https://orcid.org/0009-0007-3923-5121
Figures
Abstract
Molecular communication (MC) emerges as an encouraging concept in wireless body area nanonetworks (), which utilizes molecules as information carriers for communication between nanomachines. In this paper, we aim to define an electrical model of a molecular-based nano-transmitter to analyze the effect of the remained transmitted molecules in a fluidic medium. To this end, we will address an advection-diffusion equation with a non-zero initial condition to analyze the residual molecules’ influence the medium. Moreover, considering the energy consumption limitations of nanomachines, we will employ the derived electrical model to further investigate how nanomachines consume the energy in presence of residual molecules. Following this, to enhance the energy consumption of the nano-transmitters, the settle-time method will be proposed to tackle the negative impact of the residual molecules on energy consumption. Nevertheless, since the proposed method increases the delay at nano-transmitters, the energy-delay trade-off relation at nano-transmitters will be investigated. Then, by introducing an interruption period and a control coefficient, we control the trade-off between the energy consumption and the created delay. Finally, by considering insulin molecules as messenger molecules in our simulations, we will demonstrate that implementing short interruption periods significantly enhances energy consumption, while introducing a small amount of delay to the system. Particularly, the energy consumption is reduced by 15% and the latency is increased by 2.2 ms when 1 ms interrupt period is used for 20 mol of insulin molecule.
Citation: Kianfar G, Hosseini P, Azadi M, Abouei J, Mohammadi A (2025) Energy-delay analysis in advection-diffusion-based wireless body area networks. PLoS One 20(8): e0330744. https://doi.org/10.1371/journal.pone.0330744
Editor: Amit Kumar Goyal, Manipal Academy of Higher Education, INDIA
Received: November 15, 2024; Accepted: August 5, 2025; Published: August 29, 2025
Copyright: © 2025 Kianfar et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Data Availability: All data are in the manuscript.
Funding: This work was partially supported by the Natural Sciences and Engineering Research Council (NSERC) of Canada through the NSERC Discovery Grant RGPIN-2023-05654.
Competing interests: No authors have competing interests.
Introduction
Over the last decade, significant technological developments have revolutionized human healthcare systems. One prominent innovation in this area is the advent of wireless body area networks (WBANs), with various applications in remote patient emergency response, telemedicine, assisted living, and patient localization[1–6]. In this field, a nature-inspired communication paradigm, known as molecular communication (MC), has emerged in the realm of wireless body area nanonetworks (s) [7–10]. Unlike traditional electromagnetic (EM)-based WBANs that rely on simple nano-structured sensors, MC employs nanomachines that release molecules as information carriers, enabling communication between transmitters and receivers. While EM-based systems primarily contend with thermal noise and channel fading, MC systems face distinct challenges such as molecular diffusion and inter-symbol interference (ISI) [11].
The nanoscale structure of nanomachines and the need to minimize propagation delays in various intra-body applications make energy consumption and delay two critical factors in MC systems. Therefore, analyzing the energy-delay trade-off in advection-diffusion-based MC nanonetworks presents a promising approach to enhance the functionality and practicality of these systems in healthcare. This paves the way for energy-delay optimization, leading to the reliability and responsiveness of these systems. This particularly improves the performance of time-sensitive applications such as insulin delivery, continuous glucose monitoring, and ECGs.
The problem of delay and energy consumption control in MC systems has been investigated in several papers in the literature [12–14]. In [12], the authors investigated the trade-off between rate and delay, employing a simple network coding mechanism to enhance the rate while considering the system’s delay. The study in [14] also proposed a energy control strategy based on feedback control theory. The need for continuous feedback from the receiver to the transmitter leads to higher energy consumption in the system. However, [12] and [14] neglected the additional energy the transmitter nanomachine consumes to adjust the transmission rate and energy, respectively. This issue becomes especially problematic under strict energy consumption restrictions to adapt the transmission rate. The authors in [13] demonstrated that if the nano-transmitter releases molecules faster than the receiver’s reaction, the excess molecules remaining in the medium degrade the system’s consumption. To address this issue, they proposed an approach wherein the receiver’s feedback controls the transmission rate. Although employing adaptive transmission rates improves energy consumption, continuous feedback from the receiver to the transmitter increases complexity in MC systems and incurs extra energy consumption to adjust the transmission rate. Therefore, in [15], the authors introduced a non-adaptive pre-equalization method to reduce inter-symbol interference (ISI) for binary concentration shift keying (BCSK)-encoded sequences in a single-transmitter single-receiver diffusive system. To this aim, they considered the difference between the number of received molecules of two different types of molecules as the actual signal at the receiver side. Nevertheless, leveraging various types of molecules for transmission introduces additional complexity in terms of molecular control, synthesis, and reception.
In the literature, several modulation schemes are explored for MC, including molecular type-based, time-based, and spatial domain-based modulations [16–18]. Nevertheless, given the importance of simplicity in nanomachines, we choose BCSK modulation with a fixed transmission rate, which is often desirable for sending bits in various real-world applications. Here, the term “fixed transmission rate” does not mean that the communication system employs a fixed rate for sending all the bits; rather, it refers to the release of a specific rate of molecule concentration by the nano-transmitter to modulate bit “1” and no molecules to represent bit “0”. However, when molecules are released at a fixed rate, residual molecules can accumulate, potentially leading to increased molecule loss and longer propagation delays.
The challenges outlined above inspired us to introduce a heuristic electrical model accounting for the presence of residual molecules in the medium and indicating their impact as a generated negative current towards the nano-transmitter. The main contributions of this paper are summarized as follows:
- We utilize the solved advection-diffusion channel equation from [19, 20] to formulate an electrical model for the channel, accounting for non-zero initial conditions. This model mitigates the degradation caused by residual molecules by incorporating a dependent current source. Additionally, we introduce a new parameter, termed the “effective rate”, which signifies the actual rate of released molecules that effectively diffuse through the channel from the transmitter to the receiver nanomachine. This parameter captures the influence of residual molecules by considering their resultant negative current directed toward the transmitter nanomachines. We then prove that the negative current generated during the transmission of the
bit has a recursive relation with the negative currents caused by the preceding n–1 transmitted bits in previous time slots. It is demonstrated that the surplus of residual molecules leads to an exponential reduction in the effective rate.
- In the second part, we investigate how the remaining molecules influence the energy consumption and propagation delay. To this aim, we derive the energy model for the transmitter nanomachine by utilizing the mathematical expression of the channel with non-zero initial conditions. Subsequently, we demonstrate that the transmitter must exponentially increase the amount of released molecules through the medium to maintain a fixed transmission rate. It is also observed that the surplus of the released molecules correlates linearly with the energy consumption of the transmitter nanomachine.
- In the third section, we introduce an algorithm named “settle-time” to address the issue of exponential energy consumption escalation in the transmitter for sending n bits of information. This method primarily involves introducing interruptions between the transmission of each bit, leading to a reduction in the rate of the released molecules from the transmitter.
- Finally, we analyze the trade-off relationship between energy consumption and delay for transmission of a large number of bits. We illustrate that utilizing the relaxation time reduces the energy consumption of the transmitter at the expense of increased delay. Subsequently, we calculate and evaluate the optimal relaxation time that minimizes both energy consumption and propagation delay through simulations.
The rest of the paper is organized as follows: In section, we explain the assumptions and system model and provide the mathematical solution of the advection-diffusion equation under the non-zero initial condition. Section introduces an electrical model of the nano-transmitter in the presence of remaining molecules. The energy model of the nano-transmitter and the impact of residual molecules on its energy consumption are discussed in section. Then, we propose the “settle-time” method to enhance the energy consumption of the system in section. The trade-off relationship between energy consumption and the resulting delay at the nano-transmitter is analyzed in section. Finally, simulation results and conclusions are provided in section and section, respectively.
System model and assumptions
In this section, we formulate the mathematical expression for the channel between the transmitter nanomachine (TN) and the receiver nanomachine, denoted by NMi and NMj, respectively. We assume that the nanomachine NMi modulates bits using the Binary Concentration Shift Keying (BCSK) modulation, similar to [7, 21, 22]. In addition, it is assumed that the nanomachine NMi releases molecules at a fixed rate due to the energy consumption restrictions. Moreover, throughout this paper, the term “delay” refers to the delay incurred by propagating molecules from the transmitter to the receiver nanomachine. Thus, the delay of the transmitter and receiver circuits is neglected. To model the propagation channel from NMi to NMj during the transmission of bit, we apply the advection-diffusion expression in [7] as
in which indicates the number of particles in each unit length of the path (i, j) when
bit is released. Therefore, the number of molecules within a distance
from the TN is derived as
. Additionally, the medium parameters Rij, uij(t), and Dij(t) index the molecules’ degradation rate, the flow’s average velocity, and the medium’s diffusion coefficient, respectively. Therefore, the release rate of molecules through the path (i, j) during the transmission of
bit is obtained by Fick’s Law as follows [23]:
Considering the quantity of molecules remaining from the past transmission along the path (i, j), we face zero or non-zero initial conditions. To solve Eq (1), we focus on the non-zero initial condition as we aim to analyze the impact of remaining molecules on energy consumption and transmission delay. Assuming that uij(t) and Dij(t) remain fixed within each time slot t (i.e., and
), we express the advection-diffusion equation in Eq (1) in the Laplace domain as
where represents the Laplace transform of
. Denoting
as the Laplace transform of
in Eq (2), the rate at which molecule concentration is released from nanomachine NMi through the path (i, j) for sending
bit is obtained as follows:
Then, the solution of the differential equation in Eq (1) under zero initial condition is obtained as
where symbolizes the length of the path (i, j), and
and
represent constant coefficients. In addition, we have
,
, and
. Clearly, the solution of Eq (1) under non-zero initial condition is obtained as
where the particular solution represents the quantity of residual molecules along the path (i, j) considering the non-zero initial condition
after transmitting
bit. The value of this particular solution is calculated as
It should be noted that for
. To find the solution for constants
and
in Eq (6), we apply the boundary conditions
and
as follows:
and
Then, the constants and
are obtained as
where
Substituting Eq (10) and Eq (11) into Eq (6), the number of passing molecules during the transmission of bit through the path (i, j) under the non-zero initial condition (i.e.,
) can be written as follows:
We will employ Eq (13) to propose a novel electrical model for analyzing energy consumption and delay in a nano-transmitter under the non-zero initial condition.
Proposed electrical model of nano transmitter
In this section, we introduce an electrical model of the molecule transmission through the advective-diffusive medium under the non-zero initial condition. To this end, we initially define the concentration of molecules during the release of bit as follows:
where Sij denotes the cross-sectional area of the path (i, j). In addition, and
represent the molecular concentrations at x = 0 and
in the Laplace domain, respectively. By substituting Eq (13) and Eq (14) into Eq (4), concerning
for all initial conditions, the rate at which molecules are released is obtained as
It should be noted that we omitted the subscript “j” at the boundary x = 0 for notation simplification and replaced with
. To provide an analytical tool for representing the molecular communication process in the advection-diffusion,
and Cj(n)(s) are mapped to the electrical current Ii(n)(s) and voltage
, respectively, and we define voltage
+
. Additionally, the mathematical terms corresponding to channel properties in Eq (15), i.e., medium velocity, diffusion coefficient, and path length, can be represented as circuit impedances Z1(s) and Z2(s). As a result, the equivalent electrical circuit model of nano-transmitter NMi is introduced in Fig 1, where the values of Z1(s) and Z2(s) are obtained as
(16)
(17)
Additionally, Zout(s) and refer to the external resistance of the channel and the total current before subtracting the negative current, respectively.
Created adverse current: The adverse current models
, representing the effect of ISI created by the residual molecules from transmitting (n–1)th bit during the
time slot. Therefore, the quantity of remaining molecules that existed through the medium during sending
bit, which denotes the relationship between the adverse current and the residual molecules, is expressed as
where Tn denotes the transmission duration, and . Then, the total adverse current during the transmission of the
bit is calculated by accounting for the adverse current created by sending previous bits as
As depicted in Fig 1, the presence of remaining molecules through the medium reduces the output current by generating adverse current
. As a result, the actual rate of molecules propagating through the medium becomes lower than the rate of released molecules from the transmitter nanomachine. Therefore, to represent the actual rate of diffused molecules through the channel, we define the effective rate
as follows:
where denotes the rate of released molecules from the transmitter nanomachine for sending
bit. As observed from Eq (20), the effective rate of diffused molecules toward the nano-receiver decreases by increasing
. Moreover, as illustrated in Fig 2, the adverse current
monotonically increases with the accumulation of molecules within the medium. Therefore, to maintain a fixed effective rate, it is necessary to steadily increase the rate of released molecules from the nano-transmitter to counteract the effect of
. To this end, the transmitter is required to release additional molecules for transmitting
bit as
To demonstrate how the presence of residual molecules affects the energy consumption of the transmitter nanomachine, we investigate the energy model of the nano transmitter in the next section.
An energy model of nano-transmitter
In this part, we model the energy consumption of the transmitter nanomachine, inspired by the procedure utilized by eukaryotic cells [24–27]. In molecular cell biology, this procedure is known as exocytosis, encompassing four phases as follows (see Fig 3) [28]:
- (1). Synthesis of the messenger molecules from their building blocks.
- (2). Production of a secretory vesicle.
- (3). Carrying the secretory vesicles to the cell membrane.
- (4). Releasing the molecules via the vesicle and cell membrane fusion.
To calculate the total energy consumption Etotal, we denote the energy consumed during these four phases as Es, Ep, Ec, and Ee, respectively. Therefore, the total energy consumed by the transmitter nanomachine for sending N molecules through the medium is obtained as
where represents the ceiling operator, and Nc indexes the number of carried messenger molecules through each vesicle. By utilizing protein molecules as messengers, the synthesis energy refers to the energy required for forming peptide bonds between amino acids. In this case, the energy consumed for forming a bond between two amino acids is constant and equals 202.88 (zJ), where zeptojoule (zJ) is 10−21 Joules [29]. Therefore, the energy consumed for synthesizing one protein molecule composed of Naa amino acids is
. Similarly, the energy required for producing one vesicle mostly composed of phospholipids equals
[30]. Hence, the energy of producing vesicles is Ep = 83 ×
, where
denotes the radius of each vesicle. In addition,
and the value of Ec can be obtained as
, where runit represents the radius of the transmitter nanomachine [28,31]. Taking the above considerations into account, the amount of energy consumption in the nano-transmitter for sending the
bit is obtained as:
Considering Eq (23), we notice that the total energy consumption at the transmitter changes linearly with . In addition, we observed in section [electrical model] that the adverse current monotonically increases the number of remaining molecules in the medium. Consequently, the adverse current increases the energy consumption of the nano-transmitter exponentially. Therefore, a practical solution for minimizing the energy consumption at the nano-transmitter, named “settle-time method”, is proposed to reduce the adverse effect of the remaining molecules.
Proposed settle time method to enhance energy consumption
In this section, we propose an approach to minimize the energy consumption in the nano-transmitter through analyzing the variation of adverse current over time. To obtain the adverse current in the time domain, we employ the inverse Laplace transform of , defined in Eq (12) as follows:
Due to the complexity of calculating a closed-form expression for the adverse current , Eq (24) is numerically analyzed and depicted in Fig 4. As observed, the impact of the adverse current created from sending bit 1 reduces as time passes. Similarly, we can conclude that the number of remaining molecules is also a decreasing function of time. Therefore, the number of residual molecules during the
time interval is primarily influenced by the molecules transmitted in the more recent time slots. As depicted in Fig 5, all transmitted bits before the transmission of
bit have different effects on the amount of residual molecules during the transmission of
bit. Therefore, we ignore the effect of the previously transmitted bits with negligible impact on creating residual molecules to simplify the calculations. To this aim, we define
as the minimum value of the remaining molecules that have a considerable effect on the effective transmission rate of the
bit. Furthermore, we define the counting function of the considerable remaining molecules as follows:
where 1 ≤ k ≤ n − 1. Then, the number of bits with considerable effect on the effective transmission rate of
bit is obtained as
Therefore, using (??), Eqs (19), (??), and (??) can be written as
Thus, the energy consumption of the nano-transmitter is approximated as follows:
Eq (30) demonstrates that the practical energy consumption at the nano-transmitter for sending the bit depends on the exocytosis phases for releasing the
bit and
resulting from transmitting previous bits. Therefore, to decrease the energy consumption of the nano-transmitter, we define the settle-time Ts to apply an interruption between each two consecutive bits as
where indexes the control coefficient of the amount of the adverse current remaining through the medium after transmitting the
bit. In other words, changing ζ controls the value of
, which also impacts the total energy consumption of the nano-transmitter, as shown in Eq (23). By applying interruption Ts between each two successive bit transmissions, we have
where and
represent effective rate and the number of remaining molecules after applying the interruption Ts, respectively. Consequently, the energy consumption of the transmitter nanomachine is approximated as
Eq (34) indicates that the optimum value of occurs when the number of remaining molecules goes toward zero (i.e.,
). This situation happens when
and
. This will result in a too-large Ts, which is impractical for many real-time applications. Therefore, it is necessary to choose a proper value for ζ to have efficient energy consumption and delay. To this end, we will investigate the trade-off between energy consumption and delay via controlling ζ in the next section.
Energy-delay analysis at nano-transmitter
In this section, we analyze the trade-off between the energy consumption and created delay as a result of introduced interruptions. Depending on the nature of the application, we should tune the value of the parameter ζ. For example, the delay is more important than the energy consumption in real-time applications. However, in case we have restrictions on the consumed energy, we can set lower values to ζ and tolerate a larger delay. To this end, we initially define the delay of sending the bit as follows:
where and
denote the delay resulted from the four phases of exocytosis and the propagation delay for sending the
bit, respectively. Considering Eq (31), the total amount of delay for transmitting a packet with the length Np is calculated as
Given that the adverse current is a monotonically decreasing function over time (see Fig 4), increasing the value of ζ in Eq (36) reduces the total delay of the nano-transmitter. Conversely, as indicated in Eq (31), an increase in ζ shrinks the period Ts, leading to higher values of the adverse current
. As discussed, although decreasing Ts reduces the delay, this increases energy consumption, indicating an inherent trade-off between energy consumption and delay.
Remark 1: Based on the above arguments, there exists a linear relation between ζ and the consumed energy, whereas, any variation in the value of Ts exponentially impacts the energy consumption, as will be shown in section [simulation].
Numerical results
In this section, we numerically evaluate the energy consumption of the nano-transmitter and the created delay for sending a data packet from a nano-transmitter toward a nano-receiver. The nano-transmitter modulates the desired information to carrier molecules and prepares them in four phases of exocytosis to release them through the advective-diffusive medium. The values in Table 1 represent the physical characteristics of the nano-transmitter and the medium for insulin propagation as information carrier molecules. Since it is generally unrealistic to assume that the medium is completely free of residual molecules, all experiments in this work are conducted under non-zero initial conditions to more accurately represent practical molecular communication environments, where the presence of remaining molecules from prior transmissions is inevitable. In our work, the mathematical formulation and electrical modeling are parameterized with respect to the properties of the messenger molecule (see Eqs (1)–(7) and Table 1), and can therefore be readily adapted for other molecular species. It is worth noting that the simulations only focus on insulin as the messenger molecule, following our previous works [7, 34]. To analyze the discussed nanonetwork, we consider various scenarios for our simulations as follows:
- Scenario 1: In this scenario, we illustrate the changes in the energy consumption
for different values of remaining molecules
based on Eq (34). Furthermore, we consider different values for Ts and ζ to evaluate the influence of the settle-time method on reducing the energy consumed at the nano-transmitter.
- Scenario 2: In this scenario, we demonstrate how utilizing the settle-time method affects the delay at the nano-transmitter during the transmission of a data packet with Np bits. To this end, we consider different values of Ts and ζ in (36) and plot the total created delay at the nano-transmitter.
- Scenario 3: In the next step, the relationship between the total energy consumption and the created delay at the nano-transmitter is investigated. In this regard, we calculate the value of
for a specific amount of consumed energy using Eq (34). Then, we numerically analyze the impact of the parameter ζ and determine the corresponding created delay for a given value of energy consumption using Eq (36).
Figs 6, 7, and 8 illustrate the total energy consumption at the nano-transmitter for Scenario 1 in terms of remaining molecules, Ts, and ζ, respectively. It is observed from Fig 6 that the energy consumed at the nano-transmitter rises linearly by increasing the number of remaining molecules through the medium, as also illustrated in Eq (23). In addition, it is shown in Fig 7 that by employing the settle-time method and increasing the value of interruption period Ts, the energy consumption at the nano-transmitter exponentially decreases. This behavior can be justified by the exponential reduction of the adverse current over time (see Fig 4), leading to the exponential decrease in the energy consumption. Furthermore, as shown in Fig 8, the energy consumed at the nano-transmitter linearly increases as the parameter ζ grows. This behavior comes from the linear change in the quantity of residual molecules by changing ζ, as observed in Eq (33).
Remark 2: Figs 7 and 8 demonstrate that as the number of remaining molecules in the medium increases, applying the settle-time method significantly reduces the energy consumption at the nano-transmitter. This approach is particularly effective in practice when a large number of residual molecules remain in the medium after each bit transmission due to a low medium velocity.
Figs 9 and 10 depict the delay at the nano-transmitter for the second scenario with varying values of ζ and Ts. In Fig 9, it is shown that decreasing ζ for energy conservation exponentially increases the delay. Based on Eq (33), reducing ζ linearly decreases the number of remaining molecules and the adverse current . Consequently, considering Fig 4 and Eq (31) and Eq (36), the reduction in the adverse current
exponentially increases Ts, leading to an exponential rise in delay. Furthermore, the linear relation between Ts and the delay is illustrated in Fig 10.
Remark 3: As illustrated in Fig 7, by increasing the interruption period Ts, the medium finds enough time to counteract the effect of the remaining molecules, which reduces the energy consumed by the nano-transmitter monotonically. However, we observe from Fig 10 that the amount of the delay increases linearly by increasing the interruption period Ts. Therefore, calculating the optimum values of ζ and Ts is essential for achieving proper energy consumption and delay. To clarify the clinical applicability, the modeled molecular propagation delays (32–39 ms) are well below the maximum threshold of 50 ms for glucose regulation scenarios. In addition, the delay caused by signal processing is much lower than the mechanical delay in diffusion-advection molecular communication, and can be ignored in the total delay computation. Nevertheless, additional delays from decisionmaking algorithms may contribute to the overall system delay, which is not explicitly accounted for in the current model.
To address the problem in Remark 3 and complete our simulation results for Scenario 3, we demonstrate the relation between the consumed energy and the created delay at the nano-transmitter in Fig 11. As shown, there is an exponential relationship between the energy consumption and delay, emphasizing that lower energy consumption can be achieved by setting small values of Ts at the cost of slightly increasing the delay.
Conclusion
In this paper, we proposed a novel electrical model to capture the effect of residual molecules in the medium in an MC system. It was shown that the quantity of residual molecules after each bit transmission can be modeled as a dependent current source, generating an adverse current from the nano-receiver to the nano-transmitter. Our analysis revealed that the quantity of residual molecules resulting from the adverse current decreases monotonically over time. Additionally, we modeled the energy consumption at the nano-transmitter, demonstrating that an increase in the amount of residual molecules increases the total consumed energy at the nano-transmitter. To mitigate the negative impact of the residual molecules on energy consumption, we proposed the settle-time method and demonstrated its effectiveness through numerical results. It was shown that a trade-off exists between the length of the time interruptions in the settle-time method and the total created delay in the system, which can be controlled by adjusting the interruption period Ts and the parameter ζ. Finally, we concluded that utilizing small values of Ts can significantly reduce energy consumption, while the resulting delay at the nano-transmitter increases slightly. In this work, we performed the simulations based on Insulin molecules as the messenger molecule. Our future research involves extending the analysis and investigating how molecular properties affect the energy–delay trade-offs for a wider range of messenger molecules commonly considered for intra-body communication.
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