Mehdi Karbalayghareh
email: mehdi.karbalayghareh@utdallas.edu

Member, IEEE Farshad Miramirkhani
email: far-shad.miramirkhani@isikun.edu.tr

Student Member, IEEE Hossien B Eldeeb
email: hossien.eldeeb@ozu.edu.tr

Member, IEEE Refik Caglar Kizilirmak
email: re-fik.kizilirmak@nu.edu.kz

Senior Member, IEEE Sadiq M Sait

Fellow, IEEE Murat Uysal
email: murat.uysal@ozyegin.edu.tr

Student Member, IEEE Karbalayghareh Mehdi

King Fahd

Channel Modelling and Performance Limits of Vehicular Visible Light Communication Systems

Keywords: ), vehicular communications, ray tracing, single photon avalanche diode

come   L'archive ouverte pluridisciplinaire

everything (V2X) communication solutions enabling vehicleto-infrastructure (V2I) and vehicle-to-vehicle (V2V) connectivity. Research efforts and standardization activities on V2X have been so far centered around radio frequency (RF) technologies [START_REF] Karagiannis | Vehicular networking: A survey and tutorial on requirements, architectures, challenges, standards and solutions[END_REF][START_REF] Araniti | LTE for vehicular networking: A survey[END_REF][START_REF] Zheng | Heterogeneous vehicular networking: a survey on architecture, challenges, and solutions[END_REF]. In 2010, as an amendment of the IEEE 802.11 WiFi standard, IEEE 802.11p standard was introduced to support V2X communications in the 5.9 GHz band allocated for ITS applications [START_REF] Perallos | Intelligent Transport Systems: Technologies and Applications[END_REF]. In 2017, cellular-based V2V connectivity solution, known as LTE-V, was introduced as a part of 3GPP Release 14 [START_REF] Molina-Masegosa | LTE-V for sidelink 5G V2X vehicular communications: A new 5G technology for short-range vehicleto-everything communications[END_REF].

Since the current market penetration of V2X solutions is relatively low, the allocated RF bands are considered sufficient at the time being. However, in the near future, high interference levels can be experienced in limited RF bands particularly in high-density traffic scenarios. Channel congestion will result in longer delays and degrade the packet rate. To address such issues, visible light communications (VLC) has been proposed as an alternative vehicular access solution to RF-based V2X communications [START_REF] Yamazato | Image-sensor-based visible light communication for automotive applications[END_REF][START_REF] Uysal | Visible light communication for vehicular networking: Performance study of a V2V system using a measured headlamp beam pattern model[END_REF][START_REF] Cailean | Impact of IEEE 802.15.7 standard on visible light communications usage in automotive applications[END_REF]. VLC takes advantage of widely available light emitting diodes (LEDs) and utilizes them as wireless transmitters in addition to their primary illumination function. Since automotive manufacturers are increasingly using LED-based exterior lighting, front and back vehicle lights can serve as wireless transmitters making VLC a natural vehicular connectivity solution. In addition to interference immunity, VLC has the potential to achieve very high data rates on the order of Gbps with off-the-shelf LEDs [START_REF] Zhang | Gb/s real-time visible light communication system based on white LEDs using T-Bridge cascaded pre-equalization circuit[END_REF] or even more with custom design LEDs [START_REF] Ijaz | Experimental proof-of-concept of optical spatial modulation OFDM using micro LEDs[END_REF]. With such attractive features, vehicular VLC has received an increasing attention lately and several topics such as physical layer design, upper layer network protocols and integration with RF-based solutions for hybrid systems have been studied in the recent literature, see the survey in [START_REF] Cailean | Current challenges for visible light communications usage in vehicle applications: A survey[END_REF] and the references therein.

As in any other communication system, the propagation channel dictates the fundamental performance limits on vehicular VLC systems. Extensive studies of indoor VLC channel modelling have been already reported [START_REF] Miramirkhani | Channel modeling and characterization for visible light communications[END_REF][START_REF] Uysal | IEEE 802.15.7r1 reference channel models for visible light communications[END_REF]. However, operation in outdoor medium brings additional challenges and requires the development of dedicated vehicular VLC channels models. Initial works on vehicular VLC [START_REF] Akanegawa | Basic study on traffic information system using LED traffic lights[END_REF][START_REF] Kumar | Visible light communication for intelligent transportation in road safety applications[END_REF] assume the line-of-sight (LOS) channel model. While this model is suitable for the indoor LED luminaires with Lambertian pattern, it fails to capture the illumination characteristics of automotive low-beam and high-beam headlamps with asymmetrical intensity distributions. To address such shortcomings, a piecewise Lambertian channel model was proposed in [START_REF] Viriyasitavat | Short paper: Channel model for visible light communications using off-the-shelf scooter taillight[END_REF]. Intensity patterns for headlights were measured in experimental works [START_REF] Lee | Evaluation of visible light communication channel delay profiles for automotive applications[END_REF][START_REF] Luo | Fundamental analysis of a car to car visible light communication system[END_REF][START_REF] Luo | Performance analysis of a car-to-car visible light communication system[END_REF] and used in the performance analysis of V2V VLC systems. The work in [START_REF] Memedi | Impact of realistic light radiation pattern on vehicular visible light communication[END_REF] further presented a closed-form path loss formula as a function of distance based on data fitting to measurement results. This formula is however limited to clear weather conditions. Furthermore, since it is obtained based on data fitting, the relation of system and channel parameters to path loss is not determined.

A critical factor that needs to be further taken into account in vehicular VLC channel modelling is the road surface. In [START_REF] Lee | Evaluation of visible light communication channel delay profiles for automotive applications[END_REF], Lee et al. used Monte Carlo ray tracing and obtained vehicular VLC channel delay profiles for an asphalt road. In [START_REF] Luo | Fundamental analysis of a car to car visible light communication system[END_REF] and [START_REF] Luo | Performance analysis of a car-to-car visible light communication system[END_REF], Luo et al. developed a channel model for V2V transmissions up to 70 m over dry and wet asphalt. These works however assume fixed reflectance values for the vehicle coating (paint) and road surface. Such an assumption can be justified for infrared wavelengths; however, in the visible light band, reflectance is wavelength-dependent and this should be considered for a realistic channel modelling.

Another critical factor in the development of vehicular VLC channels is the impact of weather conditions. While there are earlier works on infrared channel modelling in the presence of different weather conditions [START_REF] Schulz | Long-term outdoor measurements using a rate-adaptive hybrid optical wireless/60 GHz link over 100 m[END_REF], these are not applicable to vehicular VLC links due to different operation wavelength. In an effort to address this, the effect of artificially generated rain and fog on the received optical signal was quantified for a red coloured taillight LED using a laboratory chamber in [START_REF] Kim | Experimental demonstration of LED-based vehicle to vehicle communication under atmospheric turbulence[END_REF] and [START_REF] Kim | Experimental demonstration of VLC-based vehicle-to-vehicle communications under fog conditions[END_REF]. In [START_REF] Elamassie | Effect of fog and rain on the performance of vehicular visible light communications[END_REF], a V2V link with a high-beam headlamp as the transmitter was considered and a path loss expression was developed for clear, rain and fog conditions. This is however valid only for short link ranges up to 20 m. In addition to that, it is obtained via data fitting; therefore, the dependence of the expression on system parameters is not clear.

In the light of above literature survey, it is obvious that there is a need for the development of a comprehensive channel model simultaneously taking into account all inherent characteristics of vehicular VLC, i.e., asymmetrical radiation pattern of headlamps, weather effects, and effect from road reflections. In an effort to address this research gap, we present a closedform path loss expression for vehicular VLC as a function of system and channel parameters. Our proposed expression builds upon the summation of geometrical loss and attenuation loss and takes into account asymmetrical patterns of light sources and geometry of V2V transmission. It is an explicit function of link distance, lateral shift between two vehicles, weather type (quantified by the extinction coefficient), transmitter beam divergence angle and receiver aperture diameter. Based on extensive non-sequential ray tracing simulations, we obtain the channel impulse responses (CIRs) for a given vehicular scenario in various weather conditions, i.e., clear, rainy and foggy, and determine correction coefficients for each weather type. In the second part of the paper, we use this path loss expression to analyze the performance of a vehicular VLC system. To enable long range transmission, a single photon avalanche diode (SPAD) is employed as the receiver. We assume that the output photon number of SPAD receiver is modelled with Poisson statistics and determine the maximum achievable link distance to ensure a pre-defined value of bit error rate (BER).

The rest of the paper is organized as follows. In Section II, we develop the path loss expression for vehicular VLC and present numerical and simulation results to demonstrate the effect of weather conditions and system parameters on the path loss. In Section III, we investigate the maximum transmission range to satisfy a targeted BER value for the V2V VLC system under consideration in various weather conditions. In Section IV, we present system performance results and discussions. Finally, we provide concluding remarks in Section V.

II. VEHICULAR VLC PATH LOSS MODEL

In this section, we first propose a closed-form path loss expression, then validate it through ray tracing simulations.

A. Path Loss Expression

We consider a single-lane road with a width of w r . We assume that two vehicles follow each other in the same lane with a possible lateral shift of d h . Each vehicle has a width of w c and they are separated from each other with a longitudinal distance of d. The two high-beam headlamps (denoted by TX1 and TX2) are adopted as wireless transmitters which are horizontally separated from each other with a distance of d tx . A single photodetector (denoted by RX), placed at the center of the back of the other vehicle, acts as a wireless receiver. It has an aperture diameter of D R and a field of view angle of FOV. L i is the propagation distance from the ith transmitter, i = 1, 2, to the receiver. The irradiance angle of the ith transmitter is given by θ i while the incident angle is defined by φ i .

The path loss consists of two main components, i.e, attenuation loss and geometrical loss. Attenuation loss is a result of scattering and absorption while the geometrical loss [START_REF] Poliak | Link budget for highspeed short-distance wireless optical link[END_REF] is a result of the fact that the transmitted beam spreads to a size larger than the receive aperture. The geometrical loss between the ith transmitter and the receiver can be expressed as

P L geo,i = 10log 10 D R (cos (θ i )) 1/ε ζL i 2 i = 1, 2 (1)
where we introduced two correction coefficients (ζ and ε) to take into account weather conditions and asymmetrical pattern of headlamp. The values of these correction coefficients are determined through ray tracing simulation study which will be discussed later.

The well-known Beer-Lambert formula defines the attenuation loss [START_REF] Uysal | Optical wireless communications: an emerging technology[END_REF]Chapter 3] and builds upon the implicit assumption that scattered photons cannot be captured by the receiver. In order to take into factor that some of the scattered rays could be received by the photodetector after some reflections, this can be modified as

P L att,i = 10log 10 exp -cL i D R ζL i ε 2 (2)
where c stands for the extinction coefficient for a specific weather type and an additional term is introduced in the negative exponential that is proportional to the geometrical propagation of the light source [START_REF] Elamassie | Performance characterization of underwater visible light communication[END_REF]. Based on ( 1) and ( 2), the path loss for the link between the ith transmitter and the receiver can be expressed as

TX1 TX2 RX d tx θ 1 ɸ 1 d h d L 1 d h 1 w c w r ɸ 2 d h 2 θ 2 L 2
P L i = 10log 10   D R (cos (θ i )) 1/ε ζL i 2 exp -cL i D R ζL i ε 2   (3)
From the geometry of layout in Fig. 1, it can be readily checked that

L i = d 2 + d 2 hi where d hi = d h ± d tx /2 and θ i = arccos (d/L i ).
Noting the use of dual transmitters and replacing L i and θ i within (3), we can obtain the overall path loss as (4).

B. Channel Modeling Approach

In the following, we conduct a simulation study in Zemax ® to validate the proposed path loss expression and determine the values of correction coefficients for the vehicular channels under consideration. Our channel modelling approach builds on non-sequential ray tracing features of Zemax ® which allows rays to propagate through the environment in any order and allows rays to be scattered and reflected back to an object that they have already encountered. This property makes nonsequential mode ideal for impulse response modelling. This method was originally adopted to model indoor VLC channels [START_REF] Miramirkhani | Channel modeling and characterization for visible light communications[END_REF][START_REF] Uysal | IEEE 802.15.7r1 reference channel models for visible light communications[END_REF] and also applied to the modelling of underwater VLC channels [START_REF] Elamassie | Performance characterization of underwater visible light communication[END_REF].

First, a 3D model of vehicular scenario (including vehicles, road etc) is constructed in Zemax ® . As inputs, we provide the specifications of vehicular light sources acting as transmitters (i.e., spectral power distribution and intensity distribution) and photodetectors placed on the vehicles for reception (i.e., location on the vehicle, aperture size and field of view). We further specify the optical characteristics of the vehicle coating and road surface by defining their wavelength-dependent reflectance values. To reflect the effects of weather conditions, we need to define the atmospheric propagation medium. Zemax ® built-in function "Bulk Scatter" allows providing the refractive index, radius and density of particles as input parameters based on Mie model [26, Chapter After simulation environment is created, non-sequential ray tracing is run to determine the CIR. Zemax ® generates an output file including the received power and path length from source to detector for each ray. These are then imported to Matlab ® to construct the CIR. Assume that N i is the number of rays received at the detector from the ith transmitter. Let P i,j and τ i,j respectively denote the power and propagation time of the jth ray, j = 1, ..., N i received from the ith transmitter. Then, the CIR can be constructed as

P L = 10log 10     1 2 2 i=1        D R (d/ d 2 + d hi 2 ) 1/ε ζ d 2 + d hi 2    2 exp   -c d 2 + d hi 2   D R ζ d 2 + d hi 2   ε/2            (4)
h sim (t) = 2 i=1
Ni j=1 P i,j δ (t -τ i,j ) where δ (t) is the Dirac delta function. The path loss can be calculated as [START_REF] Miramirkhani | A mobile channel model for VLC and application to adaptive system design[END_REF]

P L sim = 10log 10   ∞ 0 h sim (t) dt   (5)

C. Simulation Study

In our simulation study, we construct the 3D model of vehicular scenario in Fig. 1. CAD objects of vehicles are imported and processed in Zemax ® "Part Designer" which is an interface that allows to create and manipulate user-defined 3D geometries. The surface coating for two vehicles is chosen as black gloss paint.

As the high-beam headlamp of the vehicle, a commercially available white light LED (Philips Luxeon Rebel) is used [29]. This headlamp is modelled in the simulation platform by defining the relative spectral power distribution of the LED (see Fig. 2.a) and the photometric data which contains the luminous intensity in all different planes. Specifically, the photometric data is imported as an .IES file to the software. As an example, different cross sections are shown in Fig. 2.b to demonstrate the asymmetrical intensity pattern of headlamp. Blue curve shows the intensity distribution of headlamp when one looks from side (e.g., from pavement) while the red one shows the same pattern when one looks downward from top. The green curve shows the pattern when one looks downward at a 45 degrees angle. These planes are obtained through "source viewer" feature of Zemax ® which provides the directivity plots of the transmitter in different scan angles. In the simulation study, right and left headlamps with the above characteristics are placed in the front side of the first vehicle as the transmitters. A total unit (normalized) power is assumed in the simulations and the CIRs are then scaled for a given transmit power. A photodetector is placed at the back (bumper) of the other vehicle and serves as the receiver. The receiver has an aperture size of D R = 1 cm and field of view (FOV) of 180 •1 . 1 It is possible to improve signal-to-noise ratio (SNR) by narrowing the receiver FOV through the use of a proper lens. However, a narrow FOV might be problematic for mobile conditions as encountered in vehicular VLC. Therefore, we prefer a large FOV to maximize reception angle. As road type, we consider categorizations defined by the International Commission on Illumination (CIE) [START_REF] Stark | Road surface's reflectance influences lighting design[END_REF] and assume R2 type road. This defines an asphalt road consisting of 60% gravel sized larger than 10 mm. To simulate clear and foggy weathers, mixed specular and diffuse reflections are assumed while mostly specular reflections are considered for the wet road surface in rain [START_REF] Luo | Performance analysis of a car-to-car visible light communication system[END_REF].

In our simulation study, it is assumed that there is an initial separation of longitudinal distance meters between two cars. For a range of 100 meters, we obtain the CIRs through all points with 5 meters inter-distance. The simulations are conducted for clear weather, rain, moderate fog (i.e., visibility of 500 m) [START_REF] Kim | Comparison of laser beam propagation at 785 nm and 1550 nm in fog and haze for optical wireless communications[END_REF] and thick fog (i.e., visibility of 250 m) [START_REF] Kim | Comparison of laser beam propagation at 785 nm and 1550 nm in fog and haze for optical wireless communications[END_REF].

In the following, we present the proposed path loss expression versus distance for the weather conditions under consideration along with simulation results. Corresponding ζ and ε values are summarized in Table II. In addition to simulations, we include the following three benchmarks. 1) Path loss calculated based on the assumption of Lambertian source and Beer-Lambert formula: As earlier discussed, some initial works on vehicular VLC consider Lambertian source model [START_REF] Akanegawa | Basic study on traffic information system using LED traffic lights[END_REF][START_REF] Kumar | Visible light communication for intelligent transportation in road safety applications[END_REF] and Beer-lambert formula. Under these assumptions and noting that we have two light sources, we can write the path loss as [START_REF] Yamazato | Image-sensor-based visible light communication for automotive applications[END_REF] [26, Chapter 3], [START_REF] Lee | Indoor channel characteristics for visible light communications[END_REF] where m = -1/log 2 cos Φ 1/2 is the order of Lambertian emission with Φ 1/2 denoting the semi-angle of the LED. In [START_REF] Yamazato | Image-sensor-based visible light communication for automotive applications[END_REF], the first term denotes geometrical loss under the assumption of Lambertian source and the second term is the attenuation loss based on Beer-Lambert formula. 2) Path loss calculated based on the assumption of asymmetrical source and Beer Lambert formula: Combining (1) and the second term of (6), we have [START_REF] Uysal | Visible light communication for vehicular networking: Performance study of a V2V system using a measured headlamp beam pattern model[END_REF]. 3) Path loss formula introduced in [START_REF] Memedi | Impact of realistic light radiation pattern on vehicular visible light communication[END_REF] based on the curve fitting to the empirical data: This formula is given by

P L = α + 10βlog 10 1 d + γ (8)
where the coefficients α, β and γ are found via data fitting to measurement results. The experiment was conducted under clear weather conditions at night. In the experimental set-up, two vehicles follow each other in the same lane where low-beam headlamp is used as the transmitter and a receiver aperture area of is 9.8 mm × 9.8 mm (i.e., approximately 1 cm 2 ) is assumed. Clear Weather, D R =1 cm Simulation, Eq. ( 5) Lamb. Source+BL, Eq. ( 6) Asym. Source+BL, Eq. ( 7) Empirical Model, Eq. ( 8) Proposed Rain, D R =1 cm Simulation, Eq. ( 5) Lamb. Source+BL, Eq. ( 6) Asym. Source+BL, Eq. ( 7) Proposed Thick Fog, D R =1 cm Simulation, Eq. ( 5) Lamb. Source+BL, Eq. ( 6) Asym. Source+BL, Eq. ( 7) Proposed The comparison of our results with two first benchmarks will be instrumental to have insight into the contribution of asymmetrical radiation pattern to the overall path loss. The third benchmark will let us to demonstrate the accuracy of simulations through comparison with experimental measurements.

P L = 10log 10   1 2 2 i=1     πD 2 R 4 (m + 1) 2π d 2 + d 2 hi cos m (θ i ) cos (φ i )   exp -c d 2 + d 2 hi     (6)
P L = 10log 10   1 2 2 i=1   D R (cos (θ i )) 1/ε ζL i 2 exp -c d 2 + d 2 hi     (7
In Fig. 3, we present our path loss model in (4) versus distance along with simulation results and the above three benchmarks. We assume receiver aperture diameter of D R = 1 cm. First of all, it can be verified from Fig. 3.a that our proposed path loss expression in (4) maintains a good fit with empirical model in (8) which was developed only for clear weather conditions. The deviation in small distances is due to this fact that the empirical model was obtained for lowbeam headlamp while our simulation study is based on highbeam headlamp. It should be noted that the intensity pattern of low-beam headlamp is more directed towards the road surface compared to the high-beam headlamp and hence the path loss for small distances is more pronounced in the empirical model. Path Loss (dB) Path Loss (dB) Path Loss (dB)

Clear Weather D R =1 cm D R =3 cm D R =5 cm D R cm D R =15 cm Proposed (a)
Rain D R =1 cm D R =3 cm D R =5 cm D R =10 cm D R =15 cm Proposed (b)
Moderate Fog Path Loss (dB)

D R =1 cm D R =3 cm D R =5 cm D R =10 cm D R =15 cm Proposed (c)
Thick Fog D R =1 cm D R =3 cm D R =5 cm D R =10 cm D R =15 cm Proposed (d)
Fig. 4: Proposed path loss expression for different receiver aperture diameters under different weather conditions (4) has a "good fit" 2 with simulation results for all weather conditions. In contrary, the benchmarking plots based on ( 6) and ( 7) exhibit some deviations. Particularly, the path loss by Lambertian source is a significant overestimation over the actual path loss compared to an asymmetrical headlight, i.e., see the plot based on [START_REF] Yamazato | Image-sensor-based visible light communication for automotive applications[END_REF]. It is further observed that Beer-Lambert formula is a good fit with simulation results only for clear and rainy weather conditions and deviates for foggy conditions, i.e., see the plot based on [START_REF] Uysal | Visible light communication for vehicular networking: Performance study of a V2V system using a measured headlamp beam pattern model[END_REF]. Since the extinction coefficients of clear and rainy weathers are very small (i.e., c ≈ 0 m -1 ), the scattered rays can be ignored making Beer-Lambert formula valid for this case. In foggy weather conditions, the effect of scattering is significant. In this case, Beer-Lambert formula has a significant deviation from simulations. This deviation increases when the visibility decreases.

In Fig. 4, we investigate the validity of our proposed expression in (4) by varying the receiver aperture diameter. We assume D R = 1 cm, 3 cm, 5 cm, 10 cm and 15 cm. It can be verified that the path loss expression maintains a good fit for different values of receiver aperture diameters under various weather conditions. In Fig. 5, we consider different lateral shifts of d h = 1 m and 1.5 m assuming clear weather conditions and D R = 5 cm. It is observed that the lateral shift brings negligible loss with respect to perfect alignment case of d h = 0 m. 2 As a measure of data fitting accuracy, we use "R-squared (R 2 )" [START_REF] Dangeti | Statistics for machine learning[END_REF] which is said to be a "good fit" when R 2 exceeds 0.95 [START_REF] Tang | Impulse response modeling for underwater wireless optical communication links[END_REF].

III. MAXIMUM ACHIEVABLE DISTANCE

In this section, we investigate the error rate performance of vehicular VLC links and determine the achievable transmission distances in various weather conditions. We consider the use of SPAD which is known with their superior performance to detect weak signals [START_REF] Li | Nonlinear distortion in SPAD-based optical OFDM systems[END_REF][START_REF] Li | Optical OFDM with single-photon avalanche diode[END_REF]. SPADs were considered as VLC receivers in the literature for harsh propagation envi-0018-9545 (c) 2020 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TVT.2020.2993294, IEEE Transactions on Vehicular Technology ronments such as underwater [START_REF] Wang | A long distance underwater visible light communication system with single photon avalanche diode[END_REF][START_REF] Shafique | Performance analysis of single-photon avalanche diode underwater VLC system using ARQ[END_REF][START_REF] Wang | Blind detection for SPADbased underwater VLC system under P-G mixed noise model[END_REF] and gas pipelines [START_REF] Li | Single photon avalanche diode (SPAD) VLC system and application to downhole monitoring[END_REF]. However, their use was not investigated before in the context of vehicular VLC systems. Most practical VLC systems use intensity modulation and direct detection. As such, we assume that unipolar on-off keying (OOK) is employed with an average optical power of P 0 and bit time duration of T b . The power is set as P t = 0 and P t = 2P 0 , respectively for transmitting zeros and ones. Let N SP AD and C F F denote respectively the size and the fill factor (FF) of the SPAD array. The number of received photons is given by [START_REF] Karbalayghareh | Vehicular visible light communications with SPAD receivers[END_REF]

z = N SP AD (C F F ηP t h + C F F N b + N DCR ) T b ( 9
)
where h is the channel coefficient, N DCR is the dark count ratio, N b is the background light noise ratio and η = C P DE λ/h Planck v. Here, C P DE is the photon detection efficiency, λ is the wavelength, h Planck is the Planck's constant and υ is the light speed. Since we are interested in finding the maximum achievable distance, we can impose the assumption of d d hi . Therefore, (4) can be simplified as

P L = 10log 10      D R ζd 2 exp -cd D R ζd ε/2 h      (10)
Poisson distribution is widely used to statistically model the SPAD receiver. The probability mass function of the output photon numbers is given by Pr (z) = µ z e -µ /z! where µ = E (z) denotes the mean. The corresponding received photon numbers for the transmission of zeros and ones are then calculated as µ 0 = N SP AD (C F F N b + N DCR ) T b and µ 1 = N SP AD C F F ηP t hT b + µ 0 , respectively. For equally likely message signals, it can be shown that the optimum detection threshold is z th = (µ 1 -µ 0 )/ln (µ 1 /µ 0 ). BER for OOK under Poisson distribution can be then written as [START_REF] Karbalayghareh | Vehicular visible light communications with SPAD receivers[END_REF][START_REF] Sarbazi | Statistical modeling of singlephoton avalanche diode receivers for optical wireless communications[END_REF]]

P e = 1 2 ∞ z=z th +1 µ z 0 z! e -µ0 + 1 2 z th z=0 µ z 1 z! e -µ1 . (11)
Using Gaussian approximation [START_REF] Sarbazi | Photon detection characteristics and error performance of SPAD array optical receivers[END_REF], the BER expression can be expressed as

P e ≈ Q µ 1 -µ 0 √ µ 1 + √ µ 0 . (12)
In the following, we calculate the maximum achievable distance to satisfy a targeted BER. First, we need to solve h in [START_REF] Ijaz | Experimental proof-of-concept of optical spatial modulation OFDM using micro LEDs[END_REF] for d. After some mathematical manipulations on h, we obtain

c 4 (2 -ε) D R ζ ε/2 d ( 2-ε 2 ) e c 4 (2-ε) D R ζ ε/2 d ( 2 ) = c 4 (2 -ε) D R ζ ε/2 hζ 2 D 2 R ( ε-2 4 ) (13)
Noticing that the left hand side (LHS) of ( 13) is in the form of xe x , we can find its inverse by the Lambert-W function [START_REF] Shynk | Mathematical foundations for linear circuits and systems in engineering[END_REF].

Let W (xe x ) = x denote the Lambert-W function. From ( 13), we obtain d as

d =    W c 4 (2 -ε) D R ζ h ( ε-2 4 ) c 4 (2 -ε) D R ζ ε/2    2 2-ε (14)
Now, assume that P e the value of targeted BER. To solve [START_REF] Miramirkhani | Channel modeling and characterization for visible light communications[END_REF] for h in [START_REF] Ijaz | Experimental proof-of-concept of optical spatial modulation OFDM using micro LEDs[END_REF], we first solve (12) for 1 . By taking inverse Q-function from both sides of (12), we have

Q -1 P e ≈ µ 1 -µ 0 √ µ 1 + √ µ 0 . (15)
By re-arranging (15), we have

µ 1 -Q -1 P e √ µ 1 -Q -1 P e √ µ 0 + µ 0 ≈ 0 . (16)
Then, µ 1 can be obtained as

µ 1 ≈   Q -1 P e + Q -1 P e 2 +4 Q -1 P e √ µ 0 + µ 0 2   2 . (17) Based on (17)
and noting that µ 1 = N SP AD C F F ηP t hT b + µ 0 , we have [START_REF] Luo | Fundamental analysis of a car to car visible light communication system[END_REF]. By rearranging [START_REF] Luo | Fundamental analysis of a car to car visible light communication system[END_REF], h can be obtained as [START_REF] Luo | Performance analysis of a car-to-car visible light communication system[END_REF]. By replacing [START_REF] Luo | Performance analysis of a car-to-car visible light communication system[END_REF] in ( 14), the maximum achievable link distance to achieve P e can be obtained as [START_REF] Memedi | Impact of realistic light radiation pattern on vehicular visible light communication[END_REF].

IV. PERFORMANCE RESULTS AND DISCUSSIONS

In this section, we present performance results of the V2V VLC system under different weather conditions in terms of achievable distance and also investigate the effect of system parameters on the performance. Unless otherwise stated, we assume average optical power of P 0 = -50 dBm [START_REF] Li | Nonlinear distortion in SPAD-based optical OFDM systems[END_REF], receiver aperture diameter of D R = 5 cm, fill factor of C F F = 0.5 [START_REF] Fisher | A reconfigurable singlephoton-counting integrating receiver for optical communications[END_REF], array size of N SP AD = 8×8 [START_REF] Sarbazi | Photon detection characteristics and error performance of SPAD array optical receivers[END_REF] and N DCR = 7.27 kHz [START_REF] Fisher | A reconfigurable singlephoton-counting integrating receiver for optical communications[END_REF]. The simulation parameters are summarized in Table III. Since white LED headlamps are used in vehicular applications, η in ( 9) is calculated based on an average over the wide spectrum of visible band (400 nm-700 nm).

In Fig. 6, we illustrate the maximum achievable distances assuming clear weather, rain and fog for the above system configuration parameters. It is observed that a transmission range of 34.15 m is possible for clear weather at P e = 10 -6 . This slightly reduces to 33.08 m in rain. On the other hand, for foggy condition, the achievable distances are 32.12 m and 30.01 m for moderate fog and thick fog, respectively.

In Fig. 7, we investigate the effect of receiver aperture size on the maximum achievable V2V distance under the assumption of thick fog. As observed from this figure, the transmission range increases with the increase in receiver aperture diameter. This is expected because a receiver with larger aperture is able to collect more energy. Mathematically speaking, the maximum achievable distance at P e = 10 -6 is 6.39 m for D R = 1 cm. This increases to [START_REF] Luo | Fundamental analysis of a car to car visible light communication system[END_REF] should be maintained between two vehicles based on 2-second rule assuming a travelling speed of 50 km/h [46]. Therefore, the aperture size of D R = 5 cm is selected in the following. In Fig. 8, we investigate the effect of background light noise ratio on the maximum achievable V2V distance. Here, we consider thick fog and vary background light noise ratio from much smaller than dark counts (i.e., at night with N b = 0 Hz) to much bigger than dark counts during daylight (i.e., N b = 100 kHz). It is observed that as background light noise ratio increases, the maximum achievable distance decreases. For example, under the assumption of N b = 0 Hz and P e = 10 -6 , a V2V distance of 30.01 m is achievable. This reduces to 28.82 m and 25.04 m for N b = 10 kHz and N b = 100 kHz, respectively.

N SP AD C F F ηP t hT b + µ 0 ≈   Q -1 P e + Q -1 P e 2 + 4 Q -1 P e √ µ 0 + µ 0 2   2 . ( 18
) h ≈ 1 N SP AD C F F ηP t T b 1 4 Q -1 P e + Q -1 P e 2 + 4 Q -1 P e √ µ 0 + µ 0 2 -µ 0 . ( 19
)
d max ≈          W   c 4 (2 -ε) D R ζ 1 N SP AD C F F ηPtT b 1 4 Q -1 P e + Q -1 P e 2 +4 Q -1 P e √ µ 0 +µ 0 2 -µ 0 ( ε-2 4 )   c 4 (2 -ε) D R ζ ε/2          2 2-ε . ( 20
)
In Fig. 9, we investigate the effect of SPAD parameters, i.e., fill factor and array size, on the maximum achievable V2V distance under the assumption of thick fog. It is observed that as fill factor and array size increase, the transmission range increases. For N SP AD = 8 × 8 and P e = 10 -6 , a distance of 30.01 m is achievable assuming a fill factor of C F F = 0.5. This increases to 33.25 m and 39.86 m for C F F = 0.64 [START_REF] Sarbazi | Photon detection characteristics and error performance of SPAD array optical receivers[END_REF] and C F F = 1 (i.e., hypothetical case where the total array area is assumed to be active), respectively.

V. CONCLUSIONS

In this paper, we characterized the path loss of V2V VLC system as a function of weather type and system parameters. The proposed path loss expression takes into account the possibility of receiving scattered photons and asymmetrical intensity distribution of headlamps. We further used this path loss model to derive a closed-form expression for the maximum achievable V2V distance. We considered the deployment of SPADs as receivers where the output photon number is modeled with Poisson statistics. We derived the maximum

Fig. 1 :Fig. 2 :

 12 Fig. 1: Vehicle-to-vehicle scenario

Fig. 3 :

 3 Fig. 3: Proposed path loss expression and comparison with simulation results and other benchmarks. Different weather conditions are assumed while the receiver aperture size is set as D R = 1 cm

Fig. 3

 3 Fig.3further reveals out that the proposed expression in

Fig. 5 :

 5 Fig. 5: Comparison of proposed path loss expression with simulation results for different lateral shifts

  .4 m, 30.01 m, 55.87 m and 77.64 m for D R = 3 cm, D R = 5 cm, D R = 10 cm and D R = 15 cm, respectively. According to EU Member States regulations, a safe trailing distance of 28 m 0018-9545 (c) 2020 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TVT.2020.2993294, IEEE Transactions on Vehicular Technology IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY 8

Fig. 7 :Fig. 8 :Fig. 9 :

 789 Fig. 7: Maximum achievable distance assuming different receiver aperture diameters

TABLE I :

 I Characteristics of various weather types[START_REF] Uysal | Optical wireless communications: an emerging technology[END_REF] Chapter 3]

		Particle Index	Size (µm)	Density (cm -3 )	Extinction Coefficient (m -1 )
	Clear Weather	1.000277	10 -4	10 19	≈ 0
	Rain	1.33	100	0.1	≈ 0
	Moderate Fog	1.33	10	12.45	0.00782
	Thick Fog	1.33	10	24.91	0.01565
	3]. The associated values for various weather types can be
	found in Table I [26, Chapter 3].

TABLE II :

 II Correction coefficients for different weather conditions

		ζ (rad)	ε
	Clear Weather	0.1585	0.0175
	Rain	0.1598	0.0174
	Moderate Fog	0.1600	0.0172
	Thick Fog	0.1550	0.0170

TABLE III :

 III Simulation parameters[START_REF] Fisher | A reconfigurable singlephoton-counting integrating receiver for optical communications[END_REF]

	Parameters	Values
	Wavelength of light (λ)	400 nm-700 nm
	Speed of light (ν)	3 × 10 8 m/s
	Bit time duration (T b )	1 µs
	The PDE of the SPAD (C P DE )	20%
	The DCR of the SPAD (N DCR )	7.27 kHz
	The FF of the SPAD (C F F )	0.5
	Size of the SPAD (N SP AD )	8×8

  This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TVT.2020.2993294, IEEE Transactions on Vehicular Technology

		45
			Clear Weather
			Rain
			Moderate Fog
	Maximum Achievable Distance (m)	25 30 35 40	Thick Fog
		10 -9 20	10 -8	10 -7	10 -6	10 -5	10 -4
				P e	′
	Fig. 6: Maximum achievable distance under different weather conditions
	achievable V2V distance to satisfy a targeted BER value.
	Our results demonstrated that, with proper selection of system
	parameters, VLC can provide a reliable connectivity solution
	for V2V links.

0018-9545 (c) 2020 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

Authorized licensed use limited to: Ozyegin Universitesi. Downloaded on May 16,2020 at 00:53:56 UTC from IEEE Xplore. Restrictions apply.

H. B. Eldeeb was supported by the Horizon 2020 MSC ITN (VISION) under Grant 764461. Sadiq M. Sait would like to acknowledge KFUPM for support. The work of M. Uysal was supported by the Turkish Scientific and Research Council (TUBITAK) under Grant 215E311.

M.


E. Rigot and C. Vieu LAAS-CNRS, Université de Toulouse, CNRS, INSA, Toulouse, France Tissue Engineering 1 is nowadays an expanding field of research that cleverly combines living cells with supporting materials, often called biomaterials, in order to generate a functional tissue for a human organ (skin, brain, bone ….). The purpose of this activity is twofold. On the one hand, successful tissue engineering can lead to human models of tissues of very high predictive value for testing drug activity or toxicity. This could reduce drastically the use or abuse of animal testing in medical research and pharmacology. On the other hand, the tissue elaborated by engineering could be employed as an implant, that once surgically positioned in a deficient or destroyed organ inside the host is capable of regenerating the lost functions by interaction with the surrounding tissues. In this context of regenerative medicine, attempts to regenerate brain tissues, bones, cardiac muscles after trauma or stroke or to regenerate brown adipose tissues for treating obesity are the object of numerous research articles.

In this contribution we describe this specific engineering activity with a focus on the role and the function played by the material (in this case a biomaterial) which is very original compared to other fields of material science. This specificity originates from the direct interaction of the inert material with living entities (cells) that are subjected to time evolution.

The purpose is to conceive and design the right biomaterial (structure, composition, shape) with a view to monitor the transition of the so-called bioartefact (biomaterial + living cells) towards a physiological functional tissue. To this end, we need to consider its surface chemistry, its micro-scale and nano-scale topography, its 3D shape (because physiological cell microenvironments are 3D) but also its ability to co-evolve with cells along different processes such as differentiation, proliferation, colonization, vascularization... In brief, we claim in this article that the right biomaterial for tissue engineering is both a problem of material engineering and time engineering. The state of the art of this domain reveals that the material engineering aspect has been widely documented 2 . However, the consideration of the agency of time has been

Context and introduction to the question of time in tissue engineering

A functional physiological tissue is a complex mixture of living cells of different types that have attained a degree of stability often referred to in biology as homeostasis. Overtime, the cells have constructed a stable and operational architecture in adaptation to environmental conditions. This equilibrium is not a steady state but a perpetual evolution in search of a possible operational point, taking into account the environmental changes and the intrinsic time constraints of the basic mechanisms of life linked to gene expression, cell growth, cell division, cell death … At any time, we can claim that the living tissue invents a new configuration. This is adaptation. That does not mean that this new configuration is optimal in the sense of the engineer, but it represents a good fit. A key mechanism for maintaining some stability in this continuous process is linked to the various secretions of the cells that drive the collective architecture of the tissue by inducing numerous feed-back loops. All together these regulation loops monitor nutrition, waste evacuation, mechanical stability, mechanical actuations (for contractile tissues) and various functions linked to the specific tissue under consideration.

Among the secreted materials of importance are the macromolecules making the so-called Extra Cellular Matrix (ECM), such as collagen fibers or proteoglycans for example. By adjusting the chemical composition and the density of the surrounding ECM, the cells control the stiffness and porosity of their environment. They create the scaffold on which they adhere, the material through which they interact by chemical or mechanical communications. At any given moment, the physiological tissue can thus be seen as a 3D array of living cells embedded in a 3D mesh of natural biomaterials making up the surrounding matrix. The spatial position, the phenotype of each cell, the 3D configuration of the ECM scaffold, its composition and spatial variations, in brief what we call the architecture of the tissue, are the results of a long process of continuous adaptation in a stable and functional tissue. Tissue engineering i.e, designing a functional tissue starting from different living cells in culture (either cell lines or primary cells originating from the tissue under construction), thus consists in recreating this specific and complex dynamic architecture. At first glance, to construct a material scaffold mimicking the ECM grid, embedding the different cells of interests seems the best option. Unfortunately, engineers do not have the time that has been necessary to physiological processes for adjusting all the details of the cells+ECM 3D spatial configurations and reaching homeostasis in functional tissues. If we compare this approach to biomimetic design strategies3 , such as problem-based or solution-based strategies, our practice does not consist to the looking of nature as a solution or an inspiration. It consists in developing the right initial conditions for designing a 3D artificial bio-artefact capable of tissue generation during maturation. It is therefore crucial to envision this engineering project not as inventing a precise (yet out of reach) instrument capable of printing a faithful replica of the physiological tissue, but as creating an initial bioartefact capable of providing incentives for the directed generation of homeostatic functional architecture in engineered cells. The purpose is not to fabricate an electronic chip by constructing every semiconducting transistors and metallic connections in a fixed and operational architecture, but rather to position cells (progenitors, stromal, endothelial, …) together with precise guiding materials shaped in 3D in order to allow time to make the final targeted functional system: a living tissue. Nanoelectronics is not the adequate model. It is more like the art of gardening. This idea of involving time in engineered 3D structures have been popularized by the concept of 4D printing, the fourth dimension in addition to the 3 spatial ones, being time. However, we will see in the next section that current 4D printing practices do not adequately describe the agency of time as conceived in tissue engineering.

The agency of time in current 3D/4D approaches

With the popularization of additive manufacturing and the dissemination of 3D printers, new methods have emerged that nurture the sociotechnical imaginary of creating fresh and mature organs with a 3D printing system. In the scientific literature, these technologies and instruments are gathered under the umbrella of Bioprinting techniques4 . In this section we will see not only that scientists involved in Bioprinting have not considered properly time in their engineering activity but that scientists designing the so-called 4D structures have not had a clear view of the role of time.

The seminal works on bioprinting 5 for generating a tissue or an organ virtually ignored the temporal dimension of the tissues and organs. Indeed, the simple original idea consisted in acquiring high resolution 3D images (RMI, PET-Scan, X-rays tomography) of the targeted tissue then by using 3D injection printers, to reproduce in a biomimetic way with high fidelity to the model, an architecture where all the cells and extra-cellular biomaterials were placed exactly in the same position. This static approach was a scientific standoff. Not only did the 3D images of in-vivo structures not reveal essential details beyond their spatial resolution limit and the available 3D bio-printers could not produce large pieces of tissues, but it was unrealistic for more profound biological reasons. Indeed, even if it was possible to print all the details of the supporting biomaterials, together with mature cells in their right positions, this would totally inhibit the process of organogenesis which is required to obtain a functional tissue or organ. In fact, the printing machine does not print a tissue or an organ but deliver cells and biomaterials in a rigid complex and detailed structure. This artefact does not make the connections between cells, does not resume the history of each cell which is required to bring them in the right phenotypes and in equilibrium or homeostasis with the others. In other words, it is mandatory to include time and evolution at the early stage of the engineering process. Instead of generating a final structure, our alternative project therefore consists in a scenario of regeneration where the 3D-structured biomaterial triggers the evolution of a few selected and engineered cells including: stem cells, stromal cells, endothelial cells, nervous cells…. With our approach, the biomaterial is co-produced by nature and engineering.

In parallel, the community of 3D printing (not necessarily the one working on tissue engineering or regenerative medicine) has already introduced the fourth dimension of time and has popularized the name of 4D printing 6 . A 4D structure is basically a 3D artefact made of a responsive material that can react under a specific stimulus (temperature, pH, electromagnetic field, light …). Under stimulation the printed material can shrink or expand, or be locally submitted to a mechanical force, and all these effects can result in a modification of the overall 3D structure. Because the 3D structure is not static but can be commanded to change or to move, scientists manipulate temporal events and thus have proposed the phrase 4D printing that has captured the public's attention. This term is nevertheless disputable. Indeed, these fascinating structures can be better described as 3D responsive structures or actuable 3D structures. They respond on demand to an external signal in a pre-determined way. These structures are really smart but are they really 4D printed? In this design process time refers to the event triggered by the stimulus which is external to the system. The time dimension is not intrinsic to the structure itself; it is not imprinted. What engineers call 4D materials thus refers to programmed or informed matter (see figure 1 part A) shaped in 3D that can undergo a change of state in response to a triggering signal. As a matter of fact, the fourth dimension in the new experimental field of 4D printing, which is supposed to open to the time dimension, seems closely related to a kind of esthetic gesture [START_REF] Sydney Gladman | Biomimetic 4D Printing[END_REF] , or a robotic activity [START_REF] Wehner | An Integrated Design and Fabrication Strategy for Entirely Soft, Autonomous Robots[END_REF] , where time is just a sequence of programmed and predictable movements or displacements.

Biomaterials and Time in regenerative medicine : some perspectives

This paper claims that the bioprinting activity where materials mimicking the Extra Cellular Matrix are shaped in 3D together with living cells confronts bioengineering with the question of the agency of time in a configuration that has never been addressed before. Indeed, Henri Bergson brilliantly argued that time is the essential trait of living matter. In his book "Creative Evolution", he wrote: "Wherever anything lives, there is, open somewhere, a register in which time is being inscribed [START_REF] Bergson | Creative Evolution[END_REF] ." The design of bioartefacts by bioprinting, mixing engineered materials and living cells thus opens the register of time in a totally novel configuration compared to more conventional manipulation of "time" in 4D printing or when material ageing is investigated.

As seen previously, the question is really to imprint the temporal dimension in the biomaterial itself. The most obvious method for addressing this complex issue would be to create a 3D degradable supporting scaffold for the initial cells. This scaffold could keep the various seeded cells alive and drive their proliferation and differentiation at the initial moments of the culture and subsequently the maturation process of the bioartefact. Once these cells have made their own Extra Cellular Matrix, the scaffold would degrade and gradually leave the floor, allowing the cells to 'write their own history' under the operation of time. Implants, made of a biodegradable material, for example silk, are already designed to be dissolved by time. In this strategy of disappearance (see figure 1 part B), the engineered material is designed to disappear once the cells are capable of evolving, thus mimicking the phenomenon of apoptosis in living cells. Another approach would be to create non-degradable supporting scaffolds exhibiting similar properties for the seeded cells at the start of the culture but leaving enough available space for the cells to generate their own homeostatic ECM. This could be envisioned by creating complex highly porous 3D-scaffolds made of nanoscale structures providing strong support and guidance to the cells while limiting the steric hindrance of the supporting structures. Two photons laser beam writing is the typical current 3D technology that could achieve this high level of spatial resolution and should be applied in regenerative medicine in the future 10 . In this strategy of discretion (see figure 1 part C), the nano-engineered material is operational for initiating the genesis but at the end becomes negligible compared to the neo-formed extracellular matrix.

In a longer-term perspective, the 3D bioartefacts required for future regenerative medicine will involve advanced biomaterials shaped in 3D structures capable of triggering the maturation and evolution of seeded cells into a functional tissue as close as possible to a physiological one. Because time is the key process, the 3D biomaterials we need to work on, must include in their design the evolution of the bioartefact. Because cells are in a state of constant adaptation, bioengineers need to create adaptative materials (see figure 1 part D), capable to escort the perpetual reconfigurations of the cells in evolution within the bioartefact.

Interesting solutions for that can be found in the expandinging area of 4D printing. Through an ingenious combination of these ideas with the known principles of the evolution of living cells, major advances should be achieved in the near future. We could for example, imagine 3D resilient materials printed at the nanoscale by two photons laser writing and including in their structure reversible points of attachments. Because living cells evolving with in a tissue exert specific patterns of forces on their micro-environment, they could untie and re-fasten ondemand the mechanical links of their engineered scaffold. As a consequence, the engineered material would accompany the evolution of the living cells by its own temporal dimension triggered by the cellular rhythm. Careful investigations of cell nanomechanics, combined with the design of 3D nanoscale printed material with articulated pieces would be necessary to transform this idea into a reality. We have however the feeling that the required knowledges to realise this project already exists across different scientific communities. Designed true 4D biomaterials will be able to co-evolve "symbiotically" with the cells they are hosting when cultured in-vitro and also with the cells of the host, in-vivo, for implantable bioartefacts. Will it be possible for scientists to reach this level of triggering of living processes such as organogenesis? We clearly do not have the answer yet, but part of this answer resides in our capability to integrate time as a process of becoming in our biomaterial constructs.

Conclusion

In this paper we argued that Material and Time are intertwined in a radical novel configuration in the activities of tissue engineering through bioprinting. The constant adaptation of the living cells to the 3D engineered artefactual material in which they are embedded brings in conflict two different conceptions of time. Following Bergson, the material artefact, closed and engineered by our senses, manipulates a "thought time" while the living matter around it, manipulates a "lived time". This Bergsonian view of time invites engineers to invent disruptive strategies breaking with the conventional practices of the design of synthetic devices and systems. It is by thinking about time as a process of becoming that engineers will create regenerative tissues.

  1 A. Ovsianikov, A. Khademhosseini, V. Mironov, The synergy of scaffold-based and scaffold-free tissue engineering strategies, trends in Biotechnology 36(4) (2028) 348 2 E.S. Place, N.D. Evans, M.M. Stevens, Complexity in biomaterials for tissue engineering, Nature Material 8 (2009) 457 neglected or disregarded. By pointing out this deficiency, we hope to open new engineering perspectives in tissue engineering.

  5 Y. S. Zhang, R. Oklu, M.R. Dokmeci, A. Khademhosseini, Three-Dimensional Bioprinting Strategies for Tissue Engineering, Cold Spring Harb Perspect Med. 2018 Feb 1;8(2):a025718.doi: 10.1101/cshperspect.a025718. 6 S. Tibbits, The emergence of "4D printing", TED Conference (2013), https://www.ted.com/talks/skylar_tibbits_the_emergence_of_4d_printing

  10 A ; Accardo, M.C. Blatche, R. Courson, I. Loubinoux, C. Vieu, L. Malaquin, Two-photon lithography and microscopy of 3D hydrogel scaffolds for neuronal cell growth, Biomedical Physics and Engineering Express 4, 2 (2018)

Figure 1 :

 1 Figure 1: Illustration of different conception of time through scaffold engineering strategy. A: Informed matter & 4D printing B: Strategy of disappearance C: Strategy of discretion D: Adaptative Material

Badarnah, Lidia, et Usama Kadri. « A methodology for the generation of biomimetic design concepts ». Architectural Science Review 58, nᵒ 2 (3 avril 2015): 120-33. https://doi.org/10.1080/00038628.2014.922458.

I. Matai, G. Kaur, A. Seyedsalehi, A. McClinton, C.T. Laurencin, Progress in 3D bioprinting technology for tissue/organ regenerative engineering, Biomaterials 226 (2020) 119536

Acknowledgments

The authors acknowledge Bernadette Bensaude Vincent (Université Paris1 Panthéon-Sorbonne) and Viviane Quirke (Oxford Brookes University) for fruitful discussions and language improvements.