Youssef Bouaziz

Eric Royer

Guillaume Bresson

Michel Dhome

Long-term localization with map compression based on solar information

In this paper we address visual based localization in outdoor environments where the appearance changes dramatically. Such environmental changes result in a substantial transformation of the visual information of the scene, producing a significant impact on the visual based localization performance. Hence, these changes can lead to major difficulties when associating data between the current image and the landmarks in the map. One solution for this problem is to keep adding landmarks to the map in order to cover various environmental conditions. However, this solution leads to a continued growth of the map, which in turn, will result in a costly and resource-intensive localization. In this paper we present a map management approach in which we exploit information related to the sun's position to compare resemblance between the traversals in the map and maintain a diverse map that incorporates a minimum amount of data and ensures a reliable localization in different environmental conditions. We evaluated our approach on a dataset that incorporates more than 100 sequences with different environmental conditions and we compared the obtained results with a state of the art approach.

I. INTRODUCTION

Visual based localization (VBL) has become a fundamental field in robotic applications and it represents an interesting alternative to laser-based systems since it can provide an accurate localization with inexpensive setup requirements. Therefore, VBL has drawn the interest of many researchers over the past few years and has seen a growing number of real-time applications such as autonomous driving, augmented reality, robot navigation.

In this paper, we are interested in real-time VBL in outdoor environments for autonomous shuttles. In such applications, shuttles are repeatedly traversing the same path at different times. This means that they are very likely to experience many different environmental conditions which can deteriorate the localization performance even when revisiting familiar places. Day-night transition is one of the most challenging cases for VBL in outdoor environments as it engenders enormous variations in luminosity and brightness that lead to a significant difference between images captured at different times of the day. These variations can result in major difficulties when associating data between the current image and the landmarks in the map. Autonomous shuttles must deal with such environmental changes in order to provide reliable long-term localization.

Building a map that covers all environmental conditions by continuously adding landmarks to this map can help improving localization performance in changing conditions. However, this will result, also, in a ceaseless growth of the map's size that is relative to the number of traversals (experiences) performed by the shuttle. This means that after several traversals, localization will require an immense storage space to store the map and high-end CPU to find matching points between the current image and the corresponding landmarks in a vast database. In other words, realtime long-term localization will be impossible after a certain number of localization sessions. Thus, a map update strategy is required to prevent such cases and to ensure reliable realtime long-term localization.

In our previous work [START_REF] Royer | Lessons learned after more than 1000 km in an autonomous shuttle guided by vision[END_REF], we employed an autonomous shuttle for three months, totaling nearly 1500 km of autonomous travel on an industrial site. During this operation, we experienced some difficulties for long-term navigation. One of the most challenging difficulties identified in this previous work is lighting changes over the day, where the tests demonstrated that variations in lighting (caused by changes in the direction of the sunlight and the sun elevation) have more impact than long-term changes of the environment on the localization performance. This has motivated us to exploit information related to the sun position to design a map management approach with the purpose of preventing the continuous growth of the map. The sun position information are used to determine which traversals to keep and which ones to remove from the map. Traversals with correlated sun positions coordinates are considered to have similar environmental conditions. Thus, our approach consists, in a first time, in computing the sun positions related to all the traversals in the map, then, in exploiting all those computed values to classify this map into relevant and irrelevant traversals to finally produce a compressed map that incorporates a minimum number of traversals with diverse environmental conditions.

Our approach is employed on an experience-based mapping system [START_REF] Lébraly | Fast calibration of embedded non-overlapping cameras[END_REF], [START_REF] Royer | Lessons learned after more than 1000 km in an autonomous shuttle guided by vision[END_REF] that is based on key-frames and local features and is fundamentally similar to several open-source frameworks like ORB-SLAM [START_REF] Mur-Artal | Orb-slam: a versatile and accurate monocular slam system[END_REF] and Maplab [START_REF] Schneider | maplab: An open framework for research in visual-inertial mapping and localization[END_REF]. In our experiments, we use Harris corner detector [START_REF] Harris | A combined corner and edge detector[END_REF] for extracting key-points which are matched with ZNCC -Zero-mean Normalized Cross-Correlation -computed on 11×11 pixel windows around each key-point. However, our method can still be applied in the same way using other descriptors. We evaluated our approach on a new dataset recorded on our vehicle that we make available to the community and which is called IPLT1 (Institut Pascal Long-Term) dataset. We also compared the results obtained with our approach with the results obtained by another state of the art approach.

II. RELATED WORK

Considerable efforts have been made in VBL in static environments or with few minor changes, but it is only recently that localization in dynamic environments with changing environmental conditions has been addressed. Achieving reliable lifelong navigation under such environments is one of the biggest challenges for VBL.

Many works [START_REF] Murillo | Experiments in place recognition using gist panoramas[END_REF], [START_REF] Milford | Seqslam: Visual route-based navigation for sunny summer days and stormy winter nights[END_REF] have addressed this challenge by using image based matching techniques since traditional feature-based comparison methods have shown weaknesses in changing conditions. Murillo and Kosecka [START_REF] Murillo | Experiments in place recognition using gist panoramas[END_REF] proposed to improve localization in such environments by describing places with global descriptors computed on the entire images. However, this requires an exhaustive search in the map to find matches and recognize places. This means that in large-scale environments, such an operation will be very costly and will require high-end CPUs to achieve real-time performance. In order to improve localization with global images descriptors, Milford and Wyeth [START_REF] Milford | Seqslam: Visual route-based navigation for sunny summer days and stormy winter nights[END_REF] enhanced global image descriptors performance by introducing a sequential images matching technique called SeqSLAM. This technique consists of matching a sequence of current images with sequences of images in the database. SeqSLAM has significantly improved the performance of global images descriptors under harsh conditions, but it exhibits major sensitivity to changes in viewpoint.

Deep learning was employed in some approaches like [START_REF] Naseer | Semantics-aware visual localization under challenging perceptual conditions[END_REF], [START_REF] Sarlin | From coarse to fine: Robust hierarchical localization at large scale[END_REF], [START_REF] Piasco | Learning scene geometry for visual localization in challenging conditions[END_REF] to ameliorate localization performance in dynamic environments. However, these methods require high-end GPUs to achieve real-time performance and cannot provide a 6DoF pose estimation. Recently, several luminance invariant descriptors such as LIFT [START_REF] Yi | Lift: Learned invariant feature transform[END_REF], LUIFT [START_REF] Diaz-Escobar | Luift: Luminance invariant feature transform[END_REF], SOSNet [START_REF] Tian | Sosnet: Second order similarity regularization for local descriptor learning[END_REF] and REST [START_REF] Shoman | Real-to-synthetic feature transform for illumination invariant camera localization[END_REF] were proposed to improve accuracy of matching features in outdoor environments where brightness is subject to important changes. Such descriptors can achieve better performance in dynamic environment than SIFT [START_REF] Lowe | Distinctive image features from scale-invariant keypoints[END_REF] and SURF [START_REF] Bay | Surf: Speeded up robust features[END_REF] and they can be employed with other map management approaches to further improve VBL robustness under challenging conditions.

In their recent works, Bürki et al. [START_REF] Bürki | Appearance-based landmark selection for visual localization[END_REF] addressed the landmarks retrieval issue in changing conditions. They have employed information retrieval techniques from document retrieval community to fetch the landmarks that are related to the current environmental conditions. Similarly, Mac-Tavish et al. [START_REF] Mactavish | Selective memory: Recalling relevant experience for long-term visual localization[END_REF] inspired a landmark retrieval approach from recommender systems. They have proposed a collaborative filtering approach that recommends experiences according to the current environmental conditions.

In order to control the map's growth and to ensure a lifelong navigation, Muhlfellner et al. [START_REF] Mühlfellner | Summary maps for lifelong visual localization[END_REF] proposed a map management approach where they are scoring landmarks according to the number of distinct localization sessions in which they appear. Afterwards, the landmarks with the lowest scores are removed in the offline map maintenance operation. Bürki et al. [START_REF] Bürki | Map management for efficient long-term visual localization in outdoor environments[END_REF] have proposed an approach in which they have combined a landmark retrieval technique with a map management operation. In this approach, the retrieved landmarks will be updated if they lead to a successful localization, otherwise, they will be replaced by new landmarks.

In this work, we propose a map management strategy with the aim of reducing the size of the map. This strategy depends only on solar information related to each localization session. The results show that localization on a map produced by our approach was more reliable than localization on a map produced by Muhlfellner et al.'s approach [START_REF] Mühlfellner | Summary maps for lifelong visual localization[END_REF].

III. METHODOLOGY

This section addresses our proposed map update strategy. In this approach we aim to maintain a reliable map with a fixed size throughout the frequent traversals. The core idea of our approach is to produce a map that incorporates a minimum number of traversals ( N ) and able to operate in different environmental conditions. N is the number of traversals to be maintained in the map after carrying out the map management. Different values of N were chosen in the experiments section to evaluate the efficiency of our approach in different cases.

The methodology of our approach is described in the Figure 1. After each localization session, we test if the number of traversals in the map (N ) is greater than the predefined number N . If it is the case, our map management algorithm will be executed offline to reduce the size of the map. N > N means that the map contains an additional

SLAM

Fig. 1: A diagram illustrating the map management process. A new traversal is localized and added to the existing map. After finishing localization and mapping with SLAM, the map management algorithm is executed offline to check whether the map needs to be compressed or not (if N > N ). If the answer is yes, the algorithm has to act in two parts. Firstly, it has to compute the similarities between the N traversals in the map (N = N + 1). Secondly, it uses these similarity information to classify all the traversals and pick out the one that needs to be removed.

traversal (N = N + 1) which also means that it occupies additional memory space. In this case, our approach has to act offline to bound the size of the map, i.e. it has to decide which traversal has more resemblance to the others to eventually remove it.

As explained in the Figure 1, this mechanism incorporates two main parts, the similarity computation part in which we exploit information related to the sun position at each acquisition time and associate the traversals with their corresponding environmental conditions to finally compute the similarity between them with respect to the associated environmental conditions, and the classification part in which we use these similarity information to classify the traversals according to their importance and to remove the less important ones.

A. Similarity computation

In this part, we take advantage of the sun position to compare resemblance between the traversals in the map in order to determine which ones to keep in the map and which ones to remove. For each traversal in the map, we use the Astronomical Almanac's algorithm [START_REF] Michalsky | The astronomical almanac's algorithm for approximate solar position (1950-2050[END_REF] to compute the corresponding sun spherical coordinates: the Solar Elevation Angle (el) and the Solar Azimuth Angle (az) (see Figure 2). This algorithm takes as inputs the acquisition date/time and GPS latitude/longitude coordinates of each corresponding traversal. These inputs (date/time + GPS coordinates) are captured at the first frame of each traversal. This can be valid only when using short sequences like the case in this paper where we used ∼ 200 m length sequences, each one of them was recorded over ∼ 2 minutes. This means that there will be no important gap in the sun position between the start and the end of the traversal. In this paper, we tested two variants of the similarity matrix D. The first variant, D el , is built by computing the distances between the elevation angles of the traversals:

D el (i, j) = dist(el i , el j ), ∀i, j ∈ [1, N ] (1)
Where the function dist computes the angular distance between two solar elevation angles. The second variant, D az_el , is built using both of elevation and azimuth angles.

For each traversal i, we compute the Cartesian vector from the spherical coordinates:

⃗ u i =   cos az i cos el i sin az i cos el i sin el i   , ∀i ∈ [1, N ] (2)
After that, we compute the distance matrix D az_el by calculating the angle between each pair of the N Cartesian vectors:

D az_el (i, j) = |arccos( ⃗ u i • ⃗ u j )|, ∀i, j ∈ [1, N ] (3)
Figure 3 illustrates an example of the similarity matrices D el and D az_el . Both matrices were built using the same map that contains 5 traversals.

B. Classification and traversal removal

In this part, we exploit the matrix D (D el or D az_el ) built in the previous step to find out which traversal has to be removed. A hierarchical clustering [START_REF] Szekely | Hierarchical clustering via joint between-within distances: Extending ward's minimum variance method[END_REF] algorithm is used to classify the matrix D in order to select the traversal to remove. Figure 4 shows the different steps of the classification and the traversal removal: (c) In this step, we want to remove a traversal from the map while maintaining it as diverse as possible. To do so, we have to remove either the traversal i or the traversal j. Therefore, we search which one of them has more similarity to the other traversals by looking for for the minimum value in the i th and j th rows of the matrix while ignoring the diagonal and the elements with coordinates (i, j) and (j, i) (the hatched area in the figure). After finding the minimum, we remove its corresponding traversal (traversal 2 in the example). Lighting changes produced by day-night transitions can result in enormous visual gap between images recorded in the same place but at different times of the day [START_REF] Shoman | Real-to-synthetic feature transform for illumination invariant camera localization[END_REF]. This can create serious difficulties in local features matching especially using traditional features descriptors like SIFT and SURF. Our previous works [START_REF] Royer | Lessons learned after more than 1000 km in an autonomous shuttle guided by vision[END_REF] demonstrated that the hour of the day has more impact than long-term changes of the environment on the localization performance. This was demonstrated after performing localization daily over 3 months on a map recorded once at the beginning of the experiment. The results show that variations in the sun elevation and in the direction of sunlight are resulting in a more reduced number of correctly matched features than other factors like a moderate rain or a gap of 3 months.

Localization under night condition can be considered as a particular case since there are no lights provided by the sun after the astronomical night. Therefore, street lights are almost the only source of lighting which means that all sequences recorded in the night-time are visually similar. This means that keeping more than one night sequence in the map is pointless, and on the other side, removing all night sequences can lead to a serious problem and extremely impact the localization performance on night sequences. For this reason, we impose a new constraint to our algorithm: the traversal with the lowest solar elevation angle will not be nominated as the traversal to be removed (the night traversals have a negative solar elevation angle), any other night traversal will be removed. This will guarantee that the produced map will incorporate only one night traversal.

IV. EXPERIMENTS AND RESULTS

To evaluate the performance of our approach, we use our own collected dataset (IPLT dataset). This dataset contains currently more than 100 sequences and each one of them is about 200 m length. Our dataset incorporates multiple environmental conditions (day, night, dusk, rain, overcast. . . ) and in all the sequences the vehicle has followed the same path in a parking lot as shown in the Figure 5.

We were also interested in evaluating our approach on other widely used datasets like Oxford RobotCar dataset [START_REF] Maddern | 1 year, 1000km: The oxford robotcar dataset[END_REF] or NCLT dataset [START_REF] Carlevaris-Bianco | University of Michigan North Campus long-term vision and lidar dataset[END_REF], but unfortunately those datasets do not provide a great number of sequences traversing the same path which is primordial to test the efficiency of our work. Therefore, we made our dataset open to the community hoping that it will help researchers in this field to overcome this lack. Our dataset was created from recorded images of two gray-scale 100 • FOV cameras mounted on our experimental vehicle (one front and one rear camera) and wheel-odometry. We divided our dataset into 10 mapping sequences and 93 test sequences. The mapping sequences are used to build our reference map and the test sequences are used to evaluate the localization performance on the produced map with our map management approach. The 10 mapping sequences are constituted from 3 sequences recorded in a sunny condition, 3 in overcast, 2 in rain, 1 in dusk and 1 in night. In Figure 6, we present an overview of images from the 10 mapping sequences.

Fig. 6: An overview of images from the mapping sequences taken with the front camera. For each sequence we are indicating the acquisition date by a different color and symbolizing the environmental condition by a small icon (the colors are used to make easier reading Table I).

As mentioned in Section III, our approach has to limit the number of traversals used to build the reference map to a predefined number N . To do so, we keep performing SLAM on the mapping sequences, one by one, and at the end of each session we employ our map management approach to shrink the size of the map as described in Section III.

Considering that the order in which the sequences are added to the map can affect the resulting map, we tested our approach with 100, 000 different orders of the N traversals. This will result in reproducing several different N -session maps (We call an N -session map a map composed of N traversals) and the most reproduced N -session map, denoted by M * , will be used in our tests. We have obtained 4 different N -session maps using different configurations of our approach: M * el , M * az_el , ∼ M * el and ∼ M * az_el . The maps M * el and M * az_el are obtained by performing our approach on the two distance matrices D el and D * az_el respectively. We denote ∼ M * the map that has been obtained by performing our approach while imposing the constraint that ensures keeping a night sequence on the map as described in Section III-B.

In the results, we represent the average number of inliers observed in the test sequences along with the number of localization failures as criteria to evaluate the performance of the localization. Practically, we found that the localization can be considered as reliable when there are at least 30 points matched between the current image and the database, below this threshold, we count a localization failure. This is a conservative threshold to ensure the security of our autonomous vehicle [START_REF] Royer | Lessons learned after more than 1000 km in an autonomous shuttle guided by vision[END_REF].

In Figure 7, we present a comparison of the localization performance between the different N -session maps obtained by our approach (M * el , M * az_el , ∼ M * el and ∼ M * az_el ) and the map M 0 that incorporates all the 10 traversals of the mapping sequences. We also compared our approach with the map M SM generated by the Summary Maps approach proposed by Muhlfellner et al. [START_REF] Mühlfellner | Summary maps for lifelong visual localization[END_REF].

In their approach, Muhlfellner et al. are scoring the landmarks by the number of distinct sessions in which they appear and the ones with lowest scores are removed.

For each choice of N , the size of the maps generated by our approach are very close since they incorporate the same number of traversals. To ensure a fair comparison with Muhlfellner et al.'s approach, we remove from the map M SM the lowest scored landmarks until we get a map that has a similar size with the other maps. Figure 8 presents an overview of memory occupation and landmarks count of our We evaluated and compared the localization performance in the 93 sequences when using the different maps produced (M 0 , M * el , M * az_el , ∼ M * el , ∼ M * az_el and M SM ) with N = {3, 4, 5}. These sequences were manually classified into 5 different classes according to their corresponding environmental conditions: 13 sequences in "sun" class, 39 in "overcast", 12 in "rain", 17 in "dusk" and 11 in "night". The "global" class contains all 93 sequences while the "global \ {night}" class contains all the sequences excluding the night sequences (82 sequences).

For a better understanding of the results, we present in Table I The Summary Maps approach (M SM ) has shown a major weakness as we significantly decrease the size of the map. This weakness was more relevant with night sequences. This can be explained by the fact that the 10 mapping sequences include only one night sequence. Thus, landmarks of the night traversal will have a low score since they were seen only in one session. Consequently, these landmarks will be removed with the Summary Maps approach. In the opposite side, our approach has proved its capability of balancing experiences in the map. Therefore, it has managed to greatly decrease the size of the map while guaranteeing a reliable localization under different environmental conditions. According to Figure 7 and Figure 8, our approach was able to drastically reduce the size of the map (e.g. from 450MB to 185MB for N = 4) while ensuring a reliable localization under different conditions.

V. CONCLUSION

We have introduced a map management approach which computes and uses solar coordinates information to classify and reduce the number of traversals in the map in order to ensure a lifelong navigation. In the experiments part, we have compared some different variants of our method in multiple environmental conditions. The experiments have demonstrated that with our approach we were able to achieve TABLE I: Traversals included in each of the maps produced by our approach. The traversal names are colored to match the colors in the Figure 6.

Fig. 2 :

 2 Fig. 2: Solar Elevation Angle and Solar Azimuth Angle [22].

Fig. 3 :

 3 Fig. 3: Example of matrices D el and D az_el built with a map containing N = 5 traversals. The values of the matrices were normalized between 0 and 1, black color refers to 0 and white color refers to 1.

Fig. 4 :

 4 Fig. 4: Steps for classification and the selection of the traversal to remove.

Fig. 5 :

 5 Fig. 5: Example of some sequences from IPLT dataset which were recorded in a parking lot.

5 Fig. 7 :

 57 Fig. 7: Localization performance comparison on M0, M * el , M * az_el , ∼ M * el , ∼ M * az_el and the map MSM generated with the Summary Maps approach [19]. Each color refers to a map as indicated in the legend, and the boxes represent the mean +/-the standard deviation of inliers or localization failures on all the sequences of the corresponding class. Sub-figures (a), (b) and (c) represent the localization performance when choosing 3, 4 and 5 respectively as values for N .

Fig. 8 :

 8 Fig. 8: A curve in which we inspect the size of M * el , M * az_el , ∼ M * el ,

  the traversals included in each of the maps produced by our approach (M * el , M * az_el , ing to Figure7, the localization performance has globally improved when we increased the value of N . For all the choices of N , the localization performance on the map M * el with night sequences was very poor since this map does not include any night traversals. This is not the case for M * az_el which incorporates a night traversal. Both N -session maps ∼ M * el and ∼ M * az_el also include a night traversal due to the imposed constraint explained in Section III-B. According to the figure, localization under overcast and rainy conditions can be reliable even if the map does not include traversals of the same class (e.g. for N = 3, M * el contains a traversal from "rain" class while it is not the case for the other maps which is distinguishable by the inliers average. However, all of them have no localization failures except for M SM ). By increasing the value of N , the localization performance was improved in different classes for example on rainy sequences for M * az_el and ∼ M * az_el and on overcast sequences for M * el . Both maps ∼ M * el and ∼ M * az_el have shown similar performances on different classes, and they have both managed to ensure a reliable localization without any localization failure.

Université Clermont Auvergne, CNRS, SIGMA Clermont, Institut Pascal, F-63000 CLERMONT-FERRAND, FRANCE

[START_REF] Lébraly | Fast calibration of embedded non-overlapping cameras[END_REF] Institut VEDECOM, 23 bis allée des Marronniers, 78000 Versailles, France

We invite you to visit this link http://ipltuser:iplt_ro@ iplt.ip.uca.fr/datasets/ to download our dataset.

ACKNOWLEDGMENT

This work has been sponsored by the French government research program "Investissements d'Avenir" through the IMobS3 Laboratory of Excellence (ANR-10-LABX-16-01) and the RobotEx Equipment of Excellence (ANR-10-EQPX-44), by the European Union through the Regional Competitiveness and Employment program 2014-2020 (ERDF -AURA region) and by the AURA region.


Youssef Bouaziz

Eric Royer

Guillaume Bresson

Michel Dhome

A uniform distribution of landmarks for efficient map compression ⋆

Keywords: Visual-Based Navigation, Computer Vision for Transportation, Long-Term SLAM

In this paper, we address the challenge of visual-based localization in dynamic outdoor environments characterized by continuous appearance changes. These changes greatly affect the visual information of the scene, resulting in significant performance degradation in visual localization. The issue arises from the difficulty of mapping data between the current image and the landmarks on the map due to environmental variations. One approach to tackle this problem is continuously adding new landmarks to the map to accommodate diverse environmental conditions. However, this leads to map growth, which in turn incurs high costs and resource demands for localization. To address this, we propose a map management approach based on an extension of the state-of-theart technique called Summary Maps. Our approach employs a scoring policy that assigns scores to landmarks based on their appearance in multiple localization sessions. Consequently, landmarks observed in multiple sessions are assigned higher scores. We demonstrate the necessity of maintaining landmark diversity throughout map compression to ensure reliable long-term localization. To evaluate our approach, we conducted experiments on a dataset comprising over 100 sequences encompassing various environmental conditions. The obtained results were compared with those of the state-of-the-art approach, showcasing the effectiveness and superiority of our proposed method.

Introduction

Maps play a crucial role in self-driving applications, particularly in the context of high-precision localization using on-board cameras and reconstructed 3D points. However, the computational and memory limitations of mobile computing platforms pose significant challenges for real-time processing of these maps.

Therefore, effective map management techniques are essential to optimize processing power and memory usage while maintaining localization accuracy.

While substantial progress has been made in the visual SLAM community for static environments or those with minimal changes, the challenge of localization in dynamic environments with varying conditions has only recently been addressed. Achieving reliable lifelong navigation in such dynamic environments poses a major hurdle for visual SLAM. In this paper, our focus is on realtime visual-based localization in outdoor environments for autonomous shuttles. These shuttles traverse the same path repeatedly but encounter diverse environmental conditions, leading to potential degradation in localization performance, even at familiar locations.

In such scenarios, environmental changes create significant difficulties in associating data between the current image and landmarks in the map. It is imperative for autonomous shuttles to adapt to such changes in order to ensure reliable long-term localization.

One approach to improve localization performance under changing conditions is to build a map that encompasses all environmental variations by continuously adding landmarks. However, this leads to continual map growth, directly proportional to the number of shuttle traversals. Consequently, localization after multiple traversals becomes infeasible due to the excessive memory requirements to store the map and the computational demands to match points between the current image and an extensive landmark database. In essence, achieving long-term real-time localization becomes unattainable after a certain number of localization sessions.

The objective of this paper is to propose a real-time solution that addresses the map growth issue for long-term localization. To this end, we introduce a map management strategy to reduce the map size offline. The proposed strategy focuses on removing redundant data from the map, significantly reducing the computational cost of the SLAM algorithm in long-term scenarios.

Several recent works have also explored the map management problem, including studies by [START_REF] Biber | Dynamic maps for long-term operation of mobile service robots[END_REF], Mühlfellner et al. [START_REF] Mühlfellner | Summary maps for lifelong visual localization[END_REF], Dymczyk et al. [START_REF] Dymczyk | The gist of maps-summarizing experience for lifelong localization[END_REF], Burki et al. [START_REF] Bürki | Map management for efficient long-term visual localization in outdoor environments[END_REF], Krajník et al. [START_REF] Krajník | Persistent localization and life-long mapping in changing environments using the frequency map enhancement[END_REF], Halodová et al. [START_REF] Halodová | Predictive and adaptive maps for long-term visual navigation in changing environments[END_REF], among others. Some of these approaches reduce map size based on a scoring policy that suggests removing landmarks with the lowest observation rate [START_REF] Mühlfellner | Summary maps for lifelong visual localization[END_REF], [START_REF] Dymczyk | The gist of maps-summarizing experience for lifelong localization[END_REF]. Others eliminate landmarks associated with localization failures [START_REF] Bürki | Map management for efficient long-term visual localization in outdoor environments[END_REF] or those with a high incorrect matching rate [START_REF] Halodová | Predictive and adaptive maps for long-term visual navigation in changing environments[END_REF].

In this paper, we present an improvement to the Summary Maps approach proposed by Mühlfellner et al. [START_REF] Mühlfellner | Summary maps for lifelong visual localization[END_REF]. Their method scores landmarks based on the number of different localization sessions in which they appear and removes landmarks with the lowest scores in an offline process. Our improved version of Summary Maps introduces a new constraint on landmark removal, ensuring a uniform number of landmarks per traversal after map summarization (i.e., compression).

We evaluate our enhanced approach using the publicly available IPLT (Institut Pascal Long-Term) dataset [START_REF] Bouaziz | Over two years of challenging environmental conditions for localization: The iplt dataset[END_REF]. This dataset comprises 128 sequences recorded over a 16-month period, featuring a vehicle repeatedly following the same path around a parking lot with slight lateral and angular deviations. The IPLT dataset encompasses various environmental conditions, including changes in luminance, weather, seasons, and the presence of parked vehicles. Each sequence spans approximately 200 meters in length (see Fig. 1). We utilize this dataset to assess the performance of our approach and compare it against an existing state-of-the-art method. Our experiments demonstrate that our approach significantly improves localization performance, enabling successful localization even in challenging conditions.

Related work

In most real-world robotic scenarios, the ability to operate in dynamic and everchanging environments for extended periods is crucial. Simultaneous Localization and Mapping (SLAM) is a fundamental capability required in such scenarios. However, existing SLAM frameworks are often evaluated in static environments or scenes with minimal dynamic objects, such as moving people.

Recently, efforts have been made to extend the performance of localization in dynamic environments. Traditional feature-based comparison techniques in vision-based SLAM are considered unsuitable for long-term operations due to their vulnerability to changing conditions. As an alternative, Murillo and Kosecka proposed an image-based approach using the Gist representation of panoramic images to improve localization in dynamic environments [START_REF] Murillo | Experiments in place recognition using gist panoramas[END_REF]. Unlike local feature descriptors like SIFT and SURF, Gist is a global descriptor calculated using the entire image, representing an abstract scene representation. However, this approach involves an extensive search in the database to find the corresponding image, which becomes computationally expensive in large-scale environments. Milford and Wyeth introduced SeqSLAM, which enhances the performance of global image descriptors by matching sequences of images instead of individual images [START_REF] Milford | Seqslam: Visual route-based navigation for sunny summer days and stormy winter nights[END_REF]. Although SeqSLAM demonstrates impressive results on various seasonal datasets, it remains sensitive to viewpoint changes.

Mühlfellner et al. proposed Summary Maps (SM) as a map management technique, where landmarks are ranked based on the number of different localization sessions in which they appear, and low-score landmarks are removed in an offline process [START_REF] Mühlfellner | Summary maps for lifelong visual localization[END_REF]. However, SM has limitations in that landmarks in rarely visited areas receive low scores and are expected to be removed during map summarization. Dymczyk et al. addressed this bias towards frequently visited regions by designing a scoring policy that considers the expected number of trajectories to observe a landmark [START_REF] Dymczyk | The gist of maps-summarizing experience for lifelong localization[END_REF].

Krajník et al. developed a system that predicts the current state of the environment based on learned temporal patterns, constructing a new independent map in each run and integrating them into a spatio-temporal occupancy grid [START_REF] Krajník | Persistent localization and life-long mapping in changing environments using the frequency map enhancement[END_REF]. Bürki et al. improved appearance-based landmark selection by formulating a new ranking function [START_REF] Bürki | Map management for efficient long-term visual localization in outdoor environments[END_REF]. They introduced two types of sessions, "rich sessions" and "observation sessions," assuming that the environmental condition of a session performing worse than a predefined threshold is not covered in the map. They added new landmarks to cover the new encountered environmental conditions and employed offline map summarization to produce a reliable map with a fixed size. Halodová et al. extended previous work by presenting an adaptive map update scheme that removes or adds features based on their past influence on localization quality [START_REF] Halodová | Predictive and adaptive maps for long-term visual navigation in changing environments[END_REF]. They introduced an adaptive scoring policy that increments or decrements feature scores based on correct, incorrect, or unmatched matches. This strategy requires an accurate landmark retrieval technique to avoid penalizing incorrectly matched features caused by inaccurate retrievals. Bouaziz et al. proposed a keyframe retrieval technique that employs a ranking function considering factors such as Euclidean distance and underlying environmental conditions to search for keyframes with a higher number of inliers [START_REF] Bouaziz | Keyframes retrieval for robust long-term visual localization in changing conditions[END_REF].

In this paper, we present an improvement to the Summary Maps (SM) approach proposed by Mühlfellner et al. [START_REF] Mühlfellner | Summary maps for lifelong visual localization[END_REF]. Our enhanced version, named Uniform Summary Maps (USM), imposes a new constraint on landmark removal. The objective is to ensure a uniform number of landmarks in each traversal (sequence) after map summarization (compression).

Methodology

In this section, we introduce our map management approach, which extends the work proposed by Mühlfellner et al. [START_REF] Mühlfellner | Summary maps for lifelong visual localization[END_REF] called Summary Maps (SM). We have re-implemented SM on our mapping framework.

Mühlfellner et al. [START_REF] Mühlfellner | Summary maps for lifelong visual localization[END_REF] defined a landmark scoring policy that assigns scores to landmarks based on the number of different localization sessions in which they appear. Landmarks observed in multiple sessions receive high scores, indicating their value for localization. Conversely, landmarks with low scores are considered irrelevant. The Summary Maps approach utilizes these scores to summarize the map in an offline process conducted after each localization session, removing the least significant landmarks. An inherent limitation of this approach is that landmarks in rarely visited areas are assigned low scores since they are infrequently observed. Consequently, they are more likely to be removed during the map summarization process. Dymczyk et al. [START_REF] Dymczyk | The gist of maps-summarizing experience for lifelong localization[END_REF] referred to this issue as the bias towards regions that were more frequently visited and aimed to address this problem. They enhanced the scoring policy proposed in [START_REF] Mühlfellner | Summary maps for lifelong visual localization[END_REF] by considering the expected number of trajectories required to observe a landmark.

However, we have identified another limitation of the Summary Maps approach, which is somewhat similar to the aforementioned bias. This limitation can be defined as a bias towards more experienced environmental conditions. It occurs when a set of sequences with similar environmental conditions is combined with a sequence having a distinct environmental condition in the same map. For example, a collection of daytime sequences incorporated into a map with one nighttime sequence. In such cases, landmarks observed under the odd environmental condition receive low scores from the scoring policy due to their infrequent observations compared to the landmarks from the other sequences. Consequently, these landmarks are filtered out during the map summarization step. Table 1 presents an example of the bias towards more experienced environmental conditions that arises when summarizing a map containing 10 sequences using the Summary Maps approach. This map was constructed using the 10 sequences depicted in Figure 4 and includes only one nighttime sequence (the odd sequence). The table demonstrates the compression of the map using different compression ratios (1, 1.5, 2, 3, 5, and 10) with the Summary Maps approach. Compressing the map with a compression ratio r entails sorting all landmarks based on their scores and removing the lowest-scoring landmarks, representing 100 * (1 -1/r) percent of the total (e.g., removing the 50% lowest scored landmarks for r = 2). The Table shows the number of landmarks observed in each traversal after compressing the map using the Summary Maps approach at different compression ratios. The second column (r = 1) represents the initial map with no compression (100 * (1 -1/1) = 0%).

The last row represents the total number of landmarks on each map.

According to the table, a significant number of landmarks from the 7 th traversal (corresponding to the nighttime sequence) were removed after compressing the map. This demonstrates that the Summary Maps approach excludes landmarks observed in rarely experienced environmental conditions. This can pose a serious problem when localizing using nighttime sequences on the compressed map, as most of the nighttime landmarks have been filtered.

Therefore, in this paper, we propose an improvement for the Summary Maps technique called Uniform Summary Maps (USM). The goal of USM is to ensure a uniform distribution of landmarks across all traversals after map summarization. To achieve this, we introduce a constraint that imposes the distribution of a uniform number of landmarks across traversals. The algorithmic procedure for implementing USM is explained in Algorithm 1.

To further illustrate the execution of the algorithm and the effectiveness of our improved version of Summary Maps, we provide an example and compare the results with the original Summary Maps approach. Figure 3 presents an execution example of our algorithm on a map with N = 4 traversals. Each traversal l is associated with a specific number of landmarks denoted by n l land . The objective is to remove a total of n tot = 260 landmarks from the map while ensuring a uniform distribution across the traversals. Compute the number of landmarks n l land observed on each traversal l, with l ∈ [1, N ]

Algorithm 1 Uniform Summary Maps

7:

Sort the n l land landmarks of each traversal l according to the scoring policy 8:

Compute the highest number of landmarks: nmax ← max({n In the first iteration, the algorithm computes the number of landmarks n rem to remove from the traversal with the highest number of landmarks, which is the 4 th traversal.

In the second iteration, it is observed that there are two traversals, S = 2, 4, with the highest number of landmarks. Consequently, the algorithm calculates the number of landmarks to be removed n rem , and removes n rem /|S| landmarks from each traversal in S.

In the third iteration, the set S contains three traversals. The algorithm calculates the number of landmarks to be removed in this iteration as n diff ×|S| = 150. However, this value exceeds the total number of landmarks to be removed, which is n tot = 120. To ensure that we do not remove more landmarks than n tot , we use the statement min(n diff × |S|, n tot ).

Table 2 presents the results of compressing the map using our improved version of Summary Maps, while Table 1 displays the results obtained with the original Summary Maps approach.

The comparison in Table 2 reveals that after compressing the map with different compression ratios, the number of landmarks remains uniform across the different traversals. This demonstrates the effectiveness of our proposed technique in achieving a balanced distribution of landmarks while reducing the overall number of landmarks in the map.

Experiments and results

To evaluate the performance of our approach, we conducted experiments using the IPLT dataset [START_REF] Bouaziz | Over two years of challenging environmental conditions for localization: The iplt dataset[END_REF]. The IPLT dataset is a publicly available dataset that provides a diverse range of sequences for benchmarking visual localization algorithms. The dataset consists of over 100 sequences, each approximately 200 meters in length, captured under various environmental conditions including day, night, dusk, rain, and overcast. In all sequences, the vehicle follows a consistent path within a parking lot, as shown in Figure 1.

Although we acknowledge the importance of evaluating our approach on other widely used datasets such as the Oxford RobotCar dataset [START_REF] Maddern | 1 year, 1000km: The oxford robotcar dataset[END_REF] or NCLT dataset [START_REF] Carlevaris-Bianco | University of Michigan North Campus long-term vision and lidar dataset[END_REF], these datasets do not offer a substantial number of sequences traversing the same path, which is crucial for testing the effectiveness of our work. Therefore, we cannot rely on these datasets for our evaluation.

The IPLT dataset used in our study was generated using recorded images from two grayscale cameras mounted on an experimental vehicle. Each camera has a field of view of 100 • . In addition to the image data, there is also wheel odometry information for localization purposes. The dataset includes 103 sequences captured at different times of the day and under various weather conditions, such as rain, sun, and overcast. To introduce additional challenges, some sequences feature lateral and angular deviations. Out of these 103 sequences, we utilized 10 sequences to construct the global map, while the remaining 93 sequences were reserved for evaluation and analysis. We evaluated our approach on the IPLT dataset using the created global map which incorporates 10 traversals. In Figure 4, we present an overview of images from the 10 mapping sequences.

We conducted the evaluation using two different compression ratios: r = 3 and r = 2, where r represents the amount of landmark removal from the map. For instance, for r = 3, we removed 2/3 of the landmarks from the map. To assess the performance, we measured the average number of inliers observed in each test sequence, as well as the average number of localization failures per kilometer, which served as evaluation criteria.

We determined that reliable localization could be achieved when at least 30 points were matched between the current image and the database. Below this threshold, we considered it a localization failure. We have set a conservative threshold based on a study conducted by Royer et al. [START_REF] Royer | Lessons learned after more than 1000 km in an autonomous shuttle guided by vision[END_REF] to ensure the safety of the autonomous shuttle. This threshold, which we have adopted, is considered conservative and helps maintain a high level of security during the localization process.

To evaluate the impact of map compression on localization performance, we compared the localization performance on the initial global map, denoted as M 0 , which consisted of the 10 traversals, with the performance on two compressed maps: M SM generated using the Summary Maps approach [START_REF] Mühlfellner | Summary maps for lifelong visual localization[END_REF], and M U SM generated using our improved approach. In Figure 5, we present the average number of inliers per image and the average number of localization failures per kilometer observed during re-localization on these maps. In our experiments, we use Harris corner detector [START_REF] Harris | A combined corner and edge detector[END_REF] for extracting key-points which are matched with ZNCC -Zero-mean Normalized Cross-Correlation -computed on 11×11 pixel windows around each key-point. However, our method can still be applied in the same way using other descriptors. For each sequence, we provide the acquisition date and represent the environmental condition using a small icon.

The test sequences were manually classified into five different classes: "sun," "overcast," "rain," "dusk," and "night," with the "global" class containing all 93 testing sequences.

The Summary Maps approach (M SM ) exhibits a notable weakness when it comes to reducing the map size. This weakness is particularly more relevant in night sequences. The limitation arises from the fact that the 10 mapping sequences depicted in Figure 4 comprise only a single night sequence. As a result, landmarks observed during the night traversal receive a lower score compared to others, making them more likely to be removed during the map summarization process employed by the Summary Maps approach. Consequently, the nocturnal localization performance is adversely affected. This effect is evident in both subfigures (a) and (b) of Figure 5, where the number of localization failures in the night class is significant (166.6 localization failures/km with r = 3 and 49 with r = 2).

Conversely, our proposed technique, referred to as USM, effectively addresses this issue by ensuring a balanced representation of landmarks in the map after compression. As depicted in the figure, USM successfully increases the overall number of inliers and reduces the number of localization failures per kilometer, particularly during nighttime scenarios. This improvement in localization performance demonstrates the efficacy of our approach.

Discussion

Comparing the performance of our approach with existing state-of-the-art techniques poses a significant challenge. The difficulty arises from a fundamental conceptual difference between our approach and other techniques, wherein each state-of-the-art method is designed and applied within a specific mapping framework that utilizes distinct feature representations [START_REF] Bürki | Appearance-based landmark selection for visual localization[END_REF], [START_REF] Magnago | Effective landmark placement for robot indoor localization with position uncertainty constraints[END_REF]. This discrepancy in feature representation makes direct comparisons challenging, especially when comparing approaches applied to different types of SLAM frameworks, such as filter-based SLAM versus keyframe-based SLAM.

The variations in feature representation introduce complexities in evaluating and benchmarking different approaches. Each mapping framework has its own set of assumptions, algorithmic choices, and performance metrics. As a result, the performance evaluation of one approach within a specific framework may not be directly comparable to the evaluation of another approach in a different framework.

While comparing our approach with specific state-of-the-art techniques may be challenging, it is still valuable to highlight the unique aspects and contributions of our approach within the context of our chosen mapping framework. By focusing on the performance and evaluation metrics relevant to our framework, we can assess the effectiveness and efficiency of our approach compared to baseline methods or previous versions of our own approach. Additionally, conducting extensive experiments and providing thorough quantitative and qualitative analyses can help demonstrate the strengths and limitations of our approach in a more comprehensive manner.

In future research, it would be beneficial to establish standardized evaluation frameworks that can facilitate fair comparisons across different mapping approaches and frameworks. Such frameworks could define common datasets, performance metrics, and evaluation protocols, allowing for more direct comparisons between approaches. By adopting standardized evaluation practices, researchers can collectively advance the field and facilitate better understanding and comparison of different SLAM techniques.

Conclusions

In this study, we have proposed an enhancement to a state-of-the-art map management approach that leverages landmark observation information. Our improved technique introduces refinements to the scoring policy, allowing for more effective compression of maps while maintaining long-term localization performance. In our experiments, we conducted a comprehensive evaluation of our approach in various environmental conditions.

By comparing our improved technique with the original state-of-the-art approach, we have observed substantial performance gains. Our approach consistently outperformed the baseline method in terms of localization accuracy and robustness, even when operating with compressed maps. This highlights the effectiveness of our proposed enhancements and their positive impact on the overall localization performance.

Looking ahead, our future research endeavors will focus on further refining the scoring policy of our approach. We aim to incorporate additional information, such as past localization successes and failures, into the compression process. By considering a broader range of factors, we expect to enhance the efficiency and effectiveness of our map compression technique, ultimately improving the overall localization capabilities in challenging real-world scenarios.

Fig. 1 .

 1 Fig. 1. Example of 6 sequences recorded in a parking lot.

Figure 2

 2 Figure 2 illustrates the scoring function strategy proposed by Mühlfellner et al., as described in [14]. It presents an example of localization and showcases the landmarks observed in two different sessions, denoted as i and j.

Fig. 2 .

 2 Fig. 2. The landmark scoring policy proposed by Mühlfellner et al.. The figure illustrates a localization example where some landmarks (li,j) were observed in two different sessions (session i and session j) and accordingly, their scores were increased by the scoring policy.

1: Parameters: 2 :

 2 The total number of landmarks to remove: ntot 3: The number of traversals in the map: N 4: Steps: 5: repeat 6:

Fig. 3 .

 3 Fig.3. An example illustrating the execution mechanism of our proposed algorithm. The yellow colored rows designate nmax and the green colored ones designate nsmax.

Fig. 4 .

 4 Fig. 4. An overview of images from the mapping sequences taken with the front camera.For each sequence, we provide the acquisition date and represent the environmental condition using a small icon.

(a) r = 3 (b) r = 2 Fig. 5 .

 325 Fig. 5. Localization performance comparison on M0, MSM and MUSM . Each color refers to a map as indicated in the legend, and the boxes represent the mean +/the standard deviation of inliers or localization failures on all the sequences of the corresponding class. Sub-figures (a) and (b) represent the localization performance when choosing r = 3 and r = 2 respectively.

Table 1 .

 1 Bias towards more experienced environmental conditions in Summary Maps.

	Traversal	1 (no compression) 1.5	Compression ratio r 2 3	5	10
	1	140,524	113,922 100,371 75,087	50,304	20,817
	2	127,687	93,539	72,205	48,527	28,803	14,071
	3	149,065	83,821	52,025	34,912	18,913	9,324
	4	140,900	72,955	37,769	26,157	15,868	8,946
	5	122,122	97,989	86,896	55,751	29,495	14,022
	6	124,643	89,807	76,360	44,106	18,436	9,495
	7	72,044	41,333	28,777	16,310	5,092	2,709
	8	116,091	67,204	44,438	32,717	24,424	12,825
	9	127,972	96,797	78,262	51,610	34,322	16,692
	10	143,640	85,758	55,241	36,385	27,280	17,567
	total	1,264,688	843,125 632,344 421,562 252,937 126,469

Table 2 .

 2 Compressing the map with the Uniform Summary Maps.

	Traversal	1 (no compression) 1.5	Compression ratio r 2 3	5	10
	1	140,524	85,676	63,235	42,157	25,294	12,647
	2	127,687	85,676	63,235	42,157	25,294	12,647
	3	149,065	85,676	63,235	42,157	25,294	12,647
	4	140,900	85,676	63,235	42,157	25,294	12,647
	5	122,122	85,676	63,235	42,157	25,294	12,647
	6	124,643	85,676	63,235	42,157	25,294	12,647
	7	72,044	72,044	63,235	42,157	25,294	12,647
	8	116,091	85,676	63,235	42,157	25,294	12,647
	9	127,972	85,676	63,235	42,157	25,294	12,647
	10	143,640	85,676	63,235	42,157	25,294	12,647
	total	1,264,688	843,128 632,350 421,570 252,940 126,470

⋆ This work has been sponsored by the French government research program "Investissements d'Avenir" through the IMobS3 Laboratory of Excellence (ANR-10-LABX-16-01), by the European Union through the Regional Competitiveness and Employment program 2014-2020 (ERDF -AURA region) and by the AURA region.


Adrien Paschier

Alexandre Destere

Caroline Monchaud

Marc Labriffe

Pierre Marquet

Jean-Baptiste B Woillard
email: jean-baptiste.woillard@unilim.fr

Tacrolimus population pharmacokinetics in adult heart transplant patients

Keywords: tacrolimus, population pharmacokinetic, bayesian estimator, probability of target attainment, heart transplantation

 What is already known about this subject:  Tacrolimus is an immunosuppressant largely used in heart transplantation.

 Therapeutic drug monitoring is required to optimize the dose.  Only a few population pharmacokinetic models for tacrolimus have been developed in heart transplant patients 1-3 , while the others included mixed populations (HTX and other organs).

What this study adds:

 A 2 compartment with transit absorption population pharmacokinetic model was developed from full pharmacokinetics profiles in adult heart transplant patients (n=18, pk profiles= 47 and 546 concentrations).

 A bayesian estimator based on a limited sampling strategy (0h-1h-2h post administration of tacrolimus) was derived, yielding good predictive performances (AUC bias±SD = 2.72± 10.17% and imprecision = 9.9%).

 The probability of target attainment was performed for the AUC and the C0 to propose different starting doses depending on the CYP3A5 status, non expressors (0.11 mg.kg - 1 .12h -1 based on the AUC or the C0) or expressors (0.22 mg.kg -1 .12h -1 based on AUC and 0.29 mg.kg -1 .12h -1 based on C0).

Introduction

Tacrolimus is a narrow therapeutic index immunosuppressant, largely used in combination with mycophenolate mofetil in heart transplant patients [START_REF] Kobashigawa | Tacrolimus with mycophenolate mofetil (MMF) or sirolimus vs. cyclosporine with MMF in cardiac transplant patients: 1-year report[END_REF] . Tacrolimus is characterized by a large inter-individual variability [START_REF] Brunet | Therapeutic Drug Monitoring of Tacrolimus-Personalized Therapy: Second Consensus Report[END_REF] which renders mandatory therapeutic drug monitoring (TDM). The most frequently used marker for tacrolimus TDM is the trough concentration (C0) even if, its correlation with rejection is still a matter of debate [START_REF] Brunet | Therapeutic Drug Monitoring of Tacrolimus-Personalized Therapy: Second Consensus Report[END_REF][START_REF] Braithwaite | Identifying the association between tacrolimus exposure and toxicity in heart and lung transplant recipients: A systematic review[END_REF] . The interdose area under the curve (AUC) of tacrolimus is theoretically the best exposure marker and has been proposed for TDM [START_REF] Brunet | Therapeutic Drug Monitoring of Tacrolimus-Personalized Therapy: Second Consensus Report[END_REF] . No prospective randomized study has investigated the association between AUC or C0 and acute rejection in heart transplant recipients. A single study of 25 heart transplant patients treated with oral tacrolimus, showed that the AUC was significantly lower in rejection cases vs cases without rejection [START_REF] Undre | Pharmacokinetics of tacrolimus in heart transplantation[END_REF] . Nowadays, the C0 target in heart transplant patients used in the two first months post transplantation is from 15 to 20 µg.L -1 5 . Nevertheless, a study using a sample of 110 heart transplant patients has shown an increased risk of acute kidney injury within the first 2 weeks after transplantation for C0>15 µg.L - 1 8 . This led the second conference of consensus on Tacrolimus to suggest revising the C0 target. However, this has not yet been done. Currently, no AUC target has been defined in heart transplant patients.

The large variability in tacrolimus exposure is caused by several factors including the cytochrome P450 family 3 subfamily A member 5 (CYP3A5) status (with a two times lower dose requirement in CYP3A5*3 homozygous carriers [START_REF] Woillard | Tacrolimus Updated Guidelines through popPK Modeling: How to Benefit More from CYP3A Pre-emptive Genotyping Prior to Kidney Transplantation[END_REF] in comparison to carriers of at least one *1 allele (CYP3A5*1/*1 or CYP3A5*1/*3), food intake, the type of organ transplanted, the time-period post-transplantation, age and hematocrit [START_REF] Vanhove | Clinical determinants of calcineurin inhibitor disposition: a mechanistic review[END_REF] .

Currently, only a few population pharmacokinetic models (POPPK) dedicated to heart transplant patients have been developed [START_REF] Kirubakaran | Adaptation of a population pharmacokinetic model to inform tacrolimus therapy in heart transplant recipients[END_REF][START_REF] Gong | Population pharmacokinetic analysis of tacrolimus in Chinese cardiac transplant recipients[END_REF][START_REF] Han | Prediction of tacrolimus dosage in the early period after heart transplantation: a population pharmacokinetic approach[END_REF] . Other tacrolimus developed models also include lung [START_REF] Sikma | Unbound Plasma, Total Plasma, and Whole-Blood Tacrolimus Pharmacokinetics Early After Thoracic Organ Transplantation[END_REF] or liver and kidney [START_REF] Lu | Population pharmacokinetics of immediate-and prolongedrelease tacrolimus formulations in liver, kidney and heart transplant recipients[END_REF] transplant patients. In a recent work, Kirubakaran et al. [START_REF] Kirubakaran | Adaptation of a population pharmacokinetic model to inform tacrolimus therapy in heart transplant recipients[END_REF] built a POPPK based on sparse data (mostly trough concentrations) on the basis of a previously published POPPK model [START_REF] Sikma | High Variability of Whole-Blood Tacrolimus Pharmacokinetics Early After Thoracic Organ Transplantation[END_REF] to inform the effect of anti-fungal interaction [START_REF] Kirubakaran | Adaptation of a population pharmacokinetic model to inform tacrolimus therapy in heart transplant recipients[END_REF] on tacrolimus apparent clearance. None of them investigated the predicted probability of AUC target attainment using simulations to improve the first dose of tacrolimus. An accurate estimation of the interdose AUC can be obtained by combining a Maximum a posteriori Bayesian estimator (MAP-BE) derived from a POPPK model and a limited sampling strategy (LSS).

The aims of this study were: (i) to develop a POPPK model of tacrolimus in a population of heart transplant patients; (ii) to develop a MAP-BE to estimate individual PK parameters and tacrolimus exposure indices based on a limited sampling strategy (LSS); (iii) to estimate the probabilities of target attainment (PTA) for AUC and trough concentration (C0) using Monte-Carlo simulations and to propose recommendations for tacrolimus starting doses.

Material and methods

Patients and samples

Heart transplant patients of the PIGREC study receiving tacrolimus (Prograf® bid) were selected to build the model. All the patients underwent heart transplantation between 2007 and 2009 and they all received basiliximab for 3 days post transplantation. On the 4 th day, the association tacrolimus/mycophenolate mofetil was introduced. This multicentre trial, in which 7 French transplantation centres participated, was approved by the Limousin regional ethic committee and authorized by the French Drug Agency (PIGREC [Pharmacocinétique des Immunosuppresseurs chez les patients GREffés Cardiaques], EudraCT number N°2006-006832-23; ClinicalTrials.gov identifier NCT00812786). All the patients included gave their written informed consent. The initial dose of tacrolimus was 0.075 mg.kg -1 .day -1 . Forty-seven full-PK profiles (concentration data collected at T0, T20', T40', T60', T90', T2h, T3h, T4h, T6h, T8h and T9h + T12h) were collected from 18 adult heart recipients. Each patient underwent up to 4 post-transplant visits (P1=7 to 14 days: 128 concentrations, P2=1month: 126 concentrations, P3=3 month: 125 concentrations and P4=1 year after transplantation:167 concentrations). One additional blood sample was taken at D7-14 for pharmacogenetic analyses. The transplantation of any other organ before or during the PIGREC study was an exclusion criterion.

Assay

All the blood samples were analyzed in the Pharmacology, Toxicology and Pharmacovigilance Department of the Limoges University Hospital, using a validated turbulent flow chromatography-tandem mass spectrometry (TFC-MS/MS) method. Briefly, online extraction was performed at a high flow rate (1.25 ml min-1) on a Cyclone P ® , 50-mm particle size (50 ¥0.5mm I.D.) column (Cohesive technologies, MiltonKeynes, UK) in alkaline conditions using ammonium acetate and acetic acid as mobile phase. Chromatographic separation was performed in acidic conditions using a Propel C18 MS, 5 mm (50 ¥ 3.0mm I.D.) column (Cohesive technologies, Milton Keynes, UK) heated to 60°C, with a constant flow rate of 300 ml min -1 . Detection was performed using a TSQ Quantum Discovery MS/MS system (Thermo-Fisher, Les Ulis, France) equipped with an orthogonal electrospray ionization source and controlled by the Xcalibur computer program. MS/MS detection was performed in the positive ion, multiple reaction monitoring mode following two transitions for tacrolimus (m/z 821.5→768.6; m/z 821.5→786.4) and two for the internal standard ascomycin (m/z 809.3→756.4; m/z 809.3→564.4). The method was developed using ascomycin as the internal standard due to the similar molecular weight with the tacrolimus and the absence of deuterated or labeled tacrolimus at that time. This method was fully validated for tacrolimus determination in whole blood. The calibration curve used is a 1/x weighted quadratic regression to obtain the best fit across the calibration range based on the standard error of the fit and minimization of calibrator's bias. The lower limit of quantitation (LLQ) was 1 µg.L -1 and the calibration curves obtained from the LLQ up to 100 µg.L -1 yielded r 2 > 0.998. The method was found to be accurate and precise with a bias of -4.4% to 0.6% and a low coefficient of variation (CV) of -3.8% to 6.4% (CV intra-day = 14.23% and CV inter-day = 11.7 % at the LLQ). It is worth noting that all the concentrations were above the LLQ.

Genotyping

Patients' genotypes were characterized for cytochrome 3A5 (CYP3A5) rs776746 A/G (CYP3A5*3 allele) using validated TaqMan allelic discrimination assays on an ABI PRISM 7000 Sequence Detection System (Applied Biosystems, Courtaboeuf, France).

Missing data:

Missing data were imputed using the K-nearest neighbour approach using the "VIM" R package [START_REF] Kowarik | Imputation with the R Package VIM[END_REF] . The data imputations were only performed when missing data represented less than 20% of the data set. A Wilcoxon test was performed to determine if the distribution of the variables is affected by this method. Furthermore, some patients did not have a sample drawn at t12h. To circumvent this issue, the linearity of the elimination between 9-12 hours was checked after log transformation and in order to estimate the reference AUC using the trapezoidal rule, the T= 12h was then extrapolated using a mono-exponential decay from the last point available:

C(t) = C0 × e -Ke × t

Pharmacokinetic modelling

The database was randomly divided, based on the PK profiles, into a development dataset (38 PK profiles and 433 concentrations) and a validation dataset (9 PK profiles and 113 concentrations). The two populations were compared using a Wilcoxon test. The POPPK analysis was performed using a nonlinear mixed effect approach using the Stochastic Approximation Expectation-Maximization (SAEM) algorithm in MONOLIX 15 .

Structural model development

Several structural models were investigated to fit the data: one or two-compartment models with first-order elimination and with a lag time or transit compartments to describe the absorption phase. Models were parametrized in terms of apparent clearance (CL/F), apparent volume of distribution (Vd), mean transit time, transit rate constant and absorption rate. Interindividual variability was described using an exponential model. The covariance of the parameters was investigated during the modelling process. The best structural model and residual error model (comparison of additive, proportional or both) were selected based on the Bayesian information criterion (BIC) and the visual inspection of goodness-of-fit plots.

Covariate investigation

Covariates investigated on PK parameters were weight, age, hematocrit, CYP3A5, sex and the categorized time post-transplantation (7-14 days, 1-month, 3-month, 1 year).

Scatter plot and Pearson correlation coefficient were drawn between the Empirical Bayes

Estimates and the continuous covariates and boxplots and ANOVA for the categorical covariable (CYP3A5, sex and time post-transplantation). The plots of covariates vs ETA CL from the base model were drawn.

Covariates were introduced individually into the structural model following a forward inclusion and backward elimination procedure. The covariate that decreases the BIC was kept in the structural model. The clinical relevance of the covariates was also evaluated based on changes in the interpatient variability.

Categorical covariates were introduced as follows:

θ j = θ jTPV × 𝑒 θ COVi ×𝐶𝑂𝑉𝑖
Continuous covariates were tested as follows:

θ j = θ jTPV × ( 𝐶𝑂𝑉𝑖 𝐶𝑂𝑉𝑖 𝑚𝑒𝑑𝑖𝑎𝑛 ) θ COVi
θ j represent the value of the PK parameters j th , θ jTPV is the mean value of j th θ COVi a parameter estimated representing the effect of the i th covariate ( 𝐶𝑂𝑉𝑖 )

Internal validation

Goodness-of-fit plots were drawn for the final model. The final model was internally evaluated using the prediction-corrected visual predictive check (pcVPC) [START_REF] Labriffe | Population pharmacokinetics and Bayesian estimators for intravenous mycophenolate mofetil in haematopoietic stem cell transplant patients[END_REF] in the development and in the validation dataset.

Bayesian estimator

The final model developed was used as prior to develop a MAP-BE that could estimate tacrolimus AUC based on a 3-samples strategy. Several LSS using samples within the first 4 hours post-dose were investigated in the development set and compared based on the relative mean prediction error (rMPE = biais) and the root mean square error (RMSE = imprecision).

𝑟𝑀𝑃𝐸 (%):

∑ ( 𝑝𝑒𝑖 𝐴𝑈𝐶 𝑟𝑒𝑓 ) 𝑛 𝑖=1 𝑛 × 100 𝑅𝑀𝑆𝐸 (%) = √ ∑ ( 𝑝𝑒𝑖 𝐴𝑈𝐶 𝑟𝑒𝑓 ) 𝑛 𝑖=1 2 𝑛 × 100
n represents the number of pairs of estimated and measured AUC, AUCref is the calculated AUC using the trapezoidal rule and pei is the difference between the estimated and the reference AUC

As the sample at t=0h is routinely drawn for TDM, only LSS including this sample were selected.

Finally, the predictive performance of the MAP-BE based on the best LSS was tested in the validation dataset based on the RMSE and the rMPE in comparison to the reference AUC and the number of patients with a rMPE > 20%.

Simulation of dose regimens

The probability of target attainment (PTA) was determined from simulations performed in SIMULX-2021R1 17 : for both CYP3A5 genotype status (Expressor and Non expressor), 1000 simulations were drawn for different tacrolimus bid dose (2.5 mg, 5 mg, 7.5 mg, 10 mg, 15mg and 20mg) and the PTA was determined for different C0 targets (between 2.5 and 20 µg.L -1 ) and for different AUC targets (between 100 and 350 µg. h. L -1 ). To determine the AUC target in the early phase period, we used the regression equations between the C0 and the AUC developed by Saint Marcoux et al. [START_REF] Saint-Marcoux | Lessons from routine dose adjustment of tacrolimus in renal transplant patients based on global exposure[END_REF] even if we acknowledge that the correlation is not perfect (supplemental figure 1).

Nomenclature of targets and ligands

Key protein targets and ligands in this article are hyperlinked to corresponding entries in http://www.guidetopharmacology. org, the common portal for data from the IUPHAR/BPS Guide to PHARMACOLOGY [START_REF] Harding | The IUPHAR/BPS Guide to PHARMACOLOGY in 2018: updates and expansion to encompass the new guide to IMMUNOPHARMACOLOGY[END_REF] and are permanently archived in the Concise Guide to PHARMACOLOGY 2019/20. [START_REF] Alexander | THE CONCISE GUIDE TO PHARMACOLOGY 2019/20: Enzymes[END_REF][START_REF] Alexander | THE CONCISE GUIDE TO PHARMACOLOGY 2019/20: Transporters[END_REF] Results:

Patient's characteristics:

Baseline characteristics of the patients, overall and in the development and validation sets are reported in the Table 1. There were no significant differences between the 2 datasets.

Pharmacokinetics model:

Structural model:

A structural pharmacokinetics model with 2 compartments, transit absorption and a linear elimination best described the tacrolimus pharmacokinetics (supplemental Table 1). Based on one of our previous articles in renal transplant patients [START_REF] Woillard | Population pharmacokinetic model and Bayesian estimator for two tacrolimus formulations-twice daily Prograf and once daily Advagraf[END_REF] , the inter transit-compartment absorption rate (Ktr) was fixed at 5.74 h -1 to help the model to converge. The proportional error model was selected as it yielded the lowest BIC (with additive error model BIC: -3842; with proportional error model BIC : -3969).

Missing data:

There was no covariate with more than 20% of missing data (Sex= 4.2%, Age= 6.4%, CYP3A5= 0%, body weight = 8.8% and Hematocrit= 8.5%). The data imputation did not change the variable distributions as presented in supplemental Table 2 and supplemental Figure 2.

Covariate analysis:

The covariate significantly associated with the CL/F was the CYP3A5 status (supplemental Table 3) with a theta of 0.62. The addition on this covariate led to a decrease in the BIC by 19

(from -3969 to -3986).

𝐶𝑙/𝐹 = 𝑇𝑉𝐶𝐿 * 𝑒 𝑡ℎ𝑒𝑡𝑎 𝐶𝑌𝑃3𝐴5 × 𝐶𝑌𝑃3𝐴5

ThetaCYP3A5 (= 0.62) is the change in the apparent clearance in CYP3A5 expressors (CYP3A5 = 1 for expressors leading to EXP(0.62) and 0 for CYP3A5 non expressors leading to EXP(0)) ; TVCL -population mean value of apparent clearance; Cl/Fapparent clearance

The scatter plots and boxplots of the association between the covariates and the random effect of CL are presented in supplemental figure 3. Some models had similar BICs but the Relative Standard Error (%) of the kinetics parameters and the imprecision were too high.

The pharmacokinetics parameters of the final model with covariates are presented in Table 2.

The diagnostic plots for the final model are presented in Figure 1. There was an excellent correlation (r=0.98) between individual predictions and observations. The residuals of individual prediction and time were homogenously distributed around 0.

Model Evaluation

The pcVPC of the final model overlaid with the observations of the development and validation datasets are presented in Figure 2A and2B. The observed percentiles are in line with the predicted percentiles of the development dataset. In the validation set, even if they did not overlap perfectly, the observations percentiles were within the 95% predicted confident interval.

Bayesian estimation:

In the development dataset, the 0-1-2 hours post dose LSS was associated with the best performances and was selected for the evaluation in the validation dataset (Table 3). This LSS led to good performances: rMPE mean ± SD = 2.72 ± 10.17%; RMSE=9.9% (best and worst fit are presented in supplemental Figure 4). Bland Altman plot between the reference AUC and the MAP-BE LSS AUC is presented in Figure 3. No patient had a relative bias out of a +/-20% range.

Probability of target attainment as function of CYP3A5 status:

The results of the PTA for C0 and AUC are presented in Figures 4 and5, respectively. The target currently used for C0 in the early phase post heart transplantation is between 15-20 µg.L - 1 and for the AUC between 270-350 µg.h. L -1 5 . As these dose proposals are only valid for the starting dose of tacrolimus (afterwards TDM is used), only early targets post transplantation were investigated. Based on the PTA with C0 targets (considering a mean weight in France of 69 kg) [START_REF]Population totale par sexe et âge au 1er janvier 2020, France métropolitaine -Bilan démographique[END_REF] , a dose of 7.5 mg (0.11 mg.kg -1 ) for a patient CYP3A5 non expressor and 20 mg (0.29 mg.kg -1 ) for a patient expressing CYP3A5 has to be administered for 85 % of the simulated patients to reach the early phase targets (Figure 4). Similarly, for the AUC targets, a dose of 7.5 mg (0.11 mg.kg 1 ) of tacrolimus twice a day in the non expressor group and at least 15 mg (0.22 mg.kg -1 ) in the expressor group has to be administered for 85 % of the simulation to reach the early phase targets (Figure 5).

Discussion:

In this work, we developed a parametric POPPK model to describe the pharmacokinetics of tacrolimus after heart transplantation. The final model had two-compartments with transit absorption phase and first order elimination. Transit compartment models have already been used to describe the tacrolimus absorption [START_REF] Woillard | Population pharmacokinetic model and Bayesian estimator for two tacrolimus formulations-twice daily Prograf and once daily Advagraf[END_REF][START_REF] Monchaud | Population pharmacokinetic modelling and design of a Bayesian estimator for therapeutic drug monitoring of tacrolimus in lung transplantation[END_REF] while other models used a lag time [START_REF] Lu | Population pharmacokinetics of immediate-and prolongedrelease tacrolimus formulations in liver, kidney and heart transplant recipients[END_REF][START_REF] Alqahtani | Estimation of Tacrolimus Clearance in Saudi Adult Kidney Transplant Recipients[END_REF][START_REF] Andrews | A population pharmacokinetic model to predict the individual starting dose of tacrolimus in adult renal transplant recipients[END_REF] . The best LSS found in this work used 3 samples at 0, 1 hour and 2 hours post-dose. We acknowledge that the 0, 1 hour and 3 hours post dosing strategy also yields acceptable and similar results, but we chose to keep the best strategy. Several previous studies which developed LSS for tacrolimus were based on samples drawn at 0, 1 hour and 3 hours (in kidney or liver transplantation) [START_REF] Woillard | Population pharmacokinetic model and Bayesian estimator for two tacrolimus formulations-twice daily Prograf and once daily Advagraf[END_REF][START_REF] Monchaud | Population pharmacokinetic modelling and design of a Bayesian estimator for therapeutic drug monitoring of tacrolimus in lung transplantation[END_REF][START_REF] Riff | Population pharmacokinetic model and Bayesian estimator for 2 tacrolimus formulations in adult liver transplant patients[END_REF] . For example, in the Marquet et al. study [START_REF] Marquet | Clinical Pharmacokinetics and Bayesian Estimators for the Individual Dose Adjustment of a Generic Formulation of Tacrolimus in Adult Kidney Transplant Recipients[END_REF] in kidney transplant patients, the authors selected a LSS based on samples drawn at 0, 1 hour and 3 hours post dose but showed that the result with samples at 0, 1 hour and 2 hours post dose were similar.

Interestingly, in lung transplant patients a LSS based on 20 min, 2 hours and 4 hours has been previously selected showing that samples at 2 and 4 hours could be of interest in heart and lung transplant patients [START_REF] Monchaud | Population pharmacokinetic modelling and design of a Bayesian estimator for therapeutic drug monitoring of tacrolimus in lung transplantation[END_REF] .

The performance of the LSS was then evaluated in an external dataset obtained by data splitting and showed accurate performances in line with similar studies in kidney or lung transplantations (the bias and imprecision of our study are respectively of 2.7% and 9.9% vs. values between 0.1% and 2.4% for the bias or 7.7% and 9.8% for the imprecision in the other studies) [START_REF] Woillard | Population pharmacokinetic model and Bayesian estimator for two tacrolimus formulations-twice daily Prograf and once daily Advagraf[END_REF][START_REF] Monchaud | Population pharmacokinetic modelling and design of a Bayesian estimator for therapeutic drug monitoring of tacrolimus in lung transplantation[END_REF][START_REF] Medina-Aymerich | Limited sampling strategy to predict the area under the curve of tacrolimus in Mexican renal transplant pediatric patients receiving Prograf® or non-innovator formulations[END_REF] .

To help the model convergence we fixed the Ktr, based on a previous study carried out on renal transplant patients [START_REF] Woillard | Population pharmacokinetic model and Bayesian estimator for two tacrolimus formulations-twice daily Prograf and once daily Advagraf[END_REF] . Without fixing this parameter, the BIC was significantly increased (BIC= -3974).

A comparison to the PK parameters of Woillard et al. 2011 in renal transplant patients showed that in heart transplant patients, the apparent clearance was slightly decreased (from 0.37 to 0.20 L.h -1 .Kg -1 ) and the central compartment volume was strongly decreased (from 3.66 to 0.22 L.Kg -1 ) [START_REF] Woillard | Population pharmacokinetic model and Bayesian estimator for two tacrolimus formulations-twice daily Prograf and once daily Advagraf[END_REF] . Similarly, our apparent clearance is in line with the apparent clearance of other heart transplant patient models (0.27 L.h -1 .Kg -1 for Kirubakaran et al. [START_REF] Kirubakaran | Adaptation of a population pharmacokinetic model to inform tacrolimus therapy in heart transplant recipients[END_REF] , 0.22 L.h -1 .Kg -1 Gong et al. [START_REF] Gong | Population pharmacokinetic analysis of tacrolimus in Chinese cardiac transplant recipients[END_REF] and 0.21 L.h -1 .Kg -1 for Han et al. [START_REF] Han | Prediction of tacrolimus dosage in the early period after heart transplantation: a population pharmacokinetic approach[END_REF] ). The same was observed for our apparent central volume of distribution which was also strongly decreased compared to the heart transplant model (2.55 L.Kg -1 for Kirubakaran et al. [START_REF] Kirubakaran | Adaptation of a population pharmacokinetic model to inform tacrolimus therapy in heart transplant recipients[END_REF] , 11.52 L.Kg -1 for Gong et al. [START_REF] Gong | Population pharmacokinetic analysis of tacrolimus in Chinese cardiac transplant recipients[END_REF] and 12.17 L.Kg -1 for Han et al. [START_REF] Han | Prediction of tacrolimus dosage in the early period after heart transplantation: a population pharmacokinetic approach[END_REF] ).

In lung and heart transplant patients ,Sikma et al. [START_REF] Sikma | Unbound Plasma, Total Plasma, and Whole-Blood Tacrolimus Pharmacokinetics Early After Thoracic Organ Transplantation[END_REF] observed an apparent central volume of distribution around 2.99 L.Kg -1 which is also far from our value. However, even if most of the reported values for the tacrolimus apparent central volume of distribution are between 2.55 L.Kg -1 & 12.30 L.Kg -1 1-3,11,30 some studies in other organ transplant populations also reported small central compartment volume values [START_REF] Andreu | Development of a Population PK Model of Tacrolimus for Adaptive Dosage Control in Stable Kidney Transplant Patients[END_REF][START_REF] Lu Y Xia | A population pharmacokinetic study of tacrolimus in healthy Chinese volunteers and liver transplant patients[END_REF][START_REF] Co | Population pharmacokinetic analysis of tacrolimus in Mexican paediatric renal transplant patients: role of CYP3A5 genotype and formulation: Role of CYP3A5 genotype and formulation type on tacrolimus PK[END_REF] . It should be noted that the shrinkage value of the mean transit time is high (140%) but this was not a problem as we are only interested by the apparent clearance value.

In our final model, the CYP3A5 polymorphism was selected based on the tacrolimus apparent clearance. The CYP3A5 polymorphism effect is largely known and has been previously described [START_REF] Brunet | Therapeutic Drug Monitoring of Tacrolimus-Personalized Therapy: Second Consensus Report[END_REF] . This polymorphism leads to an alternative splice site in the pre-mRNA and results in a truncated enzyme [START_REF] Kuehl | Sequence diversity in CYP3A promoters and characterization of the genetic basis of polymorphic CYP3A5 expression[END_REF] . Individuals that carry at least one CYP3A5*1 allele are considered to be CYP3A5 expressors (including CYP3A5*1/*3 and CYP3A5*1/*1). These patients require a higher starting dose compared with CYP3A5*3/*3 carriers to reach the predefined target exposure early after transplantation [START_REF] Hesselink | Genetic polymorphisms of the CYP3A4, CYP3A5, and MDR-1 genes and pharmacokinetics of the calcineurin inhibitors cyclosporine and tacrolimus[END_REF] . The hematocrit has been shown to influence the apparent clearance in renal transplant patients [START_REF] Woillard | Population pharmacokinetic model and Bayesian estimator for two tacrolimus formulations-twice daily Prograf and once daily Advagraf[END_REF] and in heart transplant patients 1 , but its effect was not found to be significant in the present study, even if the range of the hematocrit values was quite large (between 26 and 47 %).

The inter-occasion variability was investigated as a categorical variable because the model was struggling to converge with the standard way to include it (estimation of IOV in addition to inter-patient variability). We had thus to hypothesis that all the PK profiles were independent, which is acceptable given the fact that the lag between the two occasions was at least two weeks.

Finally, we did not observe an effect of the categorised time post transplantation on the PK parameters.

For the PTA of C0, as expected, the expression of the CYP3A5 increased the apparent clearance of tacrolimus and higher doses were required to attain the targets compared to the non expressor group. To illustrate this effect, for a C0 target =15 µg.L -1 , a dose of 7.5 mg would lead to 80% of patients reaching the target in the non expressor group whilst only 40% would reach the target in the expressor group. In the latter group, a dose of 20 mg would be required to reach the same target. Similarly for AUC, for an AUC target = 270 µg.h.L -1 , a dose of 7.5 mg would lead in the non expressor group to a probability of target attainment of 85%. In the expressor group, this would lead only to 60%; the optimal dose being 15mg. Since the AUC is theoretically the best marker of the tacrolimus exposure [START_REF] Brunet | Therapeutic Drug Monitoring of Tacrolimus-Personalized Therapy: Second Consensus Report[END_REF] , the observed differences between the doses proposal based on the C0 or the AUC were expected. The comparison of the dose proposals between our study and the study of Woillard et al. [START_REF] Woillard | Tacrolimus Updated Guidelines through popPK Modeling: How to Benefit More from CYP3A Pre-emptive Genotyping Prior to Kidney Transplantation[END_REF] performed on renal transplant patients showed some differences as their PTA showed that a dose of 7.5 mg in the non expressor group and 15 mg in the expressor group was required to attain a target of C0= 10µg.L - 1 . The differences with the present study are most likely due to the different type of transplantation. However, the magnitude of the differences between expressors and non expressors remains similar. We chose to compare our dose proposal based on our AUC PTA to the summary of product characteristics of the tacrolimus and some differences were also observed. Indeed, the dose recommendation is 0.075 mg.kg -1 .day -1 36 to attain a AUC target of 270 µg.h.L -1 and we proposed in this study a dose of 0.11mg.kg -1 .12h -1 for a non expressor patient and a dose of 0.22 mg.kg -1 .12h -1 for an expressor patient which represents a large increase. Obviously, these proposals need to be considered only for the starting dose as after that, TDM must be used to provide dose individualisation.

However, this study has some limitations. First, it was performed with the bid formulation of tacrolimus as the once daily formulation was not available at that time. However, nowadays, the once a day formulation is also used in this population. The model developed in this article is limited to the bid formulation. Moreover, despite the fact that this study is multicentric, (seven French hospitals participated) and the data were collected under the same treatment protocol allowing generalisability of the results on a French and by extension European scale, the model is unlikely to be generalizable to other heart transplant patient populations (Chinese, American etc…). Furthermore, in our study, some PK profiles had required extrapolation of the AUC using a monoexponential decay. Before doing this extrapolation, we determined that the elimination was linear between 9-12 hours post dose. However, this extrapolation could slightly impact the trapezoidal rule AUC. Also, to determine the AUC target, we used the relation between the C0 and the AUC calculated on a kidney transplant population and their correlation coefficient was only about 0.7. Finally, to complete all kinetic profiles, some missing covariates were imputed using the K-nearest neighbours' method which imputes data based on the most similar patients. This approach is quite innovative and avoids imputing the missing data based on the population median as usually done. It is of note that a model for tacrolimus in heart transplant recipients has already been implemented in the ISBA website (https://pharmaco.chulimoges.fr). This model was developed using an Iterative Two Stage Bayesian approach in which the absorption phase was described using a double gamma distribution. Our article presents a Nonlinear Mixed Effects Models approach allowing to better evaluate the interindividual variability and additionally provides recommendations for the starting dose.

Conclusions:

In this work, we developed a POPPK model and derived a MAP-BE based on a LSS for tacrolimus in heart transplant patients that can be used for TDM based on the AUC in routine practice. We also proposed optimal first doses recommendations to reach predefined C0 or AUC depending on the CYP3A5 status.

Figure 1 :Figure 2 :Figure 3 :Figure 4 :

 1234 Figure 1: Diagnostic plots for the final model of individual (A) or population predictions (B) versus observed tacrolimus concentrations (µg.L -1 ) and weighted residuals as function of time (h) (C) or individual prediction (D) in the development population

Figure 5 :

 5 Figure 5: Proportions of simulated patients achieving different tacrolimus bid AUC (Area under the curve) targets for different dosage regimens and depending on the CYP3A5 status.

Table 1 :

 1 The characteristics of heart transplant patients receiving tacrolimus (P1= 7-14 days, P2 = 1 month, P3 = 3 month, P4= 1 year).

	Variable	Overall	Development	Validation	p-value
		Patient: 18	Patient: 11	Patient:7
		Pk profiles: 47	Pk profiles: 38	Pk profiles : 9
	Age (years)	44 [22 -63]	45 [22 -63]	37 [22-57]	0.177
	Sex (male/female)	39 (83%) / 8 (17%)	31 (82%) / 7 (18%)	8 (89%) / 1 (11%)	0.542
	Weight	66 [34 -93]	67 [36 -93]	63 [34 -83]	0.444
	Hematocrit (%)	36.5 [25.8 -47.0]	35.5 [25.8 -47.0] 37.8 [32.0-45.0]	0.341
	Tacrolimus dose (mg twice a day)	3.0 [0.5 -8.5]	3.0 [0.5 -8.5]	3.5 [1.0-8.0]	0.683
	CYP3A5 status* (expressor/ non expressor)	10 (21%)/ 37 (79%)	7 (18%) / 31 (82%)	3 (33%)/ 6 (73%)	0.437
	Post transplantation delay (P1/P2/P3/P4)	11 (23%)/ 11(23%)/ 11(23%)/ 14 (30%)	8(21%)/ 10(26%)/ 8(21%)/ 12(36%)	3(30%)/1(10%)/ 3(30%)/ 2(30%)	0.948

Continue covariate: mean[range]; Categorial covariate: n (%). P value was obtained using the Wilcoxon test. Descriptions for each PK profile were considered as independent *Number of patients in the overall population: expressor = 5 / Non expressor = 13

Table 3 :

 3 LSS comparison in the development and in the validation set : LSS -Limited sampling strategy; RMSE-Root mean square error

		LSS	RMSE (%)	Relative Mean bias (%)	Patient with bias >+/-20%
		C0.C1.C2	13.3	5.7	5
	Development	C0.C1.C3	14.1	8.0	6
	set	C0.C1.C4	14.4	8.8	5
		C0.C2.C4	17.5	9.3	10
	Validation Set	C0.C1.C2	9.9	2.7	0

Acknowledgment

The authors would like to thank Karen Poole and Jean-Paul Paschier, for the time spent on the English improvement of this article.

Funding information

No funding was received for this study.

Data availability

The data that support the findings of this study are available, on reasonable request, from the corresponding author. The data are not publicly available due to privacy or ethical restrictions.

Conflict's interest

The authors have no competing interests to declare in relation to this work.

Author contribution

AP, AD, JBW performed the analysis and wrote the paper, PM, CM and ML conceived the analysis, organized and collected data