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arXiv:1605.08311v2 [cs.IT] 3 Aug 2016
3D Stochastic Geometry Model for Large-Scale
Molecular Communication Systems
Yansha Deng∗, Adam Noel†, Weisi Guo‡, Arumugam Nallanathan∗, and Maged Elkashlan§
∗Department of Informatics, King’s College London, UK
†School of Electrical Engineering and Computer Science, University of Ottawa, Canada
‡School of Engineering, University of Warwick, UK
§School of Electronic Engineering and Computer Science, Queen Mary University of London, UK
Abstract—Information delivery using chemical molecules is an
integral part of biology at multiple distance scales and has
attracted recent interest in bioengineering and communication.
The collective signal strength at the receiver (i.e., the expected
number of observed molecules inside the receiver), resulting from
a large number of transmitters at random distances (e.g., due to
mobility), can have a major impact on the reliability and efficiency
of the molecular communication system. Modeling the collective
signal from multiple diffusion sources can be computationally
and analytically challenging. In this paper, we present the first
tractable analytical model for the collective signal strength due
to randomly-placed transmitters, whose positions are modelled
as a homogeneous Poisson point process in three-dimensional
(3D) space. By applying stochastic geometry, we derive analytical
expressions for the expected number of observed molecules at
a fully absorbing receiver and a passive receiver. Our results
reveal that the collective signal strength at both types of receivers
increases proportionally with increasing transmitter density. The
proposed framework dramatically simplifies the analysis of large-
scale molecular systems in both communication and biological
applications.
Index Terms—molecular communications, absorbing receiver,
passive receiver, stochastic geometry, interference modeling.
I. INTRODUCTION
Molecular communication via diffusion has attracted sig-
nificant bioengineering and communication engineering re-
search interest in recent years [1]. Messages are delivered
via molecules undergoing random walks [2], a prevalent phe-
nomenon in biology [3]. In fact, molecular communication
exists in nature at both the nano- and macro-scales, offering
transmit energy and signal propagation advantages over wave-
based communications [4,5]. One application example is that
swarms of nano-robots can track specific targets, such as a
tumour cells, to perform operations such as targeted drug
delivery [6]. In order to do so, energy efficient and tether-less
communications between the nano-robots must be established
in biological conditions [7], and possibly additional nano-bio-
interfaces need to be implemented [8].
Fundamentally, molecular communications involves modu-
lating information onto the property of a single or a group
of molecules (e.g., number, type, emission time). When mod-
ulating the number of molecules, each messenger node will
transmit information-bearing molecules via chemical pulses. In
a realistic environment with a swarm of robots (i.e., messenger
nodes) operating together, they are likely to transmit molecular
messages simultaneously. Due to limitations in transmitter de-
sign and molecule type availability, it is likely that many trans-
mitters will transmit the same type of information molecule.
Thus, it is important to model the collective signal strength due
to all transmitters with the same type of information molecule,
and to account for random transmitter locations due to mobility.
Existing works have largely focused on modeling: 1) the
signal strength of a point-to-point communication channel by
considering the self-interference that arises from adjacent sym-
bols (i.e., inter-symbol-interference (ISI)) at a passive receiver
[9], at a fully absorbing receiver [10], and at a reversible
adsorption receiver [11]; and 2) the collective signal strength of
a multi-access communication channel at the passive receiver
due to co-channel transmitters (i.e., transmitters emitting the
same type of molecule) with the given knowledge of their total
number and location [9].
The first work to consider randomly distributed co-channel
transmitters in 3-D space according to a spatial homogeneous
Poisson process (HPPP) is [12], where the probability density
function (PDF) of the received signal at a point location was
derived based on the assumption of white Gaussian transmit
signals. Since the receiver size was negligible, the placement
of transmitters did not need to accommodate the receiver’s
location. More importantly, only the Monte Carlo simulation,
and not particle-based simulation, was performed to verify the
derived PDF.
From the perspective of receiver type, many works have
focused on the passive receiver, which can observe and count
the number of molecules inside the receiver without interfering
with the molecules [9,12]. In nature, receivers commonly
remove information molecules from the environment once they
bind to a receptor. One example is the fully absorbing receiver,
which absorbs all the molecules hitting its surface [10, 11].
However, no work has studied the channel characteristics and
the received signal at the fully absorbing receiver in a large-
scale molecular communication system, let alone its compari-
son with that at the passive receiver.
In this paper, we model the collective signal strength at the
passive receiver and fully absorbing receiver due to a swarm
of mobile point transmitters that simultaneously emit a given
number of information molecules. Unlike [12], which focused
on the statistics of the received signal at any point location,
we focus on examining and deriving exact expressions for
the expected number of molecules observed inside two types
of receiver for signal demodulation. This is achieved using
stochastic geometry, which has been extensively used to model
and provide simple and tractable results for wireless systems
[13]. Our contributions can be summarized as follows:
1) We use stochastic geometry to model the collective signal
at a receiver in a large-scale molecular communication
system, where the receiver is either passive or fully
absorbing. We distinguish between the desired signal due
to the nearest transmitter and the interfering signal due
to the other transmitters.
2) We derive a simple closed-form expression for the ex-
pected number of molecules absorbed at the fully absorb-
ing receiver, and a tractable expression for the expected
number of molecules observed inside the passive receiver
at any time instant.
3) We define and derive tractable analytical expressions for
the fraction of molecules due to the nearest transmitter
and the fraction of molecules due to the other transmit-
ters.
4) We verify our results using particle-based simulation and
Monte Carlo simulation, which prove that the expected
number of molecules observed at both types of receiver
increases linearly with increasing transmitter density.
II. SYSTEM MODEL
We consider a large-scale molecular communication system
with a single receiver in which a swarm of point transmitters are
spatially distributed outside the receiver in R3/VΩrraccording
to an independent and HPPP Φwith density λ, where VΩrr
is the volume of receiver Ωrr. This spatial distribution, which
was previously used to model wireless sensor networks [14],
cellular networks [13] and heterogenous cellular networks [15],
has also been applied to model bacterial colonies in [16] and
the interference sources in a molecular communication system
[12]. We consider a fluid environment in the absence of flow
currents: the extension for flow currents will be treated in future
work.
At any given time instant, a number of transmitters will
be either silent or active. Thus, we define the activity prob-
ability of a transmitter that is triggered to transmit data as
ρa(0 < ρa<1). This activity probability is independent of the
receiver’slocation. Thus, the active point transmitters constitute
independent HPPPs Φawith intensities λa=λρa. Each
transmitter transmits molecular signal pulses with amplitude
NFA
tx (NPS
tx ) to the absorbing receiver (the passive receiver). We
assume the existence of a global clock such that all molecule
emissions can only occur at t= 0.
We consider two types of spherical receiver with radius rr:
1) Fully absorbing receiver [17], and 2) Passive receiver [4,
18]. To equivalently compare them, we assume both types of
receiver are capable of counting the number of information
molecules within the receiver volume at any time instant for
information decoding.
It is well known that the distance between the transmitter and
the receiver in molecular communication is the main contributor
Receiver
k th Point!
Transmitter
T
t
P
T
t
P
T
t
P
T
t
P
T
t
P
T
t
P
T
t
P
rr
Fig. 1. Illustration of a receptor receiving molecular pulse signals from point
transmitters at different distances.
to the degradation of the signal strength (i.e., the number of
molecules observed at the receiver). Instinctively, we assume
that the receiver is associated with the nearest transmitter to
obtain the strongest signal. Thus, the messenger molecules
transmitted by other active point transmitters act as interference,
which impairs the correct reception at the receiver. To measure
this impairment, we formulate the desired signal, the interfering
signal for the absorbing receiver and the passive receiver in the
following subsections.
A. Absorbing Receiver
In our proposed large-scale molecular communication sys-
tem, let us consider the center of an absorbing receiver located
at the origin. Using the Slivnyak-Mecke’s theorem [13], the
fraction FFA of molecules absorbed at the receiver until time
Tdue to an arbitrary point transmitter xat the location xwith
molecule emission occurring at t= 0 can be represented as
[17]
FFA ( Ωrr, T |kxk) = rr
kxkerfc nkxk − rr
√4DT o,(1)
where kxkis the distance between the point transmitter and the
center of the receiver where the transmitters follow a HPPP,
and Dis the constant diffusion coefficient, which is usually
obtained via experiment as in [19, Ch. 5]. The fraction FFA
uof
molecules absorbed inside the receiver until time Tdue to a
single pulse emission by the nearest active transmitter can be
represented as
FFA
u(Ωrr, T | kx∗k) = rr
kx∗kerfc nkx∗k − rr
√4DT o,(2)
where kx∗kdenotes the distance between the receiver and the
nearest transmitter,
x∗= arg min
x∈Φakxk,(3)
x∗denotes the nearest point transmitter for the receiver, and
Φadenotes the set of active transmitters’ positions.
The fraction of molecules absorbed at the receiver until
time Tdue to single pulse emissions at each active interfering
transmitter FFA
Iand that due to a single pulse emission at each
active transmitter FFA
all is represented as
FFA
I(Ωrr, T | kxk) = X
Φa/x∗
rr
kxkerfc nkxk − rr
√4DT o,(4)
and
FFA
all (Ωrr, T | kxk) = X
Φa
rr
kxkerfc nkxk − rr
√4DT o,(5)
respectively.
The expected number of molecules absorbed at the receiver
by time Tdue to all active transmitters is equivalently the
expected number of molecules absorbed at the receiver until
time T, which can be calculated as
ENFA
all (Ωrr, T )=NFA
tx FFA
all (Ωrr, T | kxk)
=NFA
tx FFA
u(Ωrr, T | kx∗k)
|{z }
EFA
u
+NFA
tx FFA
I(Ωrr, T | kxk)
|{z }
EFA
I
,(6)
where FFA
all (Ωrr, T | kxk)is given in (5), EFA
uis the fraction of
absorbed molecules at the absorbing receiver until time Tdue
to the nearest transmitter, and EFA
Iis the fraction of absorbed
molecules at the absorbing receiver until time Tdue to the
other (interfering) transmitters.
B. Passive Receiver
In a point-to-point molecular communication system with a
single point transmitter located distance kxkaway from the
center of a passive receiver with radius rr, the local point
concentration at the center of the passive receiver at time T
due to a single pulse emission by the transmitter is given as
[20, Eq. (4.28)]
C( Ωrr, T |kxk) = 1
(4πD T )3/2 exp −kxk2
4DT .(7)
The fraction of information molecules observed inside the
passive receiver with volume VΩrrat time Tis denoted as
FPS (Ωrr, T | kxk) = Z
VΩrr
C(Ωrr, T | kxk)dVΩrr,(8)
where VΩrris the volume of the spherical passive receiver.
According to (8) and Theorem 2 in [21], the fraction FPS of
information molecules observed inside the passive receiver at
time Tdue to a single pulse emission by a transmitter at time
tis derived as
FPS ( Ωrr, T |kxk) = 1
2erfrr− kxk
2√DT + erfrr+kxk
2√DT
+√DT
√πkxk"exp −(rr+kxk)2
4DT −exp −(kxk − rr)2
4DT #,
(9)
which does not assume that the molecule concentration inside
the passive receiver is uniform. This is unlike the common as-
sumption that the concentration of molecule inside the passive
receiver is uniform. Although that assumption is commonly
applied, it relies on the receiver being sufficiently far from the
transmitter (see [21]), which we cannot guarantee here since
the transmitters are placed randomly.
In the large-scale molecular communication system with a
passive receiver centered at the origin, the expected number of
molecules observed inside the receiver at time Tdue to a single
pulse emission at all active transmitters at t= 0 is given as
ENPS
all (Ωrr, T )=E(X
Φa
NPS
tx FPS ( Ωrr, T |kxk))
=NPS
tx FPS
u(Ωrr, T | kx∗k)
|{z }
EPS
u
+NPS
tx FPS
I(Ωrr, T | kxk)
|{z }
EPS
I
,
(10)
where FPS ( Ωrr, T | kxk)is given in (9), EPS
uis the fraction
of molecules observed inside the receiver at time Tdue to the
nearest transmitter, and EPS
Iis the fraction of molecules ob-
served inside the receiver at time Tdue to the other (interfering)
transmitters.
III. RECEIVER OBSERVATIONS
In this section, we first derive the distance distribution
between the receiver and the nearest point transmitter. By doing
so, we derive exact expressions for the expected number of
molecules observed inside the receiver due to the nearest point
transmitter and that due to the interfering transmitters.
A. Distance Distribution
Unlike the stochastic geometry modelling of wireless net-
works, where the transmitters are randomly located in the un-
bounded space, the point transmitters in a large-scale molecular
communication system can only be distributed outside the sur-
face of the spherical receiver. Taking into account the minimum
distance rrbetween point transmitters and the receiver center,
we derive the probability density function (PDF) of the shortest
distance between a point transmitter and the receiver in the
following proposition.
Proposition 1. The PDF of the shortest distance between any
point transmitter and the receiver in 3D space is given by
fkx∗k(x) = 4λaπx2e−λa(4
3πx3−4
3πrr
3),(11)
where λa=λρa.
Proof. See Appendix A.
Based on the proof of Proposition 1, we also derive the PDF
of the shortest distance between any point transmitter and the
receiver in 2D space in the following lemma.
Corollary 1. The PDF of the shortest distance between any
point transmitter and the receiver in 2D space is given by
fkx∗k(x) = 2λaπre−λa(πr 2−πrr
2),(12)
where λa=λρa.
B. Absorbing Receiver Observations
In this subsection, we derive a closed-form expression for the
expected number of molecules observed inside the absorbing
receiver in 3D space.
Using Campbell’s theorem, we derive the expected number
of absorbed molecules due to the nearest transmitter and that
due to the interfering transmitters until time tas
EFA
u=NFA
tx Z∞
rr
Πt
0(x) 4λaπx2e−λa(4
3πx3−4
3πrr
3)dx, (13)
and
EFA
I=NFA
tx (4πλa)2Z∞
rrZ∞
x
Πt
0(r)r2drx2e−λa(4
3πx3−4
3πrr
3)dx,
(14)
respectively.
Theorem 1. The expected net number of molecules absorbed
at the absorbing receiver in 3D space during any sampling
time interval [t, t +Tss ]is derived as
ENFA
all (Ωrr, t, t +Tss )
= 4NFA
tx √πλarrhD√πTss + 2√DrrpTss +t−√ti.
(15)
The expected number of molecules observed inside the fully
absorbing receiver in 3D space until time tis derived as
ENFA
all (Ωrr, t)= 4NFA
tx √πλarrhD√πt + 2rr√Dti.
(16)
Proof. See Appendix B.
From Theorem 1, we find that the expected number of
molecules absorbed at the absorbing receiver at time tis
linearly proportional to the density of active transmitters,
and increases with increasing diffusion coefficient or receiver
radius. As expected, we find that the expected number of
molecules absorbed at the receiver until tis always increasing
with time t.
C. Passive Receiver Observations
In the following theorem, we derive the expected number of
molecules observed inside the passive receiver in 3D space.
Using Campbell’s theorem, we derive the expected number
of observed molecules due to the nearest transmitter and that
due to the interfering transmitters at time tas
EPS
u= 4λaπN PS
tx e4
3πrr
3λaZ∞
rr
Φ (x)x2exp n−4
3πx3λaodx,
(17)
and
EPS
I=(4πλa)2e4
3πrr
3λaNPS
tx Z∞
rrZ∞
x
Φ (r)r2dr
x2e−4
3πx3λadx, (18)
respectively. In (17) and (18), Φ (r) = FPS ( Ωrr, t|r).
Theorem 2. The expected net number of molecules observed
inside the passive receiver during any sampling time interval
[t, t +Tss ]in 3D space is derived as
ENPS
all ( Ωrr, t, t +Tss |kxk)= 4NPS
tx πλa
Z∞
rr
FPS (Ωrr, t +Tss |r)r2dr −Z∞
rr
FPS (Ωrr, t|r)r2dr,
(19)
where FPS (Ωrr, t|r)is given in (9).
The expected number of molecules observed inside the pas-
sive receiver at time tin 3D space is derived as
ENPS
all (Ωrr,0, t|kxk)=
4NPS
tx πλaZ∞
rr
FPS (Ωrr, t|r)r2dr. (20)
Proof. Analogous to Appendix B without solving the integrals.
In Theorem 2, we observe that the expected number of
molecules observed inside the passive receiver also increases
proportionately with the density of active transmitters.
IV. NUMERICAL AND SIMULATION RESULTS
In this section, we examine the expected number of
molecules observed at the absorbing receiver and the passive
receiver due to simultaneous single pulse emissions at all active
point transmitters. In all figures of this section, we set the
parameters as follows: rr= 5 µm and NFA
tx =NPS
tx = 104.
In all figures, the analytical curves of the expected number
of molecules absorbed at the absorbing receiver due to all
the transmitters, the nearest transmitter, and the interfering
transmitters are plotted using Eqs. (15), (13), and (14), and
are abbreviated as “Absorbing All”, “Absorbing Nearest”, and
“Absorbing Aggregate”, respectively. The analytical curves of
the expected number of molecules observed inside the passive
receiver due to all the transmitters, the nearest transmitter, the
interfering transmitters are plotted using (19), (17), and (18),
and are abbreviated as “Passive All”, “Passive Nearest”, and
“Passive Aggregate”, respectively.
A. Particle-Based and Pseudo Simulation Validation
In Fig. 2, we set D= 80 ×10−12 m2
s, and assume that the
transmitters are placed up to R= 50 µm from the center of the
receiver at a density of λa= 10−4transmitters per µm3(i.e.,
52 average number of transmitters, including the subtraction of
the receiver volume). The receiver takes samples every Tss =
0.01 s and calculates the net change in the number of observed
molecules between samples. The default simulation time step
is also 0.01 s. Unless otherwise noted, all simulation results
were averaged over 104transmitter location permutations, with
each permutation simulated at least 10 times.
In Fig. 2, we verify the analytical expressions for the
expected net number of molecules observed during [t, t +Tss ]
at the absorbing receiver in Eq. (13) and Eq. (14), and that
inside the passive receiver in Eq. (17) and Eq. (18) by com-
paring with the particle-based simulations and the Monte Carlo
10−1 100
0
0.2
0.4
0.6
0.8
1
0.01 0.02 0.03
0
0.2
0.4
0.6
0.8
1
Absorbing Nearest
Absorbing Aggregate
Absorbing Nearest
Absorbing Aggregate
Passive Aggregate
Passive Nearest
Analytical
Particle−Based Sim.
Monte Carlo Sim.
Time (s) Time (s)
Net Number of Observed Molecules
Fig. 2. Net number of observed molecules inside the receiver as a function of
time. All curves are scaled by the maximum value of the analytical curves in
the right subplot.
TABLE II
THE S IMUL ATI ON PARA METERS A ND SCA LING VALUE S APP LIED IN FIG. 2.
Transmitter Receiver Realizations Time Scaling
Step [s] Value
Nearest Passive 10410−2149.57
Nearest Active 10410−2354.52
Aggregate Passive 10410−29.252
Aggregate Active 10310−359.42
simulations. The particle-based simulations were performed by
tracking the progress of individual particles to obtain the net
number of observed molecules during [t, t +Tss]using the
AcCoRD simulator (Actor-based Communication via Reaction-
Diffusion) [22]. The pseudo simulations rely on the Monte
Carlo simulation method, which were performed by averaging
the expected number of observed molecules due to all active
transmitters with randomly-generated location, as calculated
from Eq. (1) and Eq. (9), over 104realizations.
In the right subplot of Fig. 2, we compare passive and absorb-
ing receivers and observe the expected net number of observed
molecules during [t, t +Tss ]due to the nearest transmitter and
due to the aggregation of the interfering transmitters. In the
left subplot of Fig. 2, we lower the simulation time step to
10−4s for the first few samples of the two absorbing receiver
cases, in order to demonstrate the corresponding improvement
in accuracy. All curves in both subplots are scaled by the max-
imum value of the corresponding analytical curve in the right
subplot; the scaling values and other simulation parameters are
summarized in Table II.
1) Particle-Based Simulation Validation: Overall, there is
good agreement between the analytical curves and the particle-
based simulations in the right subplot of Fig. 2. The analytical
results for the net number of molecules observed inside the
passive receiver during [t, t+Tss]due to the nearest transmitter
is highly accurate, and even captures the net loss of molecules
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0
0.5
1
1.5
2x 105
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0
2000
4000
6000
8000
Analytical
Monte Carlo Sim.
Passive All
Absorbing All
Passive Nearest
Passive Aggregate
Absorbing Aggregate
Absorbing Nearest
Number of Observed Molecules
Time (s)
Fig. 3. Expected number of molecules observed inside the receiver as a
function of time.
observed after t= 0.1s. There is a slight deviation in the
particle-based simulation for the “passive aggregate” curve,
particularly as time approaches t= 1 s, which is primarily due
to the very low number of molecules observed at this time (note
that the scaling factor in this case is only 9.252; see Table II).
There is less agreement between the particle-based simula-
tions and the analytical expressions for the absorbing receiver,
and this is primarily due to the large simulation time step (even
though we used a smaller time step for the aggregate transmitter
case in the right subplot; see Table II). To demonstrate the
impact of the time step, the left subplot shows much better
agreement for the absorbing receiver model by lowering the
time step to 10−4s. This improvement is especially true in the
case of the nearest transmitter, as there is significant devia-
tion between the particle-based simulation and the analytical
expression for very early times in the right subplot.
2) Monte Carlo Simulation Validation: There is a good
match between the analytical curves and the Monte Carlo
simulations for the net number of molecules observed at both
types of receiver during [t, t+Tss ]due to the nearest transmitter,
which can be attributed to the large number of molecules and
the shortest distance value compared with R= 50 µm (as
shown in Table II). There is slight deviation in the Monte Carlo
simulations for the expected number of molecules observed
inside both types of receiver due to the interfering transmitters,
and this is primarily due to the restricted placement of trans-
mitters to the maximum distance R= 50 µm. In Figs. 3, better
agreement between the analytical curves and pseudo simulation
is achieved by increasing the maximum placement distance R.
Due to the extensive computational demands to simulate
such large molecular communication environments, we assume
that the particle-based simulations have sufficiently verified the
analytical models. The remaining simulation results in Fig. 3
is only generated via Monte Carlo simulation.
B. Performance Evaluation
From Fig. 2 and the scaling values in Table II, we see
that the expected net number of molecules absorbed at the
absorbing receiver is much larger than that inside the passive
receiver, since every molecule arriving at the absorbing receiver
is permanently absorbed. We also notice that the expected net
number of absorbed molecules due to the nearest transmitter is
much larger than that due to the interfering transmitters, which
may be due to a relatively low transmitter density.
Interestingly, the concurrent single pulse transmission by the
transmitters at time t= 0 results in a longer and stronger
channel response at the absorbing receiver than that at the
passive receiver. If the demodulation is based on the number of
observed molecules during each bit interval, the longer channel
response at the absorbing receiver may contribute to higher ISI
than at a passive receiver for the same bit interval, whereas its
stronger channel response may benefit signal detection.
In Fig. 3, we set the parameters: D= 120 ×10−12 m2
s,
R= 100 µm, and Tss = 0.1s. Fig. 3 plots the expected
number of molecules observed at the absorbing receiver and
the passive receiver at time t. We set the density of active
transmitters as λa= 10−3/µm3. As shown in the lower subplot
of Fig. 3, the channel responses of the receivers due to the
nearest transmitter in this large-scale molecular communication
system are consistent with those observed at the absorbing
receiver in [11, Fig. 4] and the passive receiver in [4, Fig. 2]
and [18, Fig. 1] for a point-to-point molecular communication
system.
In Fig. 3, we notice that the expected number of observed
molecules at time tdue to all the transmitters is dominated by
the interfering transmitters, rather than the nearest transmitter,
which is due to the higher density of transmitters. Furthermore,
as we might expect, the expected number of molecules observed
inside the passive receiver at time tstabilizes after t= 0.8s,
whereas that at the absorbing receiver increases linearly with
increasing time. This reveals the potential differences in optimal
demodulation and interference cancellation design for these two
types of receiver.
V. CONCLUSIONS AND FUTURE WORK
In this paper, we provided a general model for the transmitter
modelling in a large-scale molecular communication system
using stochastic geometry. The collective signal strength at a
fully absorbing receiver and a passive receiver are modelled
and examined. We derived tractable expressions for the ex-
pected number of observed molecules at the fully absorbing
receiver and the passive receiver, which were shown to increase
with transmitter density. Our analytical results were validated
through particle-based simulation and Monte Carlo simulation.
The analytical model presented in this paper can also be applied
for the performance evaluation of other types of receiver
(e.g., partially absorbing, reversible adsorption receiver, ligand-
binding receiver) in large-scale molecular communication sys-
tems by substituting its corresponding channel response.
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