Qiaoting Zhong

Guangnan Zhang

A Case-based Approach for Modelling the Risk of Driver Fatigue

Keywords: intelligent information processing, intelligent decision making, casebased reasoning, traffic safety management, driver fatigue

Fatigue-related crashes are one of the major threats to road safety worldwide. Despite the substantial work in the domain of transportation science by both the industry and academia, there are few studies in applying casebased reasoning (CBR) approach to modelling the risk of driver fatigue. This research explores the potential for fatigued driving using a database of 16,459 traffic crashes reported from 21 cities in Guangdong province, China from 2006 to 2010. The CBR system under development differentiates between fatigueddriving and non-fatigued-driving cases based on various personal and environmental traffic characteristics. The advantage of using CBR in modelling fatigued driving has been demonstrated through empirical evaluation.

Introduction

Driver fatigue has already become a leading factor contributing to traffic crashes around the world [START_REF] Zhang | Traffic accidents involving fatigue driving and their extent of casualties[END_REF]. In China, fatigued driving caused 887 (9.26%) of all highway crashes in 2011, resulting in 520 (8.1%) deaths and over RMB 37 million (10.82%) property losses [43]. To decrease the occurrences of traffic crashes and promote road safety, studying drivers' decision on driving under fatigue condition is urgent. Despite the substantial work in the domain of transportation science by both the industry and academia, there are few studies in applying case-based reasoning approach to modelling the risk of driver fatigue.

Case-based reasoning (CBR) has been used in both cognitive science and artificial intelligence [START_REF] Li | Transportation networks, case-based reasoning and traffic collision analysis: A methodology for the 21st century[END_REF], it makes use of the most similar previous cases to solve new problems [START_REF] Li | Transportation networks, case-based reasoning and traffic collision analysis: A methodology for the 21st century[END_REF][START_REF] Leake | CBR in context: the present and future. In: Leake DB (ed) Case-based reasoning, experiences, lessons, & future directions[END_REF][START_REF] Kolodner | Case-Based Reasoning[END_REF]. Since the early 1980s, CBR has been successfully applied in various fields such as legal reasoning [START_REF] Simpson | A computer model of case-based reasoning in problem solving: An investigation in the domain of dispute mediation[END_REF][START_REF] Ashley | Modeling Legal Arguments: Reasoning with Cases and Hypotheticals[END_REF][START_REF] Athakravi | Generating legal reasoning structure by answer set programming[END_REF][START_REF] Yeow | An application of case-based reasoning with machine learning for forensic autopsy[END_REF], planning [START_REF] Kolodner | Retrieval and organisation strategies in conceptual memory: A computer model[END_REF][START_REF] Hammond | CHEF: A model of case-based planning[END_REF][START_REF] Khattak | Case-based reasoning: a planning tool for intelligent transportation systems[END_REF][START_REF] Hammond | Case-based planning: Viewing planning as a memory task[END_REF][START_REF] Liu | Emergency resources demand prediction using case-based reasoning[END_REF], E-commerce [START_REF] Shimazu | Expertclerk: A conversational case-based reasoning tool for developing salesclerk agents in e-commerce webshops[END_REF][START_REF] Heras | Argue to agree: A case-based argumentation approach[END_REF], medical diagnosis [START_REF] Althoff | Case-based reasoning for medical decision support tasks: the inreca approach[END_REF][START_REF] Holt | Medical applications in case-based reasoning[END_REF][START_REF] Chattopadhyay | A casebased reasoning system for complex medical diagnosis[END_REF][START_REF] Sharaf-El-Deen | A new hybrid case-based reasoning approach for medical diagnosis systems[END_REF], incident management [START_REF] Zubair | Prediction and analysis of air incidents and accidents using case-based reasoning[END_REF][START_REF] Johnson | Using case-based reasoning to support the indexing and retrieval of incident reports[END_REF], and risk analysis [START_REF] Lu | Case-based reasoning for automated safety risk analysis on subway operation: Case representation and retrieval[END_REF][START_REF] Jin | Application of case-based reasoning for estimating preliminary duration of building projects[END_REF]. CBR is the best fit for modelling fatigued driving because of its ability to simultaneously process a large number of highly interrelated variables to arrive at a decision [START_REF] Hossain | A bayesian network based framework for real-time crash prediction on the basic freeway segments of urban expressways[END_REF][START_REF] Goh | Case-based reasoning for construction hazard identification: case representation and retrieval[END_REF]. Moreover, CBR models have significant merits as compared to statistical ones (e.g., the widely employed matched case-control logistic regression [START_REF] Abdel-Aty | Crash risk assessment using intelligent transportation systems data and real-time intervention strategies to improve safety on freeways[END_REF][START_REF] Lee | Potential real-time indicators of sideswipe crashes on freeways[END_REF]) and other artificial intelligence models (e.g., neural network) regarding the comprehensibility of output [START_REF] Goh | Case-based reasoning for construction hazard identification: case representation and retrieval[END_REF].

The novelty of the research work presented lies in the following aspects. First of all, this work explores the potential for using CBR in dealing with driver fatigue management. Secondly, this research examines a highly comprehensive set of risk factors that relate to the driver fatigue. A database of 16,459 real traffic crashes reported from 21 cities in Guangdong province, China from 2006 to 2010 is used, and a total of 50 variables in 21 categories are examined in this study. Compared with the logistic regression model that uses the same variables, our proposed CBR model had better performance in modelling fatigued driving on AUC measures in all simulations. The outcome of our proposed CBR system can help targeting the group of drivers that may intend to drive under fatigue condition, which constitutes the crucial part of driver safety management in an intelligent transportation system. This paper is organised as follows. Section 2 introduces previous relevant work from the literature. Section 3 describes the methodology. Section 4 presents the case study and empirical evaluation of our approach. Section 5 concludes the study.

Related Work

In recent years, there had been an increase in uptake of case-based reasoning (CBR) concepts in the research area of traffic control and management. For instance, Jagannathan et al. [START_REF] Jagannathan | Predicting road accidents based on current and historical spatio-temporal traffic flow data[END_REF] developed a CBR prediction system that is capable of differentiating between the accident and non-accident cases. Mounce et al. [START_REF] Mounce | A metric for patternmatching applications to traffic management[END_REF] designed a CBR system to help selection of signal timing plans. Sadek et al. [START_REF] Sadek | Case-based reasoning for real-time traffic flow management[END_REF] developed a prototype CBR routing system. Li and Zhao [START_REF] Li | A case-based reasoning approach to urban intersection control[END_REF] applied CBR to intersection control. Kofod-Petersen et al. [START_REF] Kofod-Petersen | Case-based reasoning for improving traffic flow in urban intersections[END_REF] presented a prototype implementation of a CBR system that predicts traffic flow and calculates signal plans for urban intersections. CBR methods have also been used to model the vehicle control behaviour (i.e., steer, throttle, and brake) of teen drivers [START_REF] Ontañón | Case-based prediction of teen driver behavior and skill[END_REF]. Different from the previous ones, this work aims to explore the potential for using CBR in dealing with driver fatigue management.

The risk of driver fatigue is related to a combination of situational and individual factors. The increased risk may result from a mix of biological, personal, road and environmental related factors [START_REF] Lal | A critical review of the psychophysiology of driver fatigue[END_REF]. Previous studies on detecting driver's state of fatigue have used various biological and personal indicators, such as visual indicators (e.g., face images and eye state [START_REF] Devi | Driver fatigue detection based on eye tracking[END_REF][START_REF] He | Fatigue detection using smartphones[END_REF][START_REF] Chen | A new algorithm for fatigue detection in driving[END_REF]), physiological sensor signals (e.g., EEG [START_REF] Zhang | Automated detection of driver fatigue based on entropy and complexity measures[END_REF] and ECG [START_REF] Begum | Mental state monitoring system for the professional drivers based on heart rate variability analysis and case-based reasoning[END_REF]) and driving states (e.g., accelerate, brake, shift and steer [START_REF] Wang | Fatigue driving detection system design based on driving behavior[END_REF]). However, none of them (including the CBR approach of identifying the drivers' stress state [START_REF] Begum | Mental state monitoring system for the professional drivers based on heart rate variability analysis and case-based reasoning[END_REF]) considered road and environmental features that are crucial towards drivers' intention to drive under fatigue condition. By using CBR in this research, the risk of driver fatigue is modelled based on past similar circumstances considering not only personal but also road and environmental characteristics. Moreover, most previous studies verify their models through experiments with driving simulators, whereas we validate ours by using the reliable official source of traffic crash data.

System Design

Following the work of [START_REF] Kolodner | Case-Based Reasoning[END_REF][START_REF] Li | A case-based reasoning approach to urban intersection control[END_REF], we propose a case-based system of fatigued driving that consists of three main components: representation module, retrieve module, and reuse module. Figure 1 shows a schematic overview of the system architecture. The proposed system receives traffic information, through the representation module, a target case C t and a case-base C are constructed based on such information. The retrieve module retrieves the most similar previous case from the case-base, and reuse module further manipulates the retrieved cases for the best outcome of the target case.

Case Representation

A case in the system represents a previous traffic situation [START_REF] Li | A case-based reasoning approach to urban intersection control[END_REF], and it is composed of a case number, a case description, and a case outcome. Formally, a case can be represented as follows:

-Case C = N, D, O , where,

• N is the case number;

• D is the case description;

• O is the case outcome (i.e., fatigued-driving or non-fatigued-driving);

C is used to denote the case base that contains all possible cases. The description of a case depicts the situation when the case appears [START_REF] Li | A case-based reasoning approach to urban intersection control[END_REF][START_REF] Kofod-Petersen | Case-based reasoning for improving traffic flow in urban intersections[END_REF]. Four categories of attributes are needed to describe the case. The first category contains the demographic information of the driver. The second one describes the vehicle information. The third category consists of road situation factors. The fourth category contains some environmental factors such as the weather situation. Formally, we have:

-Case description D = DD, V I, R, E , where,
• DD is the demographic information of the driver such as gender;

• V I is the vehicle information such as vehicle type;

• R is the road situation such as road type;

• E is the environmental factors such as bad weather.

For example, a simple case (No. 0) that a crash occurred on the expressway at about 8 am on Saturday was caused mainly because the fatigued driving of the male motorcyclist would be represented as 0, {male}, {motorcycle}, {expressway}, {moring, week-end} , fatigued-driving .

Case Retrieval and Reuse

The crucial part of the retrieval module is the similarity measure, which calculates the similarity between the target case and the cases in the case-base [START_REF] Jagannathan | Predicting road accidents based on current and historical spatio-temporal traffic flow data[END_REF]. For each target case, the similarity between the target case and the cases in the case-base is measured by calculating the Euclidean distance (the most widely used distance metric in CBR) between them. Given a target case t and a case c in the case-base, the similarity between both cases is:

S(t, c) = a (D t,a -D c,a ) 2
where D t,a is the normalised value of attribute a ∈ DD, V I, R, E in the target case t and D c,a is the normalised value of attribute a in the case c from the case base. Note that all the attributes are of equal importance in this paper for simplicity reasons.

With the above similarity measure, the most similar cases will be retrieved by using the K-nearest neighbour (KNN) algorithm [START_REF] Cover | Nearest neighbor pattern classification[END_REF], where K refers to the number of neighbours. The KNN algorithm, widely used in classification problems to assign objects to classes, matches each attribute in the target case to its corresponding attribute in the retrieved case [START_REF] Jagannathan | Predicting road accidents based on current and historical spatio-temporal traffic flow data[END_REF]. The predicted outcome of the target case is then obtained by the majority vote of its K nearest neighbours, i.e., the most common outcome of the K retrieved cases are reused and assigned to the target case directly.

Case Study: Based on the Traffic Crash Data in China

In this study, we use the traffic crash data for the period 2006-2010 in Guangdong Province, China. These data are extracted from the Traffic Management Sector-Specific Incident Case Data Report, the Road Traffic Accident Database of China's Public Security Department. They are the only officially available source of traffic crash data in China. Data are recorded and reported by the traffic police on-scene who conducted assessments and provided feedback immediately to the headquarters of the Traffic Management Department. These reports include characteristics of drivers, vehicle features, road conditions, the time of crashes, the environmental context for each crash and the cause of crashes such as traffic violations like driving under fatigue condition [START_REF] Zhang | Risk factors associated with traffic violations and accident severity in china[END_REF]. To experiment with the proposed approach, we set up a case-based system in which each of the samples in our datasets was a case. Table 1 shows the case structure used in the experiment where relevant binary attributes are selected based on [START_REF] Zhang | Traffic accidents involving fatigue driving and their extent of casualties[END_REF]. According to working time patterns and peoples' lifestyles in China, we classify five groups of time.

Evaluation Settings

To evaluate our proposed case-based system of drivers' risk of fatigued driving, the data set was partitioned into a training set and a test set. To be aligned with the literature (e.g., with different numbers of K were run. Ties may occur in a binary classification problem if K > 1, to avoid ties, the most similar case (K = 1), the three most similar cases (K = 3), and the five most similar cases (K = 5) were retrieved in the current study.

In comparison, the widely employed logistic regression was also conducted for each partition. The dependent variable indicated the occurrence of driving under fatigue condition in a crash, while the independent variables under consideration were the same as the CBR setting.

The performance of a computational intelligent system can be measured in many different ways. Typically, the models' predictive performances are measured regarding overall accuracy, e.g., the percentage of fatigued-driving and non-fatigued-driving classified accurately in the current research [START_REF] Han | Data Mining: Concepts and Techniques[END_REF]. However, other measures like the true positive rate (TPR, also called recall or sensitivity) tends to be the key concern when infrequent events (as fatigued driving in the current study) are to be predicted [START_REF] Weiss | Learning to predict extremely rare events[END_REF]. Complementary to TPR that measures the proportion of positives that are correctly identified as such, another primary evaluator is the false positive rate (FPR, also known as fall-out or false alarm ratio) that calculates the ratio of negatives wrongly categorised as positive. Based on pairs of TPR and FPR, a receiver operating characteristic (ROC) curve could be obtained to compare the performance of different models. We calculate the area under the ROC curve based on the work of Cantor and Kattan [START_REF] Cantor | Determining the area under the roc curve for a binary diagnostic test[END_REF], and use IBM SPSS Statistics 23 to perform all the statistical analysis.

Results and Discussion

The available data set contained 16,459 traffic crashes, of which 384 (2.33%) were fatigue-related. Among all the fatigue-related crashes, 99% of drivers were male, 88% of them had more than two years of driving experience, and 72.4% came from urban areas. Drivers that were involved in fatigue-related crashes included self-employed workers (23.7%), migrant workers (18.5%), farmers (17%), clerks (8%), and employees from other professions (32.8%). Concerning vehicle information, trucks and motorcycles constituted 56% and 16.7%, respectively. For road situation, 65.9% of these fatigue-related crashes occurred in separate lanes, and 38% occurred in expressway.

Table 2. Evaluation Results (TPR for true positive rate and FPR for false positive rate) 10:90 30:70 50:50 k=1 k=3 k=5 LR k=1 k=3 k=5 LR k=1 k=3 k=5 LR TPR 0.12 0.01 0 0 0.16 0.04 0.02 0 0.22 0.06 0.02 0 FPR 0.02 0 0 0 0.02 0.01 0 0 0.02 0 0 0 70:30 90:10 k=1 k=3 k=5 LR k=1 k=3 k=5 LR TPR 0.26 0.11 0.04 0 0.23 0.17 0.1 0 FPR 0.02 0 0 0 0.01 0.01 0 0

Considering environmental features, 51.8% of the fatigue-related crashes occurred during 00:00-06:59, and 38% took place at daytime. Seasonal distribution of the fatiguerelated crashes: 32.3%, 26.3%, 24.7%, and 16.7% occurred in the summer, winter, spring, and fall, respectively.

Figure 2 shows the comparison of the overall accuracy for all simulations. In overall, all simulations received very high accuracy in modelling fatigued driving, i.e., more than 96 percent. Logistic regression (LR) had a better performance in comparison to the proposed CBR method (no matter what the value of K) for most training/test partitions. When 70% or above of the dataset was used for training, the CBR method with K=3 achieved slightly higher accuracy than LR.

Table 2 summarises the results of true positive rate (TPR) and false positive rate (FPR) from all simulations. In all scenarios, we observed that FPR is nearly zero, indicating that both LR and the proposed CBR method have excellent performance in correctly identifying non-fatigued-driving cases. However, LR received zero TPR for all the training/test partitions, which reflects that LR is relatively biased towards the majority class in our dataset (i.e., the non-fatigued-driving case). Such bias may be caused by the imbalance situation of our dataset (i.e., only 2.33% cases were fatigue-related). Recall that TPR reflects the hits of fatigued driving reality, in such circumstances, although LR got very high accuracy and very low FPR, it is meaningless [START_REF] Weiss | Learning to predict extremely rare events[END_REF]. In contrast, our proposed CBR models with K=1 or K=3 had shown their merits by receiving non-zero TPR in all the training/test partitions.

Figure 3 illustrates the trade-off between the TPR and the FPR of each simulation. Our proposed CBR models (no matter what the value of K) had better performance than LR in modelling fatigued driving on AUC measures. Simulation of K=1 outperformed the rest as the percentage ratio of the training and test data were varied.

To sum up, we can see that the proposed CBR model with K=1 and that with K=3 have their strengths in modelling fatigued driving. To further differentiate them, we calculated the precision values in various partitions of training and test data. Precision measures the probability that a driver with a positive screening test indeed conducted fatigued driving. As shown in Fig. 4, when 10% of the data set were used for training, the K=1 model had higher precision value, whereas the K=3 model had better performance when more than 10 percent were used for training. Further empirical validation is required to study the optimal value of K for modelling fatigued driving.

Further Implications

Fatigue-related crashes are one of the major threats to road safety worldwide. The important implication of our research is: if driving under fatigue condition could be reduced or controlled successfully through the early detection, the occurrences of a crash would be reduced accordingly. When designing road traffic interventions to decrease incidences, it is well established that a change in the driver's attitude is of utmost importance. The outcome of our proposed CBR model can help targeting the group of drivers that may have an intention to drive under fatigue condition. In particular, our proposed CBR model can highlight the important features of such group of drivers in the aspects of various personal, vehicle, road, and environmental conditions. Specific countermeasures by integrating these highlighted features into a driver fatigue detection component of the intelligent transportation systems will be effective in improving traffic safety [START_REF] Paaver | Preventing risky driving: a novel and efficient brief intervention focusing on acknowledgement of personal risk factors[END_REF]. The results obtained in this study may likely be generalised to other provinces in China because the crash data in Guangdong Province and features considered are rather representative and comprehensive. The fact that Guangdong has a high percentage of residents migrating from other provinces, and is having one of the largest numbers of vehicles among all provinces in China reinforces the generalisability [START_REF] Zhang | Traffic accidents involving fatigue driving and their extent of casualties[END_REF]. Furthermore, the findings of this study can also contribute as a reference to future road safety research for other countries.

Limitations and Future Work

A live case-based reasoning system has a "4 REs" cycle (i.e., retrieve, reuse, revise, and retain) [START_REF] Aamodt | Case-based reasoning: Foundational issues, methodological variations, and system approaches[END_REF]. Since our proposed case-based system is not operating on live data, there is no component designed in the current study for the standard revise and retain process.

Our proposed system could be extended to deal with live data by adding a new part of revise and retain in the future. The advantage of using case-based reasoning in modelling fatigued driving has been demonstrated through comparing with logistic regression. In getting higher values of recall, precision and AUC, an important consideration is case attribute selection and weighting. The attributes we have chosen are based on previous work. However, it is likely that fatigued driving also occurs due to other conditions that have not yet been studied. Moreover, the contribution of each attribute to the similarity between the target case and the cases in case-base could differ as well. Thus, it is worth testing out different attributes and weighted assignments for attributes considered.

The empirical experiments show that the CBR system under development is capable of differentiating between fatigued-driving and non-fatigued-driving cases to some extent based on certain personal and environmental traffic characteristics. The evaluation of our proposed method is not exhaustive and could be expanded in several directions, for instance, to compare with other artificial intelligence techniques (e.g., neural network). It would also be worth evaluating whether or not our proposed approach could be fruitfully applied to other risky behaviours rather than the kind we have considered.

Fig. 1 .

 1 Fig. 1. Schematic Diagram Showing Steps Involved In Modelling Fatigued Driving

[ 3 ]Fig. 2 .

 32 Fig. 2. Accuracy of the different models

Fig. 3 .Fig. 4 .

 34 Fig. 3. AUC of the different models

Table 1 .

 1 Case Structure

			Case description
	Category	Feature	(Binary) Attributes
		gender	male
		age	0-24, 25-44, 45 and above
	demography	residence	urban
		driving experience 0-2 years, 3-5 years, above 5 years
		occupation	clerk, migrant worker, farmer, self-employed, other
		license condition valid
		safety condition poor
	vehicle	insured or not overloaded or not overloaded insured
		vehicle type	passenger car, truck, motorcycle
		commercial or not commercial
			express, first-class highway, second-class or below
		road type	highway, urban expressway, urban ordinary highway,
	road	traffic lane	other isolated
		road surface	dry, wet, other
		traffic control device no device, signal, sign, other
		light condition	daytime, with street lighting in the night, without street lighting in the night
		weather	rainy
	environment	weekend or not weekend
		holiday or not	holiday

time 00:00-06:59 (midnight to dawn), 07:00-08:59 (morning rush hours),12:00-13:59 (noon),17:00-19:59 (afternoon rush hours), other season spring, summer, fall, winter

Case outcome: fatigued-driving, non-fatigued-driving

Acknowledgements

This research was supported in part by the National Natural Science Foundation of China grant 71573286 and Ministry of Education Project for Humanities and Social Sciences Research (16JJDGAT006).


André Mas
email: mas@math.univ-montp2.fr

A su¢ cient condition for the CLT in the space of nuclear operators -Application to covariance of random functions

Keywords: Nuclear operators, Central limit theorem, Covariance operators, Random functions 1

We give a su¢ cient condition for the CLT to hold in the space of trace class (nuclear) operators. This condition turns out to be adapted to the asymptotic study of emprirical covariance operators of Hilbert valued random variables.

1 n n X k=1 < X k ; x > X k
Several statistical applications arise from n : functional principal component analysis, canonical covariance analysis, decomposition of gaussian curves, etc. In the case of an i.i.d. sample Dauxois, Pousse and Romain (1982) provided a complete asymptotic study of n and of its spectral elements. Silverman and [START_REF] Rice | Estimating the mean and covariance structure nonparametrically when the data are curves[END_REF] proposed estimation procedures for the mean and the covariance operator of random curves. [START_REF] Bosq | Linear processes in function spaces[END_REF] in a book devoted to the generalization of ARMA models to the in…nite-dimensional setting gave several asymptotic results for n when the X k 's are dependent variables. [START_REF] Cardot | Functional linear model[END_REF] had to study such covariance operators in the framework of a functional linear regression model. [START_REF] Mas | Weak convergence for the covariance operators of a Hilbertian linear process[END_REF] proved the asymptotic gaussianity of the vector of covariance and cross-covariance of a generalized linear process with values in a Hilbert space.

If u and v are two vectors in H; by u v we denote the rank one operator de…ned for all x in H by (u v) (x) =< u; x > v:

Let us suppose that the sample X 1 ; :::; X n is strictly stationary and denote = E (X 1 X 1 ) : In the sequel we will study weak convergence for the random operator n : It is a well-known fact that spaces of operators on Hilbert spaces may be endowed with several topologies or norms that are not equivalent if H is in…nite dimensional. It should be stressed that functional statisticians tend to match the regularity of underlying processes. Consequently the case H = L 2 [a; b] is not that common and Hilbert spaces of smoother functions such as Sobolev spaces are usually preferred.

In all the previous articles cited above, the space of operators is endowed with the Hilbert-Schmidt topology (de…ned below) for at least two reasons. First, computations are obviously simpler in this setting since n may be viewed as another Hilbert-valued random variable. Second, the central limit theorem is obtained under mild condition in this framework. This choice was made for convenience but cannot be justi…ed otherwise. It turns out that, unfortunately, the natural space is in fact the space of trace class operators. We refer to the next section for de…nition and main properties of these operator spaces. The word "natural" could be explained here by the following well-known fact (see for example Vakhania, Tarieladze, Chobanyan (1987)) : A random variable in a Hilbert space has strong moment of second order if and only if its covariance operator is a trace class operator. With mathematical symbols, this means :

E jX 1 j 2 < +1 i
is trace class:

When only "weak" moments (Pettis integral) are …nite the Hilbert-Schmidt assumption is no longer true since the covariance operator is just bounded then. In order to illustrate the situation let us switch operators to sequences. This would mean that, up to the present day, l 1 sequences were dealt only with l 2 tools. Consequently, the question of the central limit theorem in the space of trace class operators must be adressed.

As announced above, we will restrict ourselves to a su¢ cient condition for two reasons. It is probable that the necessary condition will not be obtained by usual probabilistic tools but requires very deep results related to functional analysis. Second, the condition, as enounced below, may be easily exploited in the special cases when the random operators are covariance (or cross-covariance operators).

The next paragraph outlines very basic but useful facts in operator theory.

Trace class operators

When looking for examples to illustrate general theorems on separable Banach spaces one commonly refers to spaces of functions (the L p (S; S; ) with usual notations) amongst which the l p 's or c 0 . Banach spaces of linear operators are rarely mentioned. They are however of much importance both in probability theory (especially in weak convergence theory, gaussian analysis, etc.) and in statistics where such random operators appear in quantum physics or when studying the covariance structure of random functions.

Here we will consider operators on an in…nite dimensional separable Hilbert space H and not on a Banach space. The Hilbert space setting is quite general and allows to give conditions that are easily understood. Even if L the space of bounded linear operators is not separable, it is possible to consider an important subspace : the space of compact operators, C, endowed with the ususal operator norm. For the de…nition and the main algebraic and analytic properties of C we refer to [START_REF] Dunford | Linear Operators Vol I[END_REF].

We may now de…ne the class C p C. Let T be a compact operator and k (T T ) be the k th positive eigenvalue of T T , (T denotes the adjoint operator of T ). Set k = p k ; the k th singular value of operator T; then …x p 2 N and de…ne :

C p = ( T 2 C : X k p k < +1
)

The "natural" norm on this space is de…ned for all T and for p 1 by kT k p = [ P k p k ] 1=p :Then, C p ; k k p is a separable Banach space and C p C q whenever p < q. Several other properties of C p are listed in [START_REF] Dunford | Linear Operators Vol I[END_REF].

Note also that spaces of operators are non commutative algebras, whereas algebras of functions are usually commutative and that C 2 is the space of Hilbert-Schmidt operators, which is itself a separable Hilbert space.

In the following we will focus on C 1 the space of nuclear (or trace class) operators on H endowed with nuclear norm k k 1 ; which is the thinest of all the k k p norm. In the case of a symmetric and positive operator, this norm is nothing but the sum of the eigenvalues. We will need the three following results. For further purpose we recall that the dual space of

C 1 is C 1 = L. The duality bracket between T 2 C 1 and U 2 L is hhT; U ii = trT U .
Nicole Tomszak-Jaegerman (1974) proved that C 1 is a Banach space of cotype 2. This result is crucial as far as the central limit theorem is under concern. It is a well known fact that in such a Banach space, a random variable satis…es the CLT if and only if it is pregaussian. In other words, if its covariance operator is a gaussian covariance operator. We refer to Araujo and Giné (1980) for more information about the theory of types in Banach spaces. Unfortunately this approach that may lead us to a necessary and su¢ cient condition goes beyond the scope of this article. Note anyway that conversely to the other famous cotype 2 space l 1 ; the CLT does not easily stems from the pre-gaussianity condition because C 1 is not a commutative space.

We have to change our technique and to go back to the basements of weak convergence on Banach spaces : the Prokhorov theorem. In the sequel we will consequently prove that the sequence of measures is tight, i.e ‡atly concentrated and bounded (see de Acosta (1970))

3 Main results

The i.i.d. case

In all the following T 1 ; T 2 ; :::; T n is a sequence of i.i.d. centered nuclear operators from H to H: We set S n = 1 p n P n p=1 T i : We need the following assumption : H : There exists a basis of H; say (e p ) p for which

X p q E jT 1 (e p )j 2 < +1 (1)
Proposition 1 When H holds, the sequence S n is ‡atly concentrated and bounded in probability.

Proof. In order to prove the …rst point it su¢ ces to prove that, for all " > 0;

lim m!+1 sup n P (kS n m S n m k 1 > ") = 0
where m denotes the projection on the vector space spanned by (e 1 ; :::; e m ), e i 2 H: Now we refer to Lemma 32 p.1116 of Dunford Schwartz, which provides :

kT k 1 X p jT 1 (e p )j : (2)
The previous inequality together with Markov and Jensen's inequalities imply :

P (kS n m S n m k 1 > ") 1 " X p m q E j(I m ) T 1 (e p )j 2 + X p m+1 q E jT 1 (e p )j 2 !
The second term clearly decays and Lebesgue's dominated convergence implies that the …rst term also converges to zero since for fxed p each real sequence E j(I m ) T 1 (e p )j 2 tends to zero as m tends to in…nity.

The proof of the boundedness in probability is really similar. For all strictly positive M;

P (kS n k 1 > M ) 1 M X p q E jT 1 (e p )j 2
and the term on the right tends to zero when M goes to in…nity.

Theorem 2 When H holds, the sequence S n satis…es the central limit theorem in C 1 :

Proof. The Prokhorov Theorem implies that the sequence S n is weakly relatively compact by Proposition 1. It su¢ ces now to prove that all the limits of the …nite dimensional distributions are gaussian. Lets us take T in the dual space L = C 1 . The central limit theorem on the line implies that hT; S n i = trT S n converges weakly for all T to a gaussian centered random variable with variance E (trT X 1 ) 2 = E hT; X 1 i 2 : Finally S n satis…es the central limit theorem in C 1 : The covariance operator of the gaussian limiting distribution is the covariance operator of X 1 :

Remark 3

The assumption H implies the …niteness of E kT 1 k 1 and of E kT 1 k 2 2 . Note that H is also really close to the condition that insures that a random element with values in the sequence space l 1 is pregaussian.

The m-dependent case

The result of this paragraph may be viewed as a by-product of the preceding. It seems that anyway, the development of functional techniques in statistics, especially in the analysis of time dependent processes, makes it necessary to go beyond the simple i.i.d. setting. The technique of proof is quite similar to what is done in the …nite dimensional or even the scalar case.

Let T and S denote two nuclear operators then T e S stands for the linear operator from L to C 1 de…ned by T e S (U ) = T; U S Theorem 4 Let X t be a strictly stationary m-dependent sequence of centered random trace class operators: For all p = 1; :::; m set (p) = E X 1 e X p and (p) = E X p e X 1 :

Suppose that X 1 satis…es H, then

1 p n (X 1 + ::: + X n ) w ! n!+1 G m
where G m is a centered Gaussian element in C 1 with covariance operator

m = (0) + m X j=1 ( (j) + (j))
Proof :

The scheme of the proof is quite classical. In spite of the unusual framework we will go quickly through it. In order to alleviate the computations we will also prove Theorem 4 in the case m = 1 (consequently G m becomes G 1 ). Generalization is straightforward.

The latent idea consists in using Theorem 4.2 p.25 in [START_REF] Billingsley | Convergence of probability measures[END_REF] about weak convergence for arrays of random variables in metric spaces. Take an integer k > 2 and let Y nk = n 1=2 (X 1 + ::: + X k 1 ) + (X k+1 + ::: + X 2k 1 ) + ::: + X (r 1)k+1 + ::: + X rk 1 with r = [n=k] :

Note that n 1=2 Y nk is a sum of r i.i.d. random variables each with mean zero and the covariance operator of (X 1 + ::: + X k 1 ) that is indeed :

(k 1) (0) + (k 2) ( (1) + ( 1))

The …rst elementary steps consists in proving that, for …xed k, H is full…lled for the sum X 1 + ::: + X k 1 : But, stationarity of the sequence X n yields E j(X 1 + ::: + X k 1 ) (e p )j 2 (k 1) E jX 1 (e p )j 2 :

Since X 1 satis…es H, the central limit theorem in the space of nuclear operators gives :

(n=r

) 1=2 Y nk w ! G k;1
where G k;1 is a centered Gaussian element in C 1 with the same covariance operator as above. This may be rewritten Y nk w ! k 1=2 G k;1 as n goes to in…nity. Now we have to study the weak convergence, as k tends to in…nity, of the gaussian random operator k 1=2 G k;1 : Since we are just interested in its distribution, this random variable may be written the following way :

k 1=2 G k;1 = k 2 k (G 1 + N k ) :
where N k is a centered gaussian element, independent from G 1 and with covariance op-

erator 1 k (0) : The conclusion follows k 1=2 G k;1 w ! G 1 :
We are now in position to apply Theorem 4.2 p.25 in [START_REF] Billingsley | Convergence of probability measures[END_REF]

P p nX n Y nk 1 > " P 2 4 n 1=2 r X j=1 X jk 1 > "=2 3
5 + P n 1=2 kX 1 + ::: + X n rk k 1 > "=2 r r n +1 X p=1 q E jX 1 (e p )j 2 + +1 X p=1 r k n E jX 1 (e p )j 2 + 2 k 1 n E jX 1 (e p )j 2
We used the stationarity of the process and the fact that n rk k: Since r=n 1=k; the …rst term (that does not depend on n any more) will tend to zero as k tends to in…nity and the second term as n does.

Application to empirical covariance operators

Obviously Theorem 2 may be applied to obtain a central limit theorem in trace class norm for the empirical covariance operators of i.i.d. Hilbertian random elements. The covariance operator of a random variable with strong (i.e. Bochner) second order moment is a nuclear operator. The C 1 -norm is consequently the thinest and the more natural of all standard topologies to study these special operators.

Covariance operators

Let X 1 ; :::; X n be i.i.d. centered random variables with values in H: The sequence under concern is now

S n = 1 p n n X k=1 (X k X k )
where = E (X 1 X 1 ) :

Proposition 5 Let p = E hX 1 ; e p i 2 be the sequence of the eigenvalues of ; associated to the eigenvector e p : Suppose that X p p p < +1 and E hX 1 ; e p i 4 = O 2 p ;

(3) then S n converges weakly to a centered gaussian trace class random operator G with the same covariance operator as X 1 X 1 :

Proof. First note that the condition E hX 1 ; e p i 4 = O 2 p implies that E jX 1 j 4 H < +1: We just have to prove that H holds, namely X p q E j(X 1 X 1 ) (e p )j 2 < +1 (4)

  Y nk 1 > " = 0 the conclusion of Theorem 4 will follow. Computations are similar to those leading to the tightness in the i.i.d. case.

						and if we prove
	that	lim k!+1	n lim sup	P	p	nX n

Let us take for (e p ) an eigenbasis for : We have :

) (e p )j 2 E hX 1 ; e p i 2 jX 1 j 2 q E hX 1 ; e p i 4 E jX 1 j 4 = O ( p )

hence the desired result.

Remark 6 If the random variable X 1 is gaussian the second assumption in (3) holds. Indeed we have hX 1 ; e p i = p p p where p are i.i.d gaussian centered and E hX 1 ; e p i 4 = 3 2 p . Also note that assumption H is easily checked since it su¢ ces to compute the projections of the process on a given basis of H (spline, sinusoidal, etc.) Remark 7 Theorem 4 could be applied to the covariance operator of the autoregressive process in [START_REF] Bosq | Linear processes in function spaces[END_REF] :

where the X t 's and the " t 's are H-valued and is linear from H to H: The same technique reveals fruitful in [START_REF] Mas | Weak convergence for the covariance operators of a Hilbertian linear process[END_REF]. Limiting distributions of the covariance operators (for the Hilbert-Schmidt norm) were obtained for the more general linear model :

where the a k 's are linear operators.

Cross-covariance operators

Suppose now X 1 ; :::; X n ; Y 1 ; :::Y n are all independant centered random variables with values in H: Now we set e

The random trace class operator e S n is the cross-covariance operator of the couple (X; Y ) and appears in problems connected to functional statistics (see [START_REF] Cardot | Functional linear model[END_REF], [START_REF] Bosq | Linear processes in function spaces[END_REF]). The following is an easy consequence of Theorem 2.

Proposition 8 Once more let p be the sequence of the eigenvalues of = E (X 1 X 1 ) ; associated to the eigenvector e p : Assume that : X p p p < +1 and E jY 1 j 2 < +1 then e S n converges weakly to a centered gaussian trace class random operator e N with the same covariance operator as X 1 Y 1 .


Yosu Luque

Kevin Louis

Chantal Jouanneau

Sandrine Placier

Emmanuel Esteve

Dominique Bazin

Eric Rondeau

Emmanuel Letavernier

Alice Wolfromm

Clément Gosset

† ‡|

Anna Boueilh

Maren Burbach

Perrine Frère

Marie-Christine Verpont

Sophie Vandermeersch

Dominique Langui

Michel Daudon

Vincent ‡|

†| Laurent Frochot

Mesnard

Dr Laurent Mesnard
email: laurent.mesnard@aphp.fr

Vancomycin-Associated Cast Nephropathy

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

Discovered in the 1950s, vancomycin is an old and widely prescribed drug, mostly excreted via the kidneys. [START_REF] Rodvold | Vancomycin pharmacokinetics in patients with various degrees of renal function[END_REF] Adverse effects of vancomycin, including nephrotoxicity, were initially imputed to the so-called "Mississippi mud," 2 a contaminant from the original formulations. [START_REF] Farber | Retrospective study of the toxicity of preparations of vancomycin from 1974 to 1981[END_REF] More recently and despite the purified forms now in use, the renal toxicity of vancomycin has been suspected mostly in the clinical context of ARF, which promotes its accumulation in the blood. Given that vancomycin nephrotoxicity occurs at the same time as additional well characterized renal insults (e.g., hypovolemia and coadministration of aminoglycosides among others), evidence for vancomycin nephrotoxicity has been circumstantial and sometimes hotly debated, despite ARF having been observed in 5%-25% of patients. [START_REF] Levine | Vancomycin: A history[END_REF] We report here a previously unsuspected mechanism of human vancomycin nephrotoxicity suggested by the study of nine patients. By also using different mouse models, we confirmed this mechanism of renal insult. We showed that rogue intratubular vancomycin noncrystal nanosphere-associated casts entangled with uromodulin explain the development of ARF and therefore, vancomycin nephrotoxicity.

A 56-year-old woman was referred to our department for ARF. The patient's medical history included diabetes mellitus and gastric bypass surgery for morbid obesity 2 years previously. The patient was finally diagnosed with type 2 acute myelogenous leukemia. Serum electrolytes, creatinine, and BUN were normal. Serum electrophoresis and immunofixation ruled out any superimposed monoclonal gammopathy. Immediate induction chemotherapy was started with intravenous infusions of daunorubicin, cytarabin, and gemtuzumab.

Ten days later, the patient developed pancytopenia together with fever but without any hemodynamic instability. Caspofungin and intravenous broad spectrum antibiotics, including piperacillintazobactam at 4 g four times a day, were initiated together with a vancomycin pulse of 1.5 g followed by a 3-g daily continuous vancomycin perfusion. Three days later, fever decreased, but BP increased to 160/100 mmHg. Blood analysis showed a serum creatinine increase from 0.47 at baseline to 4.23 mg/dl, confirming ARF, whereas urine analysis did not detect proteinuria or hematuria. Seventy-two hours after the first vancomycin infusion, its concentration was still measured at 87 mg/L (expected value =15-20 mg/L) (Figure 1A). At that time, vancomycin nephrotoxicity was suspected, and the drug was immediately discontinued, whereas piperacillin-tazobactam (Tazocilin) and caspofungin were maintained for an additional 10 days. At day 17, because renal failure persisted, a kidney biopsy was performed to investigate the mechanism of renal impairment and ascertain renal prognosis. Four months later, the patient eventually recovered normal renal function (serum creatinine =0.73 mg/dl).

Histologic analysis showed the absence of immune deposits or glomerular injury but severe acute tubular necrosis (ATN). Some tubular lumens contained nonspecific proteinaceous casts on light microscopy (Figure 1B). These casts were initially described as "light-voided black ovoid structures or atypical proteinaceous casts." These original formations did not show any birefringence fringe after polarization. We eventually examined the composition of these casts by scanning electron microscopy (Supplemental Figure 1, B, D, andF) and infrared microspectroscopy (Supplemental Figure 1, A, C, andE), suspecting that vancomycin might be associated with cast genesis.

Nano-to microspherical formations that corresponded with the vancomycin spectral signature (i.e., similar to a desiccated vancomycin solution deposited on control slides) were found within the casts. This signal was not detected in the surrounding tissue (despite a 6.25-mm spatial resolution) (Supplemental Figure 1). Additional analysis by transmission electron microscopy with immunogold labeling confirmed that intratubular casts are formed in the part of vancomycin aggregates that are noncrystalline spherical formations, mainly nanometric in size (100-900 nm) (Figure 1C). We developed the immunohistologic detection of vancomycin on frozen kidney (Figure 1D) and paraffin-embedded sections (Figure 1E). Moreover, vancomycin deposits colocalized with uromodulin inside the casts (Figure 1F compared with Figure 1E). Uromodulin was also found in the Bowman's space, suggesting the obstructive nature of vancomycin-associated casts (Supplemental Figure 2A). A CD68+ macrophagic infiltrate was also observed surrounding the casts and within the kidney's interstitium, suggesting that pathologic casts might induce an inflammatory process (Supplemental Figure 2B). To further confirm the pathogenicity of vancomycin-associated casts, we retrospectively examined eight additional renal biopsies with ATN that had been performed in the Table 1. Clinical characteristics of patients with the retrospective diagnosis of vancomycin-associated cast nephropathy on the basis of their renal biopsies clinical context of high-vancomycin trough levels preceding ARF (Table 1).

Characteristic Patient A B C D E F G H I Sex W M M M M W W M W Age,
To do so, we developed the staining of vancomycin from paraffin-embedded sections (Figure 2, Supplemental Figure 3). Using this technique, in each patient, we were able to detect vancomycin-associated casts in a similar location as our index patient (Figure 2, A-H). One patient who developed typical acute myeloma cast nephropathy and received intravenous vancomycin at the same time was used as a genuine negative control; vancomycin trough level was normal. His renal biopsy did not show any vancomycin staining (Figure 2I, Supplemental Figure 3), the same has been observed in other forms of ATN.

To further understand the kinetics of vancomycin-associated cast formation, injected vancomycin into mice. Observations were made between 2 hours and 2 days after the injection of vancomycin. We confirmed experimentally the nephrotoxicity of vancomycin and ruled out any pathologic role of piperacillintazobactam (Figure 3A). Mouse kidney sections were examined by either the aforementioned techniques or in vivo imaging (Figure 3, B-F). A spectral vancomycin signature was detected in pathologic tubular casts in the kidneys of vancomycin-treated mice (Figure 3, B andC, Supplemental Figure 1, E andF). We also examined the formation of vancomycin casts by in vivo imaging at day 2 (Supplemental Movies 1 and 2) and after 2 hours (Figure 3, E andF). Casts have been highlighted using a vancomycin-boron-dipyrromethene-associated dye. Vancomycin obstructive casts appear very early after the injection of vancomycin and are easily detected as early as 40 minutes postinjection. Cast distribution appears very patchy across the renal cortex (Figures 3D and4). Furthermore the staining of uromodulin was eventually found in the glomeruli (Supplemental Figure 4), confirming the obstructive nature of the casts.

We describe a previously unsuspected mechanism of vancomycin-induced ARF due to vancomycin-associated tubular cast formation. Pathologic obstructive casts are made of vancomycin nanospheres entangled with uromodulin. We showed the toxicity of vancomycin from the study of paradigmatic patients in whom serum creatinine elevation and ARF appeared immediately after infusion of vancomycin. All patients were normovolemic, and one half received vancomycin in the absence of any concomitant nephrotoxic agent. They previously had normal renal function. Kidney biopsies performed as long as 17 days after vancomycin discontinuation showed ATN and tubular casts containing vancomycin. The detection of vancomycin within the casts was confirmed by several independent methods: infrared spectroscopy, immunohistochemistry, and electron microscopy with immunogold labeling. Lastly, we replicated in mice vancomycininduced cast formation and determined intratubular obstruction by in vivo imaging. These data were also confirmed by the pathologic detection of uromodulin on glomeruli.

Published animal data suggest that vancomycin induces direct oxidative stress on proximal tubular cells. However, we evidenced a similar cast formation when examining published pictures from these studies. [START_REF] Oktem | In vivo evidences Suggesting the role of oxidative stress in pathogenesis of vancomycin-induced nephrotoxicity: Protection by erdosteine[END_REF][START_REF] Sabler | Exposure to hyperbaric oxygen intensified vancomycin-induced nephrotoxicity in rats[END_REF][START_REF] Celik | Protective effects of different antioxidants and amrinone on vancomycininduced nephrotoxicity[END_REF] However, the obstructive mechanism described in our study cannot exclude an additional and direct toxicity of vancomycin on tubular cells that has been suggested before. [START_REF] Oktem | In vivo evidences Suggesting the role of oxidative stress in pathogenesis of vancomycin-induced nephrotoxicity: Protection by erdosteine[END_REF][START_REF] Sabler | Exposure to hyperbaric oxygen intensified vancomycin-induced nephrotoxicity in rats[END_REF][START_REF] Celik | Protective effects of different antioxidants and amrinone on vancomycininduced nephrotoxicity[END_REF] Intratubular crystal-associated drug nephropathy is a well known cause of AKI and related to the intratubular precipitation of antibiotics, 8 but it has never been described with noncrystal formations, such as vancomycin.

Our patients had several risk factors that are typically associated with vancomycin nephrotoxicity: high trough concentrations, 9,10 obesity, [START_REF] Hall Rg 2nd | Empiric guideline-recommended weight-based vancomycin dosing and nephrotoxicity rates in patients with methicillin-resistant Staphylococcus aureus bacteremia: A retrospective cohort study[END_REF] concomitant administration of piperacillin-tazobactam, and in a few patients, aminoglycosides. [START_REF] Burgess | Comparison of the incidence of vancomycin-induced nephrotoxicity in hospitalized patients with and without concomitant piperacillintazobactam[END_REF][START_REF] Rybak | Nephrotoxicity of vancomycin, alone and with an aminoglycoside[END_REF] A recent study has suggested that piperacillin-tazobactam increases vancomycin-induced nephrotoxicity, but we failed to reproduce this finding experimentally in mice. [START_REF] Burgess | Comparison of the incidence of vancomycin-induced nephrotoxicity in hospitalized patients with and without concomitant piperacillintazobactam[END_REF] All of our patients had normal renal function at baseline, and their vancomycin dosage was under 4 g/d, a threshold usually associated with increased vancomycin toxicity. [START_REF] Lodise | Larger vancomycin doses (at least four grams per day) are associated with an increased incidence of nephrotoxicity[END_REF] Finally, our experimental models showed that vancomycin alone could be nephrotoxic by precipitating within tubular lumens independent of all other known risk factors, except high-trough vancomycin level, which is a prerequisite. Indeed, blood vancomycin concentration should be closely monitored.

CONCISE METHODS

More details are in Supplemental Material.

Figure 1 .

 1 Figure 1. ARF associated with nanospheric vancomycin tubular casts entangled with uromodulin. (A) Graph showing the evolution of the patient's renal function with vancomycin plasma concentrations over time. (B) Human kidney biopsy (17 days after vancomycin injection) analysis revealed nonspecific granular tubular and proteinaceous cast formation (arrows) and ATN lesions (hematoxylin and eosin staining). Scale bar, 50 mm. (C) Transmission electron microscopy analysis with immunogold labeling detecting vancomycin nanospheres in a tubular cast formation (original, magnification, 318,500), with detail on nanospheric vancomycin (labeled with 20-nm gold particles). (D) Staining with antivancomycin antibody (frozen section) revealed the specific accumulation of vancomycin in the tubular lumen. Note the absence of vancomycin in the surrounding tissue. Original magnification, 3400. (E) Serial biopsy (3-mm thick) showing one cast colocalizing vancomycin and (F) uromodulin showing the obstructive nature of vancomycin-associated casts. Horseradish peroxidase staining. Original magnification, 3600.

Figure 2 .

 2 Figure 2. Vancomycin detection in a suspected case of vancomycin-associated cast nephropathy (A to H) Vancomycin staining on paraffin-embedded renal biopsy samples from 8 patients with unexplained acute renal failure occurring in the clinical context of high trough vancomycin plasma levels. Patients did not develop circulatory shock. Concomitant high trough vancomycin plasma levels were identified retrospectively from our center's database (1999-2016). Patients' clinical characteristics (A to H) are shown in Table 1. Anti-vancomycin mouse monoclonal antibody (1:1000, Abbot, 6E44-21) has been used on 4 mm dewaxed kidney sections. Nuclei are stained with hematoxylin. A strong staining has been detected in all cases in tubular lumens, with a patchy distribution among the renal cortex. Negative control has been shown in (I). Magnification 3 200.

Figure 3 .

 3 Figure 3. Vancomycin-associated cast nephropathy confirmed in mice. (A). Following vancomycin injection (n=4 per group) mice developed acute renal failure (observation made at day 2) that is not aggravated by concomitant Tazocilin injection. *P,0.05. (B) Kidney injuries have been visible as early as two days after vancomycin injection on Masson's trichrome. Acute tubular necrosis is visible with granular material found in the tubular lumen. Scale bar: 50 mm. (C) Lower magnification showing the patchy distribution of vancomycin-associated casts. Scale bar: 50 mm. (D) Intravital confocal microscopy (enlarged field reconstructed) showing the patchy distribution of vancomycin-associated tubular casts (green) magnified by a vancomycin-linked dye (boron-dipyrromethene) at 2 hours following vancomycin injection. Scale bar: 200 mm. Control mice (E) compared to vancomycin-injected mice at day 2 post-vancomycin injection (F) observed with intravital wide-field microscopy. Tubular cast formation magnified by a vancomycin-linked dye (boron-dipyrromethene) is easily visible in (F). *P,.05. Scale bar: 100 mm.

Figure 4 .

 4 Figure 4. In vivo formation of obstructive vancomycin-associated tubular casts in mice. Intravital confocal microscopy analyses (enlarged field reconstructed) show that intratubular cast formation occurs nearly 40 minutes after vancomycin injection. The same kidney cortex area has been observed sequentially at different time points. Tubular casts are not observed before vancomycin injection (upper left panel). When vancomycin and vancomycin-linked green fluorescent dye (boron-dipyrromethene) are injected intravenously, a green fluorescence appears (upper right panel) in capillary vessels, reflecting the intravascular circulation of vancomycin. At 45 minutes (lower left panel), the first vancomycin-associated intratubular casts (green) are now visible, with their number increasing further at 90 minutes (lower right panel). Scale bar, 200 mm.

Journal of the American Society of Nephrology J Am Soc Nephrol 28: ccc-ccc, 2017

J Am Soc Nephrol 28: ccc-ccc, 2017Vancomycin Cast Nephropathy

This article contains supplemental material online at http://jasn.asnjournals.org/lookup/suppl/doi:10. 1681/ASN.2016080867/-/DCSupplemental.

ACKNOWLEDGMENTS We thank Prof. Pierre Ronco (Assistance Publique -Hôpitaux de Paris, Hôpital Tenon, Néphrologie et Dialyses, Paris, France) as well as Prof. Patrice Callard, Dr. David Buob, and Prof. Isabelle Brocheriou (Assistance Publique -Hôpitaux de Paris, Hôpital Tenon, Anatomie et cytologie pathologiques, Paris, France) for their constructive comments. We also thank the teams of the Plate-forme d'Imagerie Cellulaire Pitié Salpêtrière (Paris, France) and the Plateforme d'Imagerie et de Cytométrie de Tenon (Paris, France) and are particularly grateful to Aurélien Dauphin for the generation of confocal microscopy images.

Y.L. and L.M. had full access to all of the data in the study and take responsibility for the integrity of the data and the accuracy of the data analysis.