Tiago R Gonc

Vineeth S Varma

Salah E Elayoubi
email: salaheddine.elayoubi@centralesupelec.fr

Performance and design of robust platoons under different communication technologies

Keywords: Vehicle platoons, cooperative adaptive cruise control (CACC), wireless communication

This article studies cooperative vehicle platooning under different communications approaches and introduces a new dynamic control scheme based on the Predicted Cooperative Adaptive Cruise Control (PCACC). We start by a performance comparison of different communication technologies including VLC (Visible Light Communication) and V2V (Vehicle-to-Vehicle), with or without relaying via RSU (Road Side Unit), and assess their suitability to fully cooperative and semi-autonomous control schemes. Different from the state of the art, our main design goal is the robustness in terms of an extremely low probability of emergency braking under varying communication conditions and bursts of packet losses. In the light of the performance results, we propose an adaptive control mechanism that tunes the control gains as a function of the observed radio link quality and demonstrate, via extensive simulations, a significantly better performance compared to static control strategies.

I. INTRODUCTION

In recent years, the field of Intelligent Transportation Systems (ITS) has emerged as a research trending topic to enhance and address new traffic-system challenges [START_REF] Jia | A survey on platoon-based vehicular cyber-physical systems[END_REF]. Our particular interest in this paper is the platooning scenario which takes advantages of the particular distribution of a convoy in order to increase road capacity and to decrease fuel consumption, which is achieved by reducing the inter-vehicular distance in order to reduce the air resistance of the platoon's members. The exchange of information is crucial to the deployment of platoons as it allows taking control actions based on the most up-to-date information about the road and traffic status.

A. Paper objectives and contributions

The first objective of this paper is to evaluate different communication methods between vehicles in platoons to transmit data as illustrated in Fig. 1. The first considered technology is the Visible Light Communication (VLC), in which each vehicle in the platoon exchanges digital information through visible light with the preceding vehicle [START_REF] Ucar | IEEE 802.11p and visible light hybrid communication based secure autonomous platoon[END_REF]- [START_REF] Abualhoul | Platooning control using visible light communications: a feasibility study[END_REF]. The second method is Vehicle-to-Vehicle (V2V) which enables the communication between vehicles as long as they are in a certain range. The third method extends direct V2V communication by relaying through the network, and we label it RSUr (Road Side Unit relaying). This model involves relaying the information through a base station known as RSU in the ITS terminology or as eNodeB/gNodeB in the 3GPP terminology. In this scenario, we assume that all vehicles in the platoon are in the range of a base station that provides control information to the vehicles to complement the V2V communication. Note that there are two main V2V communication technologies for ITS, provided by IEEE 802.11p and 3GPP C-V2X. However, it is not in the scope of this paper discuss the technologies aspects, releases and differences between them. In addition to this performance comparison, the main novelty of this paper is the introduction of a dynamic control mechanism where some of the parameters of the PCACC controller are adapted based on the quality of the V2V communication channel. Packet errors are indeed inherent to all communication channels and may alter the quality of the cooperative controller. We propose a new dynamic approach that adopts the control parameters based on the observed link quality. This new controller comes with different flavours: a centralized scheme based on an offline optimization of the control parameters and a distributed controller where each vehicle adapts its own control parameters based on each communication link.

B. Positioning with respect to the state of the art

Several works considered the platooning scenario under different communication approaches. For instance, [START_REF] Vinel | Vehicle-to-vehicle communication in C-ACC/platooning scenarios[END_REF] used the 802.11p technology to evaluate the communication performance under a Cooperative Adaptive Cruise Control (CACC) controller in platoons. Likewise, [START_REF] Vukadinovic | 3GPP C-V2X and IEEE 802.11p for vehicleto-vehicle communications in highway platooning scenarios[END_REF] compared the performances of IEEE 802.11p and 3GPP C-V2X in terms of the inter-vehicular distance of the platoon. In our work, we go beyond these considerations and present an extensive performance comparison of communication modes for different control schemes, including semi-autonomous and fully cooperative control. Our primary performance metric is safety, in terms of avoiding emergency braking. This translates to robustness constraints, where the inter-vehicle distance in the platoon is set so that emergency braking is avoided in 99,999% of the cases.

As of the conception of control schemes in the presence of degraded radio link quality, [START_REF] Ploeg | Graceful degradation of cooperative adaptive cruise control[END_REF] proposed a control strategy for graceful degradation based on estimating the preceding vehicle's acceleration, while [START_REF] Lei | Evaluation of CACC string stability using SUMO, Simulink, and OMNeT++[END_REF] evaluated the impact of packet losses on CACC string stability. [START_REF] Fernandes | Platooning with IVC-enabled autonomous vehicles: strategies to mitigate communication delays, improve safety and traffic flow[END_REF] suggested different information management algorithms including one with a dynamical control parameter where they simply suggest a lower bound value for it. Our proposed dynamic controller goes far beyond a simple performance assessment under errors or a graceful degradation framework that switches to a completely local controller, as it continuously adapts to the link quality while preserving the robustness. Note that in a previous work [START_REF] Gonc ¸alves | Vehicle platooning schemes considering V2V communications: a joint communication/control approach[END_REF], we proposed a centralized design for the controller and showed that the radio link quality has a significant impact on the performance, while we present in this paper a more dynamic approach where the controller is dynamically adapted to the communication link status.

C. Organisation

The paper is organized as follows. An overview of the control aspects is presented in Section II. Section III introduces the vehicle dynamics, the platoon scenario, the simulation tool adopted and the performance evaluation for different communication methods. Section IV introduces the proposed dynamic control. Finally, we present the conclusions with considerations for future research work.

II. CONTROL AND COMMUNICATION PLATOONING SCENARIOS

This section aims to present vehicular communication technologies and their interaction with the control algorithm's performance. Note that in the control field, the term string stability is consistently used. It implies that any acceleration or braking in the first vehicle is not going to cause an amplification of the error along the tail of the platoon. In other words, as long as the first vehicle is able to avoid a collision all others will be able too. More information about mathematical definitions and conditions to ensure the string stability can be found in [START_REF] Rajamani | Vehicle dynamics and control[END_REF].

A. Fully Cooperative Control

One of the most popular platoon controllers is the slidingsurface Cooperative Adaptive Cruise Control (CACC) introduced in [START_REF] Rajamani | Design and experimental implementation of longitudinal control for a platoon of automated vehicles[END_REF], where the acceleration of the vehicle is computed based on a weighted sum of factors and its control law is given by

a i des = Ca 0 + (1 -C)a i-1 -ω 2 n (x i -x i-1 + D des ) -(2ξ -C(ξ + ξ 2 -1))ω n (v i -v i-1 ) -(ξ + ξ 2 -1)ω n C(v i -v 0 ) (1)
where D des is the desired inter-vehicular distance that we want to minimize and the index i symbolizes the vehicle index, the first vehicle (the leader) being numbered 0. (x i , v i , a i ) denote the position, velocity, and acceleration of vehicle i, respectively. The control parameters to be tuned are C, ξ and ω n . Among the parameters of this controller, an important one is C that corresponds to the weight given to the leader's acceleration in the computation of each vehicle acceleration. ξ is the controller damping ratio and ω n is the controller bandwidth.

Aiming at improvements, [START_REF] Vukadinovic | 3GPP C-V2X and IEEE 802.11p for vehicleto-vehicle communications in highway platooning scenarios[END_REF] introduced the Predictive CACC (PCACC), while [START_REF] Sybis | Communication aspects of a modified cooperative adaptive cruise control algorithm[END_REF] performed a string stability analysis for it. The PCACC controller differs from CACC by the usage of the desired acceleration of the leader and the preceding vehicle instead of the actual value. In other words, the instantaneous acceleration values in (1) are replaced by the desired ones. The PCACC is expected to be superior to the CACC because the actuation lag of the system does not affect directly the control effort which is a big limiting factor for achieving short inter-vehicle distances [START_REF] Vukadinovic | 3GPP C-V2X and IEEE 802.11p for vehicleto-vehicle communications in highway platooning scenarios[END_REF]. For simplicity, we will use the term fully cooperative control when referring to this scheme, as it includes the communication between all the vehicles in the platoon and the leader.

Note that, in all platooning systems, a basic Adaptive Cruise Control (ACC) is adopted by the leader. Its role is to establish a safe distance with the preceding vehicle that is not subject to the platooning controller.

B. Semi-Autonomous Control

Semi-autonomous control is a particular class of control that relies on the information of the preceding vehicle only, and does not utilize the leader's information. The control law is based on one vehicle look-ahead communication topology, which means that only preceding vehicle's information is required. In this case, its acceleration, velocity, and position are necessary, but without any information from the leader. In practice, this control strategy can be obtained with a simple change in one of the control parameters of the fully cooperative control, i.e. making the weight of the leader information equal to zero in [START_REF] Jia | A survey on platoon-based vehicular cyber-physical systems[END_REF]. As proven in [START_REF] Swaroop | Constant spacing strategies for platooning in automated highway systems[END_REF], this particular control law is string stable when the communication between vehicles is perfect and without delay.

III. PERFORMANCE EVALUATION OF PLATOONING

SCHEMES

Having presented the different platooning schemes, we study in this paper their performances under different communication scenarios, which will give us insights about the optimal control scheme that we will introduce next.

A. Scenarios and environment 1) Vehicle dynamics: The desired acceleration computed as the output of the CACC controller in (1) is not applied directly because of the actuation lag. Therefore, the lag is modeled as a first-order low pass filter applied to the output of the CACC control:

A i (s) = 1 τ s + 1 A i des (s) (2)
where τ is the time constant of the first-order low pass filter A i is the output, which can be interpreted as the actual vehicle acceleration, whereas A i des is the vehicle input, which can be seen as the desired acceleration. Note that, A i (s) denotes the Laplace transform of the a i time-domain variable. The idea is to approximate the dynamics of the throttle body and vehicle inertia in order to avoid instantaneous response. In this paper, we assumed a lag of τ = 0.5 s as in [START_REF] Rajamani | Vehicle dynamics and control[END_REF]. The reader can refer the upper level controller in [START_REF] Rajamani | Design and experimental implementation of longitudinal control for a platoon of automated vehicles[END_REF] for more details.

2) Platoon scenario and Robustness Criterion: The simulated scenario takes place in a four-lane highway, as illustrated in Fig. 1, with a maximum traffic density of 20 interfering vehicles/km. The system consists of a platoon of 10 automated vehicles following the leader. The vehicles of all other lanes are not in platoons (blue vehicles) and a jammer (in red) precedes the platoon leader. In the considered scenario, the speed of the jammer follows a preset sequence as in Fig. 2. The main reason for this jammer velocity profile is to study the capacity of the platoon system to avoid a collision in risky scenarios such as when the vehicle outside of the convoy suddenly applies the maximum brake capacity. Furthermore, different from the aforementioned papers that perform Monte-Carlo simulations (100 or 1000 iterations of normal conditions), we have considered a burst of packet losses. While bursts are rare events, they may occur and alter the safety of the platoon, but are not well reflected in the numerical analysis of most previous works like [START_REF] Vinel | Vehicle-to-vehicle communication in C-ACC/platooning scenarios[END_REF]- [START_REF] Cecchini | Performance comparison between IEEE 802.11p and LTE-V2V in-coverage and out-of-coverage for cooperative awareness[END_REF]. In particular, the robustness considered here is related to the following worstcase event: the jammer brakes at some time and this braking coincides with a burst of packet losses (complete interruption of the transmitted signal).

In order to be conservative, we consider long bursts of packet losses that occur with a probability of 10 -5 . Denoting by PER the probability of packet loss taking into account the channel model and packet collisions and T the time sampling interval for vehicle information, the burst size ∆ (in seconds) can be calculated as

∆ = -5/(log 10 PER)T (3)
In our simulations, we considered the sampling rate of 100 ms as advocated by the ETSI EN 302 637-2 standard.

3) Simulation tool: We used the MATLAB/Simulink environment to model the vehicle dynamics and to implement the control law. We also adopted the WLAN Toolbox of MATLAB to implement the channel configuration for an IEEE 802.11p transmission in order to obtain the Packet Error Rate (PER) taking into account the V2V fading channel aspects, the additive white Gaussian noise, the packet collisions, and other communication parameters as in Table I. The mobility behavior of vehicles is also observed in the MATLAB/Simulink environment as we consider a traffic scenario as in Fig. 1. Both fully cooperative and semi-autonomous control strategies need relative position and longitudinal velocity of the preceding vehicle so we assumed that the measurements are exact, sampled each 60 ms with 1 ms delay and done by a long-range radar as in [START_REF] Vukadinovic | 3GPP C-V2X and IEEE 802.11p for vehicleto-vehicle communications in highway platooning scenarios[END_REF]. All the vehicles in the platoon broadcast a 500 bytes Cooperative Awareness Message (CAM) each 100 ms on a 10 MHz channel bandwidth [START_REF] Alexander | Cooperative intelligent transport systems: 5.9-GHz field trials[END_REF]. The WLAN Toolbox is used to simulate the wireless environment for V2V transmissions. We adopted two different radio link models. The first one is between two consecutive vehicles and its packet error rate will be referred to as PER CV ; it uses a highway line-of-sight (LOS) profile as vehicles in the platoon are close enough. The second link models the communication between the leader and the examined vehicle with a highway non-line-of-sight (NLOS) profile since the leader is less likely to have line of sight with the considered vehicle in the platoon. Furthermore, we perform Monte-Carlo simulations to obtain the SINR (Signal to Interference and Noise Ratio) for each of the vehicles in the platoon considering Winner II Path Loss Model (B1 LOS scenario in [START_REF]1.1.2 V1.2, WINNER II channel models[END_REF]) and the interference from other vehicles. We aim to obtain conservative link reliability estimates as in [START_REF] Vukadinovic | 3GPP C-V2X and IEEE 802.11p for vehicleto-vehicle communications in highway platooning scenarios[END_REF].

B. Performance Evaluation

The focus here is to apply a longitudinal control in the platoon through different communication techniques and analyze the system stability by means of vehicle collisions in some robust and worst-case scenarios. We consider the zeroorder hold mechanism the holding strategy for the control signal during the periods of packet losses. Furthermore, in all simulations, we focus on obtaining the minimum feasible inter-vehicular distance in the platoon with a collision rate no more than 0.001 % even though we implemented a safety gap distance of 0.5 m for the emergency braking actuation to avoid collisions in practical settings.

1) Semi-Autonomous Scenario: In this first case, we adopt the semi-autonomous controller fed by VLC or V2V communication links since both of them enable a one-vehicle look-ahead communication topology. The focus is to analyze the impact of the PER in the system dynamics in terms of the minimum feasible inter-vehicular distance. In the V2V approach, the PER is related to the packet collisions and dropouts due to a poor communication link. On the other hand, in the VLC method, it can be interpreted as the measurement errors due to a poor incident angle of the light or due to rainy or foggy environments. In this scenario, we have considered a burst of losses between the leader and the preceding vehicle because it allows us to evaluate the amplification of the error along the tail of the platoon.

The column entitled semi-autonomous VLC/V2V on Table II, provides an insight into the performance in terms of the minimum feasible distance of the platoon taking into account the probability of error rate of the packet for ten vehicles with semi-autonomous control. The results indicate that the packet collisions and dropouts due to a poor communication link or incident angle of the light have a substantial impact on the system performance as a higher PER CV (PER between two consecutive vehicles) calls for higher inter-vehicular distances to avoid collisions. Note that we do not differentiate here explicitly between V2V and VLC technologies as both may have degraded qualities.

2) Fully Cooperative platoons: We now investigate the performance of fully cooperative platoons, when the platoon communicates with the leader using V2V or RSU relaying, in addition to the neighboring vehicle's communication (using V2V or VLC). Here, we also include a burst loss between the leader and the last two vehicles, due to a congested channel and path loss. This particular burst, defined in (3), was calculated based on the packet error rate of the link between the leader and the last vehicle defined as PER LLV . We aim to keep a homogeneous control strategy and small-time delay to carry the theoretical results of string stability of [START_REF] Swaroop | Constant spacing strategies for platooning in automated highway systems[END_REF].

The minimum feasible inter-vehicular distance for PCACC is also presented in Table II. Both pure V2V and RSU relaying modes are presented. When V2V is used, there is a need to keep a distance of 3.6 meters between vehicles. This distance reduces to 1.38 meters if RSU relaying is used with a high density infrastructure. This is due to the fact that the RSU selects and manages the radio resources centrally and that it is installed sufficiently high for ensuring a LOS link with vehicles on the highway. However, the advantage of RSU relaying reduces when the infrastructure becomes less dense, and direct V2V becomes better for a small RSU density.

IV. PROPOSED CONTROLLER

This section aims to introduce the proposed dynamic control scheme and to perform a case comparison over static control strategies.

A. Optimal parameters for PCACC

In contrast to existing works that assume a fixed control strategy or consider only a lower bound parameter as in [START_REF] Fernandes | Platooning with IVC-enabled autonomous vehicles: strategies to mitigate communication delays, improve safety and traffic flow[END_REF], we propose, as depicted in diagram scheme in Fig. 3, an offline optimization of the control parameters: the weight of the leader's information (C) and the desired distance (D des ), for different communication link qualities. Among the control inputs, extensive simulations demonstrate that this pair of parameters has the most substantial impact on the performance of the system. As mentioned before, the first parameter determines the weight of influence of the leader's message in the control algorithm, while the second impacts the inter-vehicular distance for the platoon. However, due to actuator lag and delay in the process, the desired distance does not correspond to the actual average inter-vehicular distance.

For any given value of PER (which is the result of the traffic density and the resulting interference) we vary the parameter weight of the leader's information (C) in a range of 0 ≤ C < 1 while minimizing the desired distance (D des ), and we register the average inter-vehicular distance (D avg ). Then, for the control input pair (C, D des ), we build an optimum lookup table as Table III that will serve as reference for the next step. Therefore, from Table III, we see that the best weight of the leader is C = 0.3 when the channel has a good quality, and is C = 0.2 for medium channel qualities. For higher error rates, the best choice is zero which corresponds to the case of semi-autonomous control. Notice that in this stage the entire platoon is subject to the same packet error rate based on the last vehicle packet error rate (PER LLV ). Furthermore, due to actuator lag and delay the string stability is not guaranteed for all platoon sizes other than those evaluated.

B. Online adaptation of the control parameters

We now move to the online adaptation of the control parameters based on the observed communication link. To summarize, we perform an offline optimization of the control parameters, above mentioned, followed by an online adaptation based on the communication link quality observed and Table III. We present two implementations of this controller: a centralized homogeneous control where the whole platoon adapts to the worst communication link, and a distributed controller where each vehicle adapts its own control parameters based on each communication link.

1) Homogeneous control: In this case, the whole platoon shares the same information that is transmitted by the leader vehicle to all members. We apply a local controller based on the packet loss observed on the communication link by the leader and the last vehicle (PER LLV ). The whole platoon adapts the control inputs corresponding to optimal values computed in the offline optimization for this PER. Thus, the whole platoon is subject to the same adaptive control procedure and we set both control parameters, weight of the leader's message and desired distance, according to its optimal values considering the worst communication link in the platoon. 2) Heterogeneous Control: An alternate approach corresponds to apply a local controller based on the packet loss observed on the communication link for each vehicle in the platoon. Therefore, based on the offline optimization mentioned before, for each vehicle, we set both control parameters according to their optimal values considering independent communication links. This implies that no additional communication resource is required for the whole platoon to exchanges the exact same information about the control law or communication conditions.

C. Simulation and case comparison

A case comparison of different implementations is performed in order to evaluate the proposed scheme over other approaches. We study the occurrence of collisions and compare the inter-vehicular distance of the platoon in a long simulation of 25 minutes in four different cases defined as follows:

• Case 1 -Static PCACC with weight of leader corresponding to C = 0.2. We apply the desired distance corresponding to the PER between the leader and the middle car in the platoon obtained under average traffic conditions. • Case 2 -Semi-autonomous (weight of leader set to zero, i.e., C = 0). • Case 3 -Homogeneous adaptive approach. We apply a uniform control in all the vehicles based on the PER between the leader and the end of the platoon (PER LLV ). • Case 4 -Heterogeneous adaptive approach. We apply a distributed controller that is based on the observed PER for each vehicle. In all cases, we consider the jammer profile as the pattern in Fig. 2, but repeated 50 times. Another important factor is the traffic density that generates interference and changes the PER for each vehicle. The traffic density varies in the range {0, 20} vehicle/km, with the packet error rate between leader to first vehicle set in the interval {0.1, 0.2} and leader to last vehicle set in the interval {0.2, 0.6} as shown in Fig. 4. Similarly, the traffic density is generated for the other vehicles in the platoon. Every 2 minutes (4 cycles), the predefined traffic density increases up to its maximum and then decreases to its minimum, and 25 minutes of traffic are simulated overall. We also apply radio error bursts as defined in (3) every 6 min of simulation.

From Table IV, we notice that Case 1 exhibited 8 collisions while cases 2, 3 and 4 had none. In the first case, the system is assumed to operate in an average traffic condition. However, it can be seen that this is not a safe approach since it does not guarantee a secure operation. In case 2, despite no collisions, there is an increase of 21% and 33% in the inter-vehicular distance when compared to cases 3 and 4, respectively. Case 4 is capable to improve approximately 10% in the inter-vehicular distance when compared to Case 3. Furthermore, it demands fewer communication resources since it adopts a distributed control approach while Case 3 uses homogeneous control that requires an exchange of the same information in the whole platoon.

To give more insights about the platoon dynamics, Fig. 5 illustrates the average inter-vehicular distance for the heterogeneous approach. It can be noticed that vehicles at the beginning are closer while vehicles at the end tend to keep larger distances due to poor communication conditions with the leader. Therefore, the proposed heterogeneous approach is demonstrated to be the best option so that platoon formation remains robust while reducing the average inter-vehicular distance.

V. CONCLUDING REMARKS This paper studies the impact of communication technologies on platooning performance and proposes novel dynamic platoon control algorithms. As of the impact of communication technologies, our results show that relaying of leader information through RSU achieves the best performance, provided that the network is carefully dimensioned for ensuring a low PER. We then proposed a dynamic controller where the quality of the communication link is continuously monitored and the control parameters updated accordingly based on the results of an offline optimization. Our simulation results show that, if the control parameters are not adapted to the channel quality, the semi-autonomous control performs best under bursts of packet losses. However, with the proposed adaptive heterogeneous control approach, using leader information results in a better performance. Our future work will target a more realistic radio link and system models in the presence of V2V relaying.

Fig. 1 .

 1 Fig. 1. Traffic scenario and illustration of the different communication approaches: VLC, V2V and RSU relaying.

Fig. 2 .

 2 Fig.2. Illustration of the jammer velocity profile adopted with 2 cycles.

FindFig. 3 .

 3 Fig. 3. Offline and online parameter election diagram.

Fig. 4 .Fig. 5 .

 45 Fig.[START_REF] Vinel | Vehicle-to-vehicle communication in C-ACC/platooning scenarios[END_REF]. Traffic density pattern adopted for the worst and the best communication link with the leader.

TABLE I COMMUNICATION

 I AND CONTROLLER PARAMETERS

	Communication	Controller
	Parameter	Value	Parameter	Value
	Simulation		PCACC (Followers)
	Duration (Tsim)	1200 s	Weight of the leader (C)	Adaptive
	Jammer profile	Check Fig. 2	Desired distance (D des )	Adaptive
	Application		Damping ratio (ξ)	2
	CAM size	500 bytes	Bandwidth (ωn)	0.5 Hz
	CAM interval	100 ms	ACC (Leader)
	PHY layer		Time gap (h)	1.4 s
	Path loss	Winner+B1 LOS	Vehicle & Highway
	Noise power	-174 dBm/Hz	Actuator lag (τ )	0.5 s
	Tx power	22.5 dBm	Vehicle length (L)	16.5 m
	MCS	QPSK, R=1/2	Max. acceleration	+2m/s 2
	Channel	Highway NLOS	Min. acceleration	-3m/s 2
	Frequency (fc)	5.9 GHz	Radar interval	60 ms
	Bandwidth (BW)	10 MHz	Lanes per direction	2
	Process delay	1 ms	Lane width	5 m
			Traffic density	20 cars/km

TABLE IV CASE

 IV COMPARISON FOR THE ONLINE IMPLEMENTATION.

		Parameters	Case 1	Case 2	Case 3	Case 4
	Controller	Weight of the leader Desired dist. (m)	0.2 0.58	0 1.03	Dynamic Dynamic	Dynamic Dynamic
		Average dist. (m)	1.21	1.67	1.38	1.26
	Outputs	Minimum dist. (m)	0.25	0.62	0.52	0.51
		Collisions	8	0	0	0


Nga Nguyen

Nicolas Huang

Jean-Louis Grossiord

Laurence Moine

Florence Agnely

Christine Vauthier
email: christine.vauthier@u-psud.fr

Van Nga Nguyen

Rheological characterization of mechanical properties of chemically crosslinked microspheres

Keywords: rheology, crosslink, hydrogels, microsphere, degradation

  L'archive ouverte pluridisciplinaire

INTRODUCTION

Hydrogels are the synthetic biomaterial class the most similar to natural living material [START_REF] Ratner | Synthetic Hydrogels for Biomedical Applications. Hydrogels for Medical and Related Applications[END_REF][START_REF] Peppas | Hydrogels in pharmaceutical formulations[END_REF]. They consist of three-dimensional crosslinked macromolecular networks with a capacity of water absorbing up to thousand times their dry weight without dissolving [START_REF] Hoffman | Hydrogels for biomedical applications[END_REF][START_REF] Kopecek | Polymer chemistry: swell gels[END_REF]. This unique property contributes to their soft consistency and biocompatibility. Therefore, they are receiving significant attention in developing biomaterials for medical applications including tissue engineering, cell encapsulation and drug delivery [START_REF] Caldorera-Moore | Micro-and nanotechnologies for intelligent and responsive biomaterial-based medical systems[END_REF][START_REF] Kopecek | Hydrogels: From soft contact lenses and implants to self-assembled nanomaterials[END_REF][START_REF] Elbert | Liquid-liquid two-phase systems for the production of porous hydrogels and hydrogel microspheres for biomedical applications: A tutorial review[END_REF][START_REF] Tajima | Preparation and functional evaluation of cell aggregates incorporating gelatin microspheres with different degradabilities[END_REF][START_REF] Garg | Scaffold: a novel carrier for cell and drug delivery[END_REF]. For many of the applications in the biomedical field, the mechanical properties of hydrogels should be appropriate.

Rheology is one of the most employed methods used to characterise gels from their formation to their intimate properties. Indeed, rheological properties are sensitive to the degree of crosslinking of the material and were widely used to follow the kinetic of crosslinking reaction used during formation of chemically crosslinked hydrogels [START_REF] Vermonden | Rheological Studies of Thermosensitive Triblock Copolymer Hydrogels[END_REF][START_REF] Moura | Rheological Study of Genipin Crosslinked Chitosan Hydrogels[END_REF][START_REF] Neamtu | The rheological behaviour of chemically crosslinked hydrogels based on poly(acrylamide)[END_REF][START_REF] Payet | Structural and rheological properties of chitosan semi-interpenetrated networks[END_REF]. Various experimental conditions are suitable to study rheological properties of hydrogel. The most used include time sweep at constant frequency and strain [START_REF] Vermonden | Rheological Studies of Thermosensitive Triblock Copolymer Hydrogels[END_REF][START_REF] Moura | Rheological Study of Genipin Crosslinked Chitosan Hydrogels[END_REF][START_REF] Neamtu | The rheological behaviour of chemically crosslinked hydrogels based on poly(acrylamide)[END_REF] or frequency sweep at constant shear stress amplitude [START_REF] Payet | Structural and rheological properties of chitosan semi-interpenetrated networks[END_REF]. Forming hydrogels can be also characterized by different rheological methods i.e., frequency sweep, strain sweep or stress sweep mode [START_REF] Khalid | Swelling properties and mechanical characterization of a semi-interpenetrating chitosan/polyethylene oxide network. Comparison with a chitosan reference gel[END_REF][START_REF] Gever | Photopolymerisation and characterisation of negative temperature sensitive hydrogels based on N,N-diethylacrylamide[END_REF]. It is noteworthy that these methods were mainly applied to characterize bulk gels while their application to monitor degradation of hydrogels including chemical crosslinks remains marginal. This is contradictory to the fact that rheological properties of hydrogels are sensitive to the degree of crosslinking and are suitable to follow gel formation during crosslinking reaction. Thus, it can be assumed that it could also be used more systematically to monitor a decrease of the crosslinking ratio occurring during degradation of a hydrolysable crosslinked hydrogel. Only a few studies have applied rheological methods to follow degradation of hydrogels caused either from a biodegradable polymer network [START_REF] Zustiak | Hydrolytically Degradable Poly(Ethylene Glycol) Hydrogel Scaffolds with Tunable Degradation and Mechanical Properties[END_REF][START_REF] Potta | Chemically crosslinkable thermosensitive polyphosphazene gels as injectable materials for biomedical applications[END_REF] or by the association of opposite charged microspheres [START_REF] Van Tomme | Degradation behaviour of dextran hydrogels composed of positively and negatively charged microspheres[END_REF]. In all cases, it was demonstrated that the rheological properties of the gels were modified during degradation and that the methods were suitable to monitor the degradation of such materials. This is an advantage over all the other methods of characterisation of polymers such as GPC, DSC, 1 H NMR which can not be applied to characterize materials made of three-dimensional polymer networks since they are not soluble in any solvent.

Rheology is a powerful tool for the characterization of bulk hydrogels. However, it is noteworthy that its application to the characterization of hydrogels under the form of nano-and microparticles has led to a very limited number of works so far [START_REF] Raquois | Microgels in coating technology: structure and rheological properties[END_REF]. The aim of our work was to develop a method based on rheological measurements that would be suitable to characterize hydrogels occurring as individual microspheres and to follow their degradation by hydrolysis of their crosslinking bonds. The first part of the work describes the set up and experimental conditions resulting into reproducible measurements of the rheological characteristics of hydrogel microspheres with different degrees of reticulation. In the second part of the work, the rheological method was applied to monitor the degradation of hydrogel microspheres containing hydrolysable crosslinking bonds.

EXPERIMENTAL

Materials

Poly(ethylene glycol methyl ether methacrylate) (PEGMMA) of number-average molecular weight 300 g/mol, poly(ethylene glycol dimethacrylate) (PEGDMA) of number-average molecular weight 575 g/mol, poly(vinyl alcohol) (PVA) (MW n 89,000-98,000 g/mol 99+% hydrolyzed) were purchased from Sigma-Aldrich (St. Louis, USA). Molar mass values were provided by supplier. Azobisisobutyronitrile (AIBN) used as polymerization initiator was obtained from Acros Organic (Geel, Belgium). Analytical grade solvents were supplied by Carlo Erba (Val de Rueil, France). All chemicals were used as purchased without further purification. Phosphate buffer saline (PBS) (58 mM, 150 mM NaCl) at pH 7.4 was used as incubation medium for the microsphere degradation study. 1 H NMR analyses were performed with a 300 MHz Bruker apparatus. Deuterium solvents are purchased from Carlo Erba.

Methods

Synthesis of ethylene glycol-co-tetralactic-co-tetraglycolic dimethacrylate (HEMA4L4G)

The chemical structure of ethylene glycol-co-tetralactic-co-tetraglycolic dimethacrylate (HEMA4L4G) is given in Figure 1.

Figure 1: Chemical structure of the hydrolysable crosslinker Ethylene glycol-co-tetralactic-cotetraglycolic dimethacrylate (HEMA4L4G) used in the synthesis of degradable microspheres

In a dry round flask containing a magnetic stir bar, lactide (8 mmol, 1.152 g), glycolide (8 mmol, 0.929 g), HEMA (4 mmol, 0.52 g) and Sn(Oct) 2 (0.025 mmol, 10 mg) were dissolved in toluene (5 mL) under an argon atmosphere. After 20h at 90°C, chloroform (5 mL) was added to dilute the reaction mixture and the resulting polymer was purified by precipitating in a large volume of petroleum ether. The precipitate was then dried under vacuum.

1 H NMR (CDCl 3 ) d (ppm): 6.13 (s, 1H, CHH=), 5.58 (s, 1H, CHH=), 5.27 (m, 1H x PD L , PLA), 4.73 (m, 2H x PD G , PGA), 4.44 (m, 2 x CH 2 , HEMA), 1.95 (s, 3H, methacrylate), 1.59 (m, 3H x PD L, PLA). According to the NMR result, the polymerization degree of lactic PD L was around 4. The same value was found for the polymerization degree of glycolic PD G .

The dried polymer obtained from the first step was then subjected to an esterification on the hydroxyl group at the end of the PLGA chain by reacting with methacryloyl chloride. The total amount of the forming macromer was dissolved in CH 2 Cl 2 (30ml) under argon in a dry flask equipped with magnetic stir bar. The content of the flask was cooled to 0°C and TEA (12 mmol, 1.6 mL) was added in order to trap HCl formed during the esterification reaction. The solution was stirred and then methacryloyl chloride (12 mmol, 1.2 mL) was added dropwise to the solution during 30 min. The stirring was continued 1h at 0°C and then 24h at room temperature. The triethylamine chloridrate salt resulting from TEA and HCl was removed by filtration and the polymer was precipitated in a large volume of petroleum ether. The crude product was then separated, dried, and kept in an argon atmosphere for further reactions.

The methacrylate functionality ƒ of the macromonomer, the polymerization degree of lactic acid and of glycolic acid (respectively PD L and PD G ) were determined from 1 H NMR spectra of the obtained products. 1 H NMR (CDCl 3 ) d (ppm): 6.22 (m, 2H x ƒ, CHH=), 5.64 (m, 2H x ƒ, CHH=), 5.18 (m, 1H x PD L , PLA), 4.82 (m, 2H x PD G , PGA), 4.31 (m, 4H, TEG), 3.64-3.70 (m, 12H, TEG), 1.97 (m, 6H x ƒ, methacrylate), 1.53 (m, 3H x PD L , PLA). PD L and PD G were around 4. The ƒ value was 97%.

Synthesis of crosslinked hydrogel microspheres

A 0.75% solution of PVA (300 mL) was introduced into a 500 mL reactor and was purged with nitrogen atmosphere for 15 min. The dispersed phase containing the crosslinking agent and a comonomer, PEGMMA, at various molar ratios were solubilized in 12 mL of toluene and degassed by bubbling nitrogen through the solution for 15 min. Non degradable microspheres were synthesised with PEGDMA as crosslinking agent at different molar ratios (expressed as percentage of the total monomers) ranging from 1 to 11 % (Table I). Degradable microspheres were synthesized using 3 % (in molar ratio regarding total amount of monomers) of HEMA4L4G as the crosslinking agent. Preparation of the microspheres was continued as followed in all cases. The dispersed phase containing the co-monomers dissolved in toluene was introduced into the aqueous phase at 30°C and agitated by means of a homemade glass 3-wing-propeller type stirrer at 250 rpm. AIBN (0.3 g) was solubilized in 2 mL of toluene and was then added into the reactor. The temperature was increased to 70°C and the reaction was allowed to proceed under stirring for 15 h at 70°C. The microspheres were washed with acetone and water before they were sieved over the following series of sieves (Inox sieve with mesh size 630, 500, 315, 100, 40 µm) (Fisher scientific, Illkirch, France). Particles retained by the sieve with a mesh size of 315 µm corresponding to microspheres with diameter ranging from 315 to 500 µm were kept for the present study. They were freeze dried immediately after preparation and purification and stored at -20°C until use.

Degradation studies

Samples of dried microspheres (0.100 ± 0.002 g) were suspended in 15 mL of PBS pH 7.4 and incubated at 37°C under lateral stirring at 100 rpm (Incubator shaker KS4000i -IKA). Three essays were performed for each time point (0, 1, 3, 5, 10, 14 and 21 days). At the different incubation times, the supernatant and the microspheres were separated for pH and rheological measurements, respectively.

Characterizations of microspheres

Morphology and size analysis

Microspheres obtained after synthesis or during the degradation process were observed by optical microscopy. The optical microscope (OLYMPUS BH2 Microscope) was equipped with leitz PL2.5/0.08 and Olympus Dplan 10 objectives and a Mightex camera. At less 25 micrographs were taken for each sample.

Particle size distribution was determined by laser diffraction on Mastersizer S apparatus (Malvern Instrument Ltd.) at 25°C. Dry beads were dispersed in water and were allowed to swell for 15 min before measurement. This time was sufficient to reach the swelling equilibrium with the type of microspheres considered in the present study. They were then introduced in the QSpec small volume sample dispersion unit. Homogenous circulation between the latter and the measurement cell was performed by means of 1000 rpm magnetically stirring. The quantity of microspheres was added in order to obtain an aperture between 5 and 10 %. Each injection was analyzed 3 times. Granulometry was analyzed using the Fraunhofer optical model. Results were presented in % volume distribution using the volume/mass moment mean diameter D [START_REF] Kopecek | Polymer chemistry: swell gels[END_REF][START_REF] Hoffman | Hydrogels for biomedical applications[END_REF] (equation 1) and the span of their size distribution (equation 2)

     d n d n i i i i D 3 4 3 , 4 (1)
      50 , 10 , 90 , v D v D v D Span   (2)
With « n i » et « d i » represented the number of particles with a define diameter, D[v, 90] is the volume diameter above which it includes 90% of the distribution, D[v, 50] is the volume diameter above which it includes 50% of the distribution, D[v, 10] is the volume diameter above which it includes 10% of the distribution.

Mass fraction measurements of microspheres

The mass fraction of the microspheres in concentrated suspensions obtained after sedimentation was determined as followed. A weighting boat was prepared with a pre-weighted piece of filter paper above which was placed a pre-weighted piece of organza. A sample of the concentrated suspension of microspheres obtained by sedimentation was placed over the organza. The mass measured, W sed , corresponded to the weight of the wet microspheres (W WM ) and the weight of the surrounded liquid. As organza showed a mesh size much below the size of the smaller microspheres and it did not absorb water, it retained the microspheres while the surrounded liquid was absorbed by the filter paper placed below. By weighting separately, the pre-weighted piece of organza with the microspheres and the pre-weighted piece of filter paper having absorbed the liquid surrounding the microspheres, it was possible to determine the weight of wet microspheres (W WM ) and that of the surrounding liquid. The mass fraction of the microspheres (ƒ wm ) expressed as a percentage can then be calculated from equation 3.

W W f Sed WM wm x100  (%wt) (3)

Determination of the Swelling ratio of the microspheres

The same procedure as described above was used to evaluate the weight of the wet microspheres (W WM ). Then the isolated wet microspheres were freeze dried and weighted again after drying to measure their dried weight (W DM ). The mass swelling ratio, Q w , was then calculated from equation 4:

W W W Q DM DM WM w   (4)

pH measurements

The pH of each supernatant obtained during degradation process was measured with a SevenMulti pH meter (Mettler Toledo) at 25°C.

Rheological measurements

The rheological properties of microspheres were evaluated by a Haake RheoStress 600 rheometer (Thermo Electron) equipped with a 35 mm plate-plate geometry. Measurements were performed at 25°C (±0.02) and regulated with a Peltier plate. A solvent trap placed on the geometry was used to prevent water evaporation during measurements. Microspheres were allowed to sediment by gravity for 30 min prior sampling. The sediment with a mass fraction in microspheres of 65 ± 3% was deposited on the inferior plate of the measurement cell.

Rheological experiments were performed using the oscillatory modes with an imposed strain. The characterization of the microspheres included two steps that were carried out successively including a shear sweep and a frequency sweep. Three rheological parameters were monitored during each measurement which included the storage (or elastic) modulus G', the loss (or viscous) modulus G'' and the phase angle δ (defined as tan δ = G''/G').

At first, a shear sweep was performed to define the stable zone (plateau) of G' and G'' moduli as a function of the applied strain (from 0.0001 to 0.1) at a constant frequency of the oscillation (1 Hz). This experiment was performed on non degradable microspheres with a crosslinking ratio ranging between 1 and 11 %. It allowed choosing a strain value in the linear viscoelastic regime which was common for all systems.

Then G', G'' moduli and phase angle δ were determined at different frequencies of the oscillation (0.01 -100 Hz) for the different types of non degradable microspheres. In the case of the degradable microspheres, the three rheological parameters were recorded on different aliquots of the same batch of microspheres at various hydrolysis times.

All measurements were performed in triplicate for each sample. Results were given as the mean value and standard deviation of G', G'' and δ which were calculated from the three determinations.

RESULTS AND DISCUSSIONS

Microsphere synthesis

Degradable and non degradable microspheres were obtained with good production yield. Indeed, 80% of the total mass of polymer produced during polymerization occurred as microspheres while only 20% were discarded as aggregates. Particles were spherical (Figure 2) and their diameter and size distribution were comprised within the range of the size of particles (315-500 µm) expected from the sieve used to isolate the microspheres after synthesis (Table I).

Swelling measurements were carried out for non-degradable microspheres with different crosslinking ratios. The swelling of the microspheres greatly depended on the crosslinking agent concentration used during the synthesis (Figure 3). Highly crosslinked microspheres contained less amount of absorbed water while the lower the crosslinking the larger was the swelling. This result agreed with what is generally observed considering bulk hydrogels [START_REF] Ibrahim | The impact of hyaluronic acid oligomer content on physical, mechanical, and biologic properties of divinyl sulfone-crosslinked hyaluronic acid hydrogels[END_REF][START_REF] Wang | Synthesis and characterization of biodegradable pH-sensitive hydrogel based on poly(ecaprolactone), methacrylic acid, and Pluronic (L35)[END_REF].

Development of the rheological method for the characterisation of the microspheres

The purpose of the work was to develop a rheological method to characterize mechanical properties of hydrogel microspheres while they remained under individual microspheres during the entire process of the measurement. This was in contrast with many works in which the properties of the hydrogel forming microspheres or nanospheres were approached by measuring the rheological properties of a bulk hydrogel of the same composition [START_REF] Hu | Preparation and properties of an injectable scaffold of poly(lactic-co-glycolic acid) microparticles/chitosan hydrogel[END_REF][START_REF] Kumachev | Highthroughput generation of hydrogel microbeads with varying elasticity for cell encapsulation[END_REF]. Preliminary experiments included the determination of the concentration in microspheres of the suspensions used for rheological measurement and determination of the gap between the plates of the measurement cell that needed to be used to provide with relevant and reproducible experimental measurements.

Concentration in microspheres in the suspension

The key parameter for rheological measurements of a suspension is that it must form a homogeneous medium. In physical studies in general and particularly in rheology, the volume fraction is frequently used to characterise the concentration of the dispersed phase in a suspension. It is an important factor which greatly influences the obtained moduli in rheological measurements [START_REF] Taylor | Rheology of Dispersions of Swollen Gel Particles[END_REF]. This dependence is not only an issue for suspensions but also for bulk hydrogels [START_REF] Raquois | Microgels in coating technology: structure and rheological properties[END_REF][START_REF] Baker | Effects of volume fraction and composition on the rheological properties of polyacrylic acid/polyacrylamide hydrogels[END_REF]. Consequently, it was important to verify that all measurements were performed at the same concentration in microspheres in the suspension throughout the study to be able to obtain relevant comparable results. The volume fraction can be calculated by the ratio of the volumes of all the particles (particles without the liquid entrapped in the void spaces subsisting between microspheres) to the total volume of the suspension (particles plus dispersing liquid). However, volume fraction is sometimes difficult to determine at it requires to know the density of the dispersed particles. This was the case with the hydrogel microspheres investigated in the present study. A related factor can be used in the case which corresponds to the mass fraction. This parameter can be measured from the determination of the weight of the wet microspheres contained in a defined weight of sample. The mass fractions measured for all the suspensions used in this study at their maximal concentration obtained after sedimentation for 30 min were 65 ± 3%. As the standard deviation was low between the different samples of microspheres, it was considered that rheological measurements were performed on suspensions at a constant mass fraction of the microspheres.

Choice of the gap between plates of the measurement cell

Sedimentation occurring during measurement is another parameter influencing results from rheological evaluations performed on microsphere suspensions. To avoid sedimentation phenomenon with the large size hydrogel microspheres dispersed in PBS, we have chosen a gap between plates of the measurement cell in which only one layer of microspheres can fit in (i.e., 300 µm). The gap was then slightly smaller than the mean size of the microspheres (microspheres sieved between 315 and 500 µm) assuming that the microspheres formed a reproducible homogenous monolayer of material between the plates of the cell measurement. This gap was also chosen to ensure that only the elasticity of the microspheres will be investigated. It was verified that the measurements were conducted in the linear regime. Thus, the microspheres were reversibly deformed as confirmed by the observations under the optical microscope.

Rheological characterization of microspheres

Shear sweep studies

First, experimental conditions to perform rheological measurements on microspheres in relevant conditions needed to be identified. It was then verified that the material deposited in the measurement cell underwent reversible deformation during shear solicitations. In general, this condition is fulfilled in the linear regime which corresponds to plateau values of G' and G'' shear moduli as a function of the strain. In a first set of experiments, rheological behaviours of non degradable microspheres were monitored under shear sweep to identify the range corresponding to the linear regime. Figure 4 presents the results of the variation of G' and G'' shear moduli for two of the non degradable microspheres used in this study (Figure 4).

Figure 4 Determination of the range of the linear regime under controlled strain in experiment with non-degradable microspheres with crosslinking molar ratio of 2% (A) and 6% (B). G'(open symbols) and G''(closed symbols) moduli were monitored as a function of strain sweeping from 0.001 to 10%. The grey zones correspond to the linear regime and the dotted lines indicate the positions of chosen controlled shear strain value (0.04%) for following experiments.

According to the results obtained from these experiments, G' and G'' moduli were stable between values of the strain ranging from 0.02 to 0.3% indicating the limits of the linear regime for all types of microspheres. The imposed strain, γ, chosen for exploring the rheological behaviour of the microspheres in the oscillatory shear mode used in the next step of the development of the method was 0.04%. It is noteworthy that this value of controlled strain is very low. It insured non-destructive micro-shear preserving microspheres from being damaged during rheological measurements. This was confirmed by a systematic observation of the microspheres by optical microscopy after rheological measurements. This controlled observation performed on the different microspheres after rheological measurements showed that all types of microspheres remained well spherical and did not exhibited any fracture (All data not shown, See for an example Figure 2 C).

Frequency sweep studies

Controlled strain mode with a constant strain at 0.04% was applied to study the viscoelastic characteristics of the non degradable microspheres prepared with crosslinking ratios ranging between 1 and 11 %. Experiments repeated three times on the same sample of microspheres but by refilling the measurement cells with a new amount of sample always gave values in a low range interval (ranging between 5 to 8 %). An example of the curves obtained showing the mean value with standard deviation of the storage (G') and loss (G'') moduli as a function of frequency was plotted in Figure 5A for the microspheres PEGDMA-1%. The two moduli G' and G'' were almost independent on the variation of frequency in the whole range of the frequency sweep. The rheological behaviour showed by the microspheres was typical of that of chemically crosslinked hydrogels. The ratio G''/G' was roughly in the order of 0.1 and appeared rather independent on the frequency. In agreement with this, the phase angle, δ, was nearly constant over the range of frequencies comprised between 0.05 and 40 Hz (Figure 5B). In those conditions, the value of the phase angle was around 8° indicating that the microspheres behaved like a gel material with nearly pure elastic properties.

As shown in Figure 6, the storage modulus, G', of the microspheres was almost independent on frequency whatever their crosslinking density was. The value of the storage modulus taken at a frequency of 10 Hz increased with the degree of crosslinking of the microspheres (Table I, Figure 7). This was also the case considering the loss modulus, G" (Figure 7B). The phase angle deduced from these measurements ranged between 3 to 8° for the different microspheres (Table I). This indicated that all microspheres showed nearly pure elastic properties.

Both G' and G'' increased linearly with the crosslinking molar ratio of the microspheres up to a crosslinking molar ratio of about 7%. Above this value, G' reached a plateau value, while G" kept increasing. The results obtained here can be discussed considering the theory of the rubber elasticity proposed by Flory [START_REF]Principle of polymer chemistry[END_REF] in which the storage modulus, G', depends directly on the number of crosslinks per elastically active chain, υ e , and on a constant characterizing the polymer solvent system, α. The relation appearing between these parameters and the storage modulus is given in equation 5.

RT G e   ' [START_REF] Caldorera-Moore | Micro-and nanotechnologies for intelligent and responsive biomaterial-based medical systems[END_REF] where R is the universal gas constant and T the temperature.

The two parameters α and υ e are related to the characteristics of a given hydrogel. According to the Flory theory, they represent the swelling capacity of the gel. As a consequence, the above relationship demonstrates the dependence between rheological moduli and swelling ratio of hydrogels. It can be pointed out that the Flory theory applied well with the microspheres considered in the present study. Both the storage modulus, G', and the swelling ratio of microspheres with degrees of crosslinking below 7% varied a lot (Figure 3). In contrast, with highly crosslinked microspheres (crosslinked density above 7%), very little differences in swelling capacities were observed and the storage modulus reached a plateau value. According to the above results the storage modulus of microspheres varied a lot with crosslinking molar ratio only with microspheres crosslinked at a maximum ratio of 7%. Considering these microspheres, an almost linear relation was found between the two parameters (Figure 7A). This suggested that degradation of microspheres through hydrolysis of the crosslink bonds can be monitored by measuring the storage modulus of the microspheres during degradation experiments performed on microspheres with an initial crosslinking molar ratio below 7%.

Rheological characterization of the degradation of hydrolysable microspheres

Degradable microspheres were synthesized using a hydrolysable crosslinker. The degradation of these microspheres was believed to take place by hydrolysis of the crosslinking bonds including lactic and glycolic acid segments occurring with contact of water. Thus, it was expected that the crosslinking density of the microspheres will be reduced during the degradation process which in turn could be monitored by measuring the storage modulus of the microspheres over time. It was also expected that the hydrolysis of the microspheres would be accompanied by the release of carboxylic acids in the incubation medium coming from the hydrolysis of the ester bonds included in the lactic and glycolic acid containing segments of the crosslinker. This effect can be monitored by measuring the pH of the incubation medium. Thus, an acidification of the incubation medium would signify that the microspheres degraded through the expected hydrolytic mechanism while a decrease of the storage modulus would indicate a loss of crosslinking bonds in the hydrogel structure of the microsphere.

Rheological measurements were performed in conditions established above on microspheres incubated in PBS. In agreement with the work presented in the first part of this paper, microspheres with a low initial crosslinking density (3%) were selected for this study.

Figure 8 presents the results of the evolution of the pH of the incubation medium of microspheres HEMA4L4G-3% monitored during the experiment. It shows a significant decrease in pH value of the dispersing medium occurring from the start of the experiment up to the 10 th day of the incubation time. The decrease of pH monitored over the first 10 days of the incubation indicated that the microspheres released acidic compounds in the incubation medium. This was in agreement with the hydrolytic mechanism expected from the chemical nature of the hydrogel composing the microspheres. After 10 days, the pH of the incubation medium remained constant indicating that the degradation of the microspheres was stopped.

Figure 8 Variation of pH of the incubation medium during incubation of the microspheres in PBS at 37°C.

Rheological measurements were performed on degradable HEMA4L4G-3% microspheres under the experimental conditions defined above: the oscillatory frequency was set to 10 Hz and the shear strain was set at 0.04% to remain in the linear regimen. Figure 9 presents the results obtained from the measurement of storage modulus, G', of the degradable microspheres HEMA4L4G-3% incubated in PBS over a period of time of 21 days.

Figure 9 Evolution of rheological properties of microspheres HEMA4L4G-3% during incubation in PBS at 37°C. A -Storage (G', open symbols) and loss (G'', closed symbols) moduli. B -Phase angle δ.

Experimental conditions for the measurement were set with an oscillatory frequency at 10 Hz and a shear strain at 0.04%. The lines were drawn to serve as guides for the eyes.

A clear decrease of G' was monitored between the start of the experiment (day 0) and day 10 where G' reached a plateau value. The loss modulus, G'', followed exactly the same tendency with pH over the same period of time. The value of the phase angle, δ, remained constant and close to 8° during the whole period of time of the experiments indicating that the hydrogel remained as an almost purely elastic material.

The fact that the storage modulus of the microspheres was decreased with time during the incubation of the microspheres in PBS was in favour with a decrease of the crosslinking density of the hydrogel forming the microspheres. The time scale in which the modification of the rheological properties of the microspheres occurred correlated well with the time scale during which the pH of the incubation medium dropped down. This last result suggested that the modification of the rheological properties monitored during the incubation of the microspheres in PBS resulted from the degradation of the hydrogel through the hydrolysis of the crosslinking bonds.

CONCLUSION

Conditions to measure rheological properties of microspheres made of a chemically crosslinked hydrogel and remaining as individual microspheres were established. It was determined that relevant and reproducible measurements could be obtained with a rheometer equipped with a parallel plate geometry in which a homogenous and continuous monolayer of microspheres is inserted in the gap between the plates. The storage moduli of the microspheres followed the theory of rubber elasticity developed by Flory. It was affected by the degree of crosslinking of the hydrogel forming the microspheres in the same crosslinking density range than that affected the swelling ratio of the microspheres. Results from this work also showed that degradation of hydrolysable crosslinked microspheres can be monitored by measuring their storage modulus over time by applying the rheological method of characterization of microspheres developed in this work. Further experiments will be conducted on hydrolysable microspheres to investigate the influence of the crosslinking molar ratio and of the nature of the degradable crosslinking bond on both the rheological properties of the microspheres and their degradation rate.

Figure 2 :

 2 Figure 2: Morphological observations by optical microscopy of microspheres PEGDMA-1% in different stages of the study. A -Before sieving. B -After sieving 315-500 µm, before rheological study. C -After sieving 315-500 µm, after rheological study. Scale bar: 400 µm

Figure 3 :

 3 Figure 3: Swelling ratio of microspheres as the function of their crosslinking molar ratio.

Figure 5

 5 Figure 5 Influence of frequency on the rheological behaviour of the microspheres PEGDMA-1%. Measurements were performed with the imposed shear strain at 0.04%. A -The storage (G') and loss (G'') moduli ; B -Phase angle δ.

Figure 6

 6 Figure 6 Determination of the storage modulus, G', of non-degradable microspheres with different crosslinking molar ratio (thick solid black line 1%, thick long dashed black line 2%, thin solid black line 2.5%, thin long dashed black line 3%, thin short dashed black line 4%, dotted black line 5%, solid grey line 6%, long dashed grey line 7%, short dashed grey line 9%, dotted grey line 11%) at the oscillatory frequency of 10 Hz and under an imposed shear strain of 0.04%.

Figure 7

 7 Figure 7 Storage modulus, G' (A), and loss modulus, G'' (B), of hydrogel microspheres with different crosslinking molar ratio.Measurements were performed with an imposed shear strain at 0.04% and at an oscillation frequency of 10Hz. The dashed line is only used as a guide to highlight the linear domains of the curve between the storage modulus and the degree of crosslinking of the microspheres.

Table I

 I Characteristics of microspheres obtained after sieving between 315 and 500 µm.

	Sample name	crosslinking molar ratio (%)	nature of crosslink	D[4,3] (µm)	Span 10 -1	G' at 10 Hz 10 3 (Pa)	δ at 10 Hz (°)
	PEGDMA-1%	1		462 ± 2	3.5 ± 0.1 1.1 ± 0.1	7.9 ± 1.8
	PEGDMA-2%	2		439 ± 2	4.0 ± 0.1	2.8 ± 0.1	5.6 ± 0.4
	PEGDMA-2.5%	2.5		418 ± 3	5.0 ± 0.1 3.8 ± 0.2	4.2 ± 0.2
	PEGDMA-3%	3		419 ± 1	4.6 ± 0.1 5.1 ± 0.7	4.3 ± 0.2
	PEGDMA-4% PEGDMA-5%	4 5	PEGDMA (non degradable)	391 ± 2 389 ± 4	2.1 ± 0.1 8.0 ± 0.4 2.3 ± 0.3 8.7 ± 0.1	3.6 ± 0.6 4.4 ± 0.4
	PEGDMA-6%	6		412 ± 2	4.8 ± 0.1 12 ± 1	4.1 ± 0.3
	PEGDMA-7%	7		423 ± 1	5.0 ± 0.1 15 ± 1	3.0 ± 0.3
	PEGDMA-9%	9		406 ± 2	5.8 ± 0.1 16 ± 1	3.8 ± 0.2
	PEGDMA-11%	11		376 ± 1	2.7 ± 0.2 17 ± 1	4.7 ± 0.6
	HEMA4L4G-3%	3	HEMA4L4G (hydrolysable)	379 ± 2	7.3 ± 0.1 4.8 ± 0.2	7.7 ± 0.7

ACKNOWLEDGEMENTS

The authors would like to thank CNRS and University of Paris Sud. The authors would also like to thank G. MEKHLOUFI for her help in performing and analysing the size of the microspheres.