Vincent Labatut
email: vlabatut@gsu.edu.tr

Hocine Cherifi
email: hocine.cherifi@u-bourgogne.fr

Accuracy Measures for the Comparison of Classifiers

Keywords: Classification, Accuracy Measure, Classifier Comparison I

The selection of the best classification algorithm for a given dataset is a very widespread problem. It is also a complex one, in the sense it requires to make several important methodological choices. Among them, in this work we focus on the measure used to assess the classification performance and rank the algorithms. We present the most popular measures and discuss their properties. Despite the numerous measures proposed over the years, many of them turn out to be equivalent in this specific case. They can also lead to interpretation problems and be unsuitable for our purpose. Consequently, the classic overall success rate or marginal rates should be preferred for this specific task.

INTRODUCTION

The comparison of classification algorithms is a complex and open problem. First, the notion of performance can be defined in many ways: accuracy, speed, cost, readability, etc. Second, an appropriate tool is necessary to quantify this performance. Third, a consistent method must be selected to compare the measured values.

Performance is most of the time expressed in terms of accuracy, which is why, in this work, we focus on this point. The number of accuracy measures appearing in the classification literature is extremely large. Some were specifically designed to compare classifiers [START_REF] Koukoulas | Introducing new indices for accuracy evaluation of classified images representing semi-natural woodland environments[END_REF], but most were initially defined for other purposes, such as measuring the association between two random variables [START_REF] Goodman | Measures of association for cross classification[END_REF], the agreement between two raters [START_REF] Cohen | A Coefficient of Agreement for Nominal Scales[END_REF] or the similarity between two sets [START_REF] Jaccard | The distribution of the flora in the alpine zone[END_REF]. Furthermore, the same measure may have been independently developed by different authors, at different times, in different domains, for different purposes, leading to very confusing typology and terminology. Besides its purpose or name, what characterizes a measure is the definition of the concept of accuracy it relies on. Concretely, this means that measures are designed to focus on a specific aspect of the overall classification results [START_REF] Foody | Status of land cover classification accuracy assessment[END_REF]. This leads to measures with different interpretations, and some do not even have any clear interpretation. Finally, the measures may also differ in the nature of the situations they can be applied to: binary vs. multiclass problem, mutually exclusive classes vs. fuzzy classes [START_REF] Jager | Measures of classification accuracy based on fuzzy similarity[END_REF], sampling design used to retrieve the data [START_REF] Stehman | Comparing thematic maps based on map value[END_REF], etc. Many different measures exist, but yet, there is no such thing as a perfect measure, which would be the best in every situation [START_REF] Stehman | Selecting and interpreting measures of thematic classification accuracy[END_REF]: an appropriate measure must be chosen according to the classification context and objectives. Because of the overwhelming number of measures and of their heterogeneity, choosing the most adapted one is a difficult problem. Moreover, it is not always clear what the measures properties are, either because they were never rigorously studied, or because specialists do not agree on the question (e.g. the question of chance-correction [START_REF] Guggenmoos-Holzmann | How Reliable Are Chance-Corrected Measures of Agreement[END_REF]). Maybe for these reasons, authors very often select an accuracy measure by relying on the tradition or consensus observed in their field. The point is then more to use the same measure than their peers rather than the most appropriate one.

In this work, we reduce the complexity of choosing an accuracy measure by restraining our analysis to a very specific but widespread, situation. We discuss the case where one wants to select the best classification algorithm to process a given data set. An appropriate way to perform this task would be to study the data properties first, then to select a suitable classification algorithm and determine the most appropriate parameter values, and finally to use it to build the classifier. But not everyone has the statistical expertise required to perform this analytic work. Therefore, in practice, the most popular method consists in sampling a training set from the considered data, building various classifiers with different classification algorithms and parameters, and then comparing their performances empirically on some test sets sampled from the same data. Finally, the classifier with highest performance is selected and used on the rest of the data.

We will not address the question of the method used to compare performances. Instead, we will discuss the existing accuracy measures and their relevance to our specific context. We will be focusing on comparing classifiers with discrete outputs (by opposition to classifier outputting real scores or probabilities). Previous works already compared various measures, but with different purposes. Some authors adopted a general, context-independent approach and numerically compared measures by studying their correlation when applied to several real-world datasets [START_REF] Caruana | Data Mining in Metric Space: An Empirical Analysis of Supervised Learning Performance Criteria[END_REF][START_REF] Liu | Comparative assessment of the measures of thematic classification accuracy[END_REF]. But as we underlined before, any comparison must be performed relatively to the application context. Other authors did focus on specific contexts, but these were different from our own. For instance, several studies of this kind exist in the remote sensing field [START_REF] Albatineh | On Similarity Indices and Correction for Chance Agreement[END_REF][START_REF] Congalton | A Review of Assessing the Accuracy of Classifications of Remotely Sensed Data[END_REF].

In the next section, we introduce the notations used in the rest of the paper. In section III, we review the main measures used as accuracy measures in the classification literature. Finally, in section IV, we compare and discuss their relevance relatively to our specific case.

II. NOTATIONS AND TERMINOLOGY

Consider the problem of estimating classes for a test set containing instances. The true classes are noted , whereas the estimated classes, as defined by the considered classifier, are noted ( ).

Most measures are not processed directly from the raw classifier outputs, but from the confusion matrix built from these results. This matrix represents how the instances are distributed over estimated (rows) and true (columns) classes:

The terms ( ) correspond to the number of instances put in class number i by the classifier (i.e. ), when they actually belong to class number j (i.e. ). Consequently, diagonal terms ( ) correspond to correctly classified instances, whereas off-diagonal terms ( ) represent incorrectly classified ones. Some measures are defined in terms of proportions rather than counts, so we additionally define the terms .

The sums of the confusion matrix elements over row i and column j are noted and , respectively. The corresponding sums of proportions, and , are defined the same way.

When considering one class i in particular, one may distinguish four types of instances: true positives (TP) and false positives (FP) are instances correctly and incorrectly classified as , whereas true negatives (TN) and false negatives (FN) are instances correctly and incorrectly not classified as , respectively. The corresponding counts are defined as , , and , respectively. The corresponding proportions are noted , , and , respectively.

III. SELECTED MEASURES

A. Nominal Association Measures

A measure of association is a numerical index, a single number, which describes the strength or magnitude of a relationship. Many association measures were used to assess classification accuracy, such as: chi-square-based measures ( coefficient, Pearson's , Cramer's , etc. [START_REF] Goodman | Measures of association for cross classification[END_REF]), Yule's coefficients, Matthew's correlation coefficient, Proportional reduction in error measures (Goodman & Kruskal's and , Theil's uncertainty coefficient, etc.), mutual information-based measures [START_REF] Finn | Use of the average mutual information index in evaluating classification error and consistency[END_REF] and others. Association measures quantify how predictable a variable is when knowing the other one. They have been applied to classification accuracy assessment by considering these variables are defined by the distributions of instances over the true and estimated classes, respectively.

In our context, we consider the distribution of instances over estimated classes, and want to measure how much similar it is to their distribution over the true classes. The relationship assessed by an association measure is more general [START_REF] Goodman | Measures of association for cross classification[END_REF], since a high level of association only means it is possible to predict estimated classes when knowing the true ones (and vice-versa). In other terms, a high association does not necessary correspond to a match between estimated and true classes. For instance, if one considers a binary classification problem, both perfect classification and perfect misclassification give the same maximal association value.

Consequently, a confusion matrix can convey both a low accuracy and a high association at the same time, which makes association measures unsuitable for accuracy assessment.

B. Overall Success Rate

Certainly the most popular measure for classification accuracy [START_REF] Witten | Data Mining: Practical Machine Learning Tools and Techniques[END_REF], the overall success rate is defined as the trace of the confusion matrix, divided by the total number n of classified instances: [START_REF] Koukoulas | Introducing new indices for accuracy evaluation of classified images representing semi-natural woodland environments[END_REF] This measure is multiclass, symmetrical, and ranges from (perfect misclassification) to (perfect classification). Its popularity is certainly due to its simplicity, not only in terms of processing but also of interpretation, since it corresponds to the observed proportion of correctly classified instances.

C. Marginal Rates

We gather under the term marginal rates a number of widely spread asymmetric class-specific measures. The TP Rate and TN Rate are both reference-oriented, i.e. they consider the confusion matrix columns (true classes). The former is also called sensitivity [START_REF] Witten | Data Mining: Practical Machine Learning Tools and Techniques[END_REF], producer's accuracy [START_REF] Congalton | A Review of Assessing the Accuracy of Classifications of Remotely Sensed Data[END_REF] and Dice's asymmetric index [START_REF] Dice | Measures of the amount of ecologic association between species[END_REF]. The latter is alternatively called specificity [START_REF] Witten | Data Mining: Practical Machine Learning Tools and Techniques[END_REF].

(2)

(3)

The correspondent estimation-oriented measures, which focus on the confusion matrix rows (estimated classes), are the Positive Predictive Value (PPV) and Negative Predictive Value (NPV) [START_REF] Witten | Data Mining: Practical Machine Learning Tools and Techniques[END_REF]. The former is also called precision [START_REF] Witten | Data Mining: Practical Machine Learning Tools and Techniques[END_REF], user's accuracy [START_REF] Congalton | A Review of Assessing the Accuracy of Classifications of Remotely Sensed Data[END_REF] and Dice's association index [START_REF] Dice | Measures of the amount of ecologic association between species[END_REF].

(4) [START_REF] Foody | Status of land cover classification accuracy assessment[END_REF] TNR and PPV are related to type I error (FP) whereas TPR and NPV are related to type II error (FN). All four measures range from to , and their interpretation is straightforward. TPR (resp. TNR) corresponds to the proportion of instances belonging (resp. not belonging) to the considered class and actually classified as such. PPV (resp. NPV) corresponds to the proportion of instances predicted to belong (resp. not to belong) to the considered class, and which indeed do (resp. do not).

D. F-measure and Jaccard Coefficient

The F-measure corresponds to the harmonic mean of PPV and TPR [START_REF] Witten | Data Mining: Practical Machine Learning Tools and Techniques[END_REF], therefore it is class-specific and symmetric. It is also known as Sørensen's similarity coefficient [START_REF] Sørensen | A method of establishing groups of equal amplitude in plant sociology based on similarity of species and its application to analyses of the vegetation on Danish commons[END_REF], Dice's coincidence index [START_REF] Dice | Measures of the amount of ecologic association between species[END_REF] and Hellden's mean accuracy index [START_REF] Hellden | A test of landsat-2 imagery and digital data for thematic mapping illustrated by an environmental study in northern Kenya[END_REF]: [START_REF] Jager | Measures of classification accuracy based on fuzzy similarity[END_REF] It can be interpreted as a measure of overlapping between the true and estimated classes (other instances, i.e. TN, are ignored), ranging from (no overlap at all to (complete overlap).

The measure known as Jaccard's coefficient of community was initially defined to compare sets [START_REF] Jaccard | The distribution of the flora in the alpine zone[END_REF], too. It is a classspecific symmetric measure defined as: [START_REF] Stehman | Comparing thematic maps based on map value[END_REF] It is alternatively called Short's measure [START_REF] Short | The landsat tutorical workbook-Basics of satellite remote sensing[END_REF]. For a given class, it can be interpreted as the ratio of the estimated and true classes intersection to their union (in terms of set cardinality). It ranges from (no overlap) to (complete overlap). It is linearly related to the F-measure [START_REF] Türk | Letter to the Editor: Map Evaluation and "Chance Correction[END_REF]:

, which is why we describe it in the same section.

E. Classification Success Index

The Individual Classification Success Index (ICSI), is a class-specific symmetric measure defined for classification assessment purpose [START_REF] Koukoulas | Introducing new indices for accuracy evaluation of classified images representing semi-natural woodland environments[END_REF]: [START_REF] Stehman | Selecting and interpreting measures of thematic classification accuracy[END_REF] The terms and correspond to the proportions of type I and II errors for the considered class, respectively. ICSI is hence one minus the sum of these errors. It ranges from (both errors are maximal, i.e. ) to (both errors are minimal, i.e. ), but the value does not have any clear meaning. The measure is symmetric, and linearly related to the arithmetic mean of TPR and PPV, which is itself called Kulczynski's measure [START_REF] Kulczynski | Die Pflanzenassociationen der Pienenen[END_REF].

The Classification Success Index (CSI) is an overall measure defined by averaging ICSI over all classes [START_REF] Koukoulas | Introducing new indices for accuracy evaluation of classified images representing semi-natural woodland environments[END_REF].

F. Agreement Coefficients

A family of chance-corrected inter-rater agreement coefficients has been widely used in the context of classification accuracy assessment. It relies on the following general formula: [START_REF] Guggenmoos-Holzmann | How Reliable Are Chance-Corrected Measures of Agreement[END_REF] Where and are the observed and expected agreements, respectively. The idea is to consider the observed agreement as the result of an intended agreement and a chance agreement. In order to get the intended agreement, one must estimate the chance agreement and remove it from the observed one.

Most authors use

, but disagree on how the chance agreement should be formally defined, leading to different estimations of . For his popular , Cohen used the product of the confusion matrix marginal proportions [START_REF] Cohen | A Coefficient of Agreement for Nominal Scales[END_REF]: . Scott's relies instead on the class proportions measured on the whole data set, noted [START_REF] Scott | Reliability of Content Analysis: The Case of Nominal Scale Coding[END_REF]:

. Various authors, including Maxwell [START_REF] Maxwell | Coefficients of agreement between observers and their interpretation[END_REF] for his Random Error (MRE), made the assumption classes are evenly distributed: .

The problems of assessing inter-rater agreement and classifier accuracy are slightly different though. Indeed, in the former, the true class distribution is unknown, whereas in the latter it is completely known. Moreover, both raters are considered as equivalent and interchangeable, whereas a classifier is evaluated relatively to a reference (true classes), whose location in the confusion matrix matters. The presented corrections for chance are defined in function of these specific traits of the inter-rater agreement problem, which means they might not be relevant in our situation.

G. Ground Truth Index

Türk's Ground Truth Index (GTI) is another chancecorrected measure, but this one was defined specially for classification accuracy assessment [START_REF] Türk | GT index: A measure of the success of predictions[END_REF]. Türk supposes the classifier has two components: one is always correct, and the other is random, which means it is correct only sometimes. A proportion of the instances are supposed to be classified by the infallible classifier, the rest being processed randomly. The coefficient is iteratively estimated and its value is considered as a measure of accuracy, interpreted as the proportion of instances the classifier will always classify correctly, even when processing other data. Both class-specific and overall versions exist, depending on whether one considers to be the same for all classes or class-specific.

The way this measure handles chance correction is more adapted to classification than the agreement coefficients [START_REF] Türk | Letter to the Editor: Map Evaluation and "Chance Correction[END_REF]. However, it has several limitations regarding the processed data: it cannot be used with less than three classes, or on perfectly classified data, and most of all it relies on the hypothesis of quasi-independence, which is rarely met in practice [START_REF] Liu | Comparative assessment of the measures of thematic classification accuracy[END_REF].

IV. DISCUSSION

A. Class Focus

As illustrated in the previous section, a measure can assess the accuracy for a specific class or over all classes. The former is adapted to situations where one is interested in a given class, or wants to conduct a class-by-class analysis of the classification results.

It is possible to define an overall measure by combining class-specific values measured for all classes, for example by averaging them, like in CSI. However, even if the considered class-specific measure has a clear meaning, it is difficult to give a straightforward interpretation to the resulting overall measure, other than in terms of combination of the classspecific values. Inversely, it is possible to use an overall measure to assess a given class accuracy, by merging all classes except the considered one [START_REF] Goodman | Measures of association for cross classification[END_REF]. In this case, the interpretation is straightforward though, and depends directly on the overall measure.

One generally uses a class-specific measure in order to distinguish classes in terms of importance. This is not possible with most basic overall measures, because they consider all classes to be equally important. Certain more sophisticated measures allow associating a weight to each class, though. However, a more flexible method makes this built-in feature redundant. It consists in associating a weight to each cell in the confusion matrix, and then using a regular (unweighted) overall measure [START_REF] Türk | Letter to the Editor: Map Evaluation and "Chance Correction[END_REF]. This method allows distinguishing, in terms of importance, not only classes, but also any possible kind of classification error.

B. Functional Relationships

Some of the measures we presented are monotonically related, and this property takes a particular importance in our situation. Indeed, our goal is to sort classifiers depending on their performance on a given data set. If two measures are monotonically related, then the order will be the same for both measures. This makes the F-measure and Jaccard's coefficient similar to us, and so are the ICSI and Kulczynski's measure. Moreover, it is well known that various combinations of two quantities can be sorted by increasing order, independently from the considered quantities: minimum, harmonic mean, geometric mean, arithmetic mean, quadratic mean, maximum [START_REF] Bullen | Handbook of Means and Their Inequalities[END_REF]. If the quantities belong to , we can even put their product at the beginning of the previous list, as the smallest combination. If we consider the presented measures, this means all combinations of the same marginal rates are the same to us. For instance, the sensitivity-precision product will always be smaller than the F-measure (harmonic mean), which in turn will always be smaller than Kulczynski's measure (arithmetic mean). It finally turns out all these measures are similar in our context: sensitivity-precision product, F-measure, Kulczynski's measure, Jaccard's coefficient and ICSI (and certainly others). Besides these combinations of TPR and PPV, this also holds for various measures corresponding to combinations of TPR and TNR, not presented here for space issues [START_REF] Witten | Data Mining: Practical Machine Learning Tools and Techniques[END_REF][START_REF] Cicchetti | High agreement but low kappa: II. Resolving the paradoxes[END_REF].

C. Range

In the classification context, one can consider two extreme situations: perfect classification (i.e. diagonal confusion matrix) and perfect misclassification (i.e. all diagonal elements are zeros). The former should be associated to the upper bound of the accuracy measure, and the latter to its lower bound.

Measure bounds can either be fixed or depend on the processed data. The former is generally considered as a favorable trait [START_REF] Zegers | A Family of Association Coefficients for Metric Scales[END_REF], because it allows comparing values measured on different data sets without having to normalize them for scale matters. Moreover, having fixed bounds makes it easier to give an absolute interpretation of the measured features.

In our case, we want to compare classifiers evaluated on the same data. Furthermore, we are interested in their relative accuracies, i.e. we focus only on their relative differences. Consequently, this trait is not necessary. But it turns out most authors normalized their measures in order to give them fixed bounds (usually or ). Note their exact values are of little importance, since any measure defined on a given interval can easily be rescaled to fit another one. Thus, several supposedly different measures are actually the same, but transposed to different scales [START_REF] Zegers | A Family of Association Coefficients for Metric Scales[END_REF].

D. Interpretation

Our goal is to compare classifiers on a given dataset, for which all we need is the measured accuracies. In other words, numerical values are enough to assess which classifier is the best on the considered data. But identifying the best classifier is useless if we do not know the criteria underlying this discrimination, i.e. if we are not able to interpret the measure. For instance, being the best in terms of PPV or TPR has a totally different meaning, since these measures focus on type I and II errors, respectively.

Among the measures used in the literature to assess classifiers accuracy, some have been designed analytically, in order to have a clear interpretation (e.g. Jaccard's coefficient [START_REF] Jaccard | The distribution of the flora in the alpine zone[END_REF]). Sometimes, this interpretation is questioned, or different alternatives exist, leading to several related measures (e.g. agreement coefficients). In some other cases, the measure is an ad hoc construct, which can be justified by practical constraints or observation, but may lack an actual interpretation (e.g. CSI). Finally, some measures are heterogeneous mixes of other measures, and have no direct meaning (e.g. the combination of OSR and marginal rates described in [START_REF] Fung | The determination of optimal threshold levels for change detection using various accuracy indices[END_REF]). They can only be interpreted in terms of the measures forming them, and this is generally considered to be a difficult task.

E. Correction for Chance

Correcting measures for chance is still an open debate. First, authors disagree on the necessity of this correction, depending on the application context [START_REF] Stehman | Comparing thematic maps based on map value[END_REF][START_REF] Türk | Letter to the Editor: Map Evaluation and "Chance Correction[END_REF]. In our case, we want to generalize the accuracy measured on a sample to the whole population. In other terms, we want to distinguish the proportion of success the algorithm will be able to reproduce on different data from the lucky guesses made on the testing sample, so this correction seems necessary.

Second, authors disagree on the nature of the correction term, as illustrated in our description of agreement coefficients. We can distinguish two kinds of corrections: those depending only on the true class distribution (e.g. Scott's and Maxwell's) and those depending also on the estimated class distribution (e.g. Cohen's and Türk's). The former is of little practical interest for us, because such a measure is linearly related to the OSR (the correction value being the same for every tested algorithm), and would therefore lead to the same ordering of algorithms. The second is more relevant, but there is still concern regarding how chance should be modeled. Indeed, lucky guesses depend completely on the algorithm behind the considered classifier. In other words, a very specific model would have to be designed for each algorithm in order to efficiently account for chance, which seems difficult or even impossible.

V. CONCLUSION

In this work, we reviewed the main measures used for accuracy assessment, from a specific classification perspective. We consider the case where one wants to compare different classification algorithms by testing them on a given data sample, in order to determine which one will be the best on the sampled population.

In this situation, it turns out several traits of the measures are not relevant to discriminate them. First, all monotonically related measures are similar to us, because they all lead to the same ordering of algorithms. This notably discards a type of chance correction. Second, their range is of little importance, because we are considering relative values.

Moreover, a whole subset of measures associating weights to classes can be discarded, because a simpler method allows distinguishing classes in terms of importance while using an unweighted multiclass measure. Concerning chance-correction, it appears it is needed for our purpose; however no existing estimation for chance seems relevant. Finally, complex measures based on the combination of other measures are difficult or impossible to interpret correctly.

Under these conditions, we advise the user to choose the simplest measures, whose interpretation is straightforward. For overall accuracy assessment, the OSR seems to be the most adapted. If the focus has to be made on a specific class, we recommend using both the TPR and PPV, or a meaningful combination such as the F-measure. A weight matrix can be used to specify differences between classes or errors.

We plan to complete this work by focusing on the slightly different case of classifiers with real-valued output. This property allows using additional measures such as the area under the ROC curve and various error measures [START_REF] Witten | Data Mining: Practical Machine Learning Tools and Techniques[END_REF].

ACKNOWLEDGMENT

The authors are grateful to the referees for their constructive comments, which helped improving the quality of this article.


Y Goulam Houssen

I Gusachenko

M.-C Schanne-Klein

J.-M Allain
email: allain@lms.polytechnique.fr

Jean-Marc Allain

Monitoring micrometer-scale collagen organization in rat-tail tendon upon mechanical strain using second harmonic microscopy

Keywords: Second Harmonic Generation, collagen, preconditioning, mechanical properties, tendon

We continuously monitored the microstructure of a rat-tail tendon during stretch/relaxation cycles. To that purpose, we implemented a new biomedical device that combined SHG imaging and mechanical testing modalities. This multi-scale experimental device enabled simultaneous visualization of the collagen crimp morphology at the micrometer scale and measurement of macroscopic strain-stress response. We gradually increased the ultimate strain of the cycles and showed that preconditioning mostly occurs in the first stretching. This is accompanied by an increase of the crimp period in the SHG image. Our results indicate that preconditioning is due to a sliding of microstructures at the scale of a few fibrils and smaller, that changes the resting length of the fascicle. This sliding can reverse on long time scales. These results provide a proof of concept that continuous SHG imaging performed simultaneously with mechanical assay allows analysis of the relationship between macroscopic response and microscopic structure of tissues.

Introduction

The relationship between microscopic macromolecular organization and macroscopic biomechanical properties is a general concern in collagenous tissues. Tendon is a model system in that respect: it is almost unidirectional and does not require biaxial loading in biomechanical studies. It is a highly structured tissue designed for force transmission from muscle to skeleton, mainly composed of type I collagen that forms fibrils of around 200 nm diameter. These fibrils assemble into fibers that further form fascicles with a crimped pattern [START_REF] Kastelic | Multicomposite Structure of Tendon[END_REF]. This hierarchical organization is responsible for the biomechanical properties of the tissue. The macroscopic mechanical properties of tendon are fairly well known [START_REF] Abrahams | Mechanical Behaviour of Tendon In Vitro -A Preliminary Report[END_REF][START_REF] Ker | Mechanics of tendon, from an engineering perspective[END_REF]. Quasi-static stretching of rat-tail tendon gives a non-linear response, with a toe region in the first few percent of strain, followed by a linear region. However, the relationship between these properties and the structure at lower scales (such as fibers or fibrils) remains a subject of strong interest [START_REF] Puxkandl | Viscoelastic properties of collagen: synchrotron radiation investigations and structural model[END_REF][START_REF] Fratzl | Cellulose and collagen: from fibres to tissues[END_REF][START_REF] Screen | Local strain measurement within tendon[END_REF][START_REF] Gupta | In situ multi-level analysis of viscoelastic deformation mechanisms in tendon collagen[END_REF]. The initial toe region is attributed to a straightening of the initially crimped collagen fibrils ( [START_REF] Rigby | The Mechanical Properties of Rat Tail Tendon[END_REF][START_REF] Viidik | Light and electron microscopic studies of collagen fibers under strain[END_REF][START_REF] Hansen | Recruitment of tendon crimp with applied tensile strain[END_REF][START_REF] Franchi | Crimp morphology in relaxed and stretched rat Achilles tendon[END_REF]. The linear region is considered as the response when the collagen fibrils are aligned in the direction of traction, so it is associated with an extension of collagen fibers combined with a sliding of the fibrils in their proteoglycan matrix [START_REF] Fratzl | Cellulose and collagen: from fibres to tissues[END_REF]. Tendons subjected to cycles of stretch/relaxation exhibit a hysteretic stress-strain relationship [START_REF] Fung | Subrupture Tendon Fatigue Damage[END_REF]. This is attributed to a rearrangement of the proteoglycan matrix around the collagen fibrils [START_REF] Puxkandl | Viscoelastic properties of collagen: synchrotron radiation investigations and structural model[END_REF][START_REF] Redaelli | Possible role of decorin glycosaminoglycans in fibril to fibril force transfer in relative mature tendons -a computational study from molecular to microstructural level[END_REF]. Further insight into these microscopic processes requires continuous monitoring of tendon microstructure during mechanical assays. In that respect, few studies using semi-continuous optical imaging of tendon stretching have been reported. Confocal microscopy has been used to track labeled cell nucleus in tendons [START_REF] Screen | Local strain measurement within tendon[END_REF]. Optical Coherence Tomography (OCT) has been used to monitor the straightening of tendon crimps with applied tensile strain [START_REF] Hansen | Recruitment of tendon crimp with applied tensile strain[END_REF]. The later technique provides images of unlabelled collagen fibrils, but with limited contrast because of its low specificity. In that context, Second Harmonic Generation (SHG) microscopy is a valuable technique to investigate the microstructure of collagenous tissues. SHG is a multiphoton mode of contrast at exactly half the excitation wavelength that is highly specific for dense noncentrosymmetric media, such as fibrillar collagen [START_REF] Freund | Connective-tissue Polarity -Optical 2nd-harmonic Microscopy, Crossedbeam Summation, and Small-angle Scattering In Rat-tail Tendon[END_REF][START_REF] Campagnola | Three-dimensional high-resolution second-harmonic generation imaging of endogenous structural proteins in biological tissues[END_REF][START_REF] Stoller | Polarization-modulated second harmonic generation in collagen[END_REF][START_REF] Zoumi | Imaging cells and extracellular matrix in vivo by using second-harmonic generation and two-photon excited fluorescence[END_REF][START_REF] Cox | 3-dimensional imaging of collagen using second harmonic generation[END_REF][START_REF] Zipfel | Live tissue intrinsic emission microscopy using multiphoton-excited native fluorescence and second harmonic generation[END_REF][START_REF] Strupler | Second harmonic imaging and scoring of collagen in fibrotic tissues[END_REF][START_REF] Deniset-Besseau | Measurement of the Second-Order Hyperpolarizability of the Collagen Triple Helix and Determination of Its Physical Origin[END_REF]. As a coherent nonlinear optical process, it is an effective structural probe of the micrometer-scale collagen macromolecular organization [START_REF] Pena | Chiroptical effects in the second harmonic signal of collagens I and IV[END_REF][START_REF] Lacomb | Phase matching considerations in second harmonic generation from tissues: Effects on emission directionality, conversion efficiency and observed morphology[END_REF][START_REF] Rivard | The structural origin of second haronic generation in fascia[END_REF]. Moreover, it is characterized by intrinsic three-dimensional (3D) resolution like two-photon excited fluorescence (2PEF). SHG microscopy hence enables 3D visualization of intact tendons by recording stacks of optically sectioned images with typically 1-3 µm axial resolution. It has been shown to provide high-contrast images of the characteristic crimped pattern of this tissue [START_REF] Stoller | Polarization-modulated second harmonic generation in collagen[END_REF][START_REF] Stoller | Quantitative second-harmonic generation microscopy in collagen[END_REF][START_REF] Williams | Interpreting second-harmonic generation images of collagen I fibrils[END_REF][START_REF] Erikson | Quantification of the second-order nonlinear susceptibility of collagen I using a laser scanning microscope[END_REF][START_REF] Legare | The role of backscattering in SHG tissue imaging[END_REF][START_REF] Gusachenko | Polarization-resolved Second Harmonic microscopy in anisotropic thick tissues[END_REF]. SHG imaging has recently been used to visualize damage in tendons after fatigue tests, and proved efficient at providing 3D maps of fiber orientations [START_REF] Fung | Second Harmonic Generation Imaging and Fourier Transform Spectral Analysis Reveal Damage in Fatigue-Loaded Tendons[END_REF]. This paper reports the development of multi-scale experiments to monitor simultaneously the microstructural organization of tendon and its macroscopic mechanical response. To that end, we perform mechanical measurements while imaging optical sections of a whole tendon fascicle using SHG microscopy and analyzed the stress-strain relationship versus the crimp morphology. To the best of our knowledge, continuous monitoring of tendon microstructure during multiple loading cycles has not been reported yet. We observe that cyclic stretch/relaxation at increasing strains leads to a shift of the toe region accompanied by an increase of the crimp spatial period. Our data confirm that the toe region is due to the straightening of the crimps, and that the linear region is due to the fibrils sliding.

The paper is organized as follows. The next section is devoted to a detailed description of the experimental methods. Then we present our mechanical measurements and SHG images for increasing cyclic stretching. We finally discuss the mechanical data through changes of the microstructure morphology.

Materials and Methods

Rat-tail tendon preparation and clamping to the testing device

The tendons were extracted from Sprague Dawley rat tails (female, 250-300 g, n = 3). The tails were kept frozen until dissection. Incisions were made at the tapered end of the tails using pliers to pull out the tendon fascicles gently. The tendons were rinsed in phosphate buffered saline (PBS) and centrifuged at 4700 rpm (3 × 10 mins) to remove all other tissue components. The tendon fascicles were stored in PBS at 4°C and used within a few days for the experiments.

Tendon fascicles were attached to the testing device as depicted in Figure 1. They were glued with cyanoacrylate onto a metallic plate (20 × 20 × 1 mm 3 ) and coiled on rods inserted in the plate, in an almost symmetric manner to limit any boundary effect due to gluing (see Fig. 1C). The fascicles were suspended vertically and allowed to rotate freely before gluing to a second plate, to minimize initial torsion. We always gently manipulated the tendon fascicles to avoid uncontrolled stretching and maintained them in PBS to prevent drying (except for the ends). The plates were then firmly clamped to small screw-tightened grips within the testing device. The distance between the grips was adjusted to maintain the tendon fascicles almost horizontal, with 1-2 mm of deflection at most. This initial deflection is due to the fascicle weight and cannot be avoided without stretching the fascicle. During all experiments, the tendon fascicles were immersed in PBS using a glass cover-slip just under the sample (see Fig. 1A andB). In some experiments, the tendon fascicle was labeled with fluorescent latex beads (1 µm diameter, L1030, Sigma-Aldrich) to monitor local strains.

Multiphoton imaging

The tendon microstructure was observed by putting the testing device in place of the stage in a custom-built laser scanning microscope (see Fig 1A andB) [START_REF] Strupler | Second harmonic imaging and scoring of collagen in fibrotic tissues[END_REF]. Tendons were illuminated in an upright geometry using circularly-polarized excitation at 860 nm from a femtosecond titanium-sapphire laser (Spectra-Physics). SHG and 2PEF signals were recorded in 2 photon-counting epi-detection channels using appropriate dichroic mirrors and spectral filters [START_REF] Strupler | Second harmonic imaging and scoring of collagen in fibrotic tissues[END_REF]. We used a 20× 0.95 NA water objective with typically 0.40 µm (lateral) × 1.6 µm (axial) resolution near the sample surface. SHG images were recorded using 200 kHz pixel rate and 0.8 µm pixel size, with typically 50 mW excitation power. We verified that we always imaged the same region of the tendon fascicle by using characteristic patterns in the SHG images as references. When necessary, we slightly shifted the microscope stage to compensate for small lateral displacements during loading cycles. Adjustment of the objective focus was usually required because of slight axial displacement of the tendon fascicle. In some experiments, z-stacks of combined SHG and 2PEF images were acquired along the full thickness of the tendon fascicle (every 2 µm). No photodamage of the tendon was observed under these experimental conditions. SHG images were analyzed using ImageJ software (W. Rasband, NIH) to estimate the crimp period (distance measurements along the tendon axis).

Mechanical testing

Mechanical tests were performed at room temperature using a uniaxial RAITH® testing device. The force was measured using a 10N load cell (XFL225D10, FGP Sensors) with 1% accuracy. Stress was obtained as the force divided by the initial cross-sectional area, assuming a cylindrical shape of the tendon fascicle. Typical diameter was 400 µm, as obtained from SHG image-stacks recorded before stretching. Strain was obtained as the ratio of the grips displacement divided by the initial length of the tendon fascicle (typically 26 mm). Displacement and force values were measured every second. The fascicles were stretched at 10 µm/s constant strain rate (around 3×10 -2 % s -1 ).

To investigate possible slipping, we compared the grips displacement to the local displacement on the tendons through the tracking of ink labels, separated by 10 mm and imaged under a binocular microscope. No significant difference was observed. However, we noticed that significant torsion could occur if the tendon fascicle torsion was not carefully minimized before gluing.

Typical loading path was composed by successive series of two stretch/relaxation cycles at the same ultimate strain (see Fig 2A). The ultimate strains of the series were progressively increased up to breaking. The traction device was immobilized during image acquisition.

Results

Stress-strain relationship

Typical strain-stress measurements are displayed in figure 2. Fig. 2A shows the loading path made up of successive series at increasing ultimate strains (2, 4, 5, 8, and 12%). Each point corresponds to a typically 40s pause for SHG imaging. Fig. 2B shows the measured stresses for the whole loading path. We observe similar stress-strain curves to those previously reported [START_REF] Rigby | The Mechanical Properties of Rat Tail Tendon[END_REF], with an initial toe-region followed by a linear part at larger strains -which can be clearly defined in our case as strains higher than 6%. The curves shift to the right as we gradually increase the ultimate strain in the series . For example, the curves for 5% ultimate strain -in blue -seem to be a translation to higher strains of the curves for 4% ultimate strain -in green. The only exception to this observation is the last curve (magenta). We attribute the backward shift at 12% to a longer pause (about 5 minutes) at 0% strain before stretching to 12 %.

The observed stresses are quite reproducible in our experiments but they are significantly lower than previously reported ones [START_REF] Hansen | Recruitment of tendon crimp with applied tensile strain[END_REF][START_REF] Sionkowska | Mechanical properties of UV irradiated rat tail tendon (RTT) collagen[END_REF][START_REF] Screen | The influence of noncollagenous matrix components on the micromechanical environment of tendon fascicles[END_REF]. The slope of the linear part gives an equivalent Young modulus of 75 MPa (considering the last stretching up to 12% strain). This quantitative discrepancy may come from uncontrolled strain application during fascicle preparation. We think however that it is most likely due to the storage in PBS solution, which was reported to lead to fascicle swelling and subsequent artificial increase of the fascicle cross-section for stress calculation [START_REF] Legerlotz | Specimens dimensions influence the measurement of material properties in tendon fascicles[END_REF]. Moreover, we corroborated that fresh tendon fascicles give similar results to previously frozen ones. Our setup also enabled us to monitor the motion of the tendon surface by imaging fluorescent beads. We verified in this way that there was no excessive torsion and that the macroscopic strain corresponded to the local strain, at least for the first stretching.

Figures 2B andC focus on the stress-strain relationship for the series with ultimate strains of 5% and 8% respectively. We also include, for clarity, the last relaxation of the previous series (in black) and the first stretch of the next series (in gray). The set of curves illustrates preconditioning: the curves shift slowly to the right, reaching lower stresses for the same strain for the second cycle. The modification of the stress-strain relationship is strongest during the first stretch of a series. The stresses in the first relaxation and in the second cycle at this ultimate strain are very close, and cannot be easily distinguished given our sensor resolution. Thus we performed only 2 cycles per series at a given ultimate strain.

To quantify more precisely the preconditioning, we measured the maximum stress reached in each cycle (Fig 2E) and the minimal strain at which we could measure a significant stress (Fig 2F). These two quantities could be defined in all the cycles, whereas we could not determine a linear part for cycles with small ultimate strain. As there are 2 cycles per series, there are 2 maximum stresses per ultimate strain in figure 2E. The maximal stress of the second cycle of a series is always smaller than that of the first cycle, illustrating preconditioning. As expected, we see that the ultimate stress increases with the ultimate strain, although this increase is not homogeneous -for example, the ultimate stresses for 4% and 5% ultimate strains are similar.

Figure 2F displays the lowest strains at which the stress is significantly different from the noise versus the ultimate strain. This quantity characterizes the onset of the toe-region of the stress-strain curve. As strain is measured with respect to the initial length of the fascicle, it can be considered as the change in the resting length of the tendon fascicle: the larger the lowest strain, the longer the fascicle. Since a series includes two stretches and two relaxations, there are four points for each ultimate strain, although in most cases the last three are indistinguishable. We call this value the limit lowest strain for a given series. It is reached after the first stretch for each series and increases linearly with the ultimate strain of the series (dotted line in Fig 2F). Linear fitting gives a slope of 0.58 and a lowest strain of 0.44%. The last point at 12 % ultimate strain is lower since we allowed the tendon fascicle to relax for 5 minutes.

Crimp evolution

In order to connect the macroscopic biomechanical behavior of tendon to its microscopic collagen organization, we paused and recorded SHG images at every point on the loading path. Typical images are displayed in Fig. 3A-E and correspond to positions emphasized by orange circles in the loading path depicted in Fig. 3F. These images show ≈ 2-3 µm thick transverse sections within the intact tendon. They display a fibrillar pattern characteristic for the collagen macromolecular organization, in agreement with previous reports [START_REF] Stoller | Quantitative second-harmonic generation microscopy in collagen[END_REF][START_REF] Williams | Interpreting second-harmonic generation images of collagen I fibrils[END_REF][START_REF] Erikson | Quantification of the second-order nonlinear susceptibility of collagen I using a laser scanning microscope[END_REF]. They do not directly show the fibers within the tendon fascicles, but correspond to interference patterns. The SHG signal indeed results from the coherent summation of the SHG response of all the fibrils within the focal volume and is more or less intense depending on the relative position and orientation of these fibrils [START_REF] Lacomb | Phase matching considerations in second harmonic generation from tissues: Effects on emission directionality, conversion efficiency and observed morphology[END_REF][START_REF] Rivard | The structural origin of second haronic generation in fascia[END_REF][START_REF] Strupler | Simulating second harmonic generation from tendon: Do we see fibrils? Biomedical Optics[END_REF].

In most of our experiments, the tendon fascicle exhibited small but well defined crimps at 0% strain (Fig. 3A). At the ultimate strain of a cycle, the fascicle was always straight, with no visible crimped pattern in intact tendons (Fig. 3B andD). In broken tendons, we observed a stretched region in the center and a crimped region near the tendon edge in the same image. During stretching, the crimps disappear at a given strain, and then reappear at higher strain during relaxation, indicating an inelastic behavior.

In order to quantify the crimp pattern along a loading path, we measured the crimp period in the SHG images. Fig. 3G shows the variation of the crimp period at 0% strain (along the dotted line for 0% in Fig. 3F). The period decreases with the cycle number, which means that the crimps are increasingly marked in SHG images (see Fig 3A, C andE) after every cycle. Note that complementary measurements of the crimp amplitude would require 3D reconstruction of the tendon fascicle by recording SHG image stacks instead of only one transverse section. This could result in confused data because of relaxation during the longer pauses which would be needed for SHG imaging.

Fig. 3H displays the crimp period at 2% strain versus the cycles with ultimate strain above 4% (see the dotted line 2% in fig. 3F). There are two points per cycle, the lowest one being always in the relaxation part.

For cycles of a given series (same ultimate strain -same color in Fig. 3), we observe that the period increases between a relaxation and the next stretch.

We note that the crimp period, shown in

Discussion

In this study, we successfully implemented continuous in situ SHG imaging of a tendon subjected to a mechanical assay. SHG microscopy advantageously provided optically sectioned images of the micrometrescale fibrillar organization of the tendon. Our SHG images showed a better contrast than polarized-light microscopy images that integrate the tendon structure over its full thickness. Moreover, the high specificity of SHG microscopy for fibrillar collagen resulted in high-contrast images compared to other 3D techniques like confocal microscopy or optical coherence tomography.

SHG images and mechanical measurements show that preconditioning is associated with a change in the stress-free configuration. It is stronger during the first stretch for a given ultimate strain, although many more cycles are required to reach the limit curve [START_REF] Abrahams | Mechanical Behaviour of Tendon In Vitro -A Preliminary Report[END_REF]. Preconditioning is associated both with a shift of the toe-region of the stress-strain curve to higher strains (Fig 2F ), and a decrease in the crimp period (Fig 3G and H). We attribute this behavior to a sliding of the collagen fibrils with respect to each other, extending the relaxed length of the tendon fascicle. Since SHG observations are sensitive to the fibril scale, our results indicate that sliding occurs at a similar or smaller scale, but they cannot distinguish between fibrillar and intra-fibrillar sliding.

However, the preconditioning is not irreversible. During an extended pause at the initial length of the tendon fascicle, we observed a decrease of the resting length (see Fig 2F). This recovery from previous preconditioning is usually attributed to a reconfiguration of the bonds between fibrils or lower-scale structures [START_REF] Vesentini | Estimation of the binding force of the collagen moleculedecorin core protein complex in collagen fibril[END_REF][START_REF] Ciarletta | A finite dissipative theory of temporary interfibrillar bridges in the extracellular matrix of ligaments and tendons[END_REF]. On a shorter time scale, we observe that the crimp period at 2% strain increases significantly between a relaxation and the next stretching of the tendon fascicle. This was not observed at 0% strain, which indicates that relaxation involves different time scales, and therefore different length scales. We cannot access faster relaxation times than a few tenths of a second, which is the time needed to adjust imaging parameters [START_REF] Gupta | In situ multi-level analysis of viscoelastic deformation mechanisms in tendon collagen[END_REF].

All these considerations indicate that preconditioning is mostly due to sliding of microstructures inside the fascicle. SHG imaging shows that a significant part of the sliding occurs between fibrils; previous works showed that sliding occurs between and inside the fibrils [START_REF] Puxkandl | Viscoelastic properties of collagen: synchrotron radiation investigations and structural model[END_REF][START_REF] Screen | Local strain measurement within tendon[END_REF][START_REF] Cheng | The micro-structural strain response of tendon[END_REF]. We also observed that the preconditioned state is temporary and can recover, at least partially, on a longer time scale [START_REF] Hansen | Recruitment of tendon crimp with applied tensile strain[END_REF].

Conclusion

We developed a new multi-scale experiment to monitor simultaneously the fibrillar collagen architecture at microscopic scale and the strain-stress relationship at macroscopic scale in biological tissues. To that end, we combined a multiphoton microscope and a mechanical device, and we performed simultaneous SHG imaging and controlled mechanical loading of rat-tail tendon. This biomechanical setup has never been reported yet to the best of our knowledge. Our study provides a proof of concept that this device enables analysis of the connection between macroscopic response and microscopic structure of tissues. We showed that mechanical preconditioning mostly arises from sliding and is reversible on long time-scales. This supports the notion that the mechanical behavior of a tendon fascicle relies on its microstructure remodeling.

Our experimental setup can be readily generalized to other mechanical assays and to any other bidimensional tissue, such as skin or cornea. SHG imaging can also be combined with 2PEF imaging to visualize either elastin fibers or labeled cells. Our method should then permit continuous analysis of 3D remodeling of connective tissues in mechanical assays and bring new valuable information to biomechanics of microstructured tissues.

Fig

  Fig 2F appears as a stair-shaped curve. The lowest strain indeed increases abruptly between the first stretch and the first relaxation in a series. Then, when the ultimate strain is increased for a new series, the first stretch has not been preconditioned, and the lowest strain does not change. Consequently, the points are almost indistinguishable and the curve in Fig 2F moves horizontally. The last point at 12 % ultimate strain is lower since we allowed the tendon fascicle to relax for 5 minutes.

  Fig 3H, evolves similarly to the lowest measurable strain, shown in Fig 2E for the same tendon fascicle. Both of them indicate that there is an irreversible deformation of the tendon fascicle during preconditioning.

Figure 3 Figure 1 :

 31 Figure 3

Figure 2 :

 2 Figure 2: Mechanical behavior of a tendon fascicle. (a) Loading path versus the image number; each color is associated with one ultimate strain. (b) Stress-strain curves for the whole loading path. (c)-(d) Stressstrain curves for a series of cycles at 5% and 8% ultimate strain, with color codes as indicated in the insert. (e) Maximum stress per cycle versus the ultimate strain. (f) Initial measurable strain versus the ultimate strain.

Figure 3 :

 3 Figure 3: SHG imaging of the tendon fascicle microstructure. (a)-(e) SHG images at the time points underlined in orange in (f); scale bars: 50 µm. (f) Loading path. (g) Evolution of the crimp spatial period at 0 % strain during the cycles. (h) Same at 2 % strain. These imaging data were recorded simultaneously with the mechanical data displayed in Figure 2, on the same tendon fascicle.

Acknowledgments

We thank Daniel Caldemaison and Viet Tran for their help in the design of the experimental setup.

Conflict of interest statement

All authors have no conflict of interest and nothing to disclose.