Keywords:

Reconstructing the chronology of mammalian evolution is a debated issue between molecule-and fossil-based inferences. A methodological limitation of molecules is the evolutionary rate variation among lineages, precluding the application of the global molecular clock. We considered 2422 first and second codon positions of the combined ADRA2B, IRBP, and vWF nuclear genes for a well-documented set of placentals including an extensive sampling of rodents. Using seven independent calibration points and a maximum-likelihood framework, we evaluated whether molecular and paleontological estimates of mammalian divergence dates may be reconciled by the local molecular clocks approach allowing local constancy of substitution rates with variations at larger phylogenetic scales. To handle the difficulty of choosing among all possible rate assignments for various lineages, local molecular clocks were based on the results of branch-length and two-cluster tests. Extensive lineage-specific variation of evolutionary rates was detected, even among rodents. Cross-calibrations indicated some incompatibilities between divergence dates based on different paleontological references. To decrease the impact of a single calibration point, estimates derived from independent calibrations displaying only slight reciprocal incompatibility were averaged. The divergence dates inferred for the split between mice and rats (~13-19 Myr) was younger than previously published molecular estimates. The most recent common ancestors of rodents, primates and rodents, boreoeutherians, and placentals were estimated to be respectively ~60, 70, 75, and 78 Myr old.

Global clocks, local clocks and quartet dating analyses suggested a Late Cretaceous origin of the crown placental clades followed by a Tertiary radiation of some placental orders like rodents.

INTRODUCTION

With the exponential growth of DNA sequence data for a variety of organisms, it becomes increasingly attractive to place an evolutionary perspective on the results obtained at the molecular level with those obtained from morphology and paleontology. However, reconstructing the chronology of mammalian diversification has been a matter of controversy.

Paleontological and molecular approaches disagree on both the tempo and mode of the early placental radiation [START_REF] Alroy | The fossil record of North American mammals: evidence for a Paleocene evolutionary radiation[END_REF][START_REF] Benton | Early origins of modern birds and mammals: molecules vs. morphology[END_REF][START_REF] Easteal | Molecular evidence for the early divergence of placental mammals[END_REF]. Fossil evidence indicates an explosive radiation of placental orders in the Paleocene some 65 million years (Myr) ago, just after the Cretaceous-Tertiary (K-T) limit. On the contrary, molecular clock approaches analyzing both mitochondrial and nuclear data support a Cretaceous origin and diversification of placentals.

Discrepancies between clocks and rocks might have their origin in the incompleteness of the fossil record or possibly in methodological shortcomings associated with the molecular approach. Several sources of error might affect the accuracy of divergence dates deduced from DNA and protein data: the number of sites analyzed [START_REF] Bromham | The power of relative rates tests depends on the data[END_REF], the number of species considered and the topology of the reference phylogeny [START_REF] Sanderson | Sources of error and confidence intervals in estimating the age of angiosperms from rbcL and 18S rDNA data[END_REF], and the molecular dating method [START_REF] Rodriguez-Trelles | A methodological bias toward overestimation of molecular evolutionary time scales[END_REF]. However, the most confounding and widespread cause is likely to be the variation of evolutionary rates, precluding the use of the global molecular clock concept [START_REF] Zuckerkandl | Divergence dates which are accurately estimated by a given calibration point. For example, the 95%-confidence interval (age ± 1.96 standard-error) for the divergence time between[END_REF]. In eukaryotic genes, lineage specific substitution rate variations have been documented for protists [START_REF] Philippe | Phylogeny of eukaryotes based on ribosomal RNA: long-branch attraction and models of sequence evolution[END_REF], plants [START_REF] Bousquet | Extensive variation in evolutionary rate of rbcL gene sequences among seed plants[END_REF], fungi [START_REF] Moncalvo | Phylogenetic relationships of agaric fungi based on nuclear large subunit ribosomal DNA sequences[END_REF], and various animals such as insects [START_REF] Huelsenbeck | Systematic bias in phylogenetic analysis: is the Strepsiptera problem solved?[END_REF], cyclostomes [START_REF] Mallatt | 28S and 18S rDNA sequences support the monophyly of lampreys and hagfishes[END_REF], or mammals [START_REF] Huchon | Variance of molecular datings, evolution of rodents, and the phylogenetic affinities between Ctenodactylidae and Hystricognathi[END_REF]. Rate variations have also been detected for mitochondrial genomes among mammals [START_REF] Gissi | Lineage-specific evolutionary rate in mammalian mtDNA[END_REF]. Two mammalian orders have long been the subject of intensive studies regarding their evolutionary rates: rodents and primates. For some molecular markers, rodents have been shown to evolve faster than other placentals [START_REF] Wu | Evidence for higher rates of nucleotide substitution in rodents than in man[END_REF][START_REF] Adkins | Molecular phylogeny and divergence time estimates for major rodent groups: evidence from multiple genes[END_REF], and to display within-order rate heterogeneities [START_REF] Huchon | Variance of molecular datings, evolution of rodents, and the phylogenetic affinities between Ctenodactylidae and Hystricognathi[END_REF]. Primates are also characterized by different rates of molecular evolution between lineages, namely faster rates in anthropoids [START_REF] Adkins | Evolution of the primate cytochrome c oxidase subunit II gene[END_REF][START_REF] Andrews | Accelerated evolution of cytochrome b in simian primates: adaptive evolution in concert with other mitochondrial proteins?[END_REF][START_REF] Andrews | Evolutionary rate acceleration of cytochrome c oxidase subunit I in simian primates[END_REF][START_REF] Liu | Episodic evolution of growth hormone in primates and emergence of the species specificity of human growth hormone receptor[END_REF].

Because changes in evolutionary rate are widespread, a "global"-clock tree cannot always be easily obtained by the removal of fast-or slow-evolving taxa without sacrificing taxon sampling [START_REF] Takezaki | Phylogenetic test of the molecular clock and linearized trees[END_REF][START_REF] Huchon | Variance of molecular datings, evolution of rodents, and the phylogenetic affinities between Ctenodactylidae and Hystricognathi[END_REF]Murphy et al. 2001b). This is the reason why a so-called local molecular clock approach has been recently proposed by [START_REF] Yoder | Estimation of primate speciation dates using local molecular clocks[END_REF]. These authors hypothesize local homogeneity of evolutionary rates with variations at larger phylogenetic scales. In a maximum likelihood (ML) framework, [START_REF] Yoder | Estimation of primate speciation dates using local molecular clocks[END_REF] attribute independent substitution rates to some fast-evolving (or slowevolving) lineages while assuming rate constancy in the others. Molecular dates are then obtained by converting branch lengths on the ML tree to time since divergence using one calibration point. This method has already been applied to derive a molecular time scale for the evolution of primates [START_REF] Yoder | Estimation of primate speciation dates using local molecular clocks[END_REF], rodents [START_REF] Huchon | From the Old World to the New World: a molecular chronicle of the phylogeny and biogeography of hystricognath rodents[END_REF] and xenarthrans [START_REF] Delsuc | The evolution of armadillos, anteaters and sloths depicted by nuclear and mitochondrial phylogenies: implications for the status of the enigmatic fossil Eurotamandua[END_REF]).

Among placentals, rodents are good candidates to test the properties of the local clock approach. Indeed, for some molecular markers, rodents have been shown to display extensive lineage-specific substitution rate heterogeneities in nuclear genes [START_REF] Huchon | Variance of molecular datings, evolution of rodents, and the phylogenetic affinities between Ctenodactylidae and Hystricognathi[END_REF][START_REF] Adkins | Molecular phylogeny and divergence time estimates for major rodent groups: evidence from multiple genes[END_REF], even between closely related taxa (e.g., [START_REF] Fieldhouse | Substitution rate variation in closely related rodent species[END_REF][START_REF] Michaux | The bushlike radiation of muroid rodents is exemplified by the molecular phylogeny of the LCAT nuclear gene[END_REF][START_REF] Michaux | Evolutionary history of the most speciose mammals: molecular phylogeny of muroid rodents[END_REF][START_REF] Rowe | Phylogenetic relationships, ecological correlates, and molecular evolution within the cavioidea (Mammalia, Rodentia)[END_REF]. Additionally, several molecular studies, based on both mitochondrial and nuclear genes, have provided a reasonably good picture of phylogenetic relationships of rodents and other placentals [START_REF] Nedbal | Molecular systematics of hystricognath rodents: evidence from the mitochondrial 12S rRNA gene[END_REF][START_REF] Nedbal | Higher-level systematics of rodents (Mammalia, Rodentia): evidence from the mitochondrial 12S rRNA gene[END_REF][START_REF] Catzeflis | Molecular systematics of hystricognath rodents: the contribution of sciurognath mitochondrial 12S rRNA sequences[END_REF][START_REF] Huchon | Molecular evolution of the nuclear von Willebrand Factor gene in mammals and the phylogeny of rodents[END_REF][START_REF] Huchon | Variance of molecular datings, evolution of rodents, and the phylogenetic affinities between Ctenodactylidae and Hystricognathi[END_REF][START_REF] Huchon | Rodent phylogeny and a timescale for the evolution of Glires: evidence from an extensive taxon sampling using three nuclear genes[END_REF][START_REF] Adkins | Molecular phylogeny and divergence time estimates for major rodent groups: evidence from multiple genes[END_REF][START_REF] Huchon | From the Old World to the New World: a molecular chronicle of the phylogeny and biogeography of hystricognath rodents[END_REF][START_REF] Debry | Phylogeny of Rodentia (Mammalia) inferred from the nuclearencoded gene IRBP[END_REF]Murphy et al. 2001a). These studies have demonstrated that rodents share an unique common ancestor, that they are composed of three major clades of mouse-like, squirrel-like, and Guinea-pig-like animals, and that their closest extant relatives are lagomorphs, followed by primates, flying-lemurs and tree-shrews. This relatively well resolved phylogeny allows its use as a molecular scaffold for dating purposes.

Furthermore, a fairly good fossil record is available for rodents, at the order (e.g., [START_REF] Hartenberger | Description de la radiation des Rodentia (Mammalia) du Paléocène supérieur au Miocène ; incidences phylogénétiques[END_REF]) and family (e.g., [START_REF] Vucetich | Review and analysis of the radiation of the South American Hystricognathi (Mammalia, Rodentia)[END_REF]) levels, providing thus a number of potentially reliable paleontological calibration points. Finally, the timing of two particular events of rodent evolution has been the matter of much controversy and represent important examples of the disagreements between the fossil and molecular dating approaches. On the one hand, the timing of rodent origins is controversial. The divergence between sciurognaths (squirrel-like rodents) and hystricognaths (Guinea-pig-like rodents) has been estimated to be 75-125 million years old by molecular dating [START_REF] Janke | The complete mitochondrial genome of the wallaroo (Macropus robustus) and the phylogenetic relationship among Monotremata, Marsupialia, and Eutheria[END_REF][START_REF] Kumar | A molecular timescale for vertebrate evolution[END_REF][START_REF] Cao | Interordinal relationships and timescale of eutherian evolution as inferred from mitochondrial genome data[END_REF][START_REF] Adkins | Molecular phylogeny and divergence time estimates for major rodent groups: evidence from multiple genes[END_REF]) whereas fossils support a rodent radiation 55 Myr ago [START_REF] Hartenberger | Description de la radiation des Rodentia (Mammalia) du Paléocène supérieur au Miocène ; incidences phylogénétiques[END_REF]. On the other hand, the age of the split between mouse and rat has been highly debated. Whereas the fossil record suggests a Mus / Rattus split around 14 Myr [START_REF] Jacobs | The evolution of murine rodents in Asia[END_REF], global molecular clocks applied to DNA or protein sequence data provide mean divergence dates of at least 33 Myr [START_REF] Nei | Estimation of divergence times from multiprotein sequences for a few mammalian species and several distantly related organisms[END_REF]), 35 Myr (Janke et al. 1994), 41 Myr [START_REF] Kumar | A molecular timescale for vertebrate evolution[END_REF], or 42 Myr [START_REF] Huchon | Variance of molecular datings, evolution of rodents, and the phylogenetic affinities between Ctenodactylidae and Hystricognathi[END_REF]. A noticeably younger molecular estimate is the 23 Myr figure of [START_REF] Adkins | Molecular phylogeny and divergence time estimates for major rodent groups: evidence from multiple genes[END_REF] based on the analysis of the growth hormone receptor gene.

The purpose of this paper is to re-evaluate the discrepancies between molecular and paleontological estimates of mammalian divergence dates, using the local molecular clock approach of [START_REF] Yoder | Estimation of primate speciation dates using local molecular clocks[END_REF] to accommodate lineage-specific variations in evolutionary rates. In order to reduce potential caveats of molecular methods, we analyzed three nuclear genes for a well-documented interordinal set of placental mammals including an extensive taxon sampling of rodent families. Nuclear genes were favored because it has been demonstrated that they contain more phylogenetic signal than mitochondrial genes (e.g., Springer et al. 2001a). Additionally, several independent calibration points-within and outside rodents-were considered in an attempt to evaluate whether the local molecular clock approach will be able to reconcile molecular data with the paleontological record.

MATERIAL & METHODS

Sequence data.

The nucleotide sequence data set used here was previously obtained by [START_REF] Huchon | Rodent phylogeny and a timescale for the evolution of Glires: evidence from an extensive taxon sampling using three nuclear genes[END_REF]. It includes 21 rodents, 19 other placental mammals and two marsupials for three A closer examination revealed that, as expected, the base composition at third codon positions was responsible for this heterogeneity. Therefore, we decided to exclude all third codon positions, and combined the three markers in order to average lineage-and genespecific rate heterogeneities. This data set of concatenated first and second codon positions represents a total of 2422 aligned sites for which stationary base composition was respected for all placental taxa.

Maximum likelihood analyses.

The General Time Reversible (GTR or REV) model of sequence evolution was chosen for nucleotides [START_REF] Lanave | A new method for calculating evolutionary substitution rates[END_REF][START_REF] Yang | Estimating the pattern of nucleotide substitution[END_REF]. Base composition was empirically estimated from the data set. Rate heterogeneity among DNA sites was described by a discrete Gamma distribution with 8 categories (Γ 8 ) [START_REF] Yang | Among-site rate variation and its impact on phylogenetic analyses[END_REF]. All analyses of local molecular clocks and corresponding datings were conducted under PAML [START_REF] Yang | PAML: a program package for phylogenetic analysis by maximum likelihood[END_REF]), version 3.1.

The highest-likelihood topology used as a reference for the dating was the one identified by [START_REF] Huchon | Rodent phylogeny and a timescale for the evolution of Glires: evidence from an extensive taxon sampling using three nuclear genes[END_REF] using the concatenation of ADRA2B, IRBP, and vWF. Optimal ML parameters were A↔C = 2.26, A↔G = 6.34, A↔T = 0.92, C↔G = 1.68, C↔T = 4.48, and G↔T = 1.00 for the substitution rate matrix, and α = 0.47 for the Gamma distribution, yielding a log-likelihood of lnL = -26054.39. This topology was compatible with the most recent studies on placental ordinal phylogenetic relationships in depicting the monophyly of Rodentia, Glires, Euarchontoglires, Laurasiatheria, Boreoeutheria, and Afrotheria (Madsen et al. 2001;Murphy et al. 2001a). Minor departures from the topology robustly identified by Murphy et al. (2001b) on 16.4 kb of nuclear and mitochondrial DNA involved only weakly supported nodes for the position of Eulipotyphla (here represented by Erinaceus), Pholidota (Manis), Dermoptera (Cynocephalus), and the location of the root of placentals (see [START_REF] Delsuc | Molecular phylogeny of living xenarthrans and the impact of character and taxon sampling on the placental tree rooting[END_REF], for a discussion of this point).

Relative rate tests.

Rate heterogeneity among taxa was detected using the method developed by [START_REF] Takezaki | Phylogenetic test of the molecular clock and linearized trees[END_REF] as implemented in the two-cluster and branch-length tests of the LINTRE package (www.bio.psu.edu/People/Faculty/Nei/Lab). The two-cluster test was a relative rate test that examined whether there is a change in substitution rate between the two descendant lineages (i.e., cluster) on each node of the bifurcating tree. The branch length test evaluated the deviation of the root-to-tip distance of each terminal taxa relative to the average root-to-tip length. Tests were conducted under the Tamura-Nei distance with α = 0.47 (the GTR distance was not available under LINTRE). Fastest-and slowest-rate branches and species were identified by the two-cluster and the branch-length tests at the stringent P < 0.01 significance level. They then served as a basis for the definition of the local molecular clocks.

Definition of local clocks.

The use of the standard likelihood ratio test for comparison of models with and without clock [START_REF] Felsenstein | Phylogenies from molecular sequences: inference and reliability[END_REF] was not appropriate here because it did not allow a comparison of two models with the same number of parameters (i.e., in our case, models including the same number of local molecular clocks). Therefore, following the pioneering approach of [START_REF] Kishino | Converting distance to time: application to human evolution[END_REF], the performances of the different clock models were evaluated using the Akaike Information Criterion (AIC; Akaike 1974), the lowest AIC values being indicative of the model that best fit the data. The AIC was calculated as -2 x lnL + 2 x (number of free parameters of the model), where lnL was the log-likelihood of the tree.

Calibration points.

We chose seven calibration points spanning the different placental clades with a special focus on rodents, following [START_REF] Huchon | Rodent phylogeny and a timescale for the evolution of Glires: evidence from an extensive taxon sampling using three nuclear genes[END_REF]: (i) Radiation of Caviomorpha at 31 Myr [START_REF] Walton | Rodents[END_REF][START_REF] Wyss | South America's earliest rodent and recognition of a new interval of mammalian evolution[END_REF]

RESULTS

Extensive lineage-specific variation of evolutionary rates.

The results of the branch length and two-cluster tests are recapitulated in Figure 1. Four taxa are evolving significantly faster than other placentals-the sloth (Bradypus), the hedgehog (Erinaceus), and two geomyoid rodents (Dipodomys, Thomomys)-, whereas eight are evolving slower-a carnivore (Felis), a perissodactyl (Equus), a primate (Homo), a flying-lemur (Cynocephalus), and four rodents (Aplodontia, Glis, Anomalurus, and Castor). For all these taxa, the ratio of the mean root-to-tip distance to its standard-error was higher than 3, leading to a highly significant branch length test (P < 0.01). The two-cluster test recorded significant differences (P < 0.01) in evolutionary rates for terminal branches-e.g., Dugong versus Procavia-as well as internal branches-e.g., the ones subtending sciuroids and glirids (Figure 1).

Setting the local molecular clocks.

The results of the branch length test (cf. The next step was to try to increase the log-likelihood of the tree by attributing new local rates to species that were initially considered as evolving according to the default rate r D = 1.00. These taxa were chosen based on the results of the two-cluster test. In order to reduce the number of parameters, we defined only one local clock-either slow or fast-per pairs of terminal branches, and chose the rate yielding the higher log-likelihood. Cynopterus, Manis, and Marmota were thus identified as slower-evolving species, and Procavia, Ochotona, Massoutiera, and Cavia as faster-evolving taxa. Local clocks for these taxa were incorporated in models with an increasing number of independent rates: model E (lnL = -26085.97), F (lnL = -26089.24), and I (lnL = -26081.32) (cf. Figure 2).

The difficulty of this multi-local clocks approach is to find a reasonable compromise between the number of parameters added-here the local clocks-and the gain of log-likelihood.

As a measure of this compromise, we followed [START_REF] Kishino | Converting distance to time: application to human evolution[END_REF] and used the AIC.

For example on model E, the log-likelihood of the model with five local clocks is lnL = -26085.97, with the following number of ML free parameters: five for the GTR model, one for the rate heterogeneity, and 45 for the branch lengths (42 taxa + 5 local clocks -2 parameters as described in [START_REF] Yoder | Estimation of primate speciation dates using local molecular clocks[END_REF]). The AIC of this model is therefore equal to 2 x 26085.97 + 2 x 51 = 52273.94. This value is the lowest found here, even relative to the AIC of the hypothesis without any clock (AIC no clock = 52282.78). Figure 2 recapitulates the relationship between the AIC values and the number of local clocks defined.

Another difficulty of the multi-local clocks approach is to choose among all the possible rate assignments for the various lineages [START_REF] Sanderson | A nonparametric approach to estimating divergence times in the absence of rate constancy[END_REF]. Because of the tremendous number of possibilities, we here evaluated only nine models (A to I) which span quite a large range in number of local clocks (2 to 18). The best model (E) found is based on 5 local clocks (Figure 2).

In this model, Massoutiera is fast-evolving, and Bradypus, Procavia, Erinaceus, Ochotona, Dipodomys and Thomomys, and Cavia are the fastest-evolving taxa, and conversely Cynopterus, Manis, Marmota and Aplodontia are slow-evolving, and Felis, Equus, Homo, Cynocephalus, Glis, Anomalurus, and Castor are the slowest-evolving taxa (Figure 3).

The ML estimate of the local clocks also provides a measure of the evolutionary rate contrast among these lineages. For the first and second codon positions of the three concatenated nuclear genes ADRA2B, IRBP, and vWF, the fastest-evolving species exhibit a 2.6-fold excess of evolutionary rates relative to the slowest ones. When 18 independent local clocks are defined (model I: Figure2), a greater excess of 3.8-fold is detected between Cavia (local rate of 1.67) and Anomalurus (0.44).

Crossed-calibrations based on seven points.

Seven calibration points are mapped on the 5-local clocks phylogram (Figure 3) in order to estimate: i) the divergence dates between the calibrating taxa, i.e., those taxa for which a paleontological age of divergence is suggested ; and ii) the age of the most recent common ancestors (MRCA) of 6 major mammalian clades, i.e., the murines (Mus versus Rattus), rodents, glires, euarchontoglires, boreoeutherians, and placentals.

Table 1 presents the results of the crossed-calibrations. To evaluate the accuracy of each calibration point-i.e., its ability to correctly estimate the true date of divergence-we searched whether a given calibration point yielded a 95% confidence interval (mean date ± 1.96 standarderror) containing the supposed splitting date of the six remaining calibration points. Crosscalibrations indicate that our calibration points are not fully congruent between them, i.e., none of them could give 95% confidence intervals that include the paleontological date of the six other calibration points (Table 1). Three calibration points (Caviomorpha at 31 Myr, Mus / Rattus at 14 Myr, and Sciuroidea at 37 Myr) could recover two paleontological dates. Two calibration points (Gliridae at 28.5 Myr and Cetartiodactyla at 63 Myr) could recover one paleontological date.

Paenungulata at 60 Myr was unable to be congruent with any of the six other calibration points.

Interestingly, some calibration points were reciprocally and accurately compatible. For example, the glirids at 28.5 Myr suggest Mus and Rattus be separated 12.8 Myr ago-the 95% confidence interval is 10.2-15.4 Myr, and contains the 14 Myr fossil estimate-, and conversely, Mus / Rattus at 14 Myr suggest Glis and Dryomys separated 31.3 Myr-the confidence interval is 26.5-36.1 Myr, and contains the 28.5 Myr fossil estimate. Three other pairs of calibration points are reciprocally compatible: caviomorphs and sciuroids, caviomorphs and lagomorphs, and Mus / Rattus and sciuroids (Table 1).

Divergence dates of mammals.

Table 2 presents the results of the local molecular-clock dating for the age of the MRCA of 6 mammalian clades. As expected from previous works indicating that divergence times depend upon the choice of calibration points [START_REF] Huchon | Variance of molecular datings, evolution of rodents, and the phylogenetic affinities between Ctenodactylidae and Hystricognathi[END_REF]Soltis et al. 2001), the age of the MRCA of murines, rodents, glires, euarchontoglires, boreoeutherians, and placentals is highly variable depending on the calibrating taxa. For example, the Gliridae calibration point suggests the MRCA of rodents be ~47 Myr old, while the Paenungulata point suggests it to be ~121 Myr old. To summarize the results based on the seven independent calibration points, we referred to the median age of the corresponding MRCA. When the seven calibration points are considered, the median ages for the MRCA of murines, rodents, glires, euarchontoglires, boreoeutherians, and placentals are ~18, 66, 73, 78, 83, and 87 Myr respectively (Table 2).

However, more recent estimates are obtained after the removal of the incompatible cetartiodactyl and paenungulate calibrations (see Table 2). In this case, the median ages of the five calibrations are respectively ~16, 60, 66, 70, 75, and 78 Myr.

DISCUSSION

Incompatibility of divergence times calculated from independent calibrations points.

A great variance of molecular estimates of divergence times has been described for mammals [START_REF] Bromham | Growing up with dinosaurs: molecular dates and the mammalian radiation[END_REF][START_REF] Huchon | Variance of molecular datings, evolution of rodents, and the phylogenetic affinities between Ctenodactylidae and Hystricognathi[END_REF]. As an illustration of this variance, molecular datings reported for the split between hystricognath and sciurognath rodents range from values as disparate as 75 to 125 Myr [START_REF] Janke | The complete mitochondrial genome of the wallaroo (Macropus robustus) and the phylogenetic relationship among Monotremata, Marsupialia, and Eutheria[END_REF][START_REF] Kumar | A molecular timescale for vertebrate evolution[END_REF][START_REF] Cao | Interordinal relationships and timescale of eutherian evolution as inferred from mitochondrial genome data[END_REF][START_REF] Adkins | Molecular phylogeny and divergence time estimates for major rodent groups: evidence from multiple genes[END_REF]). Problematically, this 50 Myr difference between two extreme estimates is of the same order of magnitude as the paleontological age of 55 Myr for modern placentals [START_REF] Hartenberger | Description de la radiation des Rodentia (Mammalia) du Paléocène supérieur au Miocène ; incidences phylogénétiques[END_REF]. In this paper, we attempted to understand the origins of the conflict between molecules and fossils by using an approach accounting for evolutionary rate variations-the ML local molecular clocks [START_REF] Yoder | Estimation of primate speciation dates using local molecular clocks[END_REF])-, and a set of seven independent calibrations spread over the placental tree.

Mean local molecular clock estimates for the age of the MRCA of murines, rodents, glires, euarchontoglires, boreoeutherians, and placentals over these seven calibrations are respectively ~19 ± 7, 71 ± 25, 79 ± 27, 84 ± 30, 89 ± 31, and 94 ± 33 Myr. The large standard-errors reflect the strong dependence of date estimates upon the choice of calibration points. In order to better understand the compatibility properties of these seven calibrations, we evaluated the level of inadequacy between paleontological (A PAL ) and molecular (A MOL ) divergence times (cf. Table 1). This inadequacy (I X / Y ) was quantified for each calibration point (e.g., node X), and for each of the 6 MRCAs under focus (e.g., Y), by the absolute difference between the date estimated by the local clocks and the age expected under paleontological hypotheses, standardized by the latter age, and expressed in percentage, i.e., by I X / Y = 100 x |A PAL (node X) -A MOL (node Y) | / A PAL (node X). If we rank the seven calibration points by decreasing order of agreement between fossils and molecules as evaluated by an increasing mean percent of inadequacy over these seven values, we have sciuroids (mean I X / Y ± standard error = 24% ± 15), caviomorphs (25% ± 16), lagomorphs and Mus / Rattus (28% ± 17), glirids (33% ± 18), cetartiodactyls (41% ± 22), and then paenungulates (100% ± 42). Clearly, datings obtained using the paenungulate-and to a lesser extent the cetartiodactyl-calibration points are at odds with estimates derived from other paleontological calibrations. This incompatibility of the cetartiodactyl and paenungulate calibrations relative to the five other points chosen within Glires may be explained by limitations of either the local clock method and / or the fossil record. (i) The taxon sampling for paenungulates is sparse in our phylogram (Figure 1), which might have affected the accuracy of branch length estimation within this clade. (ii) The cetartiodactyl calibration point is based on controversial fossil inferences. The 63 Myr value is inferred from the Mesonychia-Arctocyonia divergence [START_REF] Gingerich | Likelihood estimation of the time of origin of Cetacea and the time of divergence of Cetacea and Artiodactyla[END_REF], which assumes that Mesonychia are the sister clade of cetaceans. This hypothesis has been recently challenged by new paleontological discoveries [START_REF] Thewissen | Skeletons of terrestrial cetaceans and the relationship of whales to artiodactyls[END_REF]).

Choosing among independent calibration points

We here consider mainly calibrations within rodents, with at least one point in each of the major rodent clades [START_REF] Adkins | Molecular phylogeny and divergence time estimates for major rodent groups: evidence from multiple genes[END_REF][START_REF] Huchon | Rodent phylogeny and a timescale for the evolution of Glires: evidence from an extensive taxon sampling using three nuclear genes[END_REF]: sciuroids and glirids in the Sciuroidea + Gliridae clade, murines in the Anomaluromorpha + Castoridae + Geomyoidea + Myodonta clade, and caviomorphs in the Ctenohystrica. Together with the lagomorph point, it is clear that the rodent calibrations were characterized by greater reciprocal compatibility.

Conversely, the greatest inadequacy between molecular and paleontological ages was given by the two points outside Glires: the cetartiodactyls and the paenungulates. This observation contradicts the point of view of [START_REF] Adkins | Molecular phylogeny and divergence time estimates for major rodent groups: evidence from multiple genes[END_REF] who suggested that "the inclusion of a rodent calibration point to estimate rodent divergence dates introduces a confounding influence [on the molecular dates]".

Actually, the phylogenetic distance D X / Y between each pair (X, Y) of calibration points-expressed in substitution number per 100 sites-can be measured by the node-tonode sum of the branch lengths on the highest-likelihood phylogram (Figure 1). A positive correlation is found between this distance and the mean percent of inadequacy I X / Y between fossil and molecular dates (I X / Y = 9.7 x D X / Y -29.0 ; r 2 = 0.42 ; data not shown). In other words, an additional 9.7% of inadequacy between the expected paleontological age and the local clock estimate accumulates for each additional substitution per 100 sites separating the MRCAs under focus on the ML tree. This observation is not unexpected given that the most confounding cause for errors in the molecular dating is likely to be the variation of evolutionary rates among lineages. There are more possibilities for increasing (or decreasing) substitution rate variations to accumulate for more distant calibrating nodes-e.g., the paenungulate point is farther than the caviomorph one relative to the Mus / Rattus split-, and therefore more possibilities of obtaining reciprocally incompatible local clock estimates of divergence times. If there is a problem of over-or under-estimation with a given calibration point, its uniformly inflating or deflating impact will be the lower on the nodes located in its closest vicinity.

Setting the local molecular clocks

The divergence times obtained by our approach directly depends upon the setting of the different local rate models (A to I). The incorporation of different local rates in the tree is contingent on the interpretation of results from the branch length and two-cluster tests. With the two-cluster test, we chose to attribute local clocks to only one of the two branches displaying contrasted rates of evolution. By focusing on terminal branches, we aim to avoid overparameterization of the models. Alternative possibilities do exist such as defining local clocks on internal branches of the phylogram-e.g., those leading to several calibrating nodes (cetartiodactyls, sciuroids, glirids, and murines: cf. Figure 1). Due to the high number of potential combinations of local clocks, we only explored nine models incorporating from two to 18 local clocks, but we cannot rule out the possibility that a better fit of the data might be found by using more complex models that would give lower AIC relative to the 5-clocks model (E).

Future developments of the local clock approach might involve the construction of algorithms to explore the attribution of rates to taxa and identify the best local clock tree.

Detecting rate heterogeneities among lineages by the local clocks.

We observed a better fit of the data with less branch rates (AIC of 52273.94 with five local clocks against 52282.78 without clock). Under maximum likelihood, an independent rate is usually attributed to each branch of the phylogram, yielding 2N -3 rates for an unrooted bifurcating tree with N taxa (42 species for our data). In the particular case of the first and second codon positions of ADRA2B + IRBP + vWF, we here show that defining as few as five rates instead of 2 x 42 -3 = 81 reasonably fits the DNA sequence data. This might suggest that the 2N -3 branch parameters actually correspond to an over-parameterization of the ML model, at least in the present case.

With our 5-rates model, we identified extensive rate variation between placental lineages, but also within rodents, with detection of a three-fold contrast between the slowest and fastest evolving taxa. We incorporated an extensive taxon sampling within rodents spanning all sciurognath families and all hystricognath superfamilies. In light of our results, it is no longer accurate to suggest that rodents as a group are evolving faster than man (e.g., [START_REF] Wu | Evidence for higher rates of nucleotide substitution in rodents than in man[END_REF]. Indeed, for the three nuclear exons under focus, rodents contain both among the slowest-and fastest-evolving placental species, with respectively the scaly-tailed flying squirrel Anomalurus One should note that these local clock values should not be considered as absolute values.

Indeed, the estimate of each local rate is influenced by the choice of the others, as this can also be noticed from the results of [START_REF] Kishino | Converting distance to time: application to human evolution[END_REF]. For example, in model A (Figure 2), a single rate of 1.82 is estimated for all fast-evolving taxa. In model B, a single rate of 0.57 is estimated for all slow-evolving taxa. When the two models are combined (model C), the ML estimate of the fast rate decreases to 1.51, and the slow rate increases to 0.62. Potential caveats of using relative rate tests to detect evolutionary rate variation have been pointed out by [START_REF] Bromham | The power of relative rates tests depends on the data[END_REF]. These authors suggested that relative rate tests such as triplet relative rates or likelihood ratio tests are limited in power to detect rate variations when using short sequences. Here we used ~2400 first and second codon positions, giving us more than 90% chance of detecting a two-fold (or more) lineage-specific rate variation [START_REF] Bromham | The power of relative rates tests depends on the data[END_REF]. It is thus likely that most of the rate variation has been accounted for in our definition of local clocks. However, if moderate rate variations remain unincorporated in our local clock approach, they should have only slightly affected our date estimates.

The chronology of the diversification of placentals.

Despite the undeniable existence of gaps in the fossil record [START_REF] Easteal | Molecular evidence for the early divergence of placental mammals[END_REF], some molecular dates are clearly incompatible with paleontology. For example, regarding the age of divergence of crown placentals, two rodent calibration points yield dates that are recent (~62-68 Myr for the murine and glirid calibrations: Table 2) relative to the fossil record, which suggests that placentals were already diversified 85-90 Myr ago (e.g., [START_REF] Archibald | Late Cretaceous relatives of rabbits, rodents, and other extant eutherian mammals[END_REF]. Conversely, the date of 159 Myr for the divergence of Placentalia (paenungulate calibration: Table 2) seems ancient relative to the fossil record because stem eutherians are identified in the Early Cretaceous around 125 Myr ago [START_REF] Ji | The earliest known eutherian mammal[END_REF].

We are thus confirming previous observations of the strong dependence of molecular date estimates of cladogenesis events upon the choice of the fossil references used for calibrations [START_REF] Lee | Molecular clock calibrations and metazoan divergence dates[END_REF][START_REF] Huchon | Variance of molecular datings, evolution of rodents, and the phylogenetic affinities between Ctenodactylidae and Hystricognathi[END_REF][START_REF] Soltis | Rate heterogeneity among lineages of tracheophytes: integration of molecular and fossil data and evidence for molecular living fossils[END_REF]. In order to decrease the impact of using a single calibration point, we averaged estimates derived from 5 independent calibrations displaying only slight reciprocal incompatibility-i.e., the caviomorph, murine, glirid, sciuroid, and lagomorph points. The MRCAs of rodents, primates and rodents, boreoeutherians, and placentals were therefore estimated to be respectively ~ 60, 70, 75, and 78 Myr old.

More generally, molecular dates of divergence based on the use of a single calibration point are very crude, especially when times are computed from very recent calibrations [START_REF] Ayala | Origin of the metazoan phyla: molecular clocks confirm paleontological estimates[END_REF] or when time estimates for a group are derived from extrapolation on the times calculated for another [START_REF] Van Tuinen | Calibration of avian molecular clocks[END_REF]. It is thus attractive to simultaneously incorporate information from two independent calibration points in molecular dating, as proposed by [START_REF] Rambaut | Estimating divergence dates from molecular sequences[END_REF] in their quartet dating method. This method allows rate variation in each of the two lineages under focus. Application of quartet dating provided the following ranges of mean divergence time estimates: 55.8 Myr for the MRCA of rodents, 68.0-69.4 for the one of primates and rodents, 79.0 for the one of boreoeutherians, and 76.1-103.0 for placentals [START_REF] Eizirik | Molecular dating and biogeography of the early placental mammal radiation[END_REF]Murphy et al. 2001b;[START_REF] Huchon | Rodent phylogeny and a timescale for the evolution of Glires: evidence from an extensive taxon sampling using three nuclear genes[END_REF]. When several quartets of taxa are available to evaluate the age of a given node, local clock estimates are quite close to the most recent quartet dating estimates. Thus, molecular analyses based on global clocks, local clocks, and quartet dating seem to converge toward a Late Cretaceous origin of the crown placental clades [START_REF] Easteal | Molecular evidence for the early divergence of placental mammals[END_REF][START_REF] Huchon | Variance of molecular datings, evolution of rodents, and the phylogenetic affinities between Ctenodactylidae and Hystricognathi[END_REF][START_REF] Eizirik | Molecular dating and biogeography of the early placental mammal radiation[END_REF]Murphy et al. 2001b) followed by a Tertiary radiation of some orders like rodents [START_REF] Huchon | Rodent phylogeny and a timescale for the evolution of Glires: evidence from an extensive taxon sampling using three nuclear genes[END_REF] and chiropterans (Springer et al. 2001b).

The age of the split between mice and rats.

Among our seven phylogenetically independent calibration points, we incorporated the controversial Mus / Rattus split. Because of the consistently deeper molecular estimates of the MRCA of mouse and rat (23-42 Myr: Janke et al. 1994;[START_REF] Kumar | A molecular timescale for vertebrate evolution[END_REF][START_REF] Huchon | Variance of molecular datings, evolution of rodents, and the phylogenetic affinities between Ctenodactylidae and Hystricognathi[END_REF][START_REF] Adkins | Molecular phylogeny and divergence time estimates for major rodent groups: evidence from multiple genes[END_REF][START_REF] Nei | Estimation of divergence times from multiprotein sequences for a few mammalian species and several distantly related organisms[END_REF] relative to the fossil record (14 Myr: Jacobs and Downs 1994), the inclusion of this calibration in our analyses may have biased the estimates of the other dates towards more recent ages. However, the five less-incompatible calibration points yielded younger estimates (~13-19 Myr) relative to those previously published (Table 2). Even the more incompatible cetartiodactyl calibration provides a divergence time (22 Myr) virtually identical to the one obtained by [START_REF] Adkins | Molecular phylogeny and divergence time estimates for major rodent groups: evidence from multiple genes[END_REF] based on the DNA comparison of the growth hormone receptor gene. By contrast, the paenungulate calibration-shown to be highly incompatible with the other calibrations (see above)yielded 33 Myr (Table 2), an estimate identical to the one obtained by [START_REF] Nei | Estimation of divergence times from multiprotein sequences for a few mammalian species and several distantly related organisms[END_REF] on the concatenation of 104 protein sequences. A proposed explanation for the discrepancy between molecules and fossils for the age of the split between mice and rats is the high substitution rates of these two rodents. For example, murid rodents-the family to which mouse and rat belong-display particularly high substitution rates in their mitochondrial genomes [START_REF] Gissi | Lineage-specific evolutionary rate in mammalian mtDNA[END_REF]. This peculiarity possibly related to a change in mutational process might have obscured their phylogenetic placement for a long time [START_REF] Lin | Pika and vole mitochondrial genomes increase support for both rodent monophyly and Glires[END_REF]. Relative to previous molecular estimates, the younger dates obtained by [START_REF] Adkins | Molecular phylogeny and divergence time estimates for major rodent groups: evidence from multiple genes[END_REF], and in this paper, might be explained by a denser taxon sampling in rodents that breaks the long isolated branch leading to murids, allowing a more accurate estimation of its length.

Conclusions and perspectives.

Additional causes may be invoked to explain why different fossil calibrations can yield sometimes radically different date estimates using the local clock method. i) The choice of the genes. We combined three genetically independent nuclear markers in order to average the individual behavior of each exon ; ii) The choice of taxa. Substitution rate variations are widespread in mammalian genomes, and it is possible that rate variation would be better detected with a larger number of taxa. By decreasing the average length of terminal branches, an increased taxon sampling in a given clade will increase the accuracy of the reconstructed phylogeny [START_REF] Rannala | Taxon sampling and the accuracy of large phylogenies[END_REF]. For this reason, calibration points subtending a few isolated branches (e.g., paenungulates, sciuroids, and glirids) might be less relevant than those subtending a denser sampling (e.g., cetartiodactyls and caviomorphs). It is thus likely that the dating estimates obtained from, for example, the paenungulate calibration point, might be improved if more afrotherians were taken into account. iii) The choice of the topology. The tree we used has been confidently identified and evaluated by [START_REF] Huchon | Rodent phylogeny and a timescale for the evolution of Glires: evidence from an extensive taxon sampling using three nuclear genes[END_REF], and is congruent with those independently inferred by [START_REF] Adkins | Molecular phylogeny and divergence time estimates for major rodent groups: evidence from multiple genes[END_REF], [START_REF] Madsen | Parallel adaptative radiations in two major clades of placental mammals[END_REF], and Murphy et al. (2001a, b). We should keep in mind that the confidence intervals on molecular dates are roughly of the same order of magnitude whatever the choice of sites, taxa, and topologies [START_REF] Sanderson | Sources of error and confidence intervals in estimating the age of angiosperms from rbcL and 18S rDNA data[END_REF]. It is thus likely that the mean and standard errors on divergence dates obtained with the local clock approach are not strongly depending upon the choice of ADRA2B, IRBP, and vWF as molecular markers, a denser species representation within Glires, and the highest-likelihood phylogram as the reference topology for molecular dating purposes. However, the present study may benefit from an expanded sampling of markers among nuclear genes, and of taxa among the three other recently identified major placental clades (Laurasiatheria, Xenarthra, and Afrotheria: [START_REF] Madsen | Parallel adaptative radiations in two major clades of placental mammals[END_REF]Murphy et al. 2001a,b).

The lineage-specific variation of evolutionary rates documented in numerous mammalian genomes requires the development of powerful molecular dating methods.

Global clocks, local clocks and quartet dating are limited in their application by the assumption of rate constancy at either large or reduced phylogenetic scale. Non-or semiparametric [START_REF] Sanderson | A nonparametric approach to estimating divergence times in the absence of rate constancy[END_REF][START_REF] Sanderson | Estimating absolute rates of molecular evolution and divergence times: a penalized likelihood approach[END_REF] and Bayesian [START_REF] Thorne | Estimating the rate of evolution of the rate of molecular evolution[END_REF][START_REF] Huelsenbeck | A compound poisson process for relaxing the molecular clock[END_REF][START_REF] Kishino | Performance of a divergence time estimation method under a probabilistic model of rate evolution[END_REF][START_REF] Aris-Brosou | Effects of models of rate evolution on estimation of divergence dates with special reference to the metazoan 18S ribosomal RNA phylogeny[END_REF][START_REF] Thorne | Divergence time and evolutionary rate estimation with multilocus data[END_REF]) methods that model rate variation through time are another way to accommodate lineage-specific rate heterogeneity. The performance of these methods relies on the distribution of the parameters used to model the evolution of rates, and their use within placental mammals appears promising [START_REF] Cao | Interordinal relationships and timescale of eutherian evolution as inferred from mitochondrial genome data[END_REF]. Improvements in dating methods in concert with the increase of gene and taxon sampling in molecular studies is likely to yield more accurate divergence dates. Higher reliability of molecular dating estimates combined to new paleontological discoveries might help to reconcile rocks and clocks. according to their AIC. Open diamonds correspond to models for which the AIC values exceed the one of the model without any clock. The black diamond corresponds to the situation where the AIC is minimized (see also Figure 3). In addition to the default rate R D = 1.00, the following combinations of local clocks (R F for fast rates and R S for slow rates) have been defined under maximum likelihood: A -R F = 1.82 (a single very fast rate) ; B -R S = 0.57 (a single very slow rate) ; C (combination of models A and B) -R F = 1.51 and R S = 0.62; D -R F1 (Erinaceus) = 1.83, R F2 (Dipodomys + Thomomys) = 2.14, R F3 (Bradypus) = 1.61 (different very fast rates) ; E (cf. Figure 3

FIGURES

) -R F1 = R F2 = R F3 = R F4 (Procavia) = R F5 (Ochotona) = R F6 (Cavia) = 1.54 (a single very fast rate), R F7 (Massoutiera) = 1.24 (fast), R S2 (Felis) = R S4 (Equus) = R S5 (Homo + Cynocephalus) = R S7 (Glis) = R S8 (Anomalurus) = R

  nuclear markers: the Alpha 2B Adrenergic Receptor gene (ADRA2B ; 1170 sites), partial exon 1 of the Interphotoreceptor Retinoid Binding Protein gene (IRBP ; 1227 sites), and exon 28 of the von Willebrand Factor gene (vWF ; 1236 sites). Stationary base composition was evaluated at the 1% level of chi-square tests by comparing the nucleotide composition of each sequence to the frequency distribution assumed in the ML model. Non-stationary base composition was detected for ADRA2B, IRBP, and vWF, taken individually or in combination. The two marsupial outgroups (Macropus sp., Didelphis sp.) systematically deviated from base composition stationary assumptions whatever the data set considered. This was not the case for placental taxa, where the deviating taxa were different from one data set to an other. Dryomys nitedula, Mus musculus, and Bradypus tridactylus were heterogeneous for ADRA2B, as were Marmota monax and Orycteropus afer for IRBP, Dipodomys merriami, Thomomys talpoides, Lama sp., Physeter catodon, and Procavia capensis for vWF, and Mus musculus, Rattus norvegicus, Castor canadensis, Dryomys nitedula, Petromus typicus, Lepus crawshayi, Oryctolagus cuniculus, Physeter catodon, Orycteropus afer, Procavia capensis, and Bradypus tridactylus for the concatenated data set.

  ; (ii) Mus vs. Rattus at 14 Myr (Jacobs and Downs 1994) ; (iii) Glis vs. Dryomys at 28.5 Myr (identification of Glirinae since the Late Oligocene: Hartenberger 1994) ; (iv) Aplodontia vs. Marmota at 37 Myr (identification of Sciuridae since Late Eocene: McKenna and Bell 1997) ; (v) Ochotona vs. Leporidae at 37 Myr (identification of ochotonids since Late Eocene: McKenna and Bell 1997) ; (vi) Lama vs. other cetartiodactyls (e.g., Physeter) at 63 Myr (Gingerich and Uhen 1998) ; and (vii) Dugong vs. Procavia at 60 Myr (identification of Paenungulata since Paleocene: Gheerbrant et al. 1996). Within Glires, ages corresponded to the minimum estimates for the crown group considered. These seven paleontological calibration points were independently used on the best local clock phylogram to obtain seven sets of divergence date estimates for the main placental clades.

  Figure 1) are used to define the local molecular clocks. Nine models combining different local clocks are evaluated (Figure 2). All taxa but the faster-and slower-evolving are attributed the local default rate of R D = 1.00. For the four fasterevolving species, there are two possibilities. Either a single local clock is defined for Bradypus, Erinaceus, Dipodomys and Thomomys (the so-called model A [lnL = -26161.14]: Figure 2), or different local clocks are defined, one for the sloth, one for the hedgehog, and one for the two geomyoid rodents (model D [lnL = -26159.02]). Regarding the latter point, the definition of an unique evolutionary rate for both Dipodomys and Thomomys is obviously a more parsimonious solution relative to the definition of one independent local clock for each rodent. Moreover, the two-cluster test detected a rate variation along the branch leading to the two geomyoids relative to Castor: the same local clock is thus attributed to the ancestral branch of geomyoids and their descendants Dipodomys and Thomomys. For the eight slower-evolving taxa, the same possibilities exist, namely defining either a single local clock for all (model B [lnL = -26131.45]) or independent clocks for each taxon (model G [lnL = -26122.96]). To better describe the data, the previous models are combined in order to simultaneously consider slow and fast local clocks on the same tree: model C [lnL = -26102.76] combines A and B, and model H [lnL = -26094.70] combines D and G.

(

  local clock R S8 = 0.44: cf. Figure 2, model I) versus the pocket mouse Thomomys and the kangaroo rat Dipodomys (local clock R F2 = 1.51), and the Guinea pig (local clock R F6 = 1.67).

Figure 1 .

 1 Figure 1. Extensive nucleotide substitution rate variations in the first two codon positions of the ADRA2B + IRBP + vWF nuclear genes between placental mammals. The vertical dashed line indicates the mean value of the root-to-tip distance of the 40 placental taxa. Significantly faster-or slower-evolving species are respectively indicated by a "+" or a "-" as evidenced by the branch length test. Significantly faster-and slower-evolving branches as evidenced by the two-cluster test are respectively indicated by black arrows towards the right and white arrows towards the left. The scale unit corresponds to expected number of nucleotide substitutions per site. The log-likelihood of this tree is lnL = -26054.36, and its AIC is 52282.78. In the clocklike constrained model-with a single global clock-a significant loss of log-likelihood is observed (lnL = -26222.37 ; AIC = 52538.74).

Figure 2 .

 2 Figure 2. The relationship between the Akaike Information Criterion (AIC) and the number of local clocks. Nine different models of local clocks (A to I) have been evaluated

  S9 (Castor) = 0.59 (very slow), and R S1 (Cynopterus) = R S3 (Manis) = R S6 (Marmota + Aplodontia) = 0.73 (slow) ; F -R F = 1.55 (a very fast rate), R S = 0.63 (a very slow rate), and different intermediate rates: R F4 = 1.43, R F5 = 1.45, R F6 = 1.66, R F7 = 1.25, and R F8 (Tupaia) = 0.91 ; G -different slow rates: R S1 = 0.61, R S2 = 0.48, R S3 = 0.69, R S4 = 0.55, R S5 = 0.60, R S6 = 0.64, R S7 = 0.51, R S8 = 0.41, and R S9 = 0.51 ; H (combination of models D and G) -different fast rates: R F1 = 1.49, R F2 = 1.51, and R F3 = 1.48, and different slow rates: R S1 = 0.69, R S2 = 0.52, R S3 = 0.74, R S4 = 0.62, R S5 = 0.63, R S6 = 0.69, R S7 = 0.54, R S8 = 0.42, R S9 = 0.57 ; I -additional different fast rates: R F1 = 1.52, R F2 = 1.57, R F3 = 1.52, R F4 = 1.43, R F5 = 1.45, R F7 = 1.25, and R F6 = 1.67, and additional different slow rates: R S1 = 0.70, R S2 = 0.53, R S3 = 0.76, R S4 = 0.63, R S5 = 0.65, R S6 = 0.72, R S7 = 0.56, R S8 = 0.44, R S9 = 0.59, and R 4 = 0.92.

Figure 3 .

 3 Figure 3. Maximum likelihood phylogram with five local molecular clocks reconstructed from the first two codon positions of the ADRA2B + IRBP + vWF concatenated exons, and details of the seven fossil calibration points use for dating purposes. The local clocks (or rates: r) are indicated, with their maximum likelihood estimates. r VS , r S , r D , r F , and r VF refer respectively to very slow (tight-dashed horizontal branches), slow (large-dashed branches), default (thin branches), fast (bold branches), and very fast rates (bolder branches). The loglikelihood of this tree is lnL = -26 085.97. Stars refer to the seven calibration points that span the whole placental tree: caviomorphs, Mus / Rattus, glirids, sciuroids, lagomorphs, cetartiodactyls, and paenungulates (C 1 to C 7 from bottom to top). The most recent common ancestors of five major embedded placental clades are indicated by arrows: R for Rodentia, G for Glires (i.e., rodents + lagomorphs), E for Euarchontoglires (Glires + Primates + Dermoptera + Scandentia), Bfor Boreoeutheria (euarchontoglires + laurasiatherians), and P for Placentalia.

  calibrations and reciprocal compatibility between seven paleontological references, as inferred by local molecular clock datings based on first and second codon positions of the three concatenated exons of ADRA2B + IRBP + vWF. Divergence dates in Myr are presented along with their standard-error in parentheses. Fossil calibration ages are given between square brackets. From top to bottom lines, divergences respectively correspond to Echimys vs. Cavia, Mus vs. Rattus, Dryomys vs. Glis, Marmota vs. Aplodontia, Ochotona vs. Lepus, Lama vs. Physeter, and Dugong vs. Procavia. molecular clock datings of the age of the most recent common ancestor of six mammalian clades based on first and second codon positions of the three concatenated exons of ADRA2B + IRBP + vWF, and calibrated by seven paleontological references. Divergence dates are presented along with their standard-error in parentheses. For each of the six mammalian clades, italicized values correspond to the median of the seven divergence time estimates, and bold values correspond to the median of the five divergence age estimates after removal of the two non-compatible calibration points (cetartiodactyls and paenungulates).

ACKNOWLEDGMENTS

This work has been supported by the TMR Network "Mammalian phylogeny" (contract FMRX -CT98 -022) of the European Community to MJS, the "Genopole Montpellier Languedoc-Roussillon", and the "Action Bioinformatique inter-EPST" of the CNRS. We wish to thank the "Centre de Ressources" INFOBIOGEN (Evry, France), and Jacques Demaille and Denis Pugnère (Institut de Génétique Humaine de Montpellier, CNRS UPR 1142) for computing facilities. We thank an anonymous referee for comments on the manuscript. This is the contribution N° 2002-070 of the Institut des Sciences de l'Evolution de Montpellier (UMR 5554 -CNRS).


Ali Seyed

Elahi

Nathanael Connesson
email: nathanael.connesson@gmail.com

Yohan Payan
email: yohan.payan@univ-grenoble-alpes.fr

Disposable system for in-vivo mechanical characterization of soft tissues based on volume measurement

Keywords: Suction/Aspiration Method, Soft Tissues Characterization, Experimental Mechanics, In-Vivo Measurement, Inverse Characterization. 10.400: Mechanics of the Soft Tissues

In-vivo characterization of soft tissues is a key step toward biomechanical simulation and planning of intra-operative assisted surgery. To achieve this, aspiration method is a standard technique: tissue is aspirated through a hole while measuring the pressure and associated apex height. An inverse problem is then solved to identify the material mechanical properties.

In the literature, the apex height is usually measured using a camera, which induces design difficulties, in particular in regards on the required sterilization process for in-vivo measurements.

In this paper, the idea is to replace the apex height optical measurement by the measurement of the aspirated tissue volume. The proposed method enables to reduce the system head to a simple tube: sterilizations becomes easy and the system is disposable after use. The proposed system is thus the simplest, lightest and cheapest one could achieve. It is also to the authors knowledge the first time ever in aspiration method that the aspired volume is the extracted data.

As the data signal-to-noise ratio is the main factor impacting any applied inverse method when extracting the mechanical properties, the aim of this work is to assess and compare the experimental signal-to-noise ratio in the raw volume measurements obtained either optically or with the method proposed. Explicit results of inverse methods using volumes as input data are not presented in this paper for concision purpose.

The effects on accuracy of various experimental parameters has been investigated and quantified: the volume measurement has proved to present a same order or even 1 2 Seyed Ali Elahi, Nathanael Connesson, Yohan Payan better signal-to-noise ratio compared to optical measurements.

Introduction

Understanding and measuring human organs mechanical properties impacts medical practices [START_REF]Biomechanics of Living Organs: Hyperelastic constitutive laws for finite element modeling[END_REF] . In particular, such mechanical properties provide insights for clinical diagnosis [START_REF] Badir | Cervical softening occurs early in pregnancy: Characterization of cervical stiffness in 100 healthy women using the aspiration technique[END_REF][START_REF] Mazza | In vivo assessment of the biomechanical properties of the uterine cervix in pregnancy[END_REF] , surgical training (simulators) [START_REF] Szkely | Surgical simulators[END_REF] , tissue replacement engineering [START_REF] Gentleman | Mechanical characterization of collagen fibers and scaffolds for tissue engineering[END_REF] , and trauma research [START_REF] Snedeker | The creation of a high-fidelity finite element model of the kidney for use in trauma research[END_REF] . Since mechanical behavior of living tissues varies between invivo and ex-vivo conditions [START_REF] Hollenstein | Intraoperative mechanical characterization of human liver[END_REF] , measuring these properties in-vivo and in-situ is important.

In order to perform intra-operative measurements on humans, the procedure must be non-traumatic and operated under sterile conditions. The proposed procedure must also comply with space and time limitations in the operating room. These constraints motivated researchers to design devices with small sizes, low weights and avoiding if possible any electronic parts, so as to be able to withstand severe sterilization processes.

To in-vivo and in-situ measure the sought mechanical properties, various methods have been proposed in the literature such as indentation [START_REF] Carter | Measurements and modelling of the compliance of human and porcine organs[END_REF][START_REF] Samur | A robotic indenter for minimally invasive measurement and characterization of soft tissue response[END_REF][START_REF] Yao | Measuring the compressive viscoelastic mechanical properties of human cervical tissue using indentation[END_REF] , grasping [START_REF] Brown | In-vivo and in-situ compressive properties of porcine abdominal soft tissues[END_REF] , torsion [START_REF] Agache | Mechanical properties and Young's modulus of human skin in vivo[END_REF] , and aspiration [START_REF] Diridollou | In vivo model of the mechanical properties of the human skin under suction[END_REF][START_REF] Badir | Objective assessment of cervical stiffness after administration of misoprostol for intrauterine contraceptive insertion[END_REF] . Among these methods, aspiration/suction is the most widely used technique due to its simplicity and robustness. Several commercial devices are designed for dermatology market (Cutometer from Courage and Khazaka, Germany, or Dermaflex from Cortex Technology, Denmark). In particular, aspiration method limits relative displacement of the tissues in relation with the device and offers a good control of the applied boundary conditions.

The principle of the aspiration method consists in putting a chamber with a hole in contact with the tissue and in setting a negative pressure inside the chamber. The tissue is thus aspirated inside the chamber through the hole while aspiration pressure is measured during the test (Fig. 1). Authors usually propose to measure the aspired height using different methods such as ultrasound [START_REF] Diridollou | In vivo model of the mechanical properties of the human skin under suction[END_REF][START_REF] Hendriks | The relative contributions of different skin layers to the mechanical behavior of human skin in vivo using suction experiments[END_REF] , mechanical stops [START_REF] Badir | A novel procedure for the mechanical characterization of the uterine cervix during pregnancy[END_REF] or cameras, associated to mirrors or prisms [START_REF] Nava | In vivo mechanical characterization of human liver[END_REF][START_REF] Luboz | Linear elastic properties of the facial soft tissues using an aspiration device: towards patient specific characterization[END_REF][START_REF] Kauer | Inverse finite element characterization of soft tissues[END_REF][START_REF] Schiavone | A light sterilizable pipette device for the in vivo estimation of human soft tissues constitutive laws[END_REF][START_REF] Schiavone | In vivo measurement of human brain elasticity using a light aspiration device[END_REF][START_REF] Schiavone | LASTIC: a Light Apiration device for in vivo Soft TIssue Characterization[END_REF][START_REF] Luboz | Validation of a light aspiration device for in vivo soft tissue characterization[END_REF][START_REF] Hollenstein | Towards laparoscopic tissue aspiration[END_REF][START_REF] Weickenmeier | Suction based mechanical characterization of superficial facial soft tissues[END_REF] . Yet, as ultrasound methods are expensive and add additional experimental constraints, optical methods are usually preferred in practical cases to provide pressure versus apex height characteristic curves.

In optical methods, the process is observed by a camera, either directly (Fig. 1) or using a mirror or prism (Fig. 2a) : apex height of the aspired tissue inside the chamber is measured using recorded images. Diagram of pressure versus apex height of the aspired tissue can thus be plotted. Material properties of the tissue are then identified with these experimental data using inverse methods such as updated FE method [START_REF] Luboz | Linear elastic properties of the facial soft tissues using an aspiration device: towards patient specific characterization[END_REF] or gap method [START_REF] Florentin | Identification of the parameters of an elastic material model using the constitutive equation gap method[END_REF] . For example, such aspiration devices have been used to estimate the mechanical properties of cervix [START_REF] Badir | A novel procedure for the mechanical characterization of the uterine cervix during pregnancy[END_REF] , facial tissues [START_REF] Luboz | Linear elastic properties of the facial soft tissues using an aspiration device: towards patient specific characterization[END_REF][START_REF] Weickenmeier | Suction based mechanical characterization of superficial facial soft tissues[END_REF] , brain [START_REF] Schiavone | In vivo measurement of human brain elasticity using a light aspiration device[END_REF] or liver [START_REF] Hollenstein | Towards laparoscopic tissue aspiration[END_REF][START_REF] Hollenstein | Intraoperative mechanical characterization of human liver[END_REF] .

These examples of optical aspiration methods yet underline limitations:

(1) As a camera can not go through severe sterilization processes, its contact with tissues must be prevented, which constrained authors to propose complex designs, usually using mirrors or prisms [START_REF] Luboz | Linear elastic properties of the facial soft tissues using an aspiration device: towards patient specific characterization[END_REF][START_REF] Weickenmeier | Suction based mechanical characterization of superficial facial soft tissues[END_REF][START_REF] Nava | In vivo mechanical characterization of human liver[END_REF][START_REF] Kauer | Inverse finite element characterization of soft tissues[END_REF][START_REF] Schiavone | A light sterilizable pipette device for the in vivo estimation of human soft tissues constitutive laws[END_REF][START_REF] Schiavone | In vivo measurement of human brain elasticity using a light aspiration device[END_REF][START_REF] Schiavone | LASTIC: a Light Apiration device for in vivo Soft TIssue Characterization[END_REF][START_REF] Luboz | Validation of a light aspiration device for in vivo soft tissue characterization[END_REF][START_REF] Hollenstein | Towards laparoscopic tissue aspiration[END_REF] .

(2) The use of cameras and mirrors requires accurate relative positioning, which lead to design systems that are large and rigid [START_REF] Nava | In vivo mechanical characterization of human liver[END_REF][START_REF] Schiavone | LASTIC: a Light Apiration device for in vivo Soft TIssue Characterization[END_REF] . These constraints prevents wide and easy use of these systems during surgery.

A typical example of design evolution is presented by Schiavone et al. [START_REF] Schiavone | A light sterilizable pipette device for the in vivo estimation of human soft tissues constitutive laws[END_REF] , where a light pipette and an external camera have been first used. This system was able to meet the required severe sterilization and handling issues imposed during surgery. Yet, using an external camera induced measurements errors due to misalignment. The authors thus designed a two part Light Aspiration device for in-vivo Soft TIssue Characterization (LASTIC) [START_REF] Luboz | Validation of a light aspiration device for in vivo soft tissue characterization[END_REF][START_REF] Schiavone | LASTIC: a Light Apiration device for in vivo Soft TIssue Characterization[END_REF] (Fig. 2a). The obtained device head was a 3cm × 3cm metallic cylinder divided in two compartments with circular aspiration aperture (diameter of 12mm). The sterilizable lower compartment was a cylindrical chamber made airtight by a glass window in which a negative pressure was applied. A miniature digital camera was fixed inside an upper chamber (not sterilizable) and placed relative to the mirror so as to avoid misalignment errors.

This final LASTIC head has yet not been used in intra-operative situations due to its still relative complexity, large weight and large size compared to its aspiration aperture. In this paper, the idea is to replace the apex height optical measurement by the measurement of the aspirated tissue volume. Such a change in the method enables to remove camera, mirror and all electronics from the system head that is basically reduced to a simple aperture and is thus able to meet the required severe sterilizations (Fig 2b). The proposed system is thus the simplest, lightest and cheapest one could achieve. According to the authors, thus simplifying the system will enable a generalized use of the device.

Switching the input data from apex height to volume measurement will only slightly change the inverse method process: both apex height and aspirated tissue volume are directly related to the aspirated tissue surface shape. The mechanical properties thus identified with either input data rely on the same hypotheses (boundary conditions, material thickness and anisotropy, mechanical properties gradients due to different layers, etc.) and will provide similar results. As these inverse method results have already been published and validated in the literature [START_REF] Nava | In vivo mechanical characterization of human liver[END_REF][START_REF] Badir | A novel procedure for the mechanical characterization of the uterine cervix during pregnancy[END_REF][START_REF] Luboz | Linear elastic properties of the facial soft tissues using an aspiration device: towards patient specific characterization[END_REF][START_REF] Weickenmeier | Suction based mechanical characterization of superficial facial soft tissues[END_REF][START_REF] Hollenstein | Towards laparoscopic tissue aspiration[END_REF][START_REF] Schiavone | In vivo measurement of human brain elasticity using a light aspiration device[END_REF][START_REF] Hollenstein | Intraoperative mechanical characterization of human liver[END_REF] , such inverse identification issues are considered as beyond the scope of this paper.

Eventually, only differences in the input data signal-to-noise ratio can significantly impact the inverse analyzes. The main question to deal with is then to assess if the experimental volume measurements signal-to-noise ratio obtained optically or with the proposed method are comparable: is there a loss of measurement precision while changing of measurement mean? This question will be answered experimentally by comparing volume measurements obtained either optically or with the proposed method.

The measurement method based on aspirated volume is first thoroughly presented (section 2). This method is then compared with a specifically designed reference method and then with more classical optical measurements (sections 3 and 4). The results obtained with each setup are then analyzed with a particular attention to reproducibility, error, measurement standard deviation (section 5), and impact of different setup experimental parameters (section 6). Finally, some recommendations are provided to help users to obtain the same results quality (section 7).

Rate-based volume measurement method

During in-situ and in-vivo tests, the proposed new system is simply composed of a syringe pump, a manometer and an aspiration chamber (Fig. 3).

The volume removed by the syringe pump V total and the associated pressure P are measured during the test. Yet, the volume removed by the syringe pump V total is obviously different from the tissue volume V sample : this difference is mainly due to volume changes in the system because of the applied pressure; the system volume V system is compressible due to air expansion and elasticity of the components (connections, tubes, syringe, etc.). The total removed volume V total can therefore be seen as the sum of the aspired tissue volume V RB sample and the volume changes due to system compressibility V system :

V total (P ) = V RB sample (P ) + V system (P ) (1)
The system volume function V system (P ) is experimentally assessed during a second step by testing an undeformable material, referred to as rigid test: in this experimental configuration the aspired tissue volume V RB sample remains zero and the system volume compressibility V system can be directly measured using Eq. 1:

V system (P ) = V total rigid (P ) (2)

Combining the system volume V system (Eq. 2) and Eq. 1, the aspired tissue volume V RB sample can thus naturally be estimated by: V RB sample (P ) = V total (P ) -V system (P ) When applying the rate-based volume measurement method V RB sample during invivo experiments, the total volume V total will be measured first, using a sterilized aspiration head. The rigid test will be performed afterward, when sterilized conditions are no longer required. During this test, a clamp system can be used to avoid leakage between the aspiration head and the undeformable material. Performing this additional rigid test adds less than 5 minutes to the overall procedure.

Performance assessment test-bench

As the aim of this work is to compare optical and rate-based volume measurement signal-to-noise ratio, all results should be compared with a reference obtained in identical conditions. Three different methods and associated volume measurements will thus be performed on the same material:

(1) the proposed rate-based volume measurement method V RB sample (section 2), (2) a reference method V ref erence sample (section 3.2), (3) and a camera based method V camera sample (section 3.3). In this work, it has been chosen to perform these three measurements simultaneously on the same tested material. As no inverse characterization will be performed here, the tested material is not required to be known or to mimic accurately the tissue mechanical behavior, but only to provide a sufficient shape variation (flat to a half sphere in this case) for a pressure range of the order of magnitude observed during actual tissue mechanical characterization: the tests have been performed in a range of 0 to 50 mbars, which is approximately the range obtained while characterizing in-vivo lower lip tissues [START_REF] Luboz | Linear elastic properties of the facial soft tissues using an aspiration device: towards patient specific characterization[END_REF] .

In the performance assessment configuration, the tissue has thus been replaced with a soft thin membrane cut from a latex glove (Fig. 4). This membrane has been glued to a system providing the reference volume V ref erence sample (section 3.2). A camera has been used to take pictures of the aspired membrane during the tests and to deduce the aspired volume V camera sample (section 3.3). The provided results will thus provide the signal-to-noise ratio of the rate-based volume measurement method V RB sample and classic optical method V camera sample by using the reference volume V ref erence sample .

Rate-based volume measurement method

The method previously presented is directly applied (section 2). In particular, the rigid test in this test configuration is performed by closing the valve in the setup (Fig. 4), so that the contribution of the aspirated tissue volume is zero during the rigid test.

Reference volume measurement

A reference volume measurement device has been developed to estimate measurement errors (Fig. 4). Such a device obviously will not be used in practical cases.

This device, glued to a membrane, simply consists in a thin tube filled with water where an air bubble has been inserted (Fig. 5). Any movement of the membrane will induce movements of both the water and the bubble. Knowing the tube section S and the bubble displacement ∆L straightforwardly provides the "tissue" aspirated volume V ref erence sample :

V ref erence sample = S∆L (4)
The bubble movements ∆L have been tracked on pictures synchronized with the pressure measurement. A dark background and proper lighting have been used to help automated bubble position tracking.

Camera volume measurement method

The proposed rate-based method V RB sample should be compared to an optical based method V camera sample even if the measurements using camera will not be necessary to apply the proposed method. For this, a camera has been used to take pictures of the aspired membrane shape h(r) during the test (Fig. 6). Computing the associated volume can be easily performed by:

V camera sample = π R -R rh(r)dr ( 5
)
where r is the radius of any point M of the aspirated surface, dr is an elementary radius, h is the membrane height above point M, and R is the aperture internal radius (Fig. 6c).

A dark background and proper lighting have been used to help automated membrane shape h(r) recognition. According to Fig. 6b, some irregularities are visible in the lower part of the initial picture, which are due to the glue between the membrane and the aspiration chamber. The erroneous additional volume induced by these extra white parts is simply removed by subtracting the estimated volume of the initial picture from all other volumes during the aspiration.

Preliminary results and parameters definition

Preliminary results

An example of typical results obtained simultaneously by the three methods is presented as pressure P versus volume curves (Fig. 7).

Curves 1 to 3 are obtained using the rate-based volume measurement

method V RB

sample with a rate of 0.4ml/min, which proved to be small enough to provide quasi-static mechanical behavior:

• the total volume V total (P ) obtained with the syringe pump while aspirating the membrane is presented as Curve 2 (Eq. 1) • the measurement obtained during rigid test V total rigid (P ), valve closed, is presented as Curve 1 (Eq. 2). This volume variation is mainly due to the volume variation of air but also to the polyurethane tubes connecting the aspiration head, syringe pump and manometer. Special attention has been made to prevent any possible leakage in the connection zones.

• the sought volume V RB sample (Curve 3) is the result of the difference of the two previous curves (Eq. 3).

Curves 4 and 5 are obtained using the reference V ref erence sample and camera based methods V camera sample , respectively (Eq. 4 and 5, respectively). It should be specified here that the reference measurement system has been built using a tube of internal diameter of 0.9mm to increase the bubble movement sensitivity. The bubble position has been estimated with a precision of about ±1mm; the reference volume V ref erence sample variations are thus measured with an accuracy of about ±0.64µl.

These preliminary tests underline that all three pressure P versus volume curves (V RB sample , V ref erence sample , V camera sample ) are very similar, validating thus the three proposed methods. The signal-to-noise ratio of each method needs now to be estimated, which will be presented in section 6. reported as three thick black lines (average and average±2ST D, Fig. 9). The two curves average±2ST D will be referred to as envelope curves in the following.

The same results have been obtained on each test group (table 1) but are not presented here for concision sake.

Results and discussion

In this section the two volume measurement errors Error RB and Error camera have been analyzed and compared: averages and standard deviations of the errors have been represented for all the test groups (table 1) in two different categories:

(1) Tests with the same aspiration Aperture Diameter (AD, Section 6.1).

(2) Tests with the same Syringe Diameter (SD, Section 6.2).

Tests with the same aspiration Aperture Diameter (AD)

Rate-based results

The rate-based volume error Error RB for each test group with the same aperture diameter has been presented using only their envelope curves in Figs. 10a (AD=21mm) and 10b (AD=12.9mm). Theses results underline:

• The average rate-based errors Error RB are oscillating around zero (no measurement bias) for both aspiration Aperture Diameters AD. • The standard deviation of measurements errors can be reduced by choosing a smaller Syringe Diameter SD.

It could also be underlined here that decreasing the Syringe Diameter SD diminishes the maximal aspirated volume V total out of the system (Fig. 3) and consequently reduces the maximal pressure range.

Optical results

The envelope curves of the optical volume errors Error camera are presented for each of the aspiration Aperture Diameter AD= 21 and 12.9mm (hatched areas in Figs. 10a andb, respectively).

• The standard deviation of camera measurements errors increases with the pressure level and is of the same order of magnitude as the rate-based volume error Error RB .

It yet should be underlined here that by properly choosing the Syringe Diameter SD, the rate-based method can provide more accurate results than the optical method. This observation will encourage the use of the rate-based method rather than the optical method thanks to its simplicity and accuracy.

Results comparison

To compare the methods accuracy more fairly, the results have been evaluated at a the pressure (IP) of 18.25mbar (red vertical dashed line, Figs. 10a andb), which is the maximal pressure for which data is available for all the different tests. The extracted standard deviations at IP are presented as functions of the Syringe Diameter SD (Fig. 11). As previously mentioned, it can be underlined that: decreasing the Syringe Diameter SD reduces the error standard deviation. In particular, the Syringe Diameter SD can be chosen so as to obtain smaller errors than with the optical method. This conclusion is independent of the chosen aspiration aperture diameter or chosen pressure IP.

Tests with the same Syringe Diameter (SD)

It has been underlined previously that the rate-based measurement errors mainly depend on the Syringe Diameter SD. To emphasis the impact of the aspiration Aperture Diameter AD on the errors, the results have also been studied for similar syringe diameters (Figs. 12a to c). In each graph, the observed error curves are of the same order of magnitude whatever the Aperture Diameter AD value is. The aspiration Aperture Diameter AD has thus almost no impact on the error of the ratebased volume measurement method Error RB . The aspiration aperture diameter can thus be chosen by taking into account only criteria as space constraints or required depth analyze.

User recommendations

To apply the rate-based volume measurement method V RB sample , a user should take into account the following considerations:

• Pay a special attention to prevent any leakage in both main and rigid tests.

Before any test, applying an initial small negative pressure (in range of 1mbar) and checking the pressure stability would be advisable. This initial pressure should be very small compared to the measurements pressure range. Special fixture apparatus should be design to perform the rigid test, using, for example, a caoutchouc surface clamped on the aspiration aperture. • To avoid potential effects of the temperature on the error, the main and rigid tests should be performed in the same environment and within a short interval. • If sterilized conditions are required during the actual test, the rigid body test should be performed after the main test to avoid any possible system contamination. • Too high volume rate applied by the syringe pump can be a source of error due to viscoelastic behavior of either tissue or system: low enough volume rate should thus be used. Such a volume rate can be partially estimated during rigid tests performed before sterilization and by insuring that the volume rate has no effect on the obtained function V total rigid (P ). In this work, a volume rate of 0.4 ml/min has been selected. The same method can be applied while testing the material with equal or lower volume rate to ensure its behavior is rate independent. The presented methods and systems could also be used to estimate viscoelastic properties of materials, but this aspect is beyond the scope of this work.

Conclusions

An original method based on volume measurement has been developed and validated to perform aspiration tests on living soft tissues. Unlike previous aspiration devices, this method does not require the presence of camera, mirror, or any electronic part. It thus enable designing simpler, lighter, smaller and easily sterilizable systems compared to the previous proposed systems.

In particular, the proposed method requires two measurement steps: (1) measuring the system+tissue behavior and (2) measuring only the system behavior during a secondary test referred to as rigid test.

A performance assessment test-bench has been proposed to estimate the experimental bias and errors of the method. Classic optical method has also been implemented so as to compare each method strengths and weaknesses.

As a conclusion, the volume rate-based method proved to have similar or lower measurement error than the classic optical method provided that the syringe diameter used in the syringe pump is properly chosen and that no leakage occurs during the tests. Eventually, it has been experimentally proven that the aspiration aperture diameter has a negligible impact on the measurements error.

User recommendations have also been provided to guide any user that would be prone to use such a method.

Fig. 1 .

 1 Fig. 1. General scheme of the aspiration method.

Fig. 2 .

 2 Fig. 2. a) Design of LASTIC head and its components 23 , b) Proposed simplified design.

Fig. 3 .

 3 Fig. 3. Measurement principle of the proposed rate-based volume measurement method.

Fig. 4 .

 4 Fig. 4. Performance assessment configuration: simultaneous measurement of the aspirated tissue volume using the 3 methods: the reference method V ref erence sample , the rate-based method V RB sample

Fig. 5 .

 5 Fig. 5. Air bubble inside the spiral tube a) before and b) after the aspiration process.

Fig. 6 .

 6 Fig. 6. a) Global view of the aspiration chamber: (1) aspiration head and (2) reference measurement system and pictures used to measure the aspired volume V camera sample b) before and c) during aspiration.

Fig. 9 .

 9 Fig. 9. Comparison of 9 diagrams of rate-based errors Error RB , their average and STD for test group VT-1.

Fig. 10 .

 10 Fig. 10. Envelope curves of volume measurement errors (Error RB and Error camera ) versus pressure P for the tests with different Syringe Diameters SD and aspiration Aperture Diameters AD = a) 21mm and b) 12.9mm.

Fig. 11 .Fig. 12 .

 1112 Fig. 11. STD of volume measurement errors (Error RB and Error camera ) versus SD for the tests with various AD at Iso Pressure (IP).

Parameter definition and experimental design

The signal-to-noise ratio of the rate-based volume method depends on the inner aspiration Aperture Diameter (AD) and the Syringe Diameter (SD) associated to the syringe pump (Fig. 3). Tests have thus been performed for two different inner Aperture Diameters (AD) and three different Syringe Diameters (SD). Therefore, 6 groups of tests have been conducted. Syringe and aperture inner diameters for the 6 test groups have been summarized in table 1. As the camera volume measurements V camera sample depends only on the chosen Aperture Diameters (AD), only two measurement sets are required to perform the signal-to-noise ratio study on the camera volume measurements. and camera V camera sample methods for the 9 repetitions of test group VT-1 are presented in Fig. 8 as an example of raw data. The results show mechanical behavior differences for these successive tests, but, for each particular test, the three methods provide very similar results. This underlines that the rate-based method is able to monitor the lack of the chosen latex membrane mechanical reproducibility (attributed here to the complex local loading history and latex non linear behavior): the measurement error is lower than the mechanical behavior variations between tests. The errors of the rate-based volume measurement V RB sample and of the camera V camera sample methods have been computed using the reference volume V ref erence sample . Such errors are thus defined as:

Error camera (P ) = V camera sample (P ) -V ref erence sample (P )

For example, the experimental results of Eq. 6 (Error RB ) for test group VT-1 are presented versus pressure P in Fig. 9. In addition, the associated STandard Deviations (STD) and average of volume measurements errors Error RB have been