Quick Macroscopic Diffusion-Based Molecular Communication With Wavelike Propagation by Use of a Suspension of Relay Cells

In order to enable molecular communication on macroscopic distances without a moving medium, the model of diffusion-based information transmission with a large number of relays suspended in the medium, hence at random positions, is proposed. With an increasing number of relays, the stochastic variations approach zero and the system exhibits a quasi-deterministic behavior. Furthermore, the transmission time becomes inversely proportional to the number of relays and the impulse response independent of the channel length. Thus, the model describes a wave-like propagation, which is fundamentally different from the established flow- and diffusion-based approaches in molecular communication.


I. INTRODUCTION
M OLECULAR communication (MC) denotes data transmission by use of molecules as information carriers, thus establishing networks between nanoscale devices and biological entities, often termed the Internet of (Bio-)Nano Things or, in the case of medical applications, the Internet of Medical Things [1]. A central application scenario of MC is hence information transmission over small distances, typically in the sub-millimeter range [2], [3], [4]. On this microscale, diffusion is the dominant transport mechanism, which can be motivated by the proportionality of the Péclet number to transmission distance. Accordingly, diffusive processes in biological systems can be found over lengths between nanoand micrometers (e. g., ion channels [5] and oxygen diffusion in tissue [6], respectively). On the centimeter-and meter-scale, in contrast, flow-based material transport takes place, e. g., in the respiratory and cardiovascular system [7]. The same applies to experimental testbeds for MC, which have channel lengths between 5 mm [8] and 20 m [9]: They are nearly exclusively flow-based, both in fluidic [9], [10], [11], [12], [13], [14] and gaseous channels [15], [16], [17], [18], irrespective of the type of information carrier. Diffusion appears at most in setups with both convection and diffusion [19], [20].
The flow-based transport in macroscopic MC, however, necessitates an appropriate pump mechanism and often means confining the flow to a dedicated physical channel (see [18] and the tubes in the fluidic testbeds [9], [10], [11], [12], [13], [14]). Even then the flow profile distorts the initially compact Manuscript  injected volume, which leads to a decreasing impulse response and to increasing inter-symbol interference with increasing channel length [14]. Finally, a flowing medium itself can be undesirable when the channel is not used for data transmission only. Hence, the question is if MC on the macroscale can be realized without a flowing medium.
A comparable approach is found in the context of Ca 2 + waves, where gap junctions act as relays between cells, while Ca 2 + ions diffuse inside the cells [37], [38]. In [39], this diffusive component also extends across neighboring cells. However, these models assume identical cells in direct contact and are intended as an accurate representation of the biological process under consideration, which mostly exhibits an oscillating behavior of the Ca 2 + concentration.
In order to enable macroscopic diffusion-based communication, a multi-hop network comparable to the described approaches is needed, but with two differences: First, a significantly higher number of relays is required, in order to increase channel length by several orders of magnitude without considerably decreasing symbol rate. Furthermore, the works above assumed fixed, in most cases equal distances between the network nodes, which would require to fix them in the transmission channel. Often and particularly with a large number of nodes, this is not feasible. Instead, the relays are randomly suspended in the medium. Therefore, second, the positions of the relays are stochastically distributed. Note that, other than in mobile MC [40], these positions will be assumed fixed over time, that is, data transmission is regarded to be much quicker than the movement of the relays. Thus, the model of diffusion-based molecular communication with a large number of randomly distributed relays is proposed. This novel data transmission scheme shall enable macroscopic channel lengths, i. e., on the centimeter and meter scale, without the need for a moving medium.

II. PROPOSED MODEL
The proposed model adopts a 1-dimensional space, with the transmitter at x = 0, the receiver at x = L, and the relays randomly distributed in between. Each of these nodes is assumed to have point-shape.
At t = 0, the transmitter emits a burst of molecules, which then distribute via normal diffusion along the unbounded x-axis. If, at the position x i of relay no. i, the concentration c exceeds threshold R, a detection event is triggered, which leads to an emission of molecules after a time delay τ DE (decode-and-forward relaying). In that way, the concentration of information molecules wanders from transmitter to receiver, where the detection event corresponds to the reception of a bit '1'. Ideally, τ DE = 0 for minimum transmission time but can be > 0 in general due to the (e. g. chemical) processes inside the relay.
The molecule concentration c is conquered by Fick's law where D denotes the diffusion coefficient [41]. The contribution due to a single infinitely short emission at position x = x 0 and time t = t 0 has the solution where Θ denotes the Heaviside step function and c 0 = c(x, t|x 0 , t 0 )dx the total amount of molecules, which can be given in terms of mass, volume or number of particles.
In order to simplify mathematical treatment and to become independent of spatial and temporal scales, dimensionless quantities are introduced: With these definitions, the proposed model is defined as follows ( Fig. 1): The positions of the communication nodes are given as where x 0 = 0 and x N = 1 are the positions of the transmitter and the receiver, respectively. The indices i = 1, . . . , (N − 1) denote the relays of the N -hop process. In the following, only the propagation from transmitter to receiver will be considered, without backwards propagation, i. e., a unidirectional model. In that case, a series of detection events at times and corresponding emissions at times are obtained (t e 0 = 0). Furthermore, a refractory time τ DR is introduced that suppresses detection events at cell no. i for times t d i < t < t d i + τ DR . The corresponding times when the refractory periods end are given by Upper plot: Succession of the detection/emission events for the transmitter and the first two relays. Lower plot: Net molecule concentration ctot (blue) due to the emissions c(x, t d 2 |x i , t i ) (black) at the transmitter and relay 1 (i. e., i = 0, 1). At the depicted time t = t d 2 , i. e., the detection time at relay 2, the molecules emitted by the transmitter and relay 1 add up to the concentration ctot, which exceeds the threshold R (dashed blue) and leads to the detection event at relay 2. After τ DE , relay 2 will emit molecules at t = t e 2 ; after the refractory time τ DR , it will be susceptible to a detection event again.
With sufficiently large τ DR , backwards propagation is prevented and unidirectionality assured. Note, however, that τ DR poses a lower limit to the bit interval.
The contribution of a single emission at time t e i and position x i is derived from (2) and (3) as The full concentration is then given by the superposition of the contribution of each cell Finally, detection time t d i of the i-th relay is given by the first root of the equation The threshold R will be specified in the following by use of the relative parameter r defined by where (2πe) − 1 2 is the maximum concentration at the receiver in the non-relayed case (i. e., N = 1) and multiplication by N scales the threshold with the density of the communication nodes.
Note that, for a bidirectional model, with τ DR < ∞, a relay can have multiple detection/emission events j = 1, 2, . . . Then, (10) has to be restricted to t d i,j ≥ t r i,j with initial conditions t r i,0 = 0.

III. NUMERICAL ANALYSIS A. Methods
The model described in the previous section was implemented by use of Python with modules Numpy and SciPy in the versions 3.10.8, 1.23.5, and 1.9.3, respectively. 1) Setup Parameters: As the stochastic positions of the relay nodes lead to a variation of the target figures, each parameter setup was simulated N Sa times with different, randomly chosen positions. For that purpose, a uniform distribution within the interval ]0; 1[ was adopted. Furthermore, the time parameters τ DE = 0 and τ DR = ∞ as well as the threshold r = 0.1 were used in all cases.
2) Transmission Time and Impulse Response: The transmission time and the impulse response are defined as the detection time and the concentration at the receiver, respectively: The corresponding impulse response amplitude (IRA) is If not stated otherwise, x N = 1 was assumed.
3) Channel Length: In order to investigate the effect of channel length, x N was varied in the interval from 0 to 1.5 and the IRA was computed according to (14), while N was chosen in such way that the relay density N/x N = 10 3 was kept constant.
For comparison, a single-source diffusion (SSD) process, i. e., N = 1, was used. The maximum of its impulse response can be derived as In the proposed model, however, the number of emissions increases linearly with channel length and hence also the amount of information molecules. To provide a fair comparison between the models, the IRA of the SSD is scaled in such a way that it uses the same amount of molecules as the proposed multi-hop process for a given channel length: B. Results Fig. 2 shows the average transmission time in relation to the number of hops N (black line; N Sa = 10 4 ). The transmission time strictly decreases with an increasing number of cells and hence with relay density. For N ≳ 10, it follows a power-law behavior with exponent −1 and is thus inversely proportional to the relay density.

1) Transmission Time and Number of Relays:
The error bars in Fig. 2 indicate the standard deviation of the transmission time. This variation, due to differences in the relay positions between different realizations, strictly decreases with N for N > 10. The same applies to the relative standard deviation, i. e., the standard deviation divided by the average of the transmission time, shown in red in Fig. 2. It follows a power-law behavior with exponent −0.5 for N ≳ 10.  2) Channel Length: The IRA of the proposed model according to (14) for varying channel length is shown in Fig. 3 (solid black line; standard deviation indicated by dotted lines). It increases with the channel length until it reaches a plateau and then stays constant. This indicates that the impulse response approaches a stable form, which is then independent of the channel length.
For comparison, the IRA of the unscaled and the scaled SSD process, according to (15) and (16), are shown as solid red and dashed black line, respectively. The amplitude of the unscaled SSD process is strictly decreasing, due to the dilution of the molecules. The situation is different for the IRA of the scaled SSD, which shows a similar behavior compared to the multihop process: It is constant over the entire range of channel lengths. Its IRA, however, is only about half of the IRA of the proposed model. This is explained by the distributed emitters in the latter case, which are all closer to the receiver than the single emitter in the SSD case. Fig. 4. Impulse response at x = 1 of the proposed model (black line; N = 10 3 , N Sa = 10 2 , τ DE = 0, τ DR = ∞, r = 0.1) and of the scaled single-source diffusion process as reference (red line). Compared to the latter, the impulse response of the proposed model shows a much quicker rise to the maximum, which indicates a smaller transmission time (see Fig. 2), and a larger amplitude by about a factor 2 (see also Fig. 3).
3) Impulse Response: In Fig. 4, the impulse response at x N = 1 is shown (N = 10 3 ). The solid black line shows the average over N Sa = 10 2 realizations, the dashed lines ± the standard deviation over the ensemble. The impulse response shows a sharp increase, followed by a plateau-like phase with a flat maximum and a long, slowly falling slope. The sharp increase is explained by the emissions of the relays in the close vicinity of the receiver, whereas the falling slope comprises more and more contributions from more distant relays, while those contributions of the nearest neighbors are already decreasing.
For comparison, the impulse response of the scaled single-source diffusion process 10 3 c(1, t|0, 0) at x = 1 is shown in red. It exhibits a much larger transmission time, as indicated by the slower increase to the maximum, and about half the amplitude as already seen in Fig. 3. The two impulse responses approach each other for increasing time; hence, the proposed model shows a smaller inter-symbol interference relative to the IRA.
The variation between different realizations of the proposed model, as indicated by the standard deviation, is largest at the beginning of the impulse response, directly after the sharp increasing slope, and then diminishes and approaches zero. This can be explained by the relatively small number of contributing emissions at the beginning, where differences in the positions of the neighboring relays lead to different contributions and hence different impulse responses. At later times, with a larger number of contributing relays, differences in their positions cancel out and lead to negligible differences in the impulse response.
IV. DISCUSSION AND CONCLUSION In the preceding sections, a novel model of information transmission in MC was proposed. It is characterized by diffusion-based transport of the information molecules and a large number of relays at random positions. It hence might prove useful, e. g., for swarms of nanobots on the macroscopic scale.
The results indicate that variations between different realizations diminish with an increasing number of cells and that the system approaches a quasi-deterministic behavior. In particular, the impulse response develops a steady form that is independent of the channel length. Hence, the model provides means of reliable information transmission with virtually unlimited channel length. This propagation of an excitation wave, comparable to the conduction system of the heart, is a new form of MC next to approaches based on flow and single-source diffusion.
In addition to the non-decreasing amplitude of the impulse response, the transmission time of the proposed model was found to be inversely proportional to the number of relays. Thus, the model is able to scale diffusion-based MC from microscopic channel lengths up to macroscopic dimensions.
The proposed model is intended as the first step towards diffusion-based MC by use of a multitude of stochastically distributed relays. To this end, the model was kept simple, in order to demonstrate the characteristics and advantages mentioned above. Therefore, the model can be developed further in multiple respects: The next step is to consider not only unidirectional propagation from the transmitter to the receiver but also backwards propagation, which could lead to circulating excitation. This can be avoided by a sufficiently large refraction time τ DR , yet at the expense of a large bit interval. Inter-symbol interference, as indicated by the long tail of the impulse response, can be counteracted by appropriate mechanisms to restore the initial condition, where the emitted molecules decay, react with other molecules or are pumped back into the relays. Also, different modulation schemes have to be investigated: apart from on-off keying, molecular shift keying and different timing-based techniques are feasible; in higher dimensional space, the model could be combined with direction shift keying. Further extensions of the model comprise the transfer to propagation in 2-and 3-dimensional space, anomalous diffusion as well as molecule emission of finite duration, amplitude and supply. Finally, the effect of noise in the molecule concentration and in the parameters of the model and its future extensions described above have to be investigated. The former case will be of particular importance when the amount of information molecules is limited and only a moderate number of them is emitted by each relay. Then, the assumption of the continuous, noise-free distribution (8), which is only valid for a sufficiently large number of molecules, does not hold any more and diffusion noise due to the random propagation of the molecules has to be incorporated.
In addition to the described extensions of the model, suitable ranges and particularly optimal values of the model parameters have to be determined. In this way, the proposed model enables to identify or engineer relays for future applications of the model in molecular communication.