I Introduction
In our modern era, the amount of the required health care services is rapidly increasing. These high expectancies are going to overtake a proportional increase in health system infrastructures and professional personnel [1]. Telemedicine, which can be defined as the implementation of telecommunication technologies to provide medical services, has been introduced as a promising solution for the increasing shortages of the traditional medicine.
Telemedicine allows the measurement of biological and vital signals regardless of the boarders and distances. In the telemedicine paradigm, the biological signals are obtained through variety of the biological sensing technologies, from the simple pulse oxiometer and temperature sensors to more sophisticated electrophysiological and electrocardiology devices [2]. The gathered signals must be postprocessed, and eventually transmitted to the associated health care provider. The exterior communication hardwares are placed in order to form the pathway from the onbody devices to at a distant hospital or physicians’ station. One of the advanced applications of the telemedicine is the drug delivery which can be controlled via endtoend (E2E) communication links. The drug delivery is an engineered method to deliver drugs to their targeted locations while minimizing the undesired side effects. One of the most important drug delivery applications is gene therapy. The utilization of the drug delivery in the gene therapy allows to convey of the desired genetic information to the patient’s organism [3]. The main challenge in the gene therapy is minimizing the risk of in vivo toxicity and prolonging the lifespan of the payload. Also, the gene expression is shortlived due to the degradation of the plasmid in the nucleus [4]. Consequently, the high reliable with minimum error probability E2Etelemedicine communication links are crucial in gene therapy.
The E2E communication scenario plays a central and fundamental role in designing the telemedicine system between the health care provider and the human body. The applicability of the telemedicine crucially depends on the reliability of E2E communication links [5]. Therefore, several quality of service (QoS) metrics are defined as the measurement of the performance criteria for E2E communication, in which several reliable communication links must work together in various circumstances and media including “inner”, “intoon”, “on”, and “off” body areas.
Inspired by nature, one of the best solutions for inner body communication is to use chemical signals for carrying the information inside the human body to nanomachines through nanonetworks [6, 7]. This communication paradigm, which is called Molecular Communication (MC) [8], has several advantages in comparison to electromagnetic based and acoustic wave based communication. The advantages includes but not limited to low energy consumption, the biocompatible characteristics, and the existence of the biological receptors to serve as antenna [8]. Similar to Electromagnetic Communication (EC), different aspects of the communication have been studied in MC, such as channel modeling [9], noise and interference [10], and modulation and coding [11]. Recently, Diffusionbased Molecular Communication (DMC) has been received a significant attention among various MC propagation scenarios such as walkway or flowbased [9]. In DMC, without any additional and external energy, the molecules carrying the information propagate via Brownian motion in all available directions in a fluidic medium [12, 9]. However, the major challenge of DMC is its high limited range of the communication [13]. The intermediate nanomachines, which are serving as relays, are deployed to overcome this issue. The performance of relayassisted DMC is investigated in [14].
Intoon body (in2onbody) wireless communication is the EC between nanomachines inside the body and the wearable device on the surface of the body skin [15]. The main propagation environments of electromagnetic waves are inside and around the human body. Onbody wireless communication is the interconnection and networking between wearable devices and the gateway transceivers [15]. And ultimately, offbody wireless communication carries information from gateway transceivers to health care providers. The evaluation of the performance of the special scenario involving both the “inbody to onbody” and “onbody to offbody” electromagnetic wireless propagation links, including bit error rate (BER), energy consumption, and data rate are considered in [16].
In this study, we assume that inner nanomachines send information to a distant health care provider via DMC, innerbody, in2onbody, onbody, and offbody links as E2Etelemedicine communication. The time is divided into multiple slots with equal durations. At each time slot, a message is transmitted from inner nanomachine to the distant health care provider. In addition, the time slots are divided into several symbol durations for conveying the information in each link. The inner nanomachine sends its massage to the relay then the relay sends the received massage to the receiver nanomachine and ultimately through this sequence the massage is received by the health care system. Due to the fact that no buffer is considered in the nanomachines, EC and DMC communications should be in serial^{1}^{1}1 It means as soon as the packet arrives at an intermediate node, it is relayed over the next link towards the destination.. Therefore, BER and the delay of EC and DMC communication affects the E2EBER and the total delay. It implies that the combination of EC and DMC must be considered as an integrated communication link. In addition, some telemedicine services are imposing the limited delivery time of the command from the health care provider to the end nanomachine. It results in a compromise between the EC and DMC symbol durations. The main question we aim to answer is: How to choose the physical layer parameters, such as the symbol durations, in order to minimize the E2E BER? For answering this question, we firstly derive the analytical closedform expression of E2EBER when binary pulse shift keying (BPSK) modulation for EC and onoff keying (OOK) modulation for DMC is employed. Then, we formulate the optimization problem to determine the optimal symbol durations in EC and DMC and solve it by using the bisection algorithm. We also study the effect of the system parameters including the drift velocity and the detection threshold of DMC receiver on the performance of the E2E system.
The rest of paper is organized as follows: In Section II, the system model is described. In Section III, the channel model of each communication is formulated, and the optimization problem is formulated and solved in Section IV. The numerical results are presented in Section V, and finally, the paper is concluded in Section VI.
Ii system model
We assume the E2E ehealth communication includes molecular and electromagnetic wireless communication. In this E2E ehealth system, the information is exchanged between the health care provider and nanomachines/receptors in the body environment. The considered E2E ehealth system is shown in Fig.1. One should note that all innerbody communications are DMC and in2onbody, onbody, and offbody communications are EC. According to the imposed E2E delivery time, the total time duration of each time slot denoted by , is divided into two time durations: one for DMC denoted by , and the other one for EC denoted by (). Fig. 2 illustrates the symbol time duration and the dedicated time intervals of each communication type in the proposed system model.
Iia Innerbody communication
To improve the communication range in MC, the relayassisted DMC system with drift is employed [14]. This MC system consists of a source nanomachine, denoted by Node , which only transmits information signals, a destination nanomachine, denoted by Node , which receives moleculartype signals as information particles, and finally a relay nanomachine, denoted by Node , which is relaying moleculartype signals. Fig. 3 demonstrates the relayassisted diffusionbased molecular communication system employed in this article.
As could be seen in Fig. 2, the relay assisted molecular communication occurs at the beginning of each time slot, i.e., . The assisting relay node is transmitting and receiving in the fullduplex fashion and also we assume is divided into two intervals of equal duration. In the first interval, denoted by , the source transmits the information to the relay. Next, the relay receives this information and transmits the received information towards the destination in the second interval, denoted by . We adopt OOK modulation in which the transmitter nodes ( or ) release molecules to send information bit “1” and no molecule to send information bit “0” at the beginning of the time slot^{2}^{2}2 In the case that another modulation scheme is employed in the molecular communication, just in (14) and (23) should be adjusted accordingly.. Also, node exploits type molecules which can be detected by node , and node uses type molecules which can be detected by node . The use of two different types of molecules guarantees nearly interference free communication.
IiB Wireless communications links
EC between nanomachines inside the body (for example in the blood vessels) and the wearable device on the body skin or at most 20 mm away from it, is called in2onbody communication [16]. The wearable device communicates with node electromagnetically in its own time interval. The communication between the wearable device on the body skin and the gateway on the body or at most 20 cm away from it, is called onbody communication [16]. The gateway connects the onbody part to the offbody part via EC, in its own time interval. Finally, the offbody wireless part carries information from the gateway to the health care provider in the last time interval of the time slot. Further details of EC channel model and the corresponding BER are described in the following section.
The main parameters and notification used throughout this paper is listed in Table I.
parameter  Variable 
Drift velocity  v 
Diffusion coefficient  D 
Molecular noise mean  
Molecular noise variance 

Molecular symbol duration  
Number of molecules for sending bit1  
Distance between nodes T and D  
Detection threshold  
Distance between nanomachine and wearable device  
Distance between wearable device and gateway  
SNR of in2onbody communication  
SNR of onbody communication  
SNR of offbody communication  
Total symbol duration time 
Iii E2E bit error performance
To derive the closedform E2EBER, each type of communication described in the previous section is modeled and ultimately, E2EBER is calculated by concatenating them.
Iiia Molecular communication channel model and BER
Without loss of generality, it is assumed that nodes , , and are placed in one line and the drift velocity is in the direction of a line connecting the nodes through the blood vessel. Therefore, the MC propagation is modeled by one dimensional Brownian motion with drift. Moreover, in an environment with positive drift, the molecules are assumed to follow 1D Brownian motion with drift whose analysis does not change in a 2D or 3D environment, when the environment is isotropic [17]. Furthermore, the closedform expression for the probability of the first hitting to the receiver in a 3D environment with drift is currently intractable [18]. Consequently, according to the direction of the drift velocity and the location of the nodes, we employ the 1D propagation model. It is assumed that the flow of the MC medium neither split to nor concatenated with any other vessel’s blood flow. Otherwise, the drift of the nanomachines must be modeled precisely based on the splitting and merging scenario, which is out of the scope of this paper. Note that the propagation process is governed by both diffusion and drift due to the external active mobility. For the one dimensional Brownian motion with positive drift velocity, time at which a released molecule of type
hits the receiver for the first time, i.e., the absorption time, follows the probability density function (pdf) given by
[19, 17](1) 
where is the diffusion coefficient of molecule of type in the medium, is the distance between the transmitter and the receiver, is the medium velocity, and exp(
) is the exponential function. The cumulative distribution function (CDF) of the absorption time, which is interpreted as the probability of hitting the receiver for a given molecule within time
, is given by [17](2) 
where
is the standard CDF of Gaussian distribution.
Node detects the type molecules that are transmitted by node . In the receiver of node
, maximumaposterior probability (MAP) rule is employed for detection of the transmitted molecule
[14]. The information bit detected by node , denoted by , is given by(3) 
where is the total number of molecules absorbed by node in the source transmission time interval, i.e., , and is the detection threshold at node . After decoding, node reencodes and forwards it to node in the relay transmission time interval, i.e., . The information bit detected by node is denoted by .
follows the normal distribution as
[14](4)  
(5) 
where is the normal distribution, and are mean values and and are variance values which are derived as follows ^{3}^{3}3In this paper, the intersymbol interference is ignored. The analysis of this interference is out of the scope and is relegated as future works. [14]:
(6)  
(7)  
(8)  
(9) 
where in which . is symbol duration in DMC, and is the distance between nodes T and D. The value of can be calculated as [14]
(10) 
By using the MAP detection rule in (3), the conditional probabilities in (10) can be written as [20]
(11) 
where is the error function and . In this model, the communication occurs with no error if holds. Consequently, the error probability for the th bit is calculated as
(12) 
By exploiting the chain rule, the first term on the right side of (12) can be obtained as follows
(13)  
A similar approach can be employed to extend the second term of (12). Finally, after some manipulations, the error probability in (12) can be written as
(14) 
where denotes the detection threshold at node .
IiiB In2on body communication channel model and BER
Our goal in this subsection is to model the statistical BER of the in2onbody channel. The SNR of the link between the nanomachine and the wearable device can be expressed as
(15) 
(16) 
where is the path loss in dB at a reference distance , is the transmission power of the nanomachine, is the symbol duration of in2on body channel, is the path loss exponent, is the distance between nanomachine and the wearable device, is the power spectral density of the zero mean complex additive white Gaussian noise, and
is a normally distributed random variable that models shadowing effect
[21, 16].By considering Binary PhaseShift Keying (BPSK) modulation, the BER of the instantaneous SNR (denoted by ) is given by
(17) 
where erfc() is the complementary error function. The Average BER (ABER) is derived by averaging over which is given by
(18) 
Considering the fact that the SNR is lognormally distributed, the ABER in (18) does not have the analytical closedform solution
[16]. Therefore, the complementary error function is approximated as follows [22]:(19) 
Therefore, the ABER for the lognormal model, using the approximation in (19), is expressed as
IiiC Onbody communication channel model and BER
The path loss of the link between the wearable device and the gateway is a function of the distance, the part of the body that the wearable device is located, and the motion of the human body [16]. The best fitting distribution for these scenarios has been found to be the lognormal distribution [21]. The closedform of BER for the case that SNR is lognormally distributed with mean , and scale parameter , can be obtained in similar to (20). For the case that SNR is lognormally distributed with mean , and scale parameter , the BER can be solved in the closedform similar to (20). However, in this scenario, we have , and , where and are the transmit power of the wearable device, and the symbol duration time of onbody channel, respectively.
IiiD Offbody communication channel model and BER
Rayleigh fading channel with additive white Gaussian noise is assumed for offbody communication link. Thus, average BER of BPSK is given by [24]
(22) 
where is SNR per bit. , , , and are transmit power of the gateway device, symbol duration of offbody channel, the distance between the gateway and the access point and the path loss exponent, respectively.
IiiE E2E communication channel model and BER
In order to compute the E2EBER, we use the fact that BERs of the abovementioned communication links are independent, due to their independent physical mediums. Therefore, the total bit error probability of E2E system can be written as
(23) 
where is the total bit error probability and and are the bit error probability and bit correct probability of link , respectively. In addition, .
Iv Symbol duration optimization problem
Here, we consider the symbol duration of molecular and wireless communications as dependent variables. We assume that the time duration of each slot for E2E communication is predetermined and set to which implies the following equation holds:
(24) 
where indicates symbol duration of the entire wireless communications including onbody, in2onbody, and offbody communication.
Next, we formulate an optimization algorithm for finding the optimum time slot partitioning to achieve the best performance in E2E communication. The objective is to minimize the E2E BER of the system. Therefore, the problem is formulated as
(25) 
Initialization: 
Set lowerbound= 0, upperbound= 1 and 
Iterations: 
Step 1: h= (Lowerbound + upperbound)/2. 
Step 2: Solve the convex feasibility problem (27). 
Step 3: If (27) is feasible, upperbound= h; else lowerbound= h. 
Step 4: If upperbound  lowerbound , then stop. Otherwise, go back to Step 1. 
We also assume is divided into three intervals of equal duration associated to in2onbody, onbody and offbody communication links. One should note that the objective function in (25) is not a convex function, and hence, the optimization problem is not convex. However, it could be shown that the objective function is quasiconvex (see Fig. 8), due to the fact that its domain and all its sublevel sets are convex [25, 14]. One method to solve the quasiconvex optimization problem relies on the representation of the sublevel sets of a quasiconvex function via a family of convex inequalities, as described in [25]. For solving the optimization problem (25), we utilized the bisection method, in which, the optimal symbol duration is computed by solving a convex feasibility problem at each step. According to [25], the hsublevel set () of a function : is defined as
(26) 
where is the set of real numbers. Consequently, the new optimization problem whose constraint is the hsublevel set of the objective function in problem (25) can be written as
Find  (27)  
s.t  
The proposed algorithm based on the bisection method for solving our problem is presented in Table II where is a positive small number.
The bisection optimization solution is a function of the DMC parameters and SNR of EC where SNR of the EC is estimated at the beginning of the transmission as a transmission routine procedure, and DMC parameters are predetermined and fixed. Therefore, the bisection algorithm is run in the EC transmission side which does not have computational complexities. On the other hand, from the computational point of view, this method is a simple and robust rootfinding mathematical algorithm which is of low complexity as investigated in
[26].V Numerical Results
In this section, we present the numerical results to evaluate the error probability performance of the proposed E2E communication system. We also show how the system parameters affect the performance. We consider isomer of hexoses as the messenger molecules in the blood vessels [27]. In molecular communication, we assume that node R is placed in the middle point between nodes T and D and it emits the same number of molecules that node T transmits, to relay the information^{4}^{4}4 Without loss of generality, no channel coding is considered for EC and DMC. If a particular channel coding is engaged, the probability of bit errors must be adjusted accordingly. It is also assumed that the released molecules of type A and type B have the same diffusion coefficient in the medium. The system parameters used in the analysis and simulations are given in Table III.
The impact of the drift velocity, symbol duration, and on molecular BER performance are plotted in Figs. 4, 5, and 6, respectively. We assume that all wireless transmitters communicate with the constant transmit power and only changes. We can write the electromagnetic bit error probability denoted by as follows:
(28) 
where In E2E BER analysis, assume three regimes as follows:
Regime I: is dominant. In this case, in (23) is negligible compared to . Therefore, (23) can be written as = .
Regime II: is dominant. In this case, in (23) is negligible compared to . Therefore, (23) can be written as = .
Regime III: and are in the same range. For this scenario, there is a tradeoff between BER in MC and EC and (23) holds.
We plot the E2E BER for all of three regimes in Fig. 7. Fig.7 shows regime I when the molecular communication is dominant. Fig. 7 illustrates regime II when the electromagnetic wireless communication is dominant. Fig. 7 illustrates regime III that we can see the impact of on molecular, onbody, in2onbody, and offbody BER. As can be seen in Fig. 7, for ms, molecular BER is dominant. However, for ms, wireless BER is dominant. It implies that the optimal point exists for choosing the appropriate and . The performance of the proposed algorithm is evaluated in Fig. 8 for the chosen parameters. As can be observed, by using the proposed algorithm based on the bisection method, the global optimal molecular symbol duration and wireless symbol duration can be reached.
The E2E BER is also a function of the drift velocity, , and the detection threshold at the receiver, , in MC. In Fig. 9, we study the effect of the drift velocity values on the performance of E2E communication system. We set ms. As can be seen, if the value of increases, the E2E BER decreases, especially when the molecular BER is dominant. Moreover, the E2E performance enhances with increasing the drift velocity, and hence, low error probabilities can be achieved. In fact, when velocity increases, the released molecules are absorbed by the receiver with higher probability. As the velocity tends to infinity, i.e., , in (2) approaches to 1 [19], and consequently, no error occurs in the system. Also, in Fig. 10, we study the effect of the value of the detection threshold at the receiver, , on the performance of E2E communication system. As we can see in Fig. 4, affects the molecular BER, and afterwards, affects E2EBER nonlinearly. When the molecular BER is dominant, we can reach to minimum E2EBER by choosing an appropriate which should be considered as future work.
Vi Conclusion
In this paper, we investigated the E2E communication link consisting of the electromagnetic and molecular communication. First, we derived a closedform expression for the E2E bit error probability of concatenation of molecular and wireless electromagnetic communications. Then, we formulated the optimization problem that aims at minimizing the E2E bit error probability of the system to determine the optimal symbol durations for both molecular and wireless electromagnetic communications. In addition, we studied the impact of the parameters consisting of the drift velocity and the detection threshold at the receiver in MC. The results reveal that an adaptive system must be considered to achieve the minimum bit error rate and optimal performance for the E2E system.
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