat and electricity from a single fuel source at high efficiency and close to the consumption point. These units offer significant benefits: reduced primary energy consumption, reduced CO2 emissions, and avoidance of distribution losses due to central plant and network construction. The objective of this paper is to present a review of available renewable energy based micro-CHP systems and to focus on the biomass and solar based conversion devices. Finally, a novel hybrid renewable energy system is presented by coupling renewable energy sources, such as solar and biomass for micro-CHP.

Introduction

The building sector has a high impact on greenhouse gas emissions and final energy consumption in the European Union (EU) by being the largest energy end-use sector with a share of 41%. [START_REF] Doroudchi | Approaching net zero energy housing through integrated EV[END_REF] The micro combined heat and power (micro-CHP), or co-generation, units produce simultaneously heat and electricity from a single fuel source at high efficiency and close to the consumption point. [START_REF] Padinger | Best practise report on decentralized biomass fired CHP plants and status of biomass fired small-and micro scale CHP technologies[END_REF] Micro-CHP systems are operating in the building sector to produce on-site energy for space heating, domestic hot water and electricity demand. Due to the efficient decentralized production of heat and electricity, the overall system efficiency up to 85-90 % can be reached with the micro-CHP units. [START_REF] Martinez | Micro-combined heat and power systems (micro-CHP) based on renewable energy sources[END_REF] The micro-CHP systems can respond to the intrinsic requirements of the residential building sector, such as highly variable thermal and electrical demand loads. The highly variable loads require a short response time which can be provided by micro-CHP systems. [START_REF] Pereira | ORC based micro-cogeneration systems for residential application -A state of the art review and current challenges[END_REF] The European Parliament has defined the micro-cogeneration to be the units up to an electrical output of 50 kW. [START_REF] Martinez | Numerical Investigation of Energy Potential and Performance of a Residential Building-Integrated Solar -Chp System[END_REF] However, micro-CHP commonly refers to the units up to 15 kW of electrical power. [START_REF] Murugan | A review of micro combined heat and power systems for residential applications[END_REF] The renewable energy based micro-CHP systems are in the key role in reaching the primary energy and pollutant emissions reduction targets of the EU. [START_REF] Gonzá Lez-Pino | Modelling and experimental characterization of a Stirling engine-based domestic micro-CHP device[END_REF] In the literature, several review works have been conducted in terms of renewable energy based micro-CHP systems [START_REF] Murugan | A review of micro combined heat and power systems for residential applications[END_REF][START_REF] Modi | A review of solar energy based heat and power generation systems[END_REF]. In 2017, Martinez et al. [START_REF] Martinez | Micro-combined heat and power systems (micro-CHP) based on renewable energy sources[END_REF] published a review article of micro-CHP systems based on renewable energy sources. The work presented different micro-CHP conversion technologies and used fuels. Finally, they focused on the solar energy-based technologies and presented shortly the main micro-CHP systems that coupled renewable energy sources. In 2020, Cheekatamarla et al. [START_REF] Cheekatamarla | A Comprehensive Review and Qualitative Analysis of Micro-Combined Heat and Power Modeling Approaches[END_REF] presented a review article on the modelling of the micro-CHP system with a focus on combustion engine as a prime mover. They found out that the Annex 42 [START_REF] Beausoleil-Morrison | An Experimental and Simulation-Based Investigation of the Performance of Small-Scale Fuel Cell and Combustion-Based Cogeneration Devices Serving Residential Buildings[END_REF] model was the most used modelling approach.

This paper presents a review of the renewable energy-based micro-CHP systems. First, the available energy conversion devices are presented with renewable energy sources. As a second part, hybrid renewable energy systems are discussed, and a novel system is presented by coupling renewable energy sources.

Energy conversion devices for micro-CHP

The conversion devices for the micro-CHP systems can be divided into categories that are presented in Fig. 1. Fig. 1. The main conversion technologies for micro-CHP. [START_REF] Martinez | Micro-combined heat and power systems (micro-CHP) based on renewable energy sources[END_REF] The most commonly used technology is combustion based micro-CHP system. The external combustionbased systems have better fuel adaptability to use different renewable sources. [START_REF] Martinez | Micro-combined heat and power systems (micro-CHP) based on renewable energy sources[END_REF] The third category includes solar-based photovoltaic-thermal and fuel cell cogeneration. Additionally, thermoelectric generators (TEGs) can be counted to the third category. The TEG converts heat at high temperature into electrical power and low-temperature heat without moving parts or working fluid. [START_REF] Padinger | Best practise report on decentralized biomass fired CHP plants and status of biomass fired small-and micro scale CHP technologies[END_REF] The PVT technology is relatively new in the micro-CHP context.

In terms of micro-cogeneration, the main renewable energy sources are biomass, biofuel and solar. [START_REF] Martinez | Micro-combined heat and power systems (micro-CHP) based on renewable energy sources[END_REF] In Fig. 2 are presented the main sources of the biomass. Fig. 2. The different ways to process biomass for CHP. [START_REF] Padinger | Best practise report on decentralized biomass fired CHP plants and status of biomass fired small-and micro scale CHP technologies[END_REF] Figure 2 shows that the external combustion engines are more flexible in terms of solid biomass fuels. In terms of micro-CHP, the most used biomass fuels are wood pellets. In addition to fossil fuels, the internal combustion engines, gas turbines and fuel cells can be turned into a renewable energy source and be fuelled by biodiesel or -gas. The TEGs use hot gases from combustion or thermal gasification as a heat source.

This paper focuses on the following externally fired solid biomass-fuelled micro-CHP technologies: the Stirling engine, organic Rankine cycle, steam engine, and solar-based photovoltaic-thermal collectors.

Stirling engine

The Stirling engine (SE) belongs to the external combustion engines, and its operation happens in a closed and regenerative thermodynamic cycle. The SE consists of a piston, cylinder, displacer and regenerator. The hot flue gas from the external combustion process maintains the hot end of the cylinder at high temperature. The heat is removed to the working fluid (helium, hydrogen or air) by the hot end heat exchanger. Due to the expansion, the heated working fluid moves through the regenerator to the cold end of the cylinder and removes the heat to a sink. The cooled fluid is compressed by the displacer piston. At the same time, mechanical work is generated by the cyclic compression and expansion of the working gas acting on the piston. [START_REF] Martinez | Micro-combined heat and power systems (micro-CHP) based on renewable energy sources[END_REF][START_REF] Zabalaga | Performance analysis of a stirling engine hybrid power system[END_REF] The Stirling engines are classified into kinematic and dynamic types as shown in Fig. 3. [START_REF] Zare | Free piston Stirling engines: A review[END_REF] Fig. 3. The classification of the Stirling engines. [START_REF] Zare | Free piston Stirling engines: A review[END_REF] The SE driven micro-CHP systems are mainly based on the two principal types: kinematic or free-piston. On the market, ÖkoFEN provides biomass-fuelled free-piston SE micro-CHP systems. [START_REF] Padinger | Best practise report on decentralized biomass fired CHP plants and status of biomass fired small-and micro scale CHP technologies[END_REF] In Fig. 4 is shown a biomass-fuelled micro-CHP with an alpha type. Fig. 4. The biomass fuelled micro-CHP with an alpha type. [START_REF] Padinger | Best practise report on decentralized biomass fired CHP plants and status of biomass fired small-and micro scale CHP technologies[END_REF] From the residential application point of view, the SE based micro-CHP systems have following advantages: the simple design of the engine, low maintenance requirements, low noise and vibration, fuel flexibility, and low emission level because of the external combustion. [START_REF] Padinger | Best practise report on decentralized biomass fired CHP plants and status of biomass fired small-and micro scale CHP technologies[END_REF][START_REF] Martinez | Micro-combined heat and power systems (micro-CHP) based on renewable energy sources[END_REF][START_REF] Zabalaga | Performance analysis of a stirling engine hybrid power system[END_REF]

Organic Rankine cycle

The main components of the organic Rankine cycle (ORC) are evaporator, expander, condenser and feedpump. In the ORC an organic medium, such as Isopentane, Iso-octane, toluene or silicone oil, is used as working fluid. The produced heat from the combustion process is transferred to the evaporator through the heat transfer circuit (thermal oil or water). The high-pressure organic working fluid is vaporized in the evaporator and afterwards expanded to lower pressure in the circuit integrated turbine, which is connected to the generator to produce power. A heat sink recovers heat from the working fluid in the condenser after the expander. [START_REF] Padinger | Best practise report on decentralized biomass fired CHP plants and status of biomass fired small-and micro scale CHP technologies[END_REF][START_REF] Martinez | Micro-combined heat and power systems (micro-CHP) based on renewable energy sources[END_REF] The basic schematic of the ORC is presented in Fig. 5. The efficiency can be increased by adding a regenerator to the working fluid circuit and an economizer to the cooling circuit. [START_REF] Padinger | Best practise report on decentralized biomass fired CHP plants and status of biomass fired small-and micro scale CHP technologies[END_REF] Fig. 5. The basic schematic representation of the ORC. [START_REF] Pereira | ORC based micro-cogeneration systems for residential application -A state of the art review and current challenges[END_REF] The organic working fluid can be vaporized at lower temperatures and pressures compared to water. This enables the usage of low-temperature heat sources, such as biomass, waste heat and solar energy. [START_REF] Padinger | Best practise report on decentralized biomass fired CHP plants and status of biomass fired small-and micro scale CHP technologies[END_REF] In terms of residential building applications, the ORC micro-CHP systems have the following advantages: the fast response to the highly variable thermal load, lowcomplexity of the power cycle and low maintenance requirements. Additionally, the low activation temperatures between 75 -90 °C can be used with high efficiency. [START_REF] Padinger | Best practise report on decentralized biomass fired CHP plants and status of biomass fired small-and micro scale CHP technologies[END_REF][START_REF] Pereira | ORC based micro-cogeneration systems for residential application -A state of the art review and current challenges[END_REF] The Rankine cycle has been developed since 1980s and is well deployed in the large scale industrial CHP systems. [START_REF] Martinez | Micro-combined heat and power systems (micro-CHP) based on renewable energy sources[END_REF] However, in terms of micro-CHP systems, the main challenge of the ORC lies in the downscaling of the design and construction of the expander by maintaining still good performance. [2] Pereira et al. [START_REF] Pereira | ORC based micro-cogeneration systems for residential application -A state of the art review and current challenges[END_REF] presented a review on the ORC micro-CHP systems for residential applications and current challenges. They found that the focus should be in the ORC-evaporator design and how the primary energy is transferred to the power cycle to meet the variable thermal load. On the market, the Rank Company provides successfully microscale ORC systems with different fuels. Their micro-ORC system can be run with the inlet temperature range 90-120 °C and can work still efficiently with 50% partial load. [13]

Steam engine

The principle of the steam engine cycle is the same as for the ORC, but a steam piston engine is used to expand the steam. Figure 6 presents the operating principle of the steam piston engine. The steam runs to the engine and expands (picture a). The expansion forces the piston down (picture b) and the volume of the steam increases and pressure decreases. Finally, the piston moves back to the top, and the exhaust steam flows out of the cylinder for the heat recovery circuit. [START_REF] Padinger | Best practise report on decentralized biomass fired CHP plants and status of biomass fired small-and micro scale CHP technologies[END_REF] The main advantages of the steam engine are simplicity of the system, cheap design, long lifetime and fuel flexibility concerning biomass, biogas or solar energy. Additionally, it can operate with saturated steam and is highly insensitive to fluctuating steam quality.

However, the steam engine requires constant maintenance and is noisy. [START_REF] Padinger | Best practise report on decentralized biomass fired CHP plants and status of biomass fired small-and micro scale CHP technologies[END_REF][START_REF] Bouvenot | Dynamic model based on experimental investigations of a wood pellet steam engine micro CHP for building energy simulation[END_REF] Fig. 6. Operating principle of the steam piston engine. [START_REF] Padinger | Best practise report on decentralized biomass fired CHP plants and status of biomass fired small-and micro scale CHP technologies[END_REF] Bouvenot et al. [START_REF] Bouvenot | Dynamic model based on experimental investigations of a wood pellet steam engine micro CHP for building energy simulation[END_REF] investigated experimentally biomass-fuelled steam engine micro-CHP system presented in Fig. 7. The system was modelled and experimentally validated. The system reached the maximum electrical efficiency of 9 % and the global efficiency of 95% at full load operation. Fig. 7. Steam engine experimental set-up. [START_REF] Bouvenot | Dynamic model based on experimental investigations of a wood pellet steam engine micro CHP for building energy simulation[END_REF] Bouvier et al. [START_REF] Bouvier | Experimental characterisation of a solar parabolic trough collector used in a micro-CHP (micro-cogeneration) system with direct steam generation[END_REF] conducted an experimental characterisation of the solar micro-CHP system based on a SE. The solar parabolic trough collectors were used to produce steam for the engine, and the electrical power of 1.4 kW and thermal 18.3 kW were reached.

Photovoltaic-thermal

A photovoltaic-thermal (PVT) collector is a solar micro-CHP system which generates simultaneously heat and power from solar energy and the same installed area. It produces electricity by the PV module and useful heat by cooling the PV module with a coolant fluid, such as water. This leads to increased overall system efficiency but also to increased electrical efficiency due to the decreased operating temperature of the PV module. The PVT collectors have been widely studied in the literature [START_REF] Herrando | Technoeconomic modelling and optimisation of solar combined heat and power systems based on flat-box PVT collectors for domestic applications[END_REF][START_REF] Barbu | Analysis of a residential photovoltaic-thermal (PVT) system in two similar climate conditions[END_REF][START_REF] Kallio | Energy Analysis and Exergy Optimization of Photovoltaic-Thermal Collector[END_REF]. In Fig. 8 is presented a covered water-based PVT collector. The PVT collectors are well suited for the residential building sector for DHW production or space heating. On the market, the most common PVT collectors are flat plate type and air-or water-based that produce low-grade heat and electricity. They are glass-covered or coverless. The companies DualSun and Abora Solar are manufacturing successfully the high-efficiency flat plate water-based PVT collectors. With a solar array concentrator, the electrical and thermal efficiency of the PVT can be increased and the required installation area decreased. Additionally, the higher outlet temperatures of the cooling fluid can be reached which allows the wider range of applications. The company Naked Energy has developed a high energy-intensive PVT collector based on a vacuum tube and concentrator with fewer space requirements. They have just entered the market and are seen as a game-changer.

Hybrid renewable energy systems

The hybrid renewable energy systems (HRES) are defined as the combination of two or more renewable energy sources. [START_REF] Shivarama Krishna | A review on hybrid renewable energy systems[END_REF] The particularly interesting is the coupling the renewable energy based micro-CHP systems with each other, such as the PVT collectors with the biomass combustion. In this kind of hybrid system, the PVT collectors can be used to produce heat for DHW and electricity during the summer months and biomass combustion during the cooler period when heat demand is higher. Compared to a biomass micro-CHP alone, the proposed coupling can reduce primary energy use and CO2 emissions even more because of solar energy use.

Wang et al. [START_REF] Wang | Flexible PVT-ORC hybrid solar-biomass cogeneration systems: the case study of the university sports centre in[END_REF] studied thermos-economic feasibility of a hybrid solar-biomass cogeneration system based on flat plate type PVT collectors and biomassfuelled ORC micro-CHP system integrated with thermal storage for a sport-centre application. The system is shown in Fig. 9. They found that integration of the solar increased significantly the electrical energy production capability. Fig. 9. Working principal of the PVT-ORC micro-CHP. [START_REF] Wang | Flexible PVT-ORC hybrid solar-biomass cogeneration systems: the case study of the university sports centre in[END_REF] Kosmadakis et al. [START_REF] Kosmadakis | Experimental testing of a low-temperature organic Rankine cycle (ORC) engine coupled with concentrating PV/thermal collectors: Laboratory and field tests[END_REF] studied experimentally coupling of concentrating PVT collectors directly with the low-temperature ORC. Their tests revealed that the system can have sufficient efficiency; the coupling with PVT increases power generation, and the system is feasible.

The company ÖkoFEN provides residential hybrid renewable energy systems. The system, called Pellematic Smart XS, consists of the flat plate PVT collectors and high efficient biomass boiler with heat storage. The other provided hybrid system is called myEnergy365, which integrates a SE micro-CHP called Pellematic Condes_e, PV panels and energy storages. The produced energy is 100% renewable and enables 100% self-sufficiency. Both systems are equipped with the intelligent regulation system, which has a key role in the hybrid energy system. Replacing the PV panels by PVT panels in myEnergy365 system would increase the initial investment costs, but the energetic and environmental benefits should be further investigated.

Based on the conducted review, a novel hybrid renewable energy system is proposed. This highefficiency hybrid system consists of the ORC based micro-CHP unit fuelled by high energy-intensive PVT collectors and a biomass boiler. The benefits of the proposed system are: running the ORC is continues, 100% renewable energy and self-sufficiency, using PVT increases electricity production and reduces fuel consumption, biomass boiler ensures the heat and electricity production during less sunny periods. The system is suitable for residential use and does not require a high amount of direct solar radiation.

The proposed micro-scale system requires more investigation. The similar hybrid systems have been studied in the industrial scale by combining conventional Concentrating Solar Power (CSP) and biomass power plants. The results showed decrease in Levelized Cost of Electricity due to reduced solar field size and biomass costs. The plant capacity utilization was also increased significantly. [START_REF] Suresh | Modeling and analysis of solar thermal and biomass hybrid power plants[END_REF]

Conclusions

To reach the primary energy and emissions reduction targets of the EU, heat and power has to be produced with more efficient and environmentally friendly systems. The building sector is the largest energy enduse sector in the EU with a share of 41%. [START_REF] Doroudchi | Approaching net zero energy housing through integrated EV[END_REF] The micro-CHP systems can operate with a total efficiency of 75-95%. [START_REF] Martinez | Micro-combined heat and power systems (micro-CHP) based on renewable energy sources[END_REF] These systems are playing a key role to reduce emissions and energy use in the residential building sector.

Different technologies are available for renewable micro-CHP systems. In this review, the biomass and solar fuelled technologies are considered: Stirling engine, ORC, steam engine and PVT collectors. The literature shows that the Stirling engine and PVT micro-CHP systems are the most advanced in terms of residential applications. The ORC and SE systems have a benefit of using low-grade heat for the activation of the cycle and have a high potential for the residential applications. However, the initial investment costs should be reduced to ensure market penetration.

The hybrid energy systems are the only way to produce continuously 100% renewable energy and to reach 100% self-sufficiency with renewable energy. In conclusion, the literature and market situation shows that

Fig. 8 .

 8 Fig. 8. The water-based flat plate PVT panel. Source: DualSun

E3S Web of Conferences 294, 01004 (2021) https://doi.org/10.1051/e3sconf/202129401004 ICSREE 2021

E3S Web of Conferences 294, 01004 (2021) https://doi.org/10.1051/e3sconf/202129401004

hybrid renewable energy systems are attractive and already have an industrial reality.

In future work, as a first step, the coupling of the flat plate PVT collectors with the biomass-fuelled Stirling micro-CHP unit will be studied from energy, exergy and economic point of view. The hybrid renewable energy system will be extended with energy storages. As a next step, the proposed solar and biomass fuelled ORC-based micro-CHP system on the building sector will be studied.

The authors would like to thank Interreg V Rhin supé rieur for their support and funding of this research.


Dorin Botoc
email: dorinbotoc@yahoo.com

Monica Siroux

Alexandru Salceanu

Magnetic Refrigeration: emerging technology for sustainable refrigeration

This article presents the magnetic refrigeration technology that is based on the magnetocaloric effect, heating or cooling of a magnetic material when the applied magnetic field changes. The first part of the paper shows an introductory study of the evolution of this technology over the years, continuing with the presentation of the magnetic regenerator model, which is composed of two heat exchangers with parallel plates of magnetocaloric material, gadolinium and are separated by heat transfer agent flow channels. The benefits and advantages of this technology are described, in the context of the current stage of development.

Background and Introduction

The technology to produce artificial cold based on vapor compression was introduced on the market, on an industrial scale, more than 120 years ago, with small improvements along the way. However, it has reached a technological level where improvements and performance increases of these types of installations are unlikely. Satisfying the growing demands of refrigeration and air conditioning at the same time, the reduction of greenhouse gases has led to the advancement of research and evolution of magnetic refrigeration systems to produce artificial cold. Magnetic refrigeration is based on the magnetocaloric effect of magnetocaloric materials that have this property, which manifests itself as a change in temperature when this material is subjected to a controlled magnetic field. Magnetic properties of magnetic materials have been known since antiquity and are used in various ways to obtain benefits from magnetic fields. The density of the magnetic flux initiates the process of electromagnetic induction in the case of receiver and transmitter coils, phenomena that are widely applied. Normally, the magnetocaloric effect is the property of magnetic materials where the process of temperature change, heating/cooling of the magnetic material takes place. The effect depends on the magnetic variation when the magnetic field is applied to the magnetocaloric material. Theoretically this explanation is based on magnetic property, magnetic entropy, and temperature. The potential for temperature change of magnetic materials that have magnetocaloric effect is huge, obviously the variation of temperature can be obtained based on the properties of magnetic materials. This effect is practically reversible, and when the material is no longer under the action of the magnetic field, it returns to its original state. The magnitude of the temperature change depends in most cases on the strength of the magnetic field and to obtain maximum energy efficiency and minimize the operating costs of this installation, it is recommended that the source of the magnetic field use permanent magnets. Magnetic refrigeration uses as a working agent, solid magnetocaloric materials, namely: silicon, gadolinium, prandliu etc. These materials illustrate the unique and most strongly highlighted property, which is called the magnetocaloric effect, and is manifested by increasing and decreasing the temperature of that material when it is magnetized or demagnetized, respectively.

This process of obtaining low temperatures, lower than those of the environment has as main purpose, thermal conditioning, industrial refrigeration, obtaining low temperatures, lower than those of the environment.

The comprehensive analysis of the literature was performed in this study to understand the main developments in this field. Moreover, the phenomena were also explained in detail to develop a much clearer and a clearer idea. All research has led to the development and real understanding of magnetism, temperature, and the properties of magnetocaloric materials. In [START_REF] Pecharsky | Magnetocaloric effect and magnetic refrigeration[END_REF] it explained the magnetic effect of Gd at 294K, and another researcher experimentally demonstrated low-temperature magnetic refrigeration facilities, explained the basic mechanism of the magnetic cooling cycle, and his fellow researchers argued and explained the need to develop new thermodynamic cycles to explain with certainty the phenomena that take place in magnetic refrigeration systems. Several authors have reported in many publications on the association of the Bryton and Ericsson cycle with magnetic refrigeration, [START_REF] Kitanovski | New thermodinamic cycles for magnetic refrigeration[END_REF] AMR (Active magnetic regenerator) operating on the Brayton cycle resulted in the production of the highest cooling capacity, while regenerator operating on the Ericsson cycle is the most efficient, [START_REF] Kitanovski | New thermodinamic cycles for magnetic refrigeration[END_REF] they also mathematically explained the numerical aspects of simulation.In [START_REF] Wolf | Magnetocaloric effect and magnetic cooling near a field induced quantum critical point[END_REF] provided explanations based on the quantum of the magnetocaloric effect and proposed to increase the entropy in terms of the change in the magnetic field close to the quantum-critical points made magnetically by accurate and precise measurement of the calorimetric effect. The trend of research and publication in the field of magnetocaloric effect increased significantly until the end of the twentieth century due to the need for an energy efficient refrigeration and air conditioning system. The available scientific literature on the magnetocaloric effect can be divided into several types for specification, efficiency and relevance. These categories are based on materials, based on mathematics, experiment, numerical and thermodynamic.

The magnetocaloric effect used in magnetic refrigeration installations, has a wide use in this century of technology, namely, refrigeration systems, air conditioning, heat pumps, [START_REF] Kitanovski | Energy Applications of Magnetocaloric[END_REF] aeronautical installations with direct application in the field of medicine, automotive and installations magnetic refrigeration appliances, household, etc., [START_REF] Shirron | Applications of the magnetocaloric effect in single stage, multi stage and continuos adiabatic demagnetization refrigerators[END_REF]. Fig. 1 shows the magnetization and demagnetization of the magnetocaloric material in isothermal and adiabatic processes. Apply field ( adiabatic magnetization )

Remove field ( adiabatic demagnetization )

Fig. 1. The processes of magnetization and demagnetization of the magnetocaloric material, [START_REF] Botoc Dorin | The efficiency of magnetic refrigeration and a comparison with compressor refrigeration systems[END_REF] In [START_REF] Basso | Basics of the magnetocaloric effect[END_REF], explained the magnetocaloric effect that is established in thermodynamics; and the variation of the primary state is the magnetic moment which is implicitly related to the magnetocaloric effect and the variables as internal energy representing the increase of temperature. The entropy of the system is in fact the magnetic entropy of the magnetocaloric effect. The magnetic field H, applied from the outside leads to the adiabatic change of the temperature ∆Tad, being applied to a constant pressure of the heat transfer agent, [START_REF] Basso | Basics of the magnetocaloric effect[END_REF]. The thermodynamic demonstration of the magnetocaloric effect was made by Pecharsky, and the compression of the working agent being isothermal and the decrease of the system entropy takes place when applying the pressure and being equivalent to the magnetization of the material and the magnetic part. The working agent distends adiabatically and when the pressure decreases, the temperature decreases, and this step is equivalent to the demagnetization of the magnetocaloric material which is removed from the magnetic field H, [START_REF] Pecharsky | Giant Magnetocaloric Effect in Gd5(Si2Ge2)[END_REF]. The process is based on isothermal and adiabatic separation and the magnetic order can be seen in the figure above. With the adiabatic application of the magnetic field, there is a decrease in magnetic entropy. However, the total entropy of the entire system remains constant and as a result of this difference there is adiabatic growth, which can be explained as:

∆𝑇 = 𝑇 -𝑇 (1)
Similarly, after the isothermal application of the magnetic field, the temperature must remain constant, but nevertheless we have a decrease in the entropy of the whole system due to the magnetization effect. The difference in entropy is represented by the change in the magnetic entropy of the whole system. ∆𝑆 .

Boundary condition, modelling and simulation

Magnetocaloric refrigerators consist of a magnet, a regenerator made of a magnetocaloric material, a system that pumps a heat transfer fluid and two heat exchangers (cold (CHEX) and hot (HHEX) exchangers), [START_REF] Botoc Dorin | The efficiency of magnetic refrigeration and a comparison with compressor refrigeration systems[END_REF][START_REF] Tusek | A comprehensive experimental analysis of gadolinium active magnetic regenerators[END_REF][START_REF] Dorin | A regenerator used in the installation of magnetic refrigeration devices[END_REF]. The magnetic regenerator is the heart of the magnetic refrigeration system. We present here a magnetic regenerator 2D model. Fig. 2 shows the 2D model of the magnetic regenerator, which has a fluid channel length of 16,057 cm with a height of 3.1 cm. The geometry of the regenerator is as follows: the width and height are equal to 5.2 cm and 1.7 cm, respectively.

On either side of the regenerator are the cold (CHEX) and hot (HHEX) exchangers. The heat exchangers, CHEX and HHEX contain magnetocaloric material in the form of plates, and water was used as a heat transfer agent. The width and height of both heat exchangers are 1.8 cm and 2.5 cm, respectively, [START_REF] Tusek | Dynamic operation of an active magnetic regenerator (AMR): numerical optimization of a packed-bed AMR[END_REF][START_REF] Nakashima | A magnetic wine cooler prototyp[END_REF][START_REF] Tura | Permanent magnet magnetic refrigerator design and experimental characterization[END_REF].

The limit condition of the magnetic field used in this simulation is Ampere's law, while the condition of magnetization and demagnetization has been defined. The material used is solid in the x and y directions and the values of electrical conductivity and relative permeability were derived from the material parameters. The heat transfer between the solid material and the heat transfer agent takes place in conditions of identification of the fluid and the solid domain, and the temperature has been defined, at the initial value of 293.15 K. The normal inlet speed is 5 m / s and was applied at the input, and at the output the zero-speed limitation condition was defined, Fig. 3.

The proposed option offered by the solver is electromagnetic heating and isothermal conditions. The walls were modelled without limitation conditions. Maxwell equations can be used to represent the relationship between the magnetic field H, the temperature of the magnetocaloric material and its entropy, Ec.2.

( 𝜕𝑆(𝑇, 𝐻) 𝜕𝐻 ) = ( 𝜕𝑀(𝑇, 𝐻) 𝜕𝑇 ) (2)
For isothermal conditions, Ec. 2 can be integrated and obtained in the form of the equation below:

∆𝑆 (𝑇, ∆𝐻) = ( 𝜕𝑀(𝑇, 𝐻) 𝜕𝑇 )𝑑𝐻 (3)
Table 1. Properties of the material used in the simulation of COMSOL Multiphysics software. 3

Designation

Results and discussion

The five basic components of the magnetic regenerator system are modelled explicitly and implicitly, [START_REF] Ds Arnold | Design improvements of a permanent magnet active magnetic refrigerator[END_REF]. The component that includes the magnetocaloric material is the only one explicitly modelled, while the magnetization system, the heat exchangers, and the pumping system of the working agent are modelled by default, Fig. 2. The temperature range of the magnetic regenerator is defined as the average temperature difference of the heat transfer fluid, the outlet between the "hot" and "cold" areas and the cooling and heating powers generated are defined using the following equations: It should be noted that this study was conducted to simulate the thermal processes in the magnetic regenerator, using the boundary condition of Ampere's law which was governed by the magnetic field module based on heat transfer due to the magnetocaloric effect and as a resultincrease temperature of the magnetocaloric material, gadolinium.

As can be seen in the Fig. 4, there is magnetization nearby 𝑇 (temperature Curie) of the magnetocaloric material, gadolinium in the absence of the applied external magnetic field. The refrigeration power is calculated according to the following relation:

𝑄 ̇ = 1 𝑝 𝑀 ̇• 𝑐 , 𝑇 -𝑇 (𝑡) 𝑑𝑡 (6)
The best performances are obtained at a frequency of 0.5Hz with a fluid speed of 0.60. The COP of the active magnetic regenerator system is calculated with the formula below:

𝐶𝑂𝑃 = 𝑄 𝑊 + 𝑊 (7)
The COP (coefficient of performance) decreases with the decrease of the volume of the heat transfer agent. The simulation began with the magnetization of the magnetocaloric material, using the boundary condition of Ampere's law, which was governed by the magnetic field modulus and the next step was based on heat transfer due to the magnetocaloric effect, resulting in increased temperature of the magnetocaloric material, gadolinium, the heat is transferred to the heat transfer agent and the final stage results in the release of thermal energy, which is controlled by the laminar flow module.

Conclusions and perspective

This paper describes the benefits and advantages of magnetic refrigeration in the current context of the development stage. The second part presents a numerical model of the magnetic regenerator, which is the subject of this study and its operation is at ambient temperature. In the third part of the paper are successfully presented the results obtained, which validates the proposed model of the magnetic regenerator, which works based on the magnetocaloric effect.

Fig. 2 .

 2 Fig. 2. 2D model of the magnetic regenerator, used in COMSOL Multiphysics software modelling

Fig. 3 .

 3 Fig. 3. The mesh model of the magnetic regenerator used in the COMSOL Multiphysics software simulation.

. 4 .

 4 the results obtained in the MatLab environment are presented. The first figure represents the adiabatic temperature change of the gadolinium magnetocaloric material, used in the magnetic regenerator. The second figure shows specific magnetic entropy change, the third figure describes specific heat capacity while the last figure in this simulation shows the magnetization of gadolinium magnetocaloric material.

Fig. 4 .

 4 Fig. 4. The results obtained in the MatLab environment for adiabatic temperature change, change magnetic specific entropy, specific heat capacity and magnetization, [15].

Fig. 5 Fig. 5 .

 55 Fig. 5 Shows the evolution of the regenerator temperature, [15-17]. The temperature evolution is similar in size to that reported in the literature. The maximum temperature range reached by the model is shown and presented the temperature difference between CHEX and HHEX over a period of 220 second.

E3S Web of Conferences 294, 03001 (2021) https://doi.org/10.1051/e3sconf/202129403001 ICSREE 2021