Keywords: Fibrosis, Electronic Implant, Impedance spectroscopy, Embedded System, Medical Devices I

Relevance of impedance spectroscopy for the monitoring of implant-induced fibrosis: a preliminary study Abstract-This paper focuses on the study of fibrosis induced by an implanted medical device and explores the possibility of characterizing this process by in situ measurement of electrical impedance. The approach combines concurrent electrical and biological characterizations of fibrotic tissue, applied to electrodes implanted in an animal model. The methodology used is described. Initial results evidence the strong correlation between fibrotic tissue and electrical parameters. Identification of such electrical parameters will enable the establishment of a reliable monitoring method. INTRODUCTION Innovations in integrated circuit technologies have spurred a revolution in in vivo healthcare monitoring, thereby catalyzing a significant growth in implantable medical devices. A variety of diseases and disabilities can now be addressed using medical implants [START_REF] Mazeyrat | Wireless Communicative Stent for Follow-Up of Abdominal Aortic Aneurysm[END_REF] and prosthetics [START_REF] Lahuec | A self-powered telemetry system to estimate the postoperative instability of a knee implant[END_REF]. Active implants, such as pacemakers [START_REF] Wood | Cardiac Pacemakers From the Patient's Perspective[END_REF][START_REF] Hochberg | Visual Prosthesis[END_REF] and neurostimulators [START_REF] Fischell | Implantable neurostimulator having a data communication link[END_REF], act by electrical stimulation of targeted tissues The degradation of the electrode-living tissue interface over time has a major impact on the reliability and functionality of a wide range of implanted devices. Among the dysfunctions identified by pacemaker manufacturers, for instance, fibrosis at the myocardial tissueelectrode interface is considered a source of pacing problems as critical as battery depletion When a device is implanted, the foreign body reaction occurs which is characterised by the infiltration of inflammatory cells to the area of the implant, and this persists until the material becomes encapsulated in a dense layer of fibrotic connective tissue. The efficiency of the device, once shielded by tissue, can be severely compromised. The standard method for determining the degree of tissue reaction is via histological examination. This approach enables the visualization of specific proteins but does not enable one to follow the tissue reaction time course in a living animal [START_REF] Mcconnell | Bioimpedance modeling to monitor astrocytic response to chronically implanted electrodes[END_REF]. To assess the contribution of local tissue response to implant failure, there is therefore an urgent need to develop tools to non-invasively determine the extent, morphology and composition of local tissue response surrounding implanted devices.

Characterisation of implanted electrodes and the surrounding tissues using electrical impedance monitoring can be found in the literature. Stein et al. pioneered in vivo study on recording electrodes on mammalian nerves (cuff electrodes) [START_REF] Stein | Impedance Properties of Metal Electrodes for Chronic Recording from Mammalian Nerves[END_REF] using impedance spectroscopy. The low frequency impedance (10 Hz) reflected the capacitive component of the interface and the high frequency impedance (10 kHz) reflected the tissue resistance, based on the authors' simplified and linear equivalent circuit model. They observed that the apparent tissue resistivity increased for several weeks after implantation, which was attributed to the replacement of fluid within the cuff by connective tissue. The study of Grill et al. used the 4-point technique on subcutaneous electrodes arrays to measure more specifically the local tissue impedance and eliminate the interface impedance components [START_REF] Mcadams | BioElectrodes[END_REF], In vivo measurements were performed with a biphasic current pulse which provided a measure of DC resistivity in steady-state regime. This study was mainly focused on the tissue resistivity with time following implantation and a high resistivity was attributed to the growth of a dense tissue. Mc Connell et al. perform twoelectrode impedance spectroscopy measurements and used the well-known Cole model. They proposed a quantitative and exhaustive correlation analysis between biological and electrical markers of brain implanted electrodes [START_REF] Mcconnell | Bioimpedance modeling to monitor astrocytic response to chronically implanted electrodes[END_REF]. Although the specificity of brain tissue gives rise to a tissue reaction known as gliosis instead of fibrosis, this methodology is similar to our approach. This paper investigates the characterization of fibrosis based on electrical makers. To do so, the formation and evolution of fibrosis around implanted electrodes based on biomarkers typical of this type of tissue is also studied. Variations in the electrode impedance are observed concurrently, at different time intervals. Three mini-pigs were implanted with cuff electrodes on both carotid arteries. Electrode-tissue impedances were characterized over a period of 37 days. Histological and biological analyses were conducted, once the pigs were ethically sacrificed, on the electrodes and surrounding tissues. Then the concomitant evolution of fibrosis biological markers and impedance parameters over the period was studied.

The remainder of this paper is organized as follows. Section II describes fibrosis as well as typical biomarkers used. Section III deals with impedance spectroscopy. Section IV describes the methodology used in our experiments. Section V presents the biomarkers. Section VI provides the results obtained. Finally, Section VII concludes the paper.

II. FIBROSIS

Production of fibrotic tissue is a reparative process beginning once an in vivo device has been implanted into the body. In the first days after the device is implanted, an innate inflammatory reaction is induced, characterized by diffusion of molecules called cytokines. These molecules interfere with the nearby environment by activating a cellular proliferation phenomenon jointly with the proliferation of cells with repairing capabilities called myofibroblasts. The proliferation of myofibroblasts is strongly related to the production of an extracellular matrix that can be indirectly emphasized by visualization of dedicated proteins: fibronectin and collagenous fibers. Fibronectin aggregates the myofibroblasts through the activation of a migration step. In parallel, the collagen induces a rigidity in the new born tissue. In the normal healing process, the myofibroblasts slowly vanish once the inflammatory phenomenon stops. The tissue volume reduces and returns to close to the original structure. The fibrosis phenomenon differs from healing since the presence of activated cells (expressing α-Smooth Muscle Actin, α-SMA) is maintained over a longer duration implying a continuous accumulation of extracellular matrix (ECM). This "fibrotic" tissue is thus essentially made of threads of collagen fibrils and of myofibroblasts that encapsulate the foreign device, so that it can no longer be "seen" by the body. In the case of a chemically-stable and uncorrupted material, the thickness of the capsule stabilizes after a phase of growth [START_REF] Mcconnell | Bioimpedance modeling to monitor astrocytic response to chronically implanted electrodes[END_REF].

III. IMPEDANCE SPECTROSCOPY

When characterizing the multi-frequency impedance of an implanted electrode interface, two components can generally be identified: the electrode-tissue interface impedance dominates at low frequencies and the tissue impedance dominates at high frequencies. In a Nyquist diagram, we generally find two "depressed" semi-circular arcs whose centres lie below the real axis and whose high and low frequency intercepts with the real axis are at angles φ ≠ 90° [START_REF] Mcadams | BioElectrodes[END_REF] (Fig. 1).

Cole (1940) proposed the now much used empirical equation which describes reasonably well such "depressed" tissue impedance loci:

Z= R ∞ + (R 0 -R ∞ ) / [1 + (jωτ) α ] (1)
τ is a time constant equal to 1/2πf c and α is a fractional power dependency related to the depression angle such that = απ/2, α is a constant such that 0.5 <α<1. A simple interpretation of the above equation is in terms of a circuit where a resistance R ∞ is in series with the parallel combination of a resistance ΔR, equal to R 0 -R ∞ , and a Constant Phase Angle impedance Z CPA . Z CPA can be derived from the Cole equation and has the form:

Z CPA = 1 / Q(jω) α , ( 2
)
where Q is a measure of the magnitude of the pseudo capacitance and has units of s α /Ω or Fs α-1 [START_REF] Mcadams | BioElectrodes[END_REF]. The low frequency intercept of the electrode interface arc is denoted R ∞ on Fig. 1 and corresponds to the tissue, or extracellular fluid, resistance.

IV. IDENTIFYING FIBROSIS

An experimental approach, based on repeated in vivo impedance measurements over time, along with periodic biological characterizations, was adopted to identify physical and biological key influences in the formation and development of fibrosis. Three mini-pigs were implanted with cuff electrodes, on both carotid arteries, under anesthesia, Fig. 2. This implementation site was chosen for its practicality. Electrode impedances were characterized every three to five days over a period of 37 days. Histological and biological analyses were conducted on the electrodes and surrounding tissues at day 1 (D1) for pig #0, 15 (D15) for pig #1 and day 37 (D37) for pig #2 after the pigs were ethically sacrificed. The time scale was chosen to observe different phases of fibrosis on the electrodes. The time periods between D1 and D15, and between D15 and D37 characterize the inflammation and the chronic impact. A. Electrodes Platinum (Pt) and Stainless Steel (SS) nerve cuff (NC-5-5-4-100µm and Pt/SS-300mm) 4-contact electrodes from MicroProbes™ were chosen for this study. The stranded wires are 300 mm long. The inner cuff diameter of 5 mm is welladapted for the mini pig carotids. The four contacts denoted P 1 to P 4 , of the cuff electrodes enabled the measurement of several combinations of 2-point impedances: we chose Z 14 and Z 23 (Fig. 3). Measurements were conducted on a total of twelve cuffs implanted on the three pigs.

B. Impedance measurements

The electrode-tissue impedances were measured using the 2probes sensing method and an E5060B Agilent network analyzer. The network analyzer was calibrated before each impedance measurement. Due to considerable artifacts at frequencies above 100 kHz, measurements were restrained to a 5 Hz to 100 kHz frequency range. Over this frequency range, the semicircular arc due to the tissue resistance is not seen, only its low frequency intercept, R ∞ is observed. Therefore, the overall measured impedance was modeled as a simple resistance R ∞ in series with Z CPA . Parameters R ∞ , Q and α were extracted from the gain-phase measurements with a commercial software (ZView ™).

V. HISTOLOGICAL ANALYSIS -EVALUATION OF FIBROSIS

MARKERS

This analysis was performed on the tissues associated with the explanted cuffs. Fibrosis intensity was determined by a combination of two analyses described next.

A. Thickness of fibrotic tissue

Fibrosis intensity was evaluated by measuring thickness of this tissue associated with the cuffs. Explanted tissues were rapidly frozen and cryostat sections stained with toluidine blue, enabling the visualization of cells and extracellular matrix. Microscopic examination was performed with an optical microscope and thickness t was measured using an ocular reticule.

B. Characterization of fibrosis indicators on in vitro fibroblasts cultures

Immunohistochemistry and specific antibodies were used in order to identify three main fibrosis markers: α-SMA, fibronectin and type I collagen.

Explanted cuff-electrodes (with associated tissue or not) were incubated 24 hours in cell culture medium (DMEM-Glutamax with antibiotics). These conditioned media containing fibrosis indicators secreted in vivo were used in fibroblast cells culture and kept in 5% CO 2 in a humidified atmosphere at 37°C. To ascertain the involvement of soluble factors secreted into the cuff-electrode conditioned media in the effects observed, culture medium which had never been in contact with the cuff electrodes was used as control.

Fibroblasts were cultured on glass coverslips, fixed with 3% paraformaldehyde in phosphate buffer saline (PBS) and permeabilized with 0.1% Triton X-100 in PBS. The coverslips were saturated with 0.5% bovine serum albumine (BSA) in PBS, washed with PBS supplemented with 0.5% BSA and 0.1% Tween (rinsing buffer) and incubated with primary antibodies (AbCAM) for two hours at room temperature. They were then washed in rinsing buffer and incubated with secondary antibodies conjugated with Alexa Fluor 555 (Life Technologies) for one hour. Finally, nuclei were stained with 4',6'-diamidino-2-phénylindole (DAPI). Coverslips were mounted in Prolong Gold antifade reagent (Invitrogen) and examined with laser scanning confocal microscopy (LSM710, Zeiss). Controls in which primary antibodies were replaced with PBS were negative.

Figure 4 illustrates the detection of fibrosis biomarkers, namely thickness of fibrotic tissue (Fig. 4 (a)) and α-SMA expression in fibroblast cultures conditioned by fibrotic tissue originally surrounding the electrodes (Fig. 4 (b)). Presence of fibrosis indicators was denoted as follows: "0" absence; "1" organized marker(s) "2" marker(s) in fibrillar organization and "3" marker(s) in intense fibrillar organization. Presence of cell α-SMA, was denoted similarily ("0" absence to "3" plentiful). The overall behavior over time of the impedance parameters was reproducible. R ∞ dropped rapidly after implantation, then increased over the first two weeks and tended to stabilize until D37, a behavior consistent with the results of resistivity in [START_REF] Grill | Electrical Properties of Implant Encapsulation Tissue[END_REF]. Q and α reached their extrema between D10 and D15. Q max and α min values present a strong variability between pigs which suggests that the timing and amplitude of fibrosis reaction is subject dependent. Q and α chronograms appear to differ for Pt and SS cuffs. Similarly, the frequency dependence of the tissue resistivity was observed to depend on the type of implant in [START_REF] Grill | Electrical Properties of Implant Encapsulation Tissue[END_REF].

Histological analysis (see section V.A.) conducted at D1, D15 and D37 on the three pigs showed that fibrosis tissue thickness was maximum at D15, with a range varying from 850 µm to 1200 µm. At D37 the thickness decreased below 700 µm. Thickness variation was larger for the SS cuff. Similarly, in vitro fibrosis indicators of staining and organization (see section V.B.) were accentuated at D15 and slightly decreased at D37. Table 1 shows the detection of in vitro fibrosis markers detection from fibroblast cultures. Fibronectin (Fn) is more accentuated in time in terms of expression and organization for Pt cuffs than for SS cuffs. Expression of type I collagen (Coll) shows that the molecule organization is also more pronounced with Pt than with SS cuffs.

The timing of the changes in electrical parameters appears to be consistent with the timing of the events involved in the fibrotic process. While R ∞ and Q appear to be correlated with the fibrotic capsule thickness, Q and α could give quantitative information on the molecular organization of the fibrotic tissue.

VII. CONCLUSION

This paper involves the study of fibrosis induced by an implanted medical device. The empirical approach combines electrical and biological characterizations of fibrotic tissue, applied to electrodes implanted in an animal model. A comparative study of electrical and biological parameters collected as a function of time shows a correlation between electrical markers of fibrosis development and biological markers. Based on these preliminary results, new experiments will be performed. We expect this technique to become an efficient tool for real-time monitoring of fibrosis induced by implants.

Fig. 1 :

 1 Fig. 1: Typically observed overall impedance locus.

Fig. 2 :

 2 Fig. 2: Implementation sites of cuffs and measurements.

Fig. 3 :Fig. 4 :

 34 Fig. 3: 4-contact cuff electrode used. Inner diameter 5 mm, 100 µm contacts, 5 mm apart.

Fig. 5 :

 5 Fig. 5: In vivo impedance measurements (average values), (a) R ∞ , (b) Q, (c) α. (d) Microscopic evaluation of fibrosis intensity: thickness measurements of tissues associated with cuffs

  Lewis 1 , C. Lahuec 2 , S. Renaud 1 , E. McAdams 3 , P. Bogonez-Franco 3 , C. Lethias 4 , S. Kellouche 5 , F. Carreiras 5 , A. Pinna 8 , A. Histace 9 , M. Boissiere 5 , E. Pauthe 5 , I. Lagroye 1 , F. Soulier 6 , S. Bernard 6 , S. Binczak 7 , B. Granado 8 , P. Garda 8 , M. Terosiet 9 , A.

		3. INL, UMR5270, Univ. de Lyon 1, Lyon, France
		4. IBCP, FR3302, Univ. de Lyon 1, Lyon, France
		5. ERRMECe, EA1391, Univ. de Cergy-Pontoise, Cergy-
		Pontoise, France
		6. LIRMM, UMR5506, Univ. de Montpellier, Montpellier,
		France
		7. LE2I, UMR6306, Univ. de Bourgogne, Dijon, France
	Goguin 9 and O. Romain 9	8. LIP6, Université Paris 6, Paris, France
		9. ETIS, UMR8051, Univ. Cergy-Pontoise, Cergy-
		Pontoise, France
	1. IMS, UMR5218 Univ. de Bordeaux, Bordeaux, France
	2. TELECOM Bretagne, Brest, France

N.

TABLE 1 :

 1 IN VITRO FIBROSIS MARKERS DETECTION FROM FIBROBLAST CULTURES FOR STAINLESS (SS) AND PLATINUM (PT) ELECTRODES (FROM ABSENCE, '0', TO STRONG DETECTION, "3").

	SS/Pt	organization	Staining
		Fn	Coll	Fn	Coll	αSMA
	D1	2 / 2	1 / 2	1 / 1	1 / 3	1 / 3
	D15	2 / 2	1 / 2	2 / 2	2 / 2	2 / 1
	D37	2 / 2	0 / 1	2 / 3	1 / 1	0 / 1

ACKNOWLEDGMENT

The authors thank C. Kang and M. Bonneau from XP-MED St Germain en Laye, France, for animals preparation and surgery. This work is supported by the DEFI-SENS program of the French National Research Agency (CNRS).


Thomas Risler

Aurélien Peilloux

Jacques Prost

Homeostatic Fluctuations of a Tissue Surface

Keywords: numbers: 87.10.Mn, 05.40.Ca, 87.10.Ca, 87.18.Tt

We study the surface fluctuations of a tissue with a dynamics dictated by cell-rearrangement, celldivision, and cell-death processes. Surface fluctuations are calculated in the homeostatic state, where cell division and cell death equilibrate on average. The obtained fluctuation spectrum can be mapped onto several other spectra such as those characterizing incompressible fluids, compressible Maxwell elastomers, or permeable membranes in appropriate asymptotic regimes. Since cell division and cell death are out-of-equilibrium processes, detailed balance is broken, but a generalized fluctuationresponse relation is satisfied in terms of appropriate observables. Our work is a first step toward the description of the out-of-equilibrium fluctuations of the surface of a thick epithelium and its dynamical response to external perturbations.

A standard way by which surfaces and interfaces are brought out of equilibrium is the generation of a flux, either orthogonal or parallel to their average plane, or a combination thereof [START_REF] Cates | Soft and Fragile Matter: Nonequilibrium Dynamics, Metastability and Flow[END_REF]. A familiar example of a parallel flux is that of wind blowing on the surface of water or more generally of an interface between two immiscible fluids submitted to hydrodynamic shear [START_REF] Thiébaud | [END_REF]. In the celebrated Kardar-Parisi-Zhang model [3], which has been vastly studied by both physicists and mathematicians [4,5], the flux is orthogonal to the growing interface. A flux can also be generated by the surface itself. This situation is achieved by biological membranes, in which proteins force an ion flow from one side of the membrane to the other, the energy source being provided by the hydrolysis of adenosine 5'-triphosphate (ATP) [6][7][START_REF] Lacoste | Liposome, Lipid Bilayers and Model Membranes[END_REF]. A similar situation is obtained with a membrane catalyzing the polymerization of a biopolymer on one of its sides [START_REF] Maitra | [END_REF]. Biological membranes, however, can be brought out of equilibrium without the existence of a flux by conformational changes of some of their constituents, which break detailed balance [START_REF] Lacoste | Liposome, Lipid Bilayers and Model Membranes[END_REF]. In the case of red blood cells, for instance, this is achieved by ATP-dependent attachment and detachment of proteins that link the cytoskeleton to the phospholipids [10].

In this Letter, we propose a new way of driving an interface out of equilibrium. We consider the steady-state fluctuations of the surface of a tissue layer maintained in its homeostatic state, where cell division and cell death equilibrate on average [11]. We use generic equations derived in Ref. [12], in which stochastic cell division and cell death interfere with elastic stresses to provide effective viscoelastic equations. We neglect the mechanical role of the intercellular fluid and treat the tissue as a one-component system, which is valid for characteristic thicknesses smaller than a centimeter and time scales larger than a few minutes [13,14]. The reciprocal cou-pling between mechanical stress and cell renewal in the bulk of the tissue drives the system out of equilibrium. Despite their out-of-equilibrium nature, which breaks detailed balance and the fluctuation-dissipation relation, stochastic cell division and cell death can be described as Markovian processes [12]. Within this framework, the tissue is, therefore, expected to satisfy a generalized form of a fluctuation-response relation, which could potentially be tested experimentally [15].

We consider an isotropic and homogeneous tissue of prescribed average thickness H and infinite lateral extension resting on a rigid surface on one side and in contact with a classical fluid on the other (see Fig. 1). The fluid is of negligible viscosity and exerts a constant homogeneous pressure on the tissue, equal to the tissue homeostatic pressure [11]. Under these conditions and on time scales shorter than the characteristic evolution time of the av-erage thickness H (see the Supplemental Material [16], Sec. I), the tissue is in a steady state characterized by specific average values ρ h and σ h of the cell-number density ρ and isotropic tissue stress σ and by a vanishing cell-velocity field v α , where the index α stands for the spatial coordinates x, y, or z.

The cell-number density ρ and the cell-velocity field v α obey the continuity equation

∂ t ρ + ∂ α (ρv α ) = (k p + ξ c )ρ , (1)
where ∂ t and ∂ α denote the partial derivatives with respect to time and spatial coordinates, with summation over repeated indices. In this equation, k p = k d -k a is the global mean rate of cell production, taking into account cell division and apoptosis with rates k d and k a , and ξ c is a Gaussian white noise of zero mean. Cell division and cell death being independent stochastic processes, the variance of the noise reads

ξ c (r, t)ξ c (r , t ) = [(k d +k a )/ρ]δ(r-r )δ(t-t ).
The stress tensor σ αβ obeys the force-balance condition ∂ α σ αβ = 0. Close to the homeostatic state and to linear order, the deviation of the density from its homeostatic value δρ = ρ-ρ h is proportional to δσ = σ-σ h . The cell-production rate then reads

k p = -(1/τ i ) (δρ/ρ h ) = (1/τ i ) χ c δσ,
where τ i is a characteristic time and χ c is the shorttime tissue compressibility in the homeostatic state. The stress tensor is decomposed into an isotropic and a traceless part σ αβ = σδ αβ + σαβ , where σαα = 0. Under these conditions, the deviation δσ αβ = δσδ αβ + δσ αβ of the stress tensor with respect to its homeostatic value obeys the following Maxwell model [12]:

(1 + τ i ∂ t ) δσ = ζ∂ γ v γ -ξ (1 + τ a ∂ t ) δσ αβ = 2η ṽαβ -ξαβ . (2)
In the isotropic part, ζ = χ -1 c τ i is an effective bulk viscosity, and ξ = ζξ c is an effective bulk noise term of amplitude ϑ = ζ 2 (k d + k a )/ρ h . The remarkable absence of any compression modulus is due to the nonconservation of cell number [11,12]. In the anisotropic part, τ a , η, and ξαβ are the effective relaxation time constant, shear viscosity, and noise, a priori different from those appearing in the isotropic part, and ṽαβ =

(1/2) [∂ α v β + ∂ β v α -(2/3)∂ γ v γ δ αβ ]
is the traceless part of the velocity-gradient tensor. As done in Ref. [12], we choose for ξαβ a Gaussian white noise and write its second moment by symmetry arguments: ξαβ (r, t) ξγδ (r , t ) = θ [δ αγ δ βδ + δ αδ δ βγ -(2/3)δ αβ δ γδ ] δ(rr )δ(tt ), where θ characterizes the amplitude.

In practice, this anisotropic noise is related to the stochasticity of cell rearrangements. In general, the two noises cannot be cast in a way respecting detailed balance with a common effective temperature, and the system is out of equilibrium. One can think of such a tissue as a system in which the particle number is not conserved and the isotropic and deviatory modes are maintained at different temperatures.

To characterize the position fluctuations δH(r, t) of the tissue upper surface, we focus our attention on the two-point autocorrelation function C(rr , tt ) = δH(r, t)δH(r , t ) and on its linear response function χ to an externally applied pressure field δP e (r, t) defined as δH(r, t) = χ(rr , tt )δP e (r , t ) dr dt . We further consider the Fourier components of the different fields defined according to the following Fourier transform:

f (q, ω) = dr dt f (r, t) exp [-i(q • x + ωt)].
In a linear perturbation close to a flat interface, the different Fourier modes decouple, and we can consider the dependence on one single wave vector q without loss of generality. Close to the homeostatic state, the equation governing the cell-velocity field reads in frequency space,

η 1 ∂ α ∂ α v β + ζ 1 + 1 3 η 1 ∂ β (∂ α v α ) = ∂ α ξ 1,αβ , (3)
with

ζ 1 = ζ 1 + iωτ i , η 1 = η 1 + iωτ a , (4)
and

ξ 1,αβ = 1 1 + iωτ i ξδ αβ + 1 1 + iωτ a ξαβ . (5)
Writing the stress continuity at the upper surface with generalized Laplace pressure and negligible shear stress, we obtain in Fourier space and in the linear regime:

ζ 1 - 2 3 η 1 (iqv x + ∂ z v z ) + 2η 1 ∂ z v z + (γq 2 + κq 4 ) δH = ξ 1,zz and η 1 (iqv z + ∂ z v x ) = ξ 1,xz . (6)
Here, γ is the tissue surface tension and κ its surface bending rigidity. To these must be added the kinematic condition v z = iωδH. At its lower surface, the tissue is in contact with a hard wall, and we consider a no-slip boundary condition, such that v x = v z = 0. General expressions for the two-point autocorrelation function and the linear response function are given in the Supplemental Material [16], Sec. II. Using these results, we characterize how far from thermodynamic equilibrium the system is operating by the introduction of an effective temperature T eff equal to the thermodynamic temperature at equilibrium:

k B T eff (q, ω) = ω 2 C(q, ω) χ (q, ω) . (7)
Here, k B is the Boltzmann constant, and χ (q, ω) is the imaginary part of the response function in Fourier space. The effective temperature is, in general, frequency and wavelength dependent, and its full expression is given in the Supplemental Material [16], Sec. II. If we choose thermal noise amplitudes ϑ = 2ζk B T and θ = 2ηk B T , which define an equilibrium Maxwell model, the effective temperature equals the thermodynamic temperature T at all frequencies. More generally, when ϑ/θ = ζ/η, the effective temperature is constant and the system is analogous to an equilibrium system. We first consider the limiting case where the thickness H of the tissue is infinite or equivalently where qH 1, for which full analytic expressions can be presented here. In this limit, the functions defined above read

C(q, ω) 2 q ϑ 1 |η 1 | 2 + θ 1 |ζ 1 + 1 3 η 1 | 2 + 1 3 |η 1 | 2 ζ 1 + 4 3 η 1 2 |γq + κq 3 + 2iω η1 | 2 χ(q, ω) - 1 q 1 γq + κq 3 + 2iω η1 k B T eff (q, ω) ϑ 1 |η 1 | 2 + θ 1 ζ 1 + 1 3 η 1 2 + 1 3 |η 1 | 2 2Re η 1 ζ 1 + 1 3 η 1 ζ 1 + 4 3 η 1 * , (8)
with

ϑ 1 = ϑ/[1 + (ωτ i ) 2 ], θ 1 = θ/[1 + (ωτ a ) 2 ] and η1 = η 1 (ζ 1 + η 1 /3)/(ζ 1 + 4η 1 /3
). The star denotes the complex conjugate. Note that, due to the absence of an intrinsic length scale in the system in this case, the effective temperature has no dependence on the wave vector. In the low-frequency limit (ωτ a , ωτ i 1), these functions correspond to those of a passive compressible fluid with bulk and shear viscosities ζ and η at the temperature T eff but with the originality of a vanishing compression modulus. As already stated, the long-time compression modulus vanishes in a tissue due to the nonconservation of cell number [11,12], a feature that is impossible in a system with a conserved number of particles. Interestingly, the correspondence also holds with an incompressible fluid with effective shear viscosity η = η(ζ + η/3)/(ζ + 4η/3), since then compression modes are infinitely fast, and only viscous terms are left in the correlation and response functions. In the opposite limit (ωτ a , ωτ i 1), the expressions are analogous to those of a passive, one-component, compressible Maxwell elastomer.

In the case where the thickness H of the tissue layer is finite, as stated above, the effective temperature is both frequency and wavelength dependent. In the lowfrequency regime, it depends on wave vector only (see the Supplemental Material [16], Sec. III for its expression). When ϑ/θ = ζ/η, it can exhibit different nontrivial behaviors depending on the parameter values (see Fig. 2).

Despite the existence of a frequency-and wave-vectordependent effective temperature, the system's dynamics is Markovian. Proper observables, therefore, exist in terms of which a generalized fluctuation-response relation is satisfied [15]. We choose here to illustrate this fact in the low-frequency domain (ωτ a , ωτ i 1), where we can write a single evolution equation for δH, directly under a Markovian form. This allows us to present the explicit construction of an observable X(q, t) that obeys a fluctuation-response relation. Solving for the integration constants and using the kinematic condition at the upper surface v z = ∂ t δH, we get

∂ t δH(q, t) = -τ (q) -1 δH(q, t) + f (q, t) + ξ δH (q, t) , (9)
where τ (q) is a relaxation time constant that depends on the wave vector, f (q, t) is an external field acting on the tissue upper surface, and the noise ξ δH (q, t) is a scalar combination of integrals containing the matricial noise ξ αβ = ξδ αβ + ξαβ (see the Supplemental Material [16], Sec. IV for explicit expressions). Introducing the equal-time autocorrelation function Σ(q) of δH(q, t) in the absence of an external field as δH(q, t)δH(q , t) = Σ(q)δ(q + q ), we define X(q, t) as X(q, t) = -τ (q)Σ(q) -1 δH(q, t) .

With these definitions, we have [15] χ XX (q, ω)

-χ XX (q, -ω) = iωC XX (q, ω) , (11)
where C XX and χ XX are, respectively, the two-point autocorrelation function of the variable X and its linear response to the external field f , here written in Fourier space. The corresponding explicit expressions are given in the Supplemental Material [16], Sec. IV. Within the low-frequency domain, the regime qH 1 is particularly interesting, since the expressions of the correlation and response functions are then analogous to those of a permeable membrane with permeation con-stant λ p = H/(ζ + 4 3 η) [17]:

C(q, ω) 1 H ϑ + 4 3 θ (γq 2 + κq 4 ) 2 + ω 2 λ -2 p χ(q, ω) - 1 γq 2 + κq 4 + iωλ -1 p k B T eff 1 2 ϑ + 4 3 θ ζ + 4 3 η . ( 12
)
It is remarkable that, due to the nonconservation of cell number and the subsequent absence of long-time compression modulus, there is no lubrication regime. The leading term in the equal-time correlation function reads

C(q, t -t = 0) k B T eff γq 2 + κq 4 , (13)
and we can make direct use of the results concerning the fluctuations of equilibrium membranes [START_REF] Lipowsky | Structure and Dynamics of Membranes: I. From Cells to Vesicles/II. Generic and Specific Interactions[END_REF][START_REF] Nelson | Statistical Mechanics of Membranes and Surfaces[END_REF].

The divergence of the mean-square fluctuations of the tissue thickness is of particular interest. There are three length scales entering the problem. The first one is simply the thickness H. The second one is given by the square root of the ratio of the bending modulus over the membrane tension l κ = κ/γ. Finally, the third one, the "collision" length l c , stems from the divergence of the mean-square fluctuations of the tissue thickness as a function of lateral position [START_REF] Fisher | [END_REF]21]. It is such that δH(0, t)δH(l c , t) = H 2 . If γH 2 k B T eff , l c is proportional to the thickness H and is much smaller than l κ :

l c ∝ H κ k B T eff = l κ γH 2 k B T eff . ( 14
)
In this regime, the tissue surface fluctuations scale like distances parallel to the interface. In the tensiondominated regime where γH 2 k B T eff , l c reads

l c ∝ l κ exp πγH 2 k B T eff . (15)
In this regime, fluctuation amplitudes scale logarithmically with the distance parallel to the interface, and l c is much larger than l κ . The surface appears locally smooth, although thickness variations eventually reach values of the order of the thickness itself on length scales of order l c . A natural question concerns the possibility to observe such effects. The effective temperature in the lowfrequency, small-wave-vector regime can be estimated using Eq. (12). Keeping only the isotropic contribution in the homeostatic state, we obtain k B T eff = ζk d /ρ h . Using this expression and numerical values from the literature, we estimate that the ratio l c /l κ is typically of order one tenth to several tens and can potentially be much larger (see the Supplemental Material [16], Sec. V). This indicates that the two limiting regimes mentioned here could, in principle, be observed, and that changing the physical properties of the tissue could affect its surface structure without necessarily requiring a modification of the cell-division rate. The surface of precancerous tissues is usually rougher than that of healthy tissues, even when these tissues are not in a growth phase [START_REF] Weinberg | The Biology of Cancer[END_REF][START_REF]Pathology and Genetics of Tumours of the Breast and Female Genital Organs[END_REF]. One could speculate that some of the early cancerous mutations affect certain physical properties of the tissue, not necessarily uniquely related to cell proliferation.

In equilibrium membranes, the existence of a collision length generates an entropic repulsive interaction. In tissues, the physical meaning is likely to be very different.

In the case we discuss here, this suggests that the tissue thickness potentially vanishes. The onset of tissue disappearance is given by the characteristic time of thickness fluctuations at the scale l c :

τ d = ζ + 4 3 η l 2 c γ + κl -2 c H . ( 16
)
The fate of the tissue layer then depends on the future evolution of the region of zero thickness. This, in turns, depends on the tissue-substrate interaction and adhesion properties, and a detailed analysis of this outcome is beyond the scope of this work. Note that the linear approximation used in this Letter is valid as long as κ k B T eff . In the opposite case, the surface would appear crumpled with no angular coherence [START_REF] Nelson | Statistical Mechanics of Membranes and Surfaces[END_REF].

The tissue surface fluctuations we have studied here exhibit a surprisingly rich behavioral diversity. One can define a wave-vector-and frequency-dependent effective temperature, a clear signature of the out-of-equilibrium nature of the system. In the low-frequency regime, this effective temperature is only wave-vector dependent. As shown in Fig. 2, it can present different nontrivial behaviors, among which being monotonically varying or having an extremum at a finite wave vector. In the large-wavevector, high-frequency regime, the tissue behaves like a compressible Maxwell elastomer with nonthermal noise, whereas in the large-wave-vector, low-frequency regime, it behaves like an incompressible fluid at equilibrium. Finally, in the small-wave-vector, low-frequency regime, the mapping can be made to a fully permeable membrane with tension and curvature. This implies that, at sufficiently large scales, the tissue always has thickness fluctuations on the order of the average thickness. This could signal a finite lifetime for the tissue layer. In actual epithelial tissues, however, such an outcome is usually not observed. In this case, cell division and cell death are regulated by external chemical gradients. As a consequence, cells are globally renewed close to the basal membrane and die close to the apical surface, which leads to a well-defined thickness [START_REF] Young | Wheater's Functional Histology: A Text and Colour Atlas[END_REF][START_REF] Basan | [END_REF][26]. One could speculate that this is part of the solutions developed under evolutionary pressure to create robust epithelial tissues.

SUPPLEMENTAL MATERIAL Homeostatic Fluctuations of a Tissue Surface

Thomas Risler, 1,2 Aurélien Peilloux, 1,2 and Jacques Prost In our study, we treat the average thickness H as a constant in time. In order to justify this framework, we estimate here the characteristic time over which the average tissue thickness H evolves by diffusion, due to stochastic cell-divison and cell-death processes, with average rates k d = k a . We consider a tissue layer made of N cells of average cell-number density ρ, charateristic lateral dimension L and average thickness H. In this volume, the cell-number standard deviation in a time dt is δN = (k d + k a ) N dt. For a total volume V = H L 2 = N/ρ, the corresponding amplitude of thickness fluctuations reads:

δH 2 = (k d + k a ) H ρ L 2 dt . ( 17
)
To diffuse over distances of order H, the tissue layer therefore takes a characteristic time τ H ρ H L 2 /(k d + k a ). With a maximal rate of cell renewal of one per day, a cell-diameter of 10 µm, a tissue thickness of 100 µm, this characteristic time of the order of several hundred years for a tissue patch of one millimeter square, and several tens of thousand years for one of one centimeter square. Ignoring the temporal variations of the average thickness H is therefore perfectly valid in the framework of our study.

II. EXPRESSIONS OF THE PHYSICAL QUANTITIES FOR THE TISSUE OF FINITE THICKNESS H

The two-point autocorrelation function reads

C(q, ω) = 2 q ϑ 1 N a,1 + θ 1 N b,1 |(γq + κq 3 )D a,1 + 2iωη 1 D b,1 | 2 , ( 18
)
where

N a,1 = 2 + ζ1 2 e 8 H -1 + 4 H| ζ1 | 2 e 2 H e 4 H + 1 + 8 H 2 + 2( ζ1 + ζ * 1 ) + (1 -2 H2 )| ζ1 | 2 e 4 H +2 -2 + | ζ1 | 2 + 2 H2 [2( ζ1 + ζ * 1 ) + 3| ζ1 | 2 ] e 2 H e 4 H -1 N b,1 = 2 + ζ1 2 1 3 + | ζ1 | 2 e 8 H -1 -4 H| ζ1 | 2 5 3 + 2( ζ1 + ζ * 1 ) + | ζ1 | 2 e 2 H e 4 H + 1 +8 H -ζ1 (1 + ζ1 ) 2 + 2 ζ * 1 + (1 + 2 H2 )( ζ * 1 ) 2 + + 1 3 -ζ * 1 2 + (1 -2 H2 )| ζ1 | 2 + 2( ζ1 + ζ * 1 ) e 4 H +2 (1 + ζ1 )(2 + ζ1 ) 2 + 2 ζ * 1 + (1 + 2 H2 )( ζ * 1 ) 2 + + 1 3 -ζ * 1 -2 + | ζ1 | 2 + 2 H2 (2( ζ1 + ζ * 1 ) + 3| ζ1 | 2 ) e 2 H e 4 H -1 D a,1 = (1 + ζ1 ) (2 + ζ1 ) e 4 H -1 -4 H ζ1 e 2 H D b,1 = ζ1 (2 + ζ1 ) e 4 H + 1 + 2 2 + 2 ζ1 + (1 + 2 H2 ) ζ2 1 e 2 H , (19)
with ζ1 = ζ 1 /η 1 + 1/3 and H = qH. We also have, as in the main text,

ζ 1 = ζ 1 + iωτ i , η 1 = η 1 + iωτ a , ϑ 1 = ϑ 1 + (ωτ i ) 2 , θ 1 = θ 1 + (ωτ a ) 2 . (20)
Note that even though it is not directly visible under this form, the expression of N b,1 above is real. The linear response function reads

χ(q, ω) = - 1 q 1 γq + κq 3 + 2iωη 1 D b,1 D a,1 . (21)
We then have for the effective temperature:

k B T eff (q, ω) = ϑ 1 N a,1 + θ 1 N b,1 2Re(η 1 D * a,1 D b,1 ) . ( 22
)
We recover the thermodynamic limit of an equilibrium Maxwell model by choosing noise amplitudes that respect detailed balance, ϑ = 2ζk B T and θ = 2ηk B T . We then have k B T eff (q, ω) ≡ k B T , independent of wave vector and frequency.

In general, the effective temperature given by Eq. ( 22) above is a function of frequency and wave vector, as well as of the parameters defining the problem. In the low-and large-q limits, it has two asymptotic finite limits, which have simple expressions. The asymptotic expression in the largewave-vector regime (qH 1) is given in Eq. ( 8) of the main text. In the small-wave-vector regime (qH 1), we have

k B T eff 1 2 ϑ 1 + 4 3 θ 1 Re(ζ 1 + 4 3 η 1 )
.

In the low-frequency domain (ωτ a , ωτ i 1), the effective temperature is just a function of the wave vector q. Its asymptotic expressions read

k B T eff ϑη 2 + θ ζ + 1 3 η 2 + 1 3 η 2 2η ζ + 1 3 η ζ + 4 3 η (24)
in the large-q limit, and

k B T eff 1 2 ϑ + 4 3 θ ζ + 4 3 η (25)
in the low-q limit, as given in Eq. ( 12) of the main text.

III. EXPRESSIONS OF THE PHYSICAL QUANTITIES FOR THE TISSUE OF FINITE THICKNESS H IN THE LIMIT OF LONG TIMESCALES

In the limit of timescales much larger than τ i and τ a , where ωτ i 1 and ωτ a 1 in the frequency domain, ζ 1 and η 1 are real and reduce to the bulk and shear viscosities ζ and η, respectively. Therefore, the expressions given in section 2 above can be simplified by replacing all the variables by their real, simpler counterparts with ωτ i 1 and ωτ a 1. We then obtain

C(q, ω) = 2 q ϑN a + θN b |(γq + κq 3 )D a + 2iωηD b | 2 , (26)
where

N a = 2 + ζ 2 e 8 H -1 + 4 H ζ2 e 2 H e 4 H + 1 + 8 H 2 + 4 ζ + (1 -2 H2 ) ζ2 e 4 H +2 -2 + ζ2 + 2 H2 ζ(4 + 3 ζ) e 2 H e 4 H - 1
N b = 2 + ζ 2 1 3 + ζ2 e 8 H -
D a = (1 + ζ) (2 + ζ) e 4 H -1 -4 H ζ e 2 H D b = ζ(2 + ζ) e 4 H + 1 + 2 2 + 2 ζ + (1 + 2 H2 ) ζ2 e 2 H , (27)
with ζ = ζ/η + 1/3 and H = qH. The linear response function reads

χ(q, ω) = - 1 q 1 γq + κq 3 + 2iωη D b Da . (28)
We then have for the effective temperature:

k B T eff (q, ω) = ϑN a + θN b 2(ηD a D b ) . ( 29
)
We recover the constant thermodynamic temperature T of a fluid at equilibrium by choosing ϑ = 2ζk B T and θ = 2ηk B T .

IV. EFFECTIVE FLUCTUATION-RESPONSE RELATION IN THE LIMIT OF LONG TIMESCALES

A. Tissue of finite thickness H

The evolution equation for δH(q, t) presented in the main text in the low-frequency domain (ωτ a , ωτ i 1) reads ∂ t δH(q, t) = -τ (q) -1 δH(q, t) + f (q, t) + ξ δH (q, t) .

It contains the following functions:

τ (q) = 2η γq + κq 3 D b D a and ξ δH (q, t) = 1 2ηq N δH (q, t) D b , (31)
where D a and D b have already been introduced above in Eq. ( 27). We also have introduced

N δH (q, t) = (2 + ζ) (-2 + ζ)I 1 -2J 1 e 4 H + 2 (2 -2 H ζ)I 2 + (1 + 2 H) ζJ 2 e 3 H +2 (2 + ζ + 2 H ζ + 2(-1 + H) H ζ2 )I 1 + (-1 + 2 H) ζJ 1 e 2 H + -4(1 + H ζ))I 2 + 2(2 + ζ)J 2 e H -ζ(2 + ζ)I 1 , (32)
with

I 1 (q, t) = - q 2 0 -H dz e qz (ξ xx -ξ zz + 2iξ xy ) | z I 2 (q, t) = - qe - H 2 0 -H dz e -qz (ξ xx -ξ zz -2iξ xy ) | z (33)
and

J 1 (q, t) = q 0 -H dz ζq 2 z -H dz e qz (ξ xx -ξ zz + 2iξ xy ) | z - ζq 2 0 z dz e q(2z-z ) (ξ xx -ξ zz -2iξ xy ) | z + e qz -ξ zz + 1 -ζ iξ xy | z J 2 (q, t) = qe -H 0 -H dz ζq 2 z -H dz e q(-2z+z ) (ξ xx -ξ zz + 2iξ xy ) | z - ζq 2 0 z dz e -qz (ξ xx -ξ zz -2iξ xy ) | z + e -qz ξ zz + 1 -ζ iξ xy | z , (34)
where the noise components are understood as functions of (q, z, t). Dependencies in the coordinate z are indicated as lowerscripts. The equal-time autocorrelation function Σ(q), defined such that δH(q, t)δH(q , t) = Σ(q)δ(q + q ), is then given by

Σ(q) = ϑN a + θN b 2η(γq 2 + κq 4 )D a D b , (35)
where N a , N b , D a and D b are the functions of q introduced in Eq. (27). Finally, the observable X(q, t) defined as X(q, t) = -τ (q)Σ(q) -1 δH(q, t)

reads

X(q, t) = - 4η 2 q ϑN a + θN b D 2 b δH(q, t) . (37)
It has the following correlation and response functions in the time domain:

C XX (q, t) = 8η 3 (γ + κq 2 )(ϑN a + θN b ) D 3 b D a e -|t|/τ (q) χ XX (q, t) = - 4η 2 q ϑN a + θN b D 2 b H(t)e -t/τ (q) , (38)
where H(t) is the Heaviside step function. In the frequency domain, they read

C XX (q, ω) = 32η 4 q ϑN a + θN b D 4 b (γq + κq 3 ) 2 D 2 a + 4ω 2 η 2 D 2 b χ XX (q, ω) = - 8η 3 q ϑN a + θN b D 3 b (γq + κq 3 )D a + 2iωηD b . (39)
B. Tissue of infinite thickness H

In the limit qH 1, the expressions of the quantities defined above can be given explicitly. They read

τ (q) = 2η γq + κq 3 ζ + 1 3 η ζ + 4 3 η and ξ δH (q, t) = 1 2ηq 1 - 2 ζ I 1 - 2 ζ J 1 , (40)
with ζ = ζ/η + 1/3, and where I 1 and J 1 are the integrals given by Eqs. ( 33) and (34) in the limit qH 1. This leads to the following expression:

ξ δH (q, t) = - q 2η 0 -∞ dz z -∞ dz e qz (ξ xx -ξ zz + 2iξ xy ) | z + q 2η 0 -∞ dz 0 z dz e q(2z-z ) (ξ xx -ξ zz -2iξ xy ) | z + 1 2η 0 -∞ dz e qz 2 - ζ 2 ζ ξ xx + ζ + 2 2 ζ ξ zz + iξ xy | z . (41)
We then have

Σ(q) = 1 γq 2 + κq 4 ϑη 2 + θ ζ + 1 3 η 2 + 1 3 η 2 2η ζ + 1 3 η ζ + 4 3 η (42)
and

X(q, t) = - 4η 2 (ζ + 1 3 η) 2 ϑη 2 + θ (ζ + 1 3 η) 2 + 1 3 η 2 qδH(q, t) , (43)
which gives the following correlation and response functions in the time domain:

C XX (q, t) = 8 γ + κq 2 ζ + 1 3 η 2 ϑη 2 + θ ζ + 1 3 η 2 + 1 3 η 2 ζ + 1 3 η ζ + 4 3 η e -|t|/τ (q) χ XX (q, t) = -4q ζ + 1 3 η 2 ϑη 2 + θ ζ + 1 3 η 2 + 1 3 η 2 H(t)e -t/τ (q) , (44)
and

C XX (q, ω) = 32η 4 q ϑη 2 + θ ζ + 1 3 η 2 + 1 3 η 2 ζ + 1 3 η 4 ζ + 4 3 η 2 1 [(γq + κq 3 ) 2 + 4ω 2 η2 ] χ XX (q, ω) = - 8η 3 q ϑη 2 + θ ζ + 1 3 η 2 + 1 3 η 2 ζ + 1 3 η 3 ζ + 4 3 η 1 γq + κq 3 + 2iω η ( 45
)
in the frequency domain. As in the main text, η = η(ζ + η/3)/(ζ + 4η/3).

V. ESTIMATION OF THE COLLISION LENGTH AND TIME

We now estimate the collision length l c defined in the main text in the low-frequency, smallwave-vector domain by either Eq. ( 14) or Eq. ( 15) of the main text, depending on the value of γH 2 /(k B T eff ). In the low-frequency, small-wave-vector domain, k B T eff is given by Eq. ( 12) of the main text, and we start by estimating this quantity. To estimate the tissue bulk viscosity ζ, we use experiments where the bulk growth rate k p = k d -k a of a tissue spheroid is determined as a function of the externally applied stress thanks to a fitting procedure [27,28]. The measurement procedure leads to ζ = -(∂k p /∂P ) -1 5 10 4 Pa•day. With a cell-division rate k d of the order of one per day, we get an estimate for the isotropic elastic modulus of the spheroid in its homeostatic state Ē = ζk d of the order of 5 10 4 Pa. This is in agreement with another study, in which the shorttime responses of multicellular spheroids to an external pressure jump are measured, and which give Ē 10 4 -10 5 Pa [14]. The anisotropic counterparts of these quantities are the tissue shear viscosity η, the shear elastic modulus E and the rate of cell-neighbor exchange k r , related by E ηk r . Estimating that E is typically lower if not much lower than Ē [29,30] and that k r should typically be much larger than k d , we get that the tissue bulk viscosity ζ should be much larger than the tissue shear viscosity η. This is in agreement with experimental estimates of the shear viscosity of cellular aggregates, which give η 10 4 -10 5 Pa•s [29][30][START_REF] Beysens | Proc. Natl. Acad. Sci. U.S.A[END_REF][START_REF] Marmottant | [END_REF]. The amplitude ϑ of the isotropic noise due to cell-division and apoptosis is given in the main text. In the homeostatic state where k d = k a , it reads ϑ = 2ζ 2 k d /ρ h = 2 Ē2 /(k d ρ h ). Estimating a similar expression for the anisotropic noise amplitude θ η 2 k r /ρ h E 2 /(k r ρ h ), we see that k d k r together with Ē E or Ē E imposes ϑ θ. With these estimates, we therefore have

k B T eff 1 2 ϑ ζ = ζk d ρ h . ( 46
)
We now turn to the proper estimation of the collision length l c . We introduce the length w = γ/(ζk d ) and the linear cell dimension a with ρ h = a -3 . With a typical tissue thickness H = 10 a, we have l c l κ exp(100πw/a) in the tension-dominated regime and l c 10 l κ (w/a) in the bending-dominated regime. Within this context, the ratio l c /l κ is therefore function of the adimensional parameter w/a. A reasonable scale for a is 10 µm. Tissue surface tensions have been measured for different tissue types and using different experimental methods. The measurements give values ranging typically from a fraction up to several Millinewton per meter [29][30][START_REF] Beysens | Proc. Natl. Acad. Sci. U.S.A[END_REF][START_REF] Fung | Biomechanics: Mechanical Properties of Living Tissues[END_REF][START_REF] Foty | [END_REF][35][36][37][38][39]. With these values and ζk d 10 4 -10 5 Pa as estimated above, one finds that w/a ranges typically between 10 -4 and 10 -2 , at most 10 -1 . The lower bound of this interval corresponds to the bendingdominated regime, which leads to l c 0.1 l κ . The upper bound corresponds to the tension-dominated regime, with l c ranging from a few tens of l κ to being much larger than l κ . These order-of-magnitude estimates therefore indicate that actual tissues could be in either of these two regimes, depending on parameter values. In addition, in the tension-dominated regime, the collision length l c is exponentially dependent on the parameters characterizing the tissue. We therefore expect that small variations in some of the parameters entering the expression of l c , like the tissue surface tension γ, can have dramatic effects on the structure of the tissue surface, and that this could happen independently of alterations of the cell-division rate.

To complete our estimates, we can finally give an order of magnitude for the tissue effective temperature that we have defined. Using the values given above, one finds k B T eff 10 -11 -10 -10 J, a value that is some 10 orders of magnitude larger than thermal energy. This should however not surprise us, since the stochastic processes we study here are related to the behavior of whole cells, which are typically five orders of magnitude larger than individual molecules.

FIG. 1 :

 1 FIG.1: Illustration of the geometry of the system under study, with a schematic representation of the constitutive processes driving its out-of-equilibrium fluctuations. Cells in the tissue can be quiescent (smooth, rounded shapes), dividing (pinched, double-nucleated shape), or undergoing cell death (deformed and dotted cell and nucleus boundaries).

FIG. 2 :

 2 FIG. 2: Wave-vector dependence of the effective temperature for six different parameter sets. In all six panels, we plot the adimensional ratio Teff = 2ζkBT eff /ϑ as a function of H = qH, for different values of ϑ/θ and ζ/η. In panels (a)-(d), ϑ/θ -ζ/η = 1, and ζ/η takes the values 1/3, 2/3, 1, and 4/3 in panels (a), (b), (c), and (d), respectively. In panels (e) and (f), ϑ/θ -ζ/η = -1, and ζ/η equals 1.01 and 4/3, respectively. In panel (b), the asymptotic values in the lowand large-q limits are both equal to 3/5.

  1,2,3 1 Laboratoire Physico Chimie Curie, Institut Curie, PSL Research University, CNRS, 26 rue d'Ulm, 75005 Paris, France 2 Sorbonne Universités, UPMC Univ Paris 06, CNRS, Laboratoire Physico Chimie Curie, 75005 Paris, France 3 Mechanobiology Institute, National University of Singapore, 5A Engineering Drive 1, 117411 Singapore I. SLOW TIME EVOLUTION OF THE AVERAGE THICKNESS H


A Guizar

A Ouni

C Goursaud

C Chaudet
email: claude.chaudet@telecom-paristech.fr

J M Gorce

Quantifying the Impact of Scheduling and Mobility on IR-UWB Localization in Body Area Networks

Keywords: Body Area Networks, Mobility, positioning, Ranging, Ultra Wideband, Scheduling, Motion Capture

In the context of radiolocation in Wireless Body Area Networks (WBANs), nodes positions can be estimated through time-based ranging algorithms. For instance, the distance separating a couple of nodes can be estimated accurately by measuring the Round Trip Time of Flight of an Impulse Radio Ultra Wideband (IR-UWB) link. This measure usually relies on two or three messages transactions. Such exchanges take time and a rapid mobility of the nodes can reduce the ranging accuracy and consequently impact nodes localization process. In this paper, we quantify this localization error by confronting two broadcast-based optimized implementations of the three-way ranging algorithm with real mobility traces, acquired through a motion capture system. We then evaluate, in the same scenarios, the impact of the MAC-level scheduling of the packets within a TDMA frame localization accuracy. The results, obtained with the WSNet simulator, show that MAC scheduling can be utilized to mitigate the effect of nodes mobility.

I. INTRODUCTION

Wireless Body Area Networks (WBANs) are emerging as a key technology in many application domains, such as medicine, sports, civil security, and entertainment [START_REF] Ullah | A review of IEEE 802.15.6 MAC, PHY, and security specifications[END_REF], [START_REF] Hamalainen | On the WBAN radio channel modelling for medical applications[END_REF]. These networks are capable of collecting and transmitting information on the activity of a person, based on the analysis of the body movement.

In the context of radiolocation applications, the Impulse Radio Ultra Wideband (IR-UWB) achieves a very good performance due to its high temporal and fine multipath resolution capabilities. It provides a high precision estimation of the Time-Of-Arrival (TOA) of transmitted signal [START_REF] Xiao | A survey on impulse-radio UWB localization[END_REF]. The TOA is used by several ranging protocol like the 3-Way Ranging (3-WR) which consists in estimating the distance between a node and one of its neighbors using an exchange of three different packets: one request followed by two responses. However, depending on the wearer's movement speed, the positions of the mobile nodes may have changed before this exchange is complete, as illustrated by Figure 1. Positioning a node in a three dimensions space requires at least to measure distances with four of it neighbors. Estimation errors on all these links can easily accumulate, yielding to a coarse estimate of the node's position [START_REF] Hamie | On-body toa-based ranging error model for motion capture applications within wearable UWB networks[END_REF]. When the 3-WR procedure needs to be realized for each couple of nodes in the network, the traffic sent over the wireless medium increases rapidly with the number of nodes. In order to reduce this volume of traffic, the authors of [START_REF] Macagnano | MAC performances for localization and tracking in wireless sensor networks[END_REF] propose an algorithm called Aggregated and Broadcast (A&B) to limit the number of required handshake transactions. Each node that wants to acquire or refresh its position initiates the transaction by broadcasting a request packet to all its neighbors instead of querying a single node. After receiving multiple request packets, a node broadcasts a packet, which can contain multiple responses, and even a request. Compared to the classical peer-to-peer ranging (P2P) approach, A&B reduces the number of transmitted packets, and the delay between the acquisition of the four required distances. However, it can lead to a lower performance in presence of packet losses. An intermediate solution, peer-to-peer ranging with request broadcasting (P2P-B), which consists in broadcasting requests and transmitting responses one by one, constitutes a compromise between P2P and A&B.

This paper studies the impact of nodes mobility on the localization estimation and focuses on the MAC layer. We evaluate the impact of mobility on these various strategies using the WSNet discrete event simulator [START_REF] Chelius | [END_REF]. The nodes mobility pattern is defined by traces acquired through an experimental measurement campaign using a Vicon motion capture system. We also compare the effect on each algorithm's accuracy of the MAC-level scheduling of time slots within a TDMA frame, which defines when requests and the corresponding responses are transmitted.

The remainder of the paper is organized as follows. In Section II, we present the system model and the mobility scenario. Section III describes localization strategies and the MAC scheduling we evaluate. Section IV presents and discusses simulation results.

II. SYSTEM MODEL

A. Network Topology

We consider a single human body evolving in an indoor environment with a set of wireless devices attached to the chest, wrists, ankles and head of a body, as illustrated on Fig. 3(b). These nodes communicate through an IR-UWB physical layer. These devices operate with a single-channel frequency and are able to communicate with each other at one hop (mesh topology). We distinguish two categories of nodes. The first category regroups the anchors nodes, which are placed at the most static positions on the body (e.g. on the chest or on the back). Anchors define a Local Coordinate System (LCS). Note that the LCS is mobile and generally misaligned relatively to any Global Coordinate System (GCS). The second set of nodes comprises simple on-body nodes, whose positions are unknown and that need to be localized at least within the LCS. Localization is realized by estimating the distance between each anchors and the mobile node. In this paper, we denote by dij (t) the measured distance between node i and the anchor j at time t and by Pi (t) the estimated position of node i.

B. Ranging Estimation based on 3-WR

In this paper, the distance estimation is derived from the Time-Of-Arrival (TOA) measured for each involved packet and the 3-WR handshake protocol transactions. Figure 2 illustrates this classical exchange. The mobile node i, to be located, starts by sending a Request packet (Q ij ) to the anchor j at time T 1 . After receiving this request at time T 2 , the anchor j answers with a first Response packet (R1 ij ) at time T 3 . Node i receives this response packet at time T 4 .

Fig. 2. Three Way Ranging Protocol

Note that, at this stage, the estimated ranging can be given through 2-WR as follows:

dij (t) = 1 2 c [((T 4 -T 1 ) -(T 3 -T 2 )],
where c is the speed of radio waves (i.e. c = 3 * 10 8 m/s). Given the time resolution of this process, the clock drift, which denotes the difference between the speed of a clock circuit and the absolute time, has impact on the time of flight estimation and consequently on the ranging accuracy. Indeed, clock drifts are not identical for each circuit and two nodes will not measure exactly the same time interval between two moments. To compensate this effect, the responding node j sends a second response packet at time T 5 , which will be received by node i at time T 6 . The resulting ranging estimate is computed as follows:

dij (t) = 1 2 c [((T 4 -T 1 ) -∆t1) -((T 6 -T 4 ) -∆t2)] , (1)
where ∆t1 and ∆t2 are respectively equal to (T 3 -T 2 ) and (T 5 -T 3 ).

C. Mobility Model

To evaluate the effect of mobility on localization accuracy, we confronted the A&B and P2P-B algorithms to real mobility traces. These traces were obtained during a measurement campaign realized by the CORMORAN project carried out in June 2014 at the M2S laboratory of ENS Cachan, Bretagne, France.

The Vicon motion capture system [START_REF]Vicon[END_REF], shown in action on Fig. 3(a), utilizes 16 high frequency (100 Hz) cameras surrounding a 10 m × 6 m scene to record the position of multiple visual markers. We placed 41 markers on a human subject and 25 markers directly on the communication sensors carried by the subject, which allows us to acquire a very finegrained history of the subject movements.

In this paper, we selected two particular scenarios to evaluate the impact of mobility on localization. In the first scenario, called Yoga activity, the subject realizes a series of static positions in the same place, mimicking Yoga postures (e.g. put both feet together and hands at the sides of the body, elongate the spine and fold forward from hips with knees, etc.). In the second mobility scenario, called pedestrian walking, the subject starts moving from the middle of the scene. Then, he walks at moderate speed along a rectangular trajectory centered on the starting point. Each capture lasts 100 s. More information on this campaign is available in [START_REF] Denis | Qualitative Analysis of RSSI Behavior in Cooperative Wireless Body Area Networks for Mobility Detection and Navigation Applications[END_REF].

III. PROPOSED MAC STRATEGIES

Localization of the nodes placed on a body can easily suffer from the nodes mobility. Indeed, as nodes move constantly, their position evolves while estimating all required distances.

In this work, we only focus on the impact of nodes mobility on the localization accuracy. We therefore assume that the channel is perfect and we ignore the effect of shadowing and body obstruction. We consider, in the rest of the paper, a network composed of 8 devices, as represented on Fig. 3(b). Four fixed devices are positioned on the right of the chest (A1), the left of the chest (A2), the left hip (A3) and on the back

A. Scheduling strategies based on 3-WR handshakes

To capture the body's movement accurately, the nodes positions should be refreshed at a frequency that corresponds to the movement speed. However, a basic and sequential implementation of the three-way ranging would require a request followed by two responses per link. The resulting control traffic can increase rapidly. Besides, when evaluated consecutively, the obtained distances will not refer to the same position, leading to positioning uncertainty. It is possible to reduce these impacts by grouping requests and responses in the same packet. Two such strategies are illustrated on Fig. 4: • Nodes are localized individually (P2P-B): a node i broadcasts a single request packet Q i to all anchors.

Each anchor then sends both responses (R1 ji ) and (R2 ji ) consecutively, one anchor at a time. Broadcasting the requests has the effect to increase, the ∆t1 delay for some anchors. Besides, this delay is not uniform across the anchors. • Aggregated and Broadcast (A&B): in a first phase, all requests packets Q i are transmitted in broadcast. Then, each anchor j sends an aggregated response (R1 j ) to all nodes, followed by the second response (R2 j ). A&B allows reducing the volume of traffic and hence the size of the TDMA frame, but also results in an increase of the ∆t1 delay. However, in noisy environment with high packet-loss, its performance can be deteriorated. One may note that the P2P-B favors the ∆t1 delay (individual distance estimation), while A&B permits to obtain the four distances as fast as possible.

For these two strategies, we can derive the TDMA frame duration at the MAC layer. As depicted by Figure 5, P2P-B presents higher frame duration compared to A&B, which increases linearly with the number of nodes. This difference is explained by the number of packets sent by each protocol between m anchors and n mobile nodes. In A&B (resp. P2P-B), the nodes send n Request packets in Broadcast to the anchors. Then, the anchor nodes answer with m (resp. n.m) Response 1 packets and m (resp. n.m) Response 2 packets. Thus, we have Π A&B = n + 2.m and Π P 2P -B = n.(2m + 1) packets for A&B and P2P-B, respectively. Thus, in our scenario of 4 anchors and 4 mobile nodes, P2P-B presents a TDMA frame length of ≈ 54 ms against ≈ 18 ms for A&B. This means that for the duration of one P2P-B movement acquisition round, A&B can perform three times more the motion capture, and hence should reduce the impact of nodes mobility.

B. MAC-level slots scheduling

The strategies A&B and P2P-B both reduce the overall positioning time and save bandwidth, but delay the reception of the responses. It is however possible to play with the MAC-level scheduling (i.e. which slot of the TDMA frame is allocated to which emitter) to reduce the positioning error. The MAC layer should therefore strive to reduce the ∆t1 and ∆t2 delays as much as possible, as shown in [START_REF] Guizar | Impact of MAC scheduling on positioning accuracy for motion capture with UWB body area networks[END_REF]. For the present scenario, involving 4 anchor nodes (A 1 to A 4 ) and 4 mobile nodes (N 5 to N 8 ), we evaluate and compare four MAC-level scheduling strategies, i.e. four different permutations of the slots allocated to each mobile node in the TDMA frame:

• NS1 (N 5 , N 6 , N 7 , N 8 ), where nodes with highest (resp. lowest) mobility are positioned at the beginning (resp. end) of the TDMA frame, • NS2 (N 8 , N 5 , N 6 , N 7 ), where nodes with highest mobility are positioned in the middle of the TDMA frame, • NS3 (N 7 , N 8 , N 5 , N 6 ), where nodes with highest (resp. lowest) mobility are at the end (resp. beginning) of the TDMA frame and • NS4 (N 6 , N 7 , N 8 , N 5 ), where nodes with lowest mobility are positioned in the middle of the TDMA frame.

IV. SIMULATION AND RESULTS

A. Simulation environment

We compare the different algorithms and MAC-level scheduling strategies using the WSNet discrete-event simulator [START_REF] Chelius | [END_REF], which we modified to include simulation models that correspond to our scenarios. We implemented in WSNet a physical layer based on IEEE802.15.6 UWB [START_REF]IEEE standard for local and metropolitan area networks -part 15.6: Wireless body area networks[END_REF] in default mode (OOK modulation and 0.4875 Mb/s). At the MAC, layer we implemented a TDMA-based medium access protocol, as well as the different algorithms and scheduling approaches, detailed in Section III. Finally, we implemented a mobility model that let us exploit the mobility traces acquired during the CORMORAN project, as detailed in section II-C.

Let us consider a mobile node i that performs κ localization operations throughout the simulation. At every localization attempt happening at date t, i computes an estimate position, Pi (t) that can be compared to the reference position, which can be the position occupied by the node the beginning of the 3-WR exchange (P Ref1 (t)), at its end ((P Ref2 (t))), or at the end of the TDMA frame (P f inal (t)), as illustrated on Fig. 4. The accuracy of the localization process with respect to a given reference position is given by the root mean squared error (RMSE) of its estimate over the κ ranging operations realized throughout the simulation, expressed as follows for the reference position Ref

r ∈ {ref 1 , ref 2 , f inal}: RM SE(i, ref r ) = κ P Refr (t) -Pi (t) 2 κ
(2)

B. Impact of MAC-level scheduling on P2P-B accuracy

Let us first evaluate the impact of MAC-level slots scheduling of the different nodes on the localization accuracy. For this study, we evaluate the different parameters under a controlled scenario (e.g. Yoga) that we gradually accelerate by multiplying the speed of each on-body node multiplied by a factor 1 to 10 (1 corresponds to the real scenario).

Figure 6 shows the variation of the RMSE of the positions estimation according to the speed factor for each node MAClevel scheduling strategy presented in Section III-B with respect to the three references positions (P Ref1 , P Ref2 and P f inal ). As expected, the RMSE increases quasi-linearly with the speed factor. The distance covered during 3WR transaction increases with the node mobility, inducing more errors in the distance estimation and hence in the localization accuracy.

Figure 6(a) shows that the NS1 scheduling strategy yields to the best position estimation based on P Ref1 . As nodes with higher mobility are placed at the beginning of the TDMA frame, their position is estimated at a moment closer to the reference instant. More static nodes, positioned later, will have moved less, which mechanically leads to a better RMSE. Similarly, NS3 scheduling, in which mobile nodes are scheduled at the end of the TDMA frame better estimates positions reached by the nodes at the end of the TDMA frame (P f inal ), as shown on Fig. 6(b). Fig 6(c) represents the positioning RMSE compared to the second reference positions, which corresponds to the end of the 3-WR exchange for each node. The different MAC scheduling strategies do not show much difference. P Ref2 is therefore less sensitive to the MAClevel scheduling and should be a good candidate when no information on the individual nodes mobility is assumed.

C. Comparison of the MAC scheduling impact between P2P-B and A&B

Let us then compare the reaction of P2P-B and A&B algorithms presented in Section III-A to the slot scheduling strategy. This study relies on the same Yoga scenario as before. Estimations realized by the P2P-B and A&B are compared with the same reference positions.

Figure 7 compares the distribution of RMSE for all the allocation strategies for P2P-B and A&B. In Figure 7(a), we observe a significant gap between the different slot scheduling schemes with P2P-B when evaluating P Ref1 or P f inal . However, we can note that P Ref2 remains the best estimated position, independently of the MAC scheduling. In the case of A&B, Figure 7(b) shows that the RMSE also presents a quasi-linear increase with the movement speed. Moreover, if we compare the results between P2P-B and A&B (Fig. 7(b)), we observe that the impact of the MAC scheduling is limited for all reference positions. P Ref1 becomes the best estimated position because ∆t1 is considerably reduced. Therefore, A&B therefore tolerates more freedom with respect to the MAC scheduling.

On the basis of these results we identify the two extreme scenarios concerning the combination of the selection of the reference position and of the MAC scheduling strategy: i) the best case, where A&B (resp. P2P-B) performs the localization estimation according to P Ref1 (resp. P Ref2 ) independently on the node slot strategy and ii) the worst case, where A&B and P2P-B perform the localization estimation with respect to P f inal with mobile nodes scheduled at the very beginning of the TDMA frame NS1.

Finally, one can note that except for P Ref2 , A&B permits to reduce the average error by a factor 2. Thus, the consistency in the 4 used distances is more important than the accuracy of each individual distances.

D. Quantification of position-related errors using P2P-B and A&B under different scenarios

In the two previous sections, we examined how the average positioning error evolved in function of the MAC-level scheduling and of the aggregation algorithm. Let us now look more closely at the distribution of the positioning error under the Yoga scenario as well as under the walking scenario detailed in section II-C.

Figure 8 presents the cumulative distribution function (CDF) of the on-body nodes position estimation error under both mobility scenarios. As expected, the positioning error distribution is dependent on the mobility scenario and on the handshake transaction protocol. The Yoga scenario (characterized by a low mobility) naturally presents a lower positioning error than the pedestrian walk scenario (characterized with a higher mobility). The level of improvement obtained by using the A&B protocol in place of the P2P-B protocol depends on the considered mobility scenarios: the gap between P2P-B and A&B in the case of walk scenario is bigger than the Yoga scenario. The performance of the "raw" P2P algorithm, which does not aggregate requests in a single broadcasted packet, is very close to the performance of P2P-B, which shows that a higher gain results from grouping answers rather than requests.

In the Yoga scenario, we observe that the difference between P2P-B and A&B is small. For the worst case (Fig. 8 In the walking scenario, we notice a bigger gap between P2P-B and A&B. We observe that for the worst case (Fig. 8(a)) 90 % of error estimations are smaller than 15 mm (resp. 35 mm) for A&B (resp. P2P-B) with an intersection point at ≈ 3 mm for 40 % of estimated errors. In the best case (Fig. 8(b)), 90 % of positioning errors are smaller than 8 mm (resp. 10 mm) for A&B (resp. P2P-B) with an intersection point at ≈ 1mm for 40 % of errors.

Table I reports the average positioning error in all situations. We can observe than A&B presents a gain of ≈ 50% on the position estimation for the worst case under both mobility scenarios and a gain of 63% (resp. 90 %) for the best case under the walking (resp. Yoga) scenario.

V. CONCLUSION

In this work we have quantified the impact of mobility on the positioning estimation with based on real and precise mobility traces. We have compared different strategies of slots allocation and we have shown that when anchors answer to the nodes individually (P2P-B strategy), it is more efficient to allocate the slots in the middle of the TDMA frame to estimate the instantaneous position. Moreover, A&B yields to a better performance than P2P-B in terms of latency and accuracy, with a better estimation of the position held by the sensor when the positioning request is sent. On the other hand, we have demonstrated that the Aggregated and Broadcast (A& B) mechanism reduces the impact of the slots allocation, which allows greater flexibility in the deployment of sensors on the body, and in potentially observable movements. These results are planned to be confirmed over a realistic propagation channel. We are also extending this study to other dynamic scenarios (e.g. running) and to the context multi-BAN for group navigation.

Fig. 1 .

 1 Fig. 1. Ranging error estimation between static node and mobile node

Fig. 4 .Fig. 3 .

 43 Fig. 4. Details of the P2P-B and A&B handshakes

Fig. 5 .

 5 Fig. 5. Comparison of frame duration between P2P-B and A&B

2 Fig. 6 .Fig. 7 .

 267 Fig. 6. Impact of MAC-level scheduling on the P2P-B strategy

  (a)), 90 % of positioning errors are smaller than 10 mm (resp. 20 mm) for A&B (resp. P2P-B) with an intersection point at ≈ 4 mm for 70 % of errors. In the best case (Fig.8(b)), 90 % of positioning errors are smaller than 5 mm (resp. 6 mm) for A&B (resp. P2P-B) with an intersection point at ≈ 4 mm for 80 % of errors.

Fig. 8 .

 8 Fig. 8. Cumulative Distribution Function of the position error between P2P-B and A&B, considering a topology of 4 anchors and 4 nodes under the walking and yoga scenarios for the worst (a) and best (b) reference strategies

TABLE I AVERAGE

 I POSITION ESTIMATION ERROR IN THE WORST AND BEST CASES

ACKNOWLEDGMENT

This work has been carried out in the frame of the COR-MORAN project, which is funded by the French National Research Agency (ANR) under the contract number ANR-11-INFR-010. A part of this research has been realized at the LINCS laboratory. We thank all the involved researchers from ENS/IRISA and M2S Lab for their precious support during the measurement campaign.