Henri Boutin
email: henri.boutin@unsw.edu.au

Neville Fletcher
email: neville.fletcher@anu.edu.au

John Smith
email: john.smith@unsw.edu.au

Joe Wolfe
email: j.wolfe@unsw.edu.au

LIP MOTION, THE PLAYING FREQUENCY OF THE TROMBONE AND THE UPSTREAM AND DOWNSTREAM IMPEDANCES

We report the motion of trombone players' lips, its phase with respect to the mouthpiece pressure, the impedances of the bore and the player's vocal tract, and the frequency difference between the bore resonance and played note. The bore resonance frequency shifts very little with playing and often decreases somewhat: the effect of CO 2 can exceed that of temperature and humidity. The bore impedance is usually compliant for the note B 2 and inertive for B 3. The vocal tract impedance measured at the player's mouth is inertive for both notes. In terms of Fletcher's simple model for regeneration (JASA, 93, 2172), the results are consistent with a (+1,-1) valve for B 2 and (-1,+1) for B 3. The pressure in the mouthpiece in both cases rises before the lips separate. For B 3, where the lip motion is mainly transverse, this is consistent with the inertive load. For B 2, the substantial motion of the lips in the direction of flow provides a sweeping motion which produces the current into the bore that precedes lip opening.

INTRODUCTION

How and why do the lips of a brass player vibrate? This is an interesting question in music acoustics, with potential applications in music pedagogy and performance. Images of the motion as functions of time are an important contribution to answering the question. Various studies, including the present one, have used stroboscopy or high speed video for this purpose: Copley and Strong [START_REF] Copley | A stroboscopic study of lip vibrations in a trombone[END_REF], Yoshikawa and Muto [START_REF] Yoshikawa | Lip-wave generation in horn players and the estimation of lip-tissue elasticity[END_REF], Tarnopolsky et al [START_REF] Tarnopolsky | Vocal tract resonances and the sound of the Australian didjeridu (yidaki) i. experiment[END_REF], Campbell and colleagues [START_REF] Newton | Mechanical response measurements of real and artificial brass players lips[END_REF] and others.

In mathematical models of the vibrating lip, the acoustic impedance spectra of the upstream and downstream side of the lip are important [START_REF] Elliott | Regeneration in brass wind instruments[END_REF][START_REF] Fletcher | Autonomous vibration of simple pressure-controlled valves in gas flows[END_REF]. Simultaneous measurements of the acoustic pressure in the mouth and that in the mouthpiece during trombone playing [START_REF] Fréour | Vocal-tract influence in trombone performance[END_REF] have shown that the ratio of the former to the latter increases strongly as the players play higher, but varies among players. This increase is in part due to the way the impedance maxima of the bore decrease with increasing frequency. However, the variation among players strongly suggests that they use their vocal tracts in different ways.

Measuring the upstream impedance Z mouth -the impedance of the player's vocal tract measured near the lips -is non-trivial: the signal produced by the vibrating lip is much more powerful than the probe signal used to measure Z mouth , because the energy of the probe signal has to be divided among hundreds of different frequencies. Nevertheless, this has been reported for the trumpet [START_REF] Chen | Do trumpet players tune resonances of the vocal tract?[END_REF] and the didjeridu [START_REF] Tarnopolsky | Vocal tract resonances and the sound of the Australian didjeridu (yidaki) i. experiment[END_REF]. Measuring the downstream impedance Z bore also has challenges because: (i) the instrument operates close to an impedance peak, (ii) the impedance peaks are quite narrow and (iii) the speed of sound and thus the resonant frequency depends subtly on temperature, humidity and the concentration of CO 2 inside the bore. So it is necessary to measure Z bore under conditions close to playing so as to understand the combined effect of temperature, humidity and CO 2 . The relative timing of the lip motion and the pressure in the mouthpiece is also important. The timing of lip contact can be measured by the high frequency electrical admittance y lip between upper and lower lips [START_REF] Wolfe | Acoustical coupling between lip valves and vocal folds[END_REF][START_REF] Fréour | Development of an electrolabiograph embedded in a trombone mouthpiece for the study of lip oscillation mechanisms in brass instrument performance[END_REF].

In order to explain how the lip motion is driven at different playing frequencies, the objectives of the present study are: to investigate the effects of temperature, humidity and concentration of CO 2 on the bore resonances, to measure the mouth and bore impedances during playing, and to compare the lip contact with the pressure in the mouthpiece.

MATERIAL AND EXPERIMENTAL SETUP

We report the measurements of the impedance of the trombone bore, the impedance of the player's vocal tract, the pressure in the mouthpiece and a variable related to the opening area between the player's lips. These signals were measured using three different experimental setups described in the next paragraphs. Six amateur players were experimental subjects.

The trombone and the mouthpiece

All the players played same modified trombone (Yamaha model YBL 321). This instrument has a valve that converts it from the tenor range to the bass range, but the valve was left in the tenor position. In all experiments, the slide was kept in the same position (first position: all the way in) and the tuning slide was extended 18 mm. The instrument was clamped to a lab bench but mechanically isolated from it with foam.

The players involved in these experiments were asked to play up to five of the lowest notes above the pedal note: B 2 (nominally 117 Hz), F3 (175 Hz), F 3 (233 Hz), D4 (294 Hz) and F4 (349 Hz). However, in this brief report, only the data for two notes are analysed: B 2 (nominally 117 Hz) and B 3 (233 Hz).

The mouthpiece was replaced with an experimental mouthpiece (Figure 1) having the same cup volume (11.6 cm 3 ), diameter (2.53 cm) and throat geometry as the original mouthpiece. The mouthpiece is transparent with two flat glass windows to allow high-speed video and stroboscopic observation of the motion of the lips. In a hole of diameter 2.3 mm in one side is sealed an Endevco piezoresistive pressure transducer (model 8507C-2). Two brass electrodes are set into the rim of the mouthpiece, so that they contact the face above the upper lip and below the lower. They are connected to the input of an electroglottograph (EGG, model EG2-PCX2) from Glottal Enterprises.

The bore impedance

For the temperature studies, the capillary method is used to measure the bore impedance Z bore so that it could be sealed to the mouthpiece very quickly. The impedance head comprises an acoustic current source with high impedance and a microphone. The current source is similar to that used by Wolfe et al [START_REF] Wolfe | Acoustic impedance spectra of classical and modern flutes[END_REF] and comprises a truncated cone concentric with a conical hole, but separated from it by three 100 µm wires placed at 120 • angles. The source and a microphone (B&K model 4944A) are located side by side in the plane against which the trombone mouthpiece is clamped. The impedance head is calibrated with a semiinfinite pipe of length 142 m and of diameter 7.8 mm.

For calibration and measurements, a loudspeaker generates a broadband acoustic current at the reference plane. This contains frequency components ranging between 50 Hz and 600 Hz with a resolution of 0.67 Hz (= 44.1 kHz/2 16 ). During a calibration iteration, the amplitudes of each of the spectral components of this current are equalized [START_REF] Wolfe | Acoustic impedance spectra of classical and modern flutes[END_REF].

The vocal tract impedance

The method used to measure the vocal tract impedance of the trombone player is the three microphone technique using non-resonant loads, described by Dickens et al [START_REF] Dickens | Improved precision in acoustic impedance measurements by using calibration loads without resonances[END_REF]. Its advantage over the capillary method for this application is that a more powerful probe signal is possible, and this is helpful in the presence of the broadband noise measured close to the player's lips. We use our smallest impedance head (of outer diameter 4.8 mm) to minimise the perturbation of the player, who is asked to hold this pipe between the lips at the corner of the mouth during playing, while orienting it so that the measurement plane is between the upper and lower teeth and behind the central portion of the lips. A flange on the head The modified mouthpiece equipped with two brass electrodes connected to an electroglottograph (EGG) to measure lip contact after Wolfe and Smith [START_REF] Wolfe | Acoustical coupling between lip valves and vocal folds[END_REF], and Freour [START_REF] Fréour | Development of an electrolabiograph embedded in a trombone mouthpiece for the study of lip oscillation mechanisms in brass instrument performance[END_REF]], a piezoresistive pressure transducer connected to the mouthpiece cup. Its design follows Ayers [START_REF] Ayers | Alternative mouthpiece design for viewing the lip reed in motion[END_REF]: the two glass windows provide undistorted front view and side view of the lip motion. at 25 mm from the measurement plane limits the maximum insertion into the mouth. The probe signal contains 34 cycles of length 2 16 points (about 1.49 s), so that the measurements can last up to 50 s. 820 frequencies spaced at 0.67 Hz cover the range 50 Hz to 600 Hz.

The pressure in the mouthpiece, the lip motion and their relative phase

The pressure transducer in the mouthpiece, flush with the wall of the cup, is connected to a bridge amplifier and then to a sound card (MOTU, model Firewire 828). The sound card gain was calibrated using a known voltage source (Topward model 8120) of variable frequency. Then we replaced the transparent back plate of the mouthpiece by a plug in epoxy resin, equipped with a reference microphone (B&K 4944A). The mouthpiece was then driven with a broadband signal over the range [50 Hz, 1 kHz] and the ratio between the spectrum of signals from the pressure transducer and the reference microphone recorded. In the following experiments, with the transparent window replaced, this ratio determines the pressure in the mouthpiece cup from the transducer signal.

The EGG connected to the brass electrodes provides a signal negatively correlated with the electrical admittance between the two lips. It has no DC component, is maximum when the lips are open and minimum when they are closed. Thus it is correlated with and approximately in phase with the opening area between the lips, so we call it OV (t) (for open variable). In order to evaluate the delay introduced by the EGG we connected a voltage-controlled resistance at its input. The delay (0.1 ms between 100 Hz and 1 kHz) is not negligible compared to the period of the notes played (≈ 10 ms), but this difference can be added when comparing the phase difference between the pressure in the mouthpiece and the lip opening area as given by the outputs of the piezotransducer and the EGG.

TEMPERATURE AND GAS EFFECTS ON THE RESONANCES OF THE BORE

As the trombonist starts to play, the temperature, the humidity and the concentration of CO 2 all increase in the bore. How do the impedance peaks vary with playing? We measured the input impedance of the trombone before the instrumentalist plays, and after he has played for 3 s, 10 s, 30 s and 4 min. Between measurements, the air inside the bore was flushed, and the bore impedance was measured again. During the 3 s and 10 s experiments, the player was asked to take a breath and to play a single sustained note. These experiments reproduce the playing conditions met by the player when his instrument is initially dry and at ambient temperature (26.8 • C and relative humidity 57% in this instance). In the longer experiments (30 s and 4 min) the performer was asked to play several long notes and to inhale at will. The impedance head, coated with a thin layer of petroleum jelly, was sealed to the mouthpiece rim, within 3 s of the end of the playing and the impedance measured for successive cycles over the following 50 s. This protocol was followed once or twice with every player. The impedance spectra were sampled at 0.67 Hz. Around the peaks, these data were interpolated with a fitted cubic function, giving an estimated frequency resolution of 0.1 Hz. The mean curves corresponding to each playing duration are shown in Figure 2. For all playing conditions, the second, third and fourth impedance peaks fall within the ranges 113.0 -113.9 Hz, 170.9 -172.1 Hz, and 227.7 -229.4Hz. These are the peaks used to play B 2, F 3 and B 3 respectively. For each, we note that the peak frequencies and amplitudes decrease after playing for 3 s. After 10 s and 30 s of playing, the frequencies are slightly greater than after 3 s and the amplitudes almost unchanged. After 4 min, the peak frequencies return approximately to their initial values, while the amplitudes decrease further. The same trend was observed on the 9 peaks between 50 Hz and 600 Hz. The changes in their amplitudes and frequencies are plotted on Figure 3.

The repeated measurements made before playing on different days are always located at less than 0.26% from their average frequency, with a standard deviation of 0.09%. Figure 3 shows that, after playing for 3 s and 10 s, the peak frequencies decrease in average by 0.48% and 0.62% respectively. Because humidity and temperature both increase the speed of sound, we conclude that, in these experiments, increased concentration in CO 2 in the bore more than compensates for the increases in temperature and humidity. This is not surprising: 3 s is easily enough time for the player to replace the air in the bore, but probably not enough time for the air to warm the instrument. Rises in water concentration are limited by the temperature.

In the 30 s-experiment, the players inhaled at least once between notes. So the CO 2 ceased increasing. This would explain why the peak frequencies stopped decreasing. In the 4 minute experiment, the players breathed several times as well, and there was time for temperature and thus water concentration to rise. Here, resonant frequencies differed little from the dry, ambient conditions: an average increase of 0.02%. The decrease in the amplitude of the impedance peaks can possibly be attributed to humidity. We expect that gas composition would have very little effect on the characteristic impedance ρc/A. However, as suggested by Coltman [START_REF] Coltman | Acoustical losses in wet instrument bores (l)[END_REF], air near 100% humidity could, in a standing wave, evaporate and condense during each cycle, which would increase losses. This could explain why after playing 4 min, the minima in Z bore increase in magnitude (by 3.3% in average) while the maxima decrease (by 3.8% in average).

THE VOCAL TRACT IMPEDANCE

Among the previous studies focusing on the brass players' lips, two different lip motions in different ranges are reported [START_REF] Copley | A stroboscopic study of lip vibrations in a trombone[END_REF][START_REF] Yoshikawa | Lip-wave generation in horn players and the estimation of lip-tissue elasticity[END_REF]. In the lower register, components of motion parallel and perpendicular to the flow have similar amplitudes, while in the high register the parallel component is much smaller. A simple model by Fletcher [START_REF] Fletcher | Autonomous vibration of simple pressure-controlled valves in gas flows[END_REF] associates qualitatively different kinds of auto-oscillatory lip valve motion with different values of reactive components of the upstream and downstream impedances.

Measurements of the upstream impedance spectrum used a small impedance head inserted in the corner of the mouth with its end lying between upper and lower teeth and close behind the lips. None of the players reported any difficulty to play the notes in this study (B 2 and B 3) with the impedance head between the teeth and some could comfortably play several notes above these. We asked the players to play long notes, then to take another breath when they were out of air, and to start again. Each measurement lasted 30 s and allowed the subjects to play usually between 2 and 3 notes of 10 s. The experiment was carried out several times with each performer playing the lower pitch note B 2 and then the higher pitch note B 3. For each vocal tract configuration, the impedance curves obtained were fairly similar; two of them are displayed on Figure 4.

The players reported that they raised their tongue while playing a high pitch note. This would be expected to raise the characteristic impedance, could explain why Z mouth is typically higher when the player is playing B 3. (An analogous result was observed by Wolfe et al [START_REF] Wolfe | The acoustics of wind instruments-and of the musicians who play them[END_REF], using a mechanical trombone-playing machine with an outwards opening valve.) Finally, we removed the impedance probe from the player's mouth and asked each player to play notes (B 2 and B 3) for about 10 s. We recorded the sound pressure in the mouthpiece, p mp , and calculated its spectrum over a 1 s interval, in the sustained period, after the attack. We compare it with the bore resonances measured after 10 s, since it is approximately the duration of the notes played. Henceforth, Z bore refers to Z 10s . f p varies somewhat for different performers and samples. However, for all six players, f p is always located above the bore resonance when the note played is B 2 and below when it is B 3. (These inequalities are true for all of the Z bore spectra measured in this study under the various conditions described above.) From Fletcher's simple model [START_REF] Fletcher | Autonomous vibration of simple pressure-controlled valves in gas flows[END_REF], a required condition for a valve to auto-oscillate is:

Im W p1 µ(σ 2 Z 2 -σ 1 Z 1 ) √ 2ρp 1 + W x(Z 1 + Z 2 ) > 2πf p × k, ( 1
)
where W is the valve width perpendicular to the flow, x the static component of the valve opening and p1 the static pressure upstream of the valve. ρ is the air density, k the attenuation coefficient of the valve and f p its playing frequency. µ is proportional to the valve flap area divided by the effective mass involved in the oscillation. The pair (σ 1 , σ 2 ) = (±1, ±1) describes how the valve operates: σ 1 = +1 if a positive pressure upstream of the valve tends to open it further, and -1 if it tends to close it. σ 2 is similarly defined with respect to the pressure downstream of the valve. Finally Z 1 is the impedance 'looking into' the upstream side of the valve in the direction of the load, and Z 2 'looking into' the downstream side of the valve. Here they correspond to Z mouth and Z bore respectively. This simplified model assumes that the valve channel is negligible length, the static pressure downstream is zero and that the valve flaps never collide with each other and undergo simple harmonic motion. In addition it neglects the inertance of air inside the channel and the flow generated by the flap motion. Even though these simplifying assumptions are not satisfied by the lip valve, the model is still qualitatively helpful in discussing whether a vibration mode can auto-oscillate or whether it is driven by an other mode.

Our measurements show that the imaginary parts of the impedances (X mouth and X bore ) have magnitudes comparable with the real parts (R mouth and R bore ) at the playing frequencies, cf. Figure 5. This means that the losses in the vocal tract and the bore and its radiation impedance cannot be neglected. Thus (1) leads to the following conditions:

σ 2 X bore -σ 1 X mouth > α(X mouth + X bore ) × .. ..(σ 2 R bore -σ 1 R mouth ), ( 2 )
with α = W x √ 2ρ pmouth +W x(R mouth +R bore ) (3)
Typical values of X and R at both playing frequencies studied are obtained from the measurements of Figure 5. This involves removing the very narrow peaks at the harmonics of the note being played and interpolating and smoothing the real and imaginary parts of the vocal tract impedance with a Savitsky-Golay filter: Since R mouth , R bore and x are positive, α > 0 in (2). In the case of B 2, -X bore > X mouth > 0. Then the only condition for (2) to be satisfied with any positive values of R mouth is (σ 1 , σ 2 )=(+1, -1). This is the regeneration mode for an outward swinging valve, or swinging door valve.

Z bore Z mouth (M P a.m -3 s) (M P a.m -3 s) fp R bore X bore R mouth X
For B 3, X bore > X mouth > 0. Then, a (-1, +1) valve always satisfies the auto-oscillation condition with any positive value of the real part R mouth . By giving plausible order of magnitude to the model parameters, p1 = 2kPa, W = 1cm and x1 = 1mm, the simple model predicts that a (-1,-1) valve could also auto-oscillate. But this also depends on the real part of Z mouth . As a result, in the higher range, our results and this model together suggest that the brass player's lips probably do not operate as a (+1,+1) valve but more likely as a (-1,+1).

THE TIME SIGNALS

In the next experiment, we investigate the phase relationship between the pressure in the mouthpiece p mp and the motion of the lips. As detailed in Section 2, the contact electrodes on the experimental mouthpiece allow us to measure the AC component of the electrical admittance between the lips y lip . During the previous experiment, as we asked the performers to play B 2 and B 3, we measured y lip and the pressure in the mouthpiece p mp simultaneously. The signal shown on Figure 6 is inversely related to this admittance: it is a minimum when the contact area between the lips is maximum and vice versa. For this reason, we call this the open variable OV (t). Copley and Strong [START_REF] Copley | A stroboscopic study of lip vibrations in a trombone[END_REF] show that the lips of a trombonist close completely during each cycle, as do those of a hornist [START_REF] Yoshikawa | Lip-wave generation in horn players and the estimation of lip-tissue elasticity[END_REF]. This was also observed by high speed video in the present study. During this closed phase, air flow between the lips is zero, y lip is maximum and OV is a minimum. Figure 6 shows an example of OV (t) and p mp (t). For each of the played notes, the pressure in the mouthpiece starts to increase before the lips begin to open and reaches its minimum value just before the lips close completely. This feature was observed for each note played and each performer. The air flow between the lips is zero when the lips are closed. When the lips are open, however, it is not simple to relate the flow to the lip opening or to OV .

For the higher played note B 3, the impedance measurements show that Z bore is inertive at the playing frequency (Figure 4), so one would naïvely expect the flow into the mouthpiece to lag the pressure p mp . (In practice, the flow between the lips is broadly related to lip opening, but the motion is complicated by higher harmonics, whose phase relations may in principle be different from that of the fundamental.)

In contrast, for the lower note B 2, Z bore is compliant at the playing frequency and so the flow in the mouthpiece U bore is expected to lead the mouthpiece pressure p mp . This raises an obvous question: how can the flow into the mouthpiece begin while the lips are still closed? Yoshikawa and Muto [START_REF] Yoshikawa | Lip-wave generation in horn players and the estimation of lip-tissue elasticity[END_REF] provided images from side viewing of the lip motion in a transparent horn mouthpiece. They show that the upper lip moves outward significantly in the lower range and that the lips separate after they have moved forward into the mouthpiece.

Video images taken in the present project give similar results: for B 2, the lip motion has a large component in the direction of the flow, and this motion leads the transverse motion in phase. In contrast, for B 3 the motion is almost entirely transverse. For B 2, this sweeping motion generates a positive flow into the mouthpiece before the lips separate and before pressure begins increasing in the mouthpiece. In this preliminary account, we do not report the magnitudes of this 'sweeping' current, but simple calculations show that its magnitude is of the same order as the total acoustic flow into the mouthpiece. In contrast, for the higher note B 3, the lip motion is almost entirely perpendicular to the flow, so U bore essentially equals the air flow between the lips, and this current can start to increase only when the lips open.

CONCLUSIONS

These observations suggest that, for B 2, the lips operate as an outwards opening valve and for B 3, as a sideways sliding valve. Fletcher's simple model [START_REF] Fletcher | Autonomous vibration of simple pressure-controlled valves in gas flows[END_REF] predicts the regeneration conditions for such valves in terms of the acoustical impedance upstream and downstream of the lip valve. The upstream impedance, Z mouth depends on the configuration of the player's vocal tract and, at the playing frequencies reported here, it increases when the player raises his tongue.

According to the curves of Figure 2, the frequency and amplitude of the peaks in Z bore depend only weakly on the duration of playing, probably because of compensating effects. So the changes in the frequency of these peaks are rather smaller than the displacement of the playing frequency from the peak in Z bore .

Further measurements carried out with six performers showed that Z bore is usually compliant when the note played is B 2 and inertive when the note played is B 3. According to a simple theoretical model of valve [START_REF] Fletcher | Autonomous vibration of simple pressure-controlled valves in gas flows[END_REF], this change of sign in the imaginary part of Z bore explains why the lip valve cannot have the same operating mode for both played notes. The model predicts that the lip valve is likely to oscillate as a (+1,-1) valve in the low range and as a (-1,+1) valve in the high range.

Plots of p mp (t) and OV (t) (Figure 6) showed that, during playing, the pressure in the mouthpiece leads the lip opening. This result is explained for the higher note, when Z bore is inertive, by flow between the lips. For the lower note, however, this phase relation requires that the flow into the bore has a large component that begins before the lips separate. This is explained by the sweeping action of the lip: for the lower note, the lips move forwards into the mouthpiece before they separate, generating a flow whose magnitude is of the same order as the acoustic flow into the bore.

This preliminary report presents only some of the data in the early part of this study. The conference presentation will have further data on more players, a greater range of notes, and a larger set of techniques.

Figure 1 .

 1 Figure1. The modified mouthpiece equipped with two brass electrodes connected to an electroglottograph (EGG) to measure lip contact after Wolfe and Smith[START_REF] Wolfe | Acoustical coupling between lip valves and vocal folds[END_REF], and Freour[START_REF] Fréour | Development of an electrolabiograph embedded in a trombone mouthpiece for the study of lip oscillation mechanisms in brass instrument performance[END_REF]], a piezoresistive pressure transducer connected to the mouthpiece cup. Its design follows Ayers[START_REF] Ayers | Alternative mouthpiece design for viewing the lip reed in motion[END_REF]: the two glass windows provide undistorted front view and side view of the lip motion.

Figure 2 .

 2 Figure 2. This graph shows several measurements of the impedance of the bore of the trombone equipped with the experimental mouthpiece when the slide is in 1 st position and the trigger valve is not depressed. The three peaks in magnitude are the 2 nd , 3 rd and 4 th resonances of the bore.The corresponding notes are labelled on the graph. Z dry is the average curve obtained before playing; Z 3s , Z 10s , Z 30s and Z 4min are the average curves obtained after the musicians played the notes they wished for these durations. All measurements were carried out the same day. The temperature and the relative humidity at the time were respectively 26.8 • C and 57%.

Figure 3 .

 3 Figure3. This plot displays the relative deviations (in %) of the impedance peak amplitudes and frequencies, after the musician played for 3s, 10s, 30s and 4min. The center of each ellipse is the mean of all measurements in each condition and the semi-axes show the standard deviations.

Figure 4 .

 4 Figure 4. This plot shows the magnitude of the trombone impedance (after playing 10 s) and that of the vocal tract impedance of the musician while playing a B 2, tongue in low position, and while playing a B 3, tongue in high position. Both measurements used an impedance head (4.8mm-diameter) introduced in the mouth of the player. The narrow peaks on the curves of vocal tract impedance are artefacts due to the sound of the fundamental playing frequency and, for B 2, its second harmonic.

6 Figure 5 .

 65 Figure 5. This plot shows the real and imaginary parts of the bore impedance (after playing 10s) and of the vocal tract impedance of the musician while adopting two different vocal tract positions: tongue in low position to play a B 2 and in higher position to play a B 3.

Figure 6 .

 6 Figure 6. The pressure in the mouthpiece p bore (t) and the open variable OV (t) (OV is inversely related to the electrical admittance between the upper and lower lip and is a maximum when the lips are separated and a minimum when they are in contact). (a) the musician played a lower pitch (B 2) note, and (b), a higher note (B 3)

Table 1 .

 1 Typical values of the real and imaginary parts of Z

	mouth

bore and Z mouth at both playing frequencies 115.7 Hz (B 2) and 227.4 Hz (B 3)

Acknowledgments

We thank the Australian Research Council for support and our volunteer subjects.


Grégory Miras

Alex Boulton

Natalie Kübler

Jean-Paul Narcy-Combes

Andrew Linn

Danielle Candel

Jacqueline Léon

Keywords: French, affiliate, AFLA, Linguistique appliquée

that it was created as far back as 1964, at the same time as AILA. The landscape of French research may explain why the tag "Linguistique Appliquée" is not as widespread as its equivalents in some other scientific cultures.

A brief history of AFLA 1

The Association Française de Linguistique Appliquée 2 (AFLA) and The Association Internationale de Linguistique Appliquée 3 (AILA) were both created in 1964 after the first international conference in Nancy (October 26-31 st ) on "Semantic information in linguistics and in machine translation". The year 1964 is a keystone in Nancy (France) because many linguists were recruited creating a "euphoric atmosphere" (Carton 2015: 29) increased by numerous interdisciplinary connections. Moreover, it was the year when third-generation computers were released allowing more extensive data collection and analysis. AFLA was officially created on 21 st December 1964 by Guy Capelle as a non-profit association under the French law of 1901 (Figure 1). AFLA was probably created a month earlier as it is mentioned that this receipt of registration cancels the one delivered on 6 th November 1964. This law defines an association as "a convention by which two or more people permanently share their knowledge or activity in another goal than sharing profit". It is, however, accepted that the institutionalisation of applied linguistics, in France, started in 1958 with the creation of the "Centre de linguistique appliquée" in Besançon, followed by "Études de linguistique appliquée", a journal launched by Bernard Quemada (Candel 2011: 99). Later, "La revue français de linguistique appliquée" published its first issue in 1996. For a detailed analysis of these two journals, the reader is referred to [START_REF] Candel | Les « Linguistique appliquée[END_REF]. The institutionalisation of applied linguistics (AL) is a recent phenomenon even if linguists had confronted language theories and the teaching and learning of languages long before 1964. Linn et al. (2011: 2) recognize that, from the early beginning, Applied Linguistics included various domains such as the ones mentioned above as well as others such as automatic translation. This institutionalisation of the discipline at the time of its creation had two major and specific characteristics (Linn et al. 2011: 8):

1. An international scope: this aspect was completed with the creation of AILA; 2. A strong national anchorage: in France, it started with vocabulary studies and lexicology. [START_REF] Linn | Présentation: Linguistique appliquée et disciplinarisation[END_REF] suggest other contextualised needs in the USA and other Englishspeaking areas, and in Germany and Scandinavian areas.

Specifically in France, Carton (2015: 27-9) highlights three factors that explain the emergence of Applied Linguistics:

1) Substantial cultural changes during the 60's leading to a desire for innovation.

2) The joint development of linguistics theories and communication technologies inducing intellectual emulation. 3) An institutional context favourable to change in French universities allowing an improved visibility and acknowledgment of language studies.

From the beginning, the statutes of AFLA mentioned that the association aims at "promoting and coordinating research in all the domains of applied linguistics and encouraging interdisciplinary and international cooperation in these domains" (article 1). All AFLA executive boards and presidents have tried to implement this objective in their policy. Table 1 shows some of the AFLA presidents since its creation.

2006-2012

Geoffrey Williams

2012-2017

Jean-Paul Narcy-Combes

2017today(2018)

Grégory Miras

It is important to mention that a precise history of AFLA is hard to trace as it had to face serious political and human tensions leading, on the one hand, to the creation of the Association des chercheurs et enseignants didacticiens des langues étrangères (Acedle) in 1989 and, on the other, to the resignation of a number of the members of the executive board and of the organizing committee on 8 th December 2001. Candel and Feuillet (2015: 35-36) mention three historical steps for Applied Linguistics in France and AFLA:

1) 1958-1966: nine early crucial years for Applied Linguistics in terms of national and international institutionalisation. 2) 1990-1996: a warning and what came afterwards when during the 1993 AILA world congress in Amsterdam, AFLA was threatened with disaffiliation due to its lack of activity. 3) 2008-2012: a gradual resumption of activity within AFLA, first at a national level and then at international level when several members of AFLA met during the 2014 AILA world congress in Brisbane (Australia).

Two key elements have to be taken into account when it comes to understanding the role and the contextualised situation of applied linguistics at international and national levels. First of all, applied linguists should find solutions to real world problems (Brumfit 1977). At the time, social needs were related to contemporary international migrations in the 19 th century and then to the fall of colonial empires in the 1960s leading to high demand in foreign language learning (Linn et al. 2011: 9). Secondly, Applied Linguistics is constitutively pluri-and interdisciplinary, and thus is linked to related domains or disciplines such as pedagogy and sociology.

According to [START_REF] Candel | Linguistique appliquée » : Parcours définitoires et lexicographiques[END_REF] who examined the term "applied linguistics" in dictionaries and in seminal texts:

The scope of applied linguistics appears quite rich, but it is also perceived as being composite, and this has led to negative judgments, concerning the denomination as well as the designated area. Finally, the definitions have reached a measure of stability, the applied side continues to outweigh the theoretical one, and the discipline continues to evolve. (Candel 2011: 113) For Candel, this highlights the vitality of applied linguistics. Despite that fact, AFLA and its members have experienced a number of challenges; the association is currently undergoing a promising process of revitalisation and has an important role to play in a newly redefined field of applied linguistics in contemporary French research.

AFLA's renewed conception of AL 4

As mentioned by Danielle Candel above, the term appliquée (applied) in linguistique appliquée (Applied Linguistics), in France, may hold negative judgments that partly explain why the field is not as strong as it is in other countries and in scientific literature published in English. The second contextualized element, that has to be taken into consideration, is the French specificity of the links between applied linguistics and "didactique des languescultures" (teaching and learning of languages and cultures). [START_REF] Chevalier | Combats pour la linguistique[END_REF] summarize these tensions:

The notion of "linguistique appliquée" was soon perceived as pejorative in a country where only fundamental research is promoted, no matter its quality, and which is suspicious of what comes from the field. The notion was related to prestigious disciplines […]; but teachers felt ignored. (Chevalier and Encrevé 2006: 95-96) In France, Applied Linguistics, seen as a network of researchers, has had to face a shift with the old but ongoing debate over the claim of disciplinary specificity, with "Didactique des langues-(cultures)" relying on a diachronic perspective and a synchronic theorization which are more than a simple "application" suggested by the term "applied" linguistics (Chiss 2009: 129). This paper will not further develop the contextual links between Fundamental Linguistics, Applied Linguistics and Didactique des langues-cultures in France; the reader is referred to other sources such as [START_REF] Chiss | Sciences du langage et didactique des langues : une relation privilégiée[END_REF].

Following the words of Michael Halliday, in Léon (2011: 79), who considers "Applied Linguistics as an evolving theme", Condamines and Narcy-Combes (2015: 1) suggest that the concept of Applied Linguistics could be revisited in order to alleviate historical tensions. One of the key points is terminological, implying that instead of considering "Applied" Linguistics, the focus should be on the notions of située (situated) and/or impliquée (involved/engaged/committed). In their view, and as an extension of what Brumfit (1977) called real world problems, situated science implies a perspective in which research is not simply applied to a project but in which it is part of the project where the two are interconnected and modify each other so that the project may go forward. Bonneuil and Joly (2013: 19) theorized this point concerning the interconnected relations between science and society. This point leads to the fact that science is situated by socio-political constraints [START_REF] Callon | La science telle qu'elle se fait[END_REF]; among others) and a contemporary episteme (cf. uncertainty and complexity according to Narcy-Combes, 2010). This does not imply that research is a completely relative process, as [START_REF] Klein | Allons-nous liquider la science ? Galilée et les indiens[END_REF] mentions it in the field of physics:

The sociology of science is right to insist on the importance of context in the way it builds itself. Yet, should it be concluded that scientific results are radically relativistic. Certainly not. (Klein 2008: 75) According to Condamines and Narcy-Combes (2015: 2), "an idealistic scientific pathway would start from a distanciation leading to a reflexion/description linked to one or several theories and allowing authentic actions". Action can lead to "engagement" which is a step further than Situated Linguistics and needs to question researchers as citizens. Distanciation, research methodologies and epistemological responsibility provide checks and balances to ensure objectivity. Objectivity is not the opposite of subjectivity, nor is it neutral when conducting research. It depends on how the researcher considers himself or herself "committed", in terms of ethics, individual posture and deontology [START_REF] Corijn | Éléments d'un projet pour Ixelles[END_REF]. A balance obviously has to be found between commitment and scientific distanciation, but it would be difficult to say there is no hidden political agenda in research on education, for example.

Researchers are becoming increasingly aware of their potential political involvement, which may be a good sign.

Following on from this, the two authors suggest that Linguistics is a situated science, and thus question the relevance of the opposition "Applied Linguistics vs. Linguistics" (Condamines & Narcy-Combes 2015: 13) and maybe even "vs. Didactique des languescultures".

Fields of interest

The different domains of Applied Linguistics covered by AFLA members are highly varied: language acquisition, contrastive analysis, speech analysis, L1 and Lx teaching and learning, child language, professional language, minority and regional languages, lexicology, language policy, language disorders, translation, terminology, automated language-processing, corpus-based approaches, digital technologies, etc.

Considering the landscape of language studies in France, AFLA has developed strong and deep links with partner associations in order to take into account the diversity of languages, teacher professional networks and other transdisciplinary complementary domains (see Table 2). The intention is for other partner associations to join during our next annual meeting in 2018, fulfilling our goal to better represent all the fields of Applied Linguistics. Sign languages deserve a mention with Trait d'Union or the Association Française des Interprètes et Traducteurs en Langue des Signes (AFILS), along with research into English learning and teaching with Association pour la Recherche en Didactique de l'Anglais et en Acquisition (ARDAA) or on language studies with Association des Sciences du Langage (ASL) and biligualism with Association pour le Développement de l'Enseignement Bi/plurilingue (ADEB).

Table 2: Official Partner associations in 2017 lingualism, didactics, lexicography, corpus linguistics, terminology, translation studies, computational linguistics, variation, etc. However, these disciplines often share similar research fields and methodologies. Transdisciplinarity and multidisciplinarity are trademarks of the 21 st century, as can be seen in the emergence of so many multi-or transdisciplinary fields, including examples which combine "humanities" and "pure sciences". Applied Linguistics, because of the variety of fields which it is involved in, has followed the inexorable development of this process of hybridization. Furthermore, the practice of Applied Linguistics has come to involve not only the application of theoretical knowledge, but also the emergence of new fields of investigation, which then feed back into current debates within the language studies. The aim of the TRELA 2015 international conference that followed the CRELA conference in Nancy 2013, which questioned the notion of Linguistique Appliquée in France nowadays, was to allow researchers and other practitioners in the different fields of Applied Linguistics to discuss and debate issues relating to common areas of research, pooling ideas on these topics from a multidisciplinary or transdisciplinary perspective.

AFLA has a number of upcoming events in 2018, 2019 and 2023. In 2018, our annual meeting will focus on how language studies can respond to social needs. Participants will talk about their research and its links to social needs through public or private institutions such as practicing as an otolaryngologist, applying linguistics in businesses, doing forensic linguistics, building a gender-neutral French language and questioning the role of the linguist in minority languages such as French Sign Language. The aim will be to foster a dynamic trend regarding the possible continuum of engagement/commitment of researchers in social needs (or at least to make them more visible). This first step will pave the way for our 2019 annual event combined with the AILA EBIC meeting, followed by the 2023 AILA world congress in Lyon (France) entitled "Diversity and social cohesion in a globalized world: Moving toward more engaged language studies".

-Leading AILA Research Networks (ReN):

2013-2016 (convenor: Jean-Paul Narcy-Combes): "Sociocultural theory and emergentism". The scope was to question how current trends in SLA research are largely concerned with the socio-cultural contexts of language acquisition/learning and have therefore rediscovered the importance of Vygtskyan thinking. Even if we agree on the major aspects of this approach, there is still a theoretical gap with regard to the overarching concept of learning which recent research on emergentist views of learning processes might help to fill. The work of the ReN aims at modelling the mental and developmental processes of second language acquisition/learning within social interaction and at conducting empirical research in this field. Some of the participants are engaged in a common research project based on a computer-enhanced transnational, multilingual and multicultural learning environment. The exchange between this project and the larger circle of ReN members is promising for both sides.

2018-2020 (convenor: Shona Whyte): "Crosslinguistic perspectives on the terminology of language teaching and learning research in French and English". The scope of the ReN is to analyse how the crosslinguistic L2 studies terminology network seeks to contribute to collaboration across French-speaking and English-speaking scholarly communities by offering a forum for participants to review, clarify, and update terms in second language acquisition, second and foreign language teaching, educational linguistics, and language education across the two languages and cultures. It may be possible to extend the project to additional languages (German, Italian, Spanish) if progress is significant and a second term for the network seems fruitful.

Conclusion

This brief history of "Applied Linguistics" and "Linguistique appliquée" in France should not be taken as an attempt to dwell on the past but should help us to look forward to the future of our field. As a (non-)conclusion, Stegu (2011) may be right when he says that:

Applied Linguistics is nothing of the above [an autonomous (meta)discipline or a more-or-less autonomous gathering of (sub)disciplines] or it is all of the above: it is nothing but a summary of all that was said or written about it and of all that will be said or written. (Stegu 2011: 138) However, situated science such as Applied Linguistics is a way for researchers to take part in the knowledge-based society. This role is even more important given the challenges our societies and our world are facing in this beginning of the 21 st century.

Figure 1 :

 1 Figure 1: AFLA's declaration of administrative creation in 1964.

Table 1 :

 1 AFLA presidents (years and names in bold are confirmed).

	Dates	Names
	1962-?	Antoine Culioli
	?	Hélène Huot
	2000-2001	Isabelle Leglise
	?	Annie Delaveau

This section draws on a paper written by Anne Condamines and Jean-Paul Narcy-Combes entitled "La linguistique appliquée comme science située" (Applied linguistics as a situated science") published in Cultures de recherche en linguistique appliquée in 2015. To read the full paper: https://hal.archives-ouvertes.fr/hal-01286390 (14/12/2017).

History and epistemology in language studies

The upcoming years will be devoted to defining a small number of themes that will reorganize the association not in terms of subdomains but in terms of transdisciplinary research responding to social needs and contemporary challenges.

Activities

AFLA members have been rethinking the notion of Applied Linguistics in many ways beyond our borders, including:

-Participating in AILA conferences and EBIC meetings:

2014 AILA world congress in Brisbane (Australia) and 2017 AILA world congress in Rio (Brazil).

-Publishing papers and coordinating scientific issues, recent examples being:

Miras, G. (ed., 2017). Emergentism. RDLC -Les cahiers de l'Acedle. 14-1. (Available onlinehttps://rdlc.revues.org/1082).

Carton, F., Narcy-Combes, J.-P., Narcy-Combes, M.-F., Toffoli, D. (2016, eds.). Cultures de recherche en linguistique appliquée 5 . Paris: Riveneuve éditions.

-Organizing national conferences:

Table 3 shows a few historical examples of congresses for which AFLA has been involved. The last two conferences in 2013 and 2015 focused on Research Cultures in Applied Linguistics (CRELA, Nancy) and Areas of Research in Applied Linguistics (TRELA, Paris). Both considered that Applied Linguistics in the 21 st century is a rich and varied discipline, with many sub-domains. Each of these has its own research tradition, often associated with the particular countries in which it developed: language acquisition/learning, bi-and multi-


Damien ; Voyer
email: voyer@eigsi.fr

Sergio Corridore

Annabelle Collin

Riccardo Scorretti

Clair Poignard

Numerical modeling of floating potentials in electrokinetic problems using an asymptotic method

Keywords: 72 Numerical methods Floating potential, asymptotic method, nonlinear problem, 4-electrode system

teaching and research institutions in France or abroad, or from public or private research centers.

I. INTRODUCTION

HE concept of floating potential is encountered in electrokinetic problems when isolated conductors of high conductivity are embedded in a dielectric or low conductive medium [START_REF] Konrad | The Finite Element Modeling of Conductors and Floating Potentials[END_REF]. One can mention for example earthing systems where metal rods are buried into the ground [START_REF] Sabariego | Perturbation Finite Element Method for the Analysis of Earthing Systems with Vertical Rods[END_REF] or 4-electrode systems that are often used to measure the impedance of biological tissues [START_REF] Ivorra | In vivo electrical conductivity measurements during and after tumor electroporation: conductivity changes reflect the treatment outcome[END_REF]. The direct solution of this kind of problems has two main drawbacks: this approach requires the meshing of the interior of the conductors and the high contrast of conductivities leads to numerical issues in the matrix inversion [START_REF] Amann | Simulation of floating potentials in industrial applications by boundary element methods[END_REF]. Replacing the electrodes by perfect conductors provides a reasonable approximation. The resulting problem leads to an additional degree of freedom related to the isopotential value on the surface of the perfect conductor. However, the solution can be computed rigorously without modification of the matrix system using superposition of solutions [START_REF] Wu | Comparison Study of Finite Element Methods to Deal With Floating Conductors in Electric Field[END_REF][START_REF] Yu | Efficient 3-D Extraction of Interconnect Capacitance Considering Floating Metal Fills With Boundary Element Method[END_REF] or iterative techniques [START_REF] Sabariego | Perturbation Finite Element Method for the Analysis of Earthing Systems with Vertical Rods[END_REF][START_REF] Aiello | FEM-DBCI Solution of Open-Boundary Electrostatic Problems in the Presence of Floating Potential Conductors[END_REF].

In this paper, we show that a more accurate approximation can be achieved thanks to an asymptotic expansion of the potential. Asymptotic expansions are formal series with respect to a small parameter (here the ratio of conductivities), which provide an accurate and rigorous approximation of the solution (here the potential). In the case of floating potentials, the zero order solution (the first term of the expansion) is shown to coincide with the perfectly conductor case. The potential is then approached one step further with the first order correction (second term of the expansion). It implies two successive problems defined respectively in the ambient medium and in the high conductive inclusions, which avoids describing the high contrast of conductivities in a single problem.

The paper is organized as follows: the formal asymptotic expansion of the solution to the electrokinetic problem is given in Section II. In section III, the numerical convergence with respect the conductivity ratio is studied in a 2D-toy configuration. The method is then applied to a 3D example that shows the consistency of the method, while a direct computation leads to unsatisfactory results. Finally, a nonlinear case is proposed in section IV in the 2D configuration.

II. FORMULATION OF THE ELECTROKINETIC PROBLEM

We consider the model of a 4-electrode system dedicated to measure the impedance of a biological tissue. In this device, a current is injected through both external electrodes inside the biological tissue. In the same time, both internal electrodes are connected to a high impedance in the external circuit; consequently, no current is supposed to flow through them. Internal electrodes are said to be passive while external ones are said to be active. The impedance of the biological tissue is then calculated measuring the potential difference induced between the internal passive electrodes that do not suffer from to voltage drop due to the electrode/tissue interfacial impedance [START_REF] Ivorra | In vivo electrical conductivity measurements during and after tumor electroporation: conductivity changes reflect the treatment outcome[END_REF].

The general model of a symmetric 4-electrode system with two pairs of active and passive electrodes is shown in Figure 1. The initial problem to solve reads:

where  g is the border for the active electrode,  out the external border of the biological tissue and  sym the symmetry plane.  e is the conductivity of the biological tissue defined inside the domain  e and  i the conductivity of the passive electrodes defined inside the domains  i.

The solution for  e and  i can be expanded using formal series with respect to the parameter  =  e / i << 1:

where the terms  k are constant and { i k ,  e k } solve a differential problem set respectively in  i and  e . Identifying the terms with the same power in , we infer the cascade of problems. The first problem to solve appears to be perfect conductor case. The couple ( e 0  0 ) is the solution to

The solution ( e 0  0 ) is computed by superposing two solutions of problems where the homogeneous Dirichlet condition is set respectively on the boundary  i of the passive electrode and on the boundary  g of the active electrode. More precisely, consider the two following problems:

Then  e 0 =  +  0  in  e and  i 0 =  0 in  i with or equivalently
To improve the zero order approximation, the terms { i 1 ,  e 1   1 } satisfy similar problems as (3) with { i 0 ,  e 0 } as source terms:

The numerical problem (7a) is solved using a Lagrange multiplier to ensure the Gauge condition. Instead of solving (7b), one solves

The constant  1 is given by substituting  by  in [START_REF] Yu | Efficient 3-D Extraction of Interconnect Capacitance Considering Floating Metal Fills With Boundary Element Method[END_REF]. The first order correction is then given by  e 1 = +  1  in  e .

III.NUMERICAL RESULTS

A. Linear case

A 2D model of a symmetric 4-electrode system is first introduced in order to perform the convergence study. The problem is solved using the finite element method with a coarse, medium and fine meshes (see Fig. 2; note that the dimensions are realistic). The number of degree of freedoms (dof) for solving directly the initial problem ( 1) is 4 300, 67 000 and 1 060 000 for the coarse, medium and fine meshes. The above potential { i k ,  e k }, k=0,1 are easier to compute since there is no high contrast in the conductivities and only a single domain is involved in the different problems ( i or  e ). In particular, the number of dof in the domain  e (resp. in the domain  i ) is 4 100 (respectively 200), 65 000 (respectively 2 000) and 1 020 000 (respectively 40 000) for the coarse, medium and fine meshes. A 4-electrode device is designed to measure the resistance R defined as the ratio of the potential difference U between the passive electrode and the current I flowing through the active

{ -∇.(𝜎 𝑒 ∇𝜇) = 0 𝑖𝑛 Ω 𝑒 𝜇 |∂Ω 𝑖 = 1 𝜇 |Γ 𝑔 ∪ Γ 𝑠𝑦𝑚 = 0 𝜎 𝑒 ∂ 𝑛 𝜇 |Γ 𝑜𝑢𝑡 = 0 (4𝑎) (4𝑏) { -∇.(𝜎 𝑒 ∇𝜆) = 0 𝑖𝑛 Ω 𝑒 𝜆 |Γ 𝑔 = 𝑉 λ |∂Ω 𝑖 ∪ Γ 𝑠𝑦𝑚 = 0 𝜎 𝑒 ∂ 𝑛 𝜆 |Γ 𝑜𝑢𝑡 = 0 { 𝜑 𝑒 = 𝜑 0 𝑒 + 𝜀 𝜑 1 𝑒 + ⋯ 𝜑 𝑖 = 𝛼 0 + 𝜀( 𝜑 1 𝑖 + 𝛼 1 ) + ⋯ { -∇ • (𝜎 𝑒 ∇𝜑 𝑒 ) = 0 𝑖𝑛 Ω 𝑒 𝜑 𝑒|Γ 𝑔 = 𝑉 𝜑 𝑒|Γ 𝑠𝑦𝑚 = 0 𝜎 𝑒 ∂ 𝑛 𝜑 𝑒|Γ 𝑜𝑢𝑡 = 0 -∇ • (𝜎 𝑖 ∇𝜑 𝑖 ) = 0 𝑖𝑛 Ω 𝑖 𝜑 𝑖|∂Ω 𝑖 = 𝜑 𝑒|∂Ω 𝑖 𝜎 𝑒 ∂ 𝑛 𝜑 𝑒|∂Ω 𝑖 = 𝜎 𝑖 ∂ 𝑛 𝜑 𝑖|∂Ω 𝑖 (1)
(2) electrodes. Since the passive electrodes are not perfect conductors, the mean potential at the surface of the passive electrodes is considered to compute the resistance:

{ -∇.(𝜎 𝑖 ∇𝜑 1 𝑖 ) = 0 𝑖𝑛 Ω 𝑖 ∂ 𝑛 𝜑 1 𝑖 |∂Ω 𝑖 = ∂ 𝑛 𝜑 0 𝑒| ∂Ω 𝑖 ∫ ∂Ω 𝑖 𝜑 1 𝑖 = 0 (7𝑎) { -∇ • (𝜎 𝑒 ∇𝜑 1 𝑒 ) = 0 𝑖𝑛 Ω 𝑒 𝜑 1 𝑒| Γ 𝑔 ∪ Γ 𝑠𝑦𝑚 = 0 𝜎 𝑒 ∂ 𝑛 𝜑 1 𝑒| Γ 𝑜𝑢𝑡 = 0 𝜑 1 𝑒| ∂Ω 𝑖 = 𝜑 1 𝑖 |∂Ω 𝑖 + 𝛼 1 with 𝛼 1 such that ∫ ∂Ω 𝑖 𝜎 𝑒 ∂ 𝑛 𝜑 1 𝑒 = 0 𝑑 (7𝑏) { -∇ • (𝜎 𝑒 ∇𝜒) = 0 𝑖𝑛 Ω 𝑒 𝜒 |∂Ω 𝑖 = 𝜑 1 𝑖 |∂Ω 𝑖 𝜒 |Γ 𝑔 ∪ Γ 𝑠𝑦𝑚 = 0 𝜎 𝑒 ∂ 𝑛 𝜒 |Γ 𝑜𝑢𝑡 = 0 (7𝑐) (3)
{ -∇.(𝜎 𝑒 ∇𝜑 0 𝑒 ) = 0 𝑖𝑛 Ω 𝑒 𝜑 0 𝑒| Γ 𝑔 = 𝑉 𝜑 0 𝑒| Γ 𝑠𝑦𝑚 = 0 𝜎 𝑒 ∂ 𝑛 𝜑 0 𝑒| Γ 𝑜𝑢𝑡 = 0 𝜑 0 𝑒| ∂Ω 𝑖 = 𝛼 0 with 𝛼 0 such that ∫ ∂Ω 𝑖 𝜎 𝑒 ∂ 𝑛 𝜑 0 𝑒 𝑑𝑠 = 0 (5) 𝛼 0 = -∫ ∂Ω 𝑖 𝜎 𝑒 ∂ 𝑛 𝜆 𝑑𝑠 / ∫ ∂Ω 𝑖 𝜎 𝑒 ∂ 𝑛 𝜇 𝑑𝑠 (6) 𝛼 0 = -∫ Ω 𝑒 𝜎 𝑒 ∇𝜆∇𝜇 𝑑𝑥/∫ Ω 𝑒 𝜎 𝑒 |∇𝜇| 2 𝑑𝑥 Page 2
where the factor 2 is introduced because of the symmetry in the initial problem.

Using the asymptotic expansion, R is approached by In (9), the leading term of R depends only on the couple ( e 0  0 ) while ( e 0  0 ) and ( i 1  1 ) are involved at the second order of accuracy. The next paragraph deals with this numerical convergence study. The resistance R is computed using (8) for a tissue with a constant conductivity  e. Fig. 3 reports the relative error of the asymptotic development with respect to  at the zero and first orders. Since the exact solution is not analytical, the reference value is chosen as the numerical solution of the direct computation with the fine mesh. One observes that when the asymptotic method is performed with the fine mesh, the relative error on R decreases for the zero order computation following the rate  Adding the first order correction, the relative error decreases following a rate slightly lower than  2 . Those observations are in agreement with the theoretical result given by (9). However, when the mesh is coarser, the accuracy of the finite element solution is not sufficient enough and the error due to the mesh is greater than the error due to the truncation of the asymptotic expansion. Thus, the relative error on R does not decrease under 3.5 10 -3 (respectively 2.1 10 -4 ) for the coarse (respectively medium) mesh in Fig. 3: this corresponds exactly to the relative error on R obtained with the direct computation performed on the same mesh.

A 3D model of the symmetric 4-electrode system has been simulated in order to get closer to the real device. The mesh shown Fig. 4 is such that there are 3 200 000 dof to solve the initial problem; the number of dof in the domain  e (respectively in the domain  i ) is 1 200 000 (2 100 000). In these conditions, the computation time to solve both zero and first order problems with the asymptotic method is reduced by 20% compared to the time required to solve the direct problem. The simulation performed when  = 10 -1 gives a relative error on R of 5.0 10 -3 for the zero order and 5.5 10 -3 for the first order; the error decreases to 1.1 10 -3 for the zero order and 5.6 10 -4 for the first order when  = 10 -2 . The comparison cannot be performed further because of the limitation of the numerical resources. Moreover, the matrix of the initial problem becomes numerically singular due to the high contrast of conductivities and the numerical inversion is not accurate.

B. Nonlinear case

When the 4-electrode system is used to monitor the tissue impedance during electroporation, the problem becomes nonlinear since the tissue conductivity depends on the electric field amplitude. In this section, a sigmoid function is introduced to model the dependence of the conductivity  e with respect to the electric field as it is generally considered in the electroporation phenomenon [START_REF] Breton | Nonlinear steady-state electrical current modeling for the electropermeabilization of biological tissue[END_REF]: relative error on R zero order, coarse mesh zero order, medium mesh zero order, fine mesh first order, coarse mesh first order, medium mesh first order, fine mesh where the parameters  0 ,  1 , a; b and d depend on the type of biological tissue. When electroporation occurs, the tissue conductivity increases in the area between the electrodes (see Fig. 3

𝑅 = 𝑈 𝐼 = 2 |∂Ω 𝑖 | ∫ ∂Ω 𝑖 𝜑 𝑖 𝑑𝑠 ∫ Γ 𝑔 𝜎 𝑒 ∂ 𝑛 𝜑 𝑒 = 2𝑉 |∂Ω 𝑖 | ∫ ∂Ω 𝑖 𝜑 𝑖 𝑑𝑠 ∫ Ω 𝑒 𝜎 𝑒 |∇𝜑 𝑒 | 2 𝑑𝑥 + ∫ Ω 𝑖 𝜎 𝑖 |∇𝜑 𝑖 | 2 𝑑𝑥 (8) 𝑅 = 2𝑉 ∫ Ω 𝑒 𝜎 𝑒 |∇𝜑 0 𝑒 | 2 𝑑𝑥 { α 0 + 𝜀 ( α 1 -α 0 2∫ Ω 𝑒 ∇𝜑 0 𝑒 ∇𝜑 1 𝑒 + ∫ Ω 𝑖 |∇𝜑 1 𝑖 | 2 ∫ Ω 𝑒 |∇𝜑 0 𝑒 | 2 𝑑𝑥 ) } + 𝑂(𝜀 2 ) (9)
𝜎 𝑒 (∇𝜑 𝑒 ) = 𝜎 0 + 𝜎 1 1 + 𝑑 𝑒𝑥𝑝 ( -|∇𝜑 𝑒 | -𝑎 𝑏 ) (10)

top).

Considering a nonlinear conductivity  e ( e ) in (1), it should not be possible to rigorously formulate the zero and first order of the asymptotic solution as proposed in section II. However, since iterative techniques are necessary used to solve a nonlinear problem, the asymptotic development can be applied at each iteration. In this paper, Picard's algorithm [START_REF] Breton | Nonlinear steady-state electrical current modeling for the electropermeabilization of biological tissue[END_REF] is used to compute the solution  =  e +  i in the nonlinear problem. From the solution  k-1 computed at the iteration k-1, the problems (4a)-(4b) are solved with  e ( k-1 ) to calculate the intermediate solution  e+i k =  e k +  i k . Then, a relaxation factor K k is applied to optimize the contribution of the intermediate solution  e+i k to the solution  k =  k-1 + K k ( e+i k - k-1 ) computed at the iteration k; note that K k is corrected at each iteration according to how the solution is approached in order to accelerate and stabilize the convergence [START_REF] Breton | Nonlinear steady-state electrical current modeling for the electropermeabilization of biological tissue[END_REF].

Picard's algorithm has been performed to solve the nonlinear problem for both the direct solution and the asymptotic solution at the zero order, in the case where the conductivity  i = 10 3   is set for the passive electrodes. In order to make a mindful comparison, the 2D model has been chosen with the fine mesh. The algorithm stops when the relative error between solutions  k-1 and  k reaches 10 -8 : 23 iterations are required for the asymptotic approach and 33 iterations for the direct solution. The relative error on R is 1.2 10 -3 with the asymptotic development at the zero order, which is 22 times larger than the error obtained in the linear case. Fig. 5 shows that the relative error on the conductivity between the direct solution and the asymptotic approach reaches 7.10 -3 in the electroporated region.

IV.CONCLUSION

In this paper, an asymptotic approach has been proposed to solve the problem of floating potentials that present a high conductivity compared to the ambient medium. The convergence study shows that this approach enables to improve the accuracy of the solution where perfect conductors are assumed. Compared to the direct solution, the advantage of the asymptotic approach is that less numerical resources are required since the floating conductors are meshed in a distinct problem. Another advantage of the asymptotic approach is that it avoids numerical artifacts that appear with the direct solution when the contrast of conductivity is too large. Nonlinear problems can also be solved numerically using the asymptotic method: a realistic example shows that the asymptotic solution at the zero order introduces a larger error compared to the linear case.

ACKNOWLEDGEMENT

This research has been carried out thanks to the support of the research projects DYNAMO and NUMEP granted by Plan Cancer. This work lies in the scope of the Inria Associate team NUM4SEP.

Figure 1 :

 1 Figure 1: different domains considered in a symmetric 4-electrode device

Fig. 2 :

 2 Fig.2: 2D model of the 4-electrode system with a coarse mesh. The diameter for the active (resp. passive) electrodes is 0.45 mm (resp. 0.35 mm) and the separation distance between negative and positive electrodes is 4.6 mm (resp. 2.2 mm).

Fig. 3 :

 3 Fig.3: Convergence of the asymptotic approach at the zero and first orders for three different meshes with the 2D model

Fig. 4 :

 4 Fig.4: 3D model of the symmetric 4-electrode system with the same dimensions as in Fig. 2. The height of the active (respectively passive) electrodes is 6.3 mm (respectively 5.2 mm).

Fig. 5 :

 5 Fig.5: (top) Distribution of the conductivity computed using the asymptotic approach (bottom) Relative error on the conductivity between the asymptotic approach and the direct solution. V = 138 Volts in (1) and the tissue parameters are  0 = 0.1 S/m,  1 = 0.3 S/m, a= (E 0 +E 1 )/2; b = (E 1 -E 0 )/c, c = 8 and d = 10 with E 0 = 460 V/cm and E 1 = 700 V/cm.


Floriane Gidel

Damien Voyer

Clair Poignard

In Silico Electrical Modeling of Cell Aggregates

Keywords: Electrical Modeling of Cell Aggregates, Cell Networks, Homogenization, Numerical Characterization of Cell Spheroids, Multiscale Modeling

The paper deals with two different approaches to model cell aggregates submitted to an electric stimulation, namely the equivalent circuit approach and the theoretical homogenization. For each approach, the effective impedance of the cell aggregate is given, enabling a comparison between the different models. Regarding the circuit approach, a variability in the electric parameters of the circuit in series is known to provide anomalous relaxation similar to a constant phase element model. For lognormal distribution of the parameters, a new link between the effective impedance and both arithmetic and geometric means is given. The second approach deals with the theoretical -but periodic-homogenization approach. The idea is to consider the sample as a periodic aggregate composed of a large number of cells. In each cell the electric potential is governed by the electroquasistatic model. The formal two-scale analysis leads to the so-called bidomain model, enabling a novel definition of the tissue impedance, generalizing the Maxwell-Garnett formula to cells with any geometrical configuration and without any dilution assumption. Interestingly, the microscale cell organization is shown to impact the effective impedance of the sample, linking the cell and the tissue properties.

I. INTRODUCTION

T HE response of tissues and cell aggregates to electric stimulation still suffers from a lack of accurate modeling, even though the phenomenon is quite well described by the physics at the cell scale, from the linear regime (low amplitude pulse excitation) to the electroporation phenomenon [START_REF] Foster | Dielectric properties of tissues and biological materials: a critical review[END_REF], [START_REF] Fear | A novel equivalent circuit model for gapconnected cell[END_REF], [START_REF] Debruin | Modeling electroporation in a single cell. I. effects of field strength and rest potential[END_REF], [START_REF] Leguèbe | Conducting and permeable states of cell membrane submitted to high voltage pulses: Mathematical and numerical studies validated by the experiments[END_REF]. With the constantly increasing interest for bioimpedancemetry, in particular for cancer detection and treatment, such a lack of knowledge has to be overcome. Interestingly, 3D multicellular spheroids are mesoscale biological models -between the cell and the tissue scales -that mimic the tissue organization. They enable to overcome the 2D restriction of standard cell cultures and they provide an in vitro tissue model, which is very interesting especially in biological oncology. Understanding the electrical properties of 3D spheroids is a crucial challenge to provide a better understanding of the influence of electric field on tissues. 3D multicellular spheroids raise important theoretical and computational challenges, from theoretical homogenization approaches to high performance computing since they are made of several tens of thousands of cells.

The aim of the present paper is to compare cell circuits and homogenization approaches, and to provide a generalized impedance formula for cell aggregates. Section II presents the simple equivalent circuit approach, which provides a rough understanding of the electrical of cell assemblies with an explanation of the anomalous relaxation measured by bioimpedancemetry. As shown below, the anomalous relaxation is linked to a variability of the electrical properties of the circuit. The rigorous homogenization approach is then presented in Section III to exhibit the global behavior of the cell aggregates, accounting for the geometry of the microstructure i.e the cell geometry -and thus the tissue anisotropy. A Manuscript received XXXX; revised XXXX. Corresponding author: C. Poignard (email: clair.poignard@inria.fr).

new homogenized formula of tissue impedance is proposed, generalizing the so-called Maxwell-Garnett formula to cells with any geometrical configurations and without any dilution assumption. The paper proposes thus a complementing approach to the full numerical approach previously proposed in [START_REF] Guittet | A Voronoi Interface Approach to Cell Aggregate Electropermeabilization[END_REF], [START_REF] Mistani | A parallel Voronoi-based approach for mesoscale simulations of cell aggregate electropermeabilization[END_REF]. It is worth noting that this study is restricted to the linear regime.

II. THE EQUIVALENT CIRCUIT APPROACH

A. Equivalent circuit of single cell Equivalent circuits are often used to model the electrical properties of biological cell (see for instance [START_REF] Foster | Dielectric properties of tissues and biological materials: a critical review[END_REF], [START_REF] Fear | A novel equivalent circuit model for gapconnected cell[END_REF]). The membrane is seen as a capacitance in parallel with a resistance. The cell cytoplasm is a conductive medium modeled by a resistance in series with the membrane. This cell circuit is set in parallel with the extracellular domain, seen as a conductive medium (see Fig 1a). Using standard electric circuit laws, the

C m ∂ t V m + 1 R m + 1 R i V m = 1 R i (U L -U R ). (1)
In time-harmonic regime, the effective admittance of the whole circuit Y C := I/(U L -U R ) (see Fig. 1a) depends on the pulsation ω as

Y C (ω) = 1 R e + 1 R i 1 1 + Z m (ω)/R i , (2)
with Z m (ω) = R m /(1+jωR m C m ), the membrane impedance. At low frequency, the cell membrane impacts the admittance, while the membrane is transparent at high frequency:

Y C ∼ ω→0 1 R e + 1 R i + R m ∼ 1 R e , (3a)
Y C ∼ ω→∞ 1 R e + 1 R i , (3b)
since the membrane resistance is very high (R m R i , R e ).

B. Cell circuit networks in series

Recent investigations have addressed the micro / macroscopic relation thanks to equivalent circuit approaches to understand the anomalous relaxation measured in the experiments of tissue sample or cell aggregates impedancemetry. The idea consists in considering the spheroid (or the tissue) as a set of N 1 unit cells (U.Cs) denoted by

(C k ) k=1,••• ,N , whose dielectric parameters are (R k e , R k i , Z k m (ω)).
The cells are set in series, each U.C. being described by the equivalent circuit of Fig. 1a. The equivalent impedance Z eq of such assemblies is a complex-value function of the pulsation ω given by

eq (ω) = 1 N N k=1 1 1 R k e + 1 R k i + Z k m (ω) . (4)
The factor 1/N is set to keep the size of the tissue sample constant. Indeed, adding more details -thus more circuitsin the tissue description should not increase the global size of the sample. If all the cells are identical, the equivalent impedance of the sample is nothing but the inverse of the U.C. admittance [START_REF] Collin | Mathematical analysis and 2-scale convergence of a heterogeneous microscopic bidomain model[END_REF]. As mentioned by Zorn [START_REF] Zorn | Logarithmic moments of relaxation time distributions[END_REF], adding a variability in the electrical parameters leads to a global anomalous relaxation phenomenon, showing that the sample can be modeled as a Constant Phase Element (CPE) with parameters α and β, see Fig. 1b. Joris' thesis [START_REF] Joris | Microimpedance spectroscopy for local characterization of tissue structures[END_REF] investigated the link between large circuits made of slightly varying cell parameters with CPE. Here we show that for R e and R i following a lognormal distribution in the same range, the arithmetic mean of R e gives the low frequency behavior of the sample while the geometric means of R e and R i are linked to the high frequency limit of the sample impedance. More precisely, Fig. 3a-3b shows a simulation with a circuit made of N =1 million of U.Cs C in series. Lognormal distributions with mean value and standard deviation of (60kΩ, 80%), (40kΩ, 50%) and (200pF, 50%) are chosen for R e , R i and C m respectively. Due to the high value of membrane resistance R m , its influence is neglected. The numerical results show that the low frequency impedance is linked to the arithmetic mean of R k e while the high frequency impedance amplitude is given by the geometric mean of the parameters. Therefore, from the knowledge of the resistances distribution, one can obtain the formula for the equivalent outer and inner resistances R e and R i thanks to the arithmetic and geometric mean operators denoted by M a and M g . From (3), R e = M a (R k e ) N k=1 and 1

R i = 1 M g (R k i ) N k=1 + 1 M g (R k e ) N k=1 - 1 R e . (5)
Then the corresponding CPE circuit of Fig. 1b with α = 0.91, and β = 5.9 × 10 -10 fits with the circuit in series for the lognormal distribution of the U.C parameters.

Letting N going to infinity, and assuming that the electrical parameters of the circuit (R e , R i , R m , C m ) are characterized by specific probabilistic distribution functions denoted respectively by ψ e , ψ i , ψ r m , ψ c m and mean values R e , R i , R m , C m , we infer the following expression of the tissue impedance:

Z eq (ω) = R 4 + Ψ(s) 1 R e s 1 + 1
R i s 2 + R m s 3 1 + jωR m C m s 4
ds, [START_REF] Franzone | Degenerate evolution systems modeling the cardiac electric field at micro-and macroscopic level[END_REF] where

Ψ : s ∈ R 4 + → ψ e (s 1 )ψ i (s 2 )ψ r m (s 3 )ψ c m (s 4 ).
The above formula is an extension of the approach proposed by Zorn [START_REF] Zorn | Logarithmic moments of relaxation time distributions[END_REF] to explain the anomalous relaxation of simpler RC circuits. Note that the link between α and β of the CPE model and the distribution probabilities is still an open question.

III. IMPEDANCE OF CELL SPHEROIDS

As shown above, the cell circuit approach enables accounting for the cell variability of the tissue, but the geometrical aspects are not considered. Therefore these 0D models cannot describe the tissue anisotropy, and partial differential equations of the electric potential have to be considered.

A. Membrane voltage of 3D-cell in time-harmonic regime

Consider now the 3D model of a single cell whose membrane is denoted by Γ m . The membrane capacitance and conductance per surface unit are denoted by κ m and γ m . Let σ i and σ e be the (scalar) conductivities of the cell interior denoted by O i and the extracellular medium O e respectively. The TMV is denoted by

U |Γ m = U | Γ + m -U | Γ - m .
It satisfies the following equations set on Γ m in the time-harmonic regime:

(jωκ m + γ m ) U |Γ m = σ i ∂ n U | Γ - m = σ e ∂ n U | Γ + m , (7)
where where ω is the pulsation, U is a complex-valued harmonic function in O i and O e , and n the normal vector to Γ m outwardly directed. Note that the impedance corresponding to the PDE model given by ( 7) is a pseudo-differential operator which links Dirichlet and Neumann traces on the membrane.

B. Periodic homogenization

Two-scale analysis [START_REF] Allaire | Homogenization and two-scale convergence[END_REF] is a powerful tool to derive the homogenized electrical model of a tissue of characteristic size L composed of N 1 small U.C.s, each composed of a biological cell embedded in an ambient medium, as schematised in Fig. 2. Roughly speaking, the idea of homogenization consists in transposing the microstructure modelhere the cell scale model given by ( 7))-to the macroscopic scale thanks to series expansion of the solution in term of the scaling parameter ε = L/N . From 1kHz to several MHz, the rescaled complex membrane conductivity σ * m defined by σ * m (ω) = L (jωκ m + γ m ), is such that εσ * m = O(1). Several papers have been devoted to the homogenization for syncytial tissues, a problem close to our but with the scaling σ * m = O(ε) in [START_REF] Collin | Mathematical analysis and 2-scale convergence of a heterogeneous microscopic bidomain model[END_REF], [START_REF] Franzone | Degenerate evolution systems modeling the cardiac electric field at micro-and macroscopic level[END_REF], [START_REF] Neu | Homogenization of syncytial tissues[END_REF]. Such scaling is relevant for cardiac application, but it fails to describe the impedance of a biological tissue in the frequency range 1kHz to 10MHz. Following the same theoretical approach, the solution of ( 7) set in the domain Ω presents a spatial oscillating profile due to the alternation of extracellular and intracellular medium. The keypoint of the two-scale analysis consists in splitting between the rapidly oscillating variable and the "slow" variable, thanks to the following expansion of U ε solution to [START_REF] Gidel | Linking the transmembrane voltage of cell to the local electric field in tissue by a homogenization approach[END_REF] in formal series:

" = L/N ⌦ O " e O " i " m @⌦ Characteristic length: L 1 (a) Y Yi Ye Y 1 1 (b)
U ε (x) = k≥0 ε k U k e (x, x/ε), ∀x ∈ O ε e , k≥0 ε k U k i (x, x/ε), ∀x ∈ O ε i , (8)
where for any k ≥ 0, the functions U k e and U k i are defined for (x, y) ∈ Ω × Y, and are periodic in y = x/ε. Any derivative in x makes appear a sum of terms in power of ε, which have to be combined with the formal series. For instance in O e ,

∇ x U ε e = k≥0 ε k-1 ∇ y U k e + ε k ∇ x U k e . (9)
Identifying the terms with the same power in ε leads to elementary problems, which make appear the influence of the microstructure. The reader may refer to [START_REF] Gidel | Linking the transmembrane voltage of cell to the local electric field in tissue by a homogenization approach[END_REF] for more details.

1) Effective complex conductivity of the sample

The identification process leads to the first-order correctors W 1 e and W 1 i , which are harmonic vector-fields defined in Y e and Y i respectively, with vanishing mean value in Γ Y -to ensure their uniqueness-and such that they satisfy on

Γ Y σ e ∇ y W 1 e | + Γ Y -σ i ∇ y W 1 i | - Γ Y • n = (σ i -σ e )n, (10a) εσ * m (ω) W |Γ Y = σ i ∇ y W 1 i | - Γ Y • n + n . ( 10b
)
Periodic conditions are imposed on ∂Y. The above correctors are used to obtain the effective conductivities of the cell and the extracellular phases denoted respectively by σ e and σ i :

σ e = σ e |Y e | Id + 1 |Y e | Ye ∇ y W 1 e dy , (11a)
σ i = σ i |Y i | Id + 1 |Y i | Yi ∇ y W 1 i dy . ( 11b
)
Note that even though the conductivities σ e and σ i are scalar, anisotropy of the effective conductivity may appear due to the geometry of the microstructure. Then, the homogenized problem in the limit "ε goes to 0" reads as a two-phase problem (U e , U i ) solution of the coupled problem

∇ • σ e ∇U e + σ i ∇U i = 0, (12a)
σ m (ω)v = εσ m (ω) W 1 e (x/ε)∇U e -W 1 i (x/ε)∇U i -ε∇ • σ i ∇U i , (12b)
where v = U e -U i is the homogenized membrane potential and the complex conductivity of the membranes given by

σ m (ω) = σ * m (ω)|Γ Y |.
Boundary conditions similar to the exact problem are imposed on ∂Ω. Interestingly, from (12a) one defines the effective conductivity of the tissue E as

E = σ e Id + M e + |Y i | (σ i M i /σ e -Id -M e ) , (13)
where M e,i = 1 |Ye,i| Ye,i ∇ y W e,i dy and the effective admittance of the sample is then given by Y (ω) = S E (ω), where the ratio S/ is form factor, S being the surface of one electrode and the distance between the two electrodes. The formula is a geometrical generalization of (4), in the case of identical cells where the anisotropy is taken into account.

2) Link with the Maxwell-Garnett formula Maxwell-Garnett formula has been derived under the assumption of dilute suspension of spherical balls of conductivity σ * , without accounting for the thin membrane. The wellknown formula for effective conductivity is

σ * = σ e (1 + M + M 2 ), M = d|Y i | σ * -σ e σ * + (d -1)σ e , (14)
for d = 2, 3. This formula has been used for spherical cells by replacing σ * by a pseudo effective cell complex conductivity 1/(1/(σ * m )+1/σ i ), however this approach is not rigorous due to the jump of the potential across the membrane. Indeed, the effective cell conductivtiy is a Fourier multiplier and not just a scalar (see for more details [START_REF] Gidel | Linking the transmembrane voltage of cell to the local electric field in tissue by a homogenization approach[END_REF]). Note that if the membrane becomes very conductive, the TMV is very small and the first approximation of the above formula under dilution assumption is the Maxwell-Garnett formula.

3) Numerical simulations The parameters for the numerical simulations are given in Table I. These parameters are set in accordance with the literature [START_REF] Fear | A novel equivalent circuit model for gapconnected cell[END_REF], [START_REF] Foster | Dielectric properties of tissues and biological materials: a critical review[END_REF], [START_REF] Joris | Microimpedance spectroscopy for local characterization of tissue structures[END_REF]. Using [START_REF] Zhao | Tumor cell characterization and classification based on cellular specific membrane capacitance and cytoplasm conductivity[END_REF], the effective impedance is computed for spherical cells with volume fractions φ i := |Y i | = 0.12 and 0.37. We compare the numerical results with the standard Maxwell-Garnett formula in 2D and 3D. At high dilution (φ i = 0.12) Maxwell-Garnett formula provides barely an approximation of the sample impedance, but without accounting for the anisotropy. At lower dilution, the discrepancy between standard Maxwell-Garnett formula, which have been derived for homogeneous small inclusions and our approach is quite high as shown in Fig. 3c-3d. The impedance phases and amplitudes are in same range as the results of section II, thus confirming the analogy between the homogenized admittance ( 13) and the circuit impedance (4). Z MG for i =0.12, d=2 Z El x for i =0.12 Z El y for i =0.12 Z MG for i =0.37, d=3 Z S for i =0.37, in 3D

(c) Homogenization Z MG for i =0.12, d=2 Z El x for i =0.12 in 2D Z El y for i =0.12 in 2D Z MG for i =0.37, d=3 Z S for i =0.37, in 3D

IV. CONCLUSION

Two different approaches to model cell aggregates have been presented above. The equivalent circuit approach has been shown to exhibit the main electrical features of the cell aggregates, including the anomalous relaxation. Interestingly, for lognormal distributions, the low frequency limit of the aggregate impedance is the mean of the resistances R k e , while the high frequency limit involves the geometric mean of both R k e and R k i . The main interest of this approach lies in its simplicity. In particular, due to the small number of parameters, the identification of the CPE model parameters is feasible. Therefore, CPE models are interesting for the calibration with experimental measurement. However the geometry of the microstructure cannot be obtained.

The main result of this paper consists in deriving a new mixture formula for cells, thanks to the rigorous homogenization approach accounting for cell geometry and thus describing the tissue anisotropy. Interestingly, our formula is a generalized Maxwell-Garnett formula type for cell aggregates in any geometrical configuration and without any dilution assumption. The current main drawback of the proposed homogenization lies in the assumption of periodicity, which prevents accounting for cell variability. Forthcoming investigation will consist in deriving a stochastic homogenization approach to account for the cell variability, and thus for the anomalous relaxation of the cell aggregates. In addition, under hypothesis of dilution, the explicit formula of the solution to [START_REF] Leguèbe | Conducting and permeable states of cell membrane submitted to high voltage pulses: Mathematical and numerical studies validated by the experiments[END_REF] will be derived and compared to data measurements in a future work.

These two approaches have to be complemented with a purely numerical approach, which consists in solving the cell scale PDE [START_REF] Gidel | Linking the transmembrane voltage of cell to the local electric field in tissue by a homogenization approach[END_REF] in a large number of cells. A first impressive result has been obtained by Mistani et al. in the context of electroporation [START_REF] Mistani | A parallel Voronoi-based approach for mesoscale simulations of cell aggregate electropermeabilization[END_REF], where the time-dependent nonlinear model of Leguèbe et al. [START_REF] Leguèbe | Conducting and permeable states of cell membrane submitted to high voltage pulses: Mathematical and numerical studies validated by the experiments[END_REF] has been simulated on 30 thousands of cells.

Fig. 1 .

 1 Fig. 1. (a): Equivalent circuit of the unit cell C composed of a cell and its surrounding ambiant medium. (b): Equivalent circuit for cell aggregates including the Constant Phase Element (CPE) to account for cell membranes and variability.

Fig. 2 .

 2 Fig. 2. (a): The cell aggregate sample Ω composed of N 1 unit cells. (b): the rescaled unit cell, Y.

Fig. 3 .

 3 Fig. 3. Top: Amplitude (3a) and phase (3b) of the equivalent impedance Zeq given by (4), and comparison with the CPE of parameter α = 0.91, β = 5.9× 10 -10 . Bottom: Amplitude (3c) and phase (3d) of Z = (1/Y α)α=x,y,z for the cell ellipse in 2D (Z El x , Z El y ) of volume fraction φ i = |Y i |/|Y| = 0.12 and a 3D sphere of volume fraction φ i = |Y i |/|Y| = 0.37 denoted by Z S . Comparison with the corresponding Maxwell-Garnett formula (14) in blue.

ACKNOWLEDGMENT

This research has been supported by the Plan Cancer project NUMEP PC201615 of the Plan Cancer, and by the Inria Associate team NUM4SEP led by C.P. F.G. is French Young Talent 2019 -L'Oréal-UNESCO for Women in Science.