Advanced Target Detection via Molecular Communication
Reza Mosayebi, Wayan Wicke, Vahid Jamali, Arman Ahmadzadeh, Robert Schober, Masoumeh Nasiri-Kenari
aa r X i v : . [ c s . ET ] M a y Advanced Target Detection via MolecularCommunication
Reza Mosayebi † , Wayan Wicke ‡ , Vahid Jamali ‡ , Arman Ahmadzadeh ‡ ,Robert Schober ‡ , and Masoumeh Nasiri-Kenari †† Sharif University of Technology, Tehran, Iran; ‡ University of Erlangen-Nuremberg, Erlangen, Germany
Abstract —In this paper, we consider target detection in suspi-cious tissue via diffusive molecular communications (MCs). Ifa target is present, it continuously and with a constant ratesecretes molecules of a specific type, so-called biomarkers , into themedium, which are symptomatic for the presence of the target.Detection of these biomarkers is challenging since due to thediffusion and degradation, the biomarkers are only detectable inthe vicinity of the target. In addition, the exact location of thetarget within the tissue is not known. In this paper, we proposeto distribute several reactive nanosensors (NSs) across the tissuesuch that at least some of them are expected to come in contactwith biomarkers, which cause them to become activated. Uponactivation, an NS releases a certain number of molecules of asecondary type into the medium to alert a fusion center (FC),where the final decision regarding the presence of the target ismade. In particular, we consider a composite hypothesis testingframework where it is assumed that the location of the target andthe biomarker secretion rate are unknown, whereas the locationsof the NSs are known. We derive the uniformly most powerful(UMP) test for the detection at the NSs. For the final decisionat the FC, we show that the UMP test does not exist. Hence, wederive a genie-aided detector as an upper bound on performance.We then propose two sub-optimal detectors and evaluate theirperformance via simulations.
I. I
NTRODUCTION
During the past years, diffusion-based molecular commu-nication (MC) systems have received significant attentionas a candidate for the design of bio-inspired nanonetworks,because of their size scale, biocompatibility, and low energyconsumption [1]. Bio-inspired nanonetworks have many po-tential applications, especially in healthcare and environmentalmonitoring [1].One of the fundamental challenges in healthcare monitoringis the problem of early detection of signs of an anomaly inthe body, like the presence of a tumor [2], which we referto as target detection [3]. In particular, for tumor detection,direct detection of the tumor cells themselves is difficult sincetheir size is small and their locations are unknown. Instead,a significant body of research has been devoted to detectingprotein molecules, referred to as biomarkers , that are secretedby the tumor cells into blood vessels and tissue [4], [5].However, detection of these biomarkers is also challengingsince due to diffusion and degradation, the biomarkers mightbe detectable only in the vicinity of the target. In addition, theexact location of the target within the tissue and the secretionrate of the biomarkers are in general not known. Detection ispossible if a sensor with the ability to detect these biomarkersis placed in the vicinity of the target. Due to recent advances innanotechnology, one interesting approach to detect biomarkersis to employ engineered nanosensors (NSs) [4], [6]. Target detection in MC systems is different from targetdetection in wireless sensor networks because of the signal-dependent noise in the MC channel and the possibility ofreactions between molecules. In the MC literature, the problemof anomaly detection was considered in [7]–[12]. In [7]–[9]the use of mobile nanosensors (MNSs) is proposed to detectthe presence of anomaly in the vasculature. In particular, in [7]and [8], it is assumed that MNSs move through the vasculatureand gather at the target location by binding to the target. In ourrecent work [9], anomaly detection using MNSs is proposedwhere the MNSs are activated if they come in contact withbiomarkers. The MNSs move through the vasculature and arethen collected by a fusion center (FC), which decides on thepresence of anomaly. In [10]–[12] employing fixed NSs isproposed for target detection in body tissue. In particular, in[10], [11], the channel between the NSs and the FC is assumedto be an additive white Gaussian noise channel; while in [12]a Poisson signal-dependent noise channel is considered.In this paper, similar to [10]–[12], we assume that multipleNSs are placed on the surface of a suspicious tissue, whichwe refer to as the surveillance area , along with an FC.However, we adopt a more realistic receiver model comparedto [10]–[12] for the NSs and the FC, respectively, namely ageneral reactive receiver model [13]. Similar to [9], we assumethat a target continuously releases biomarkers into the tissue,including the surveillance area. If a biomarker reaches an NS,it may react with the receptors on the surface of the NS andthus activate them. If the number of activated receptors of anNS exceeds a threshold, it will secrete a certain number ofmolecules of a secondary unique type that is detectable bythe reactive FC into the environment. Unlike [12], we makethe realistic assumption that both the location of the possibletarget and its biomarker secretion rate are unknown. However,the locations of the NSs are assumed to be known. The maincontributions of this paper can be summarized as follows:1) We derive an analytical expression for the probabilitymass function (PMF) of the number of activated receptorson the surface of an NS which is a function of thetarget location and the continuous biomarker secretionrate. We validate the result with particle-based simulationof Brownian motion and the reactive receiver.2) Next, we derive the optimal hard detection scheme forthe NSs and show that it corresponds to a uniformly mostpowerful (UMP) test. The UMP test is a test that, withoutknowledge of unknown parameters, performs equal to theoptimal Neyman-Pearson detector [14] that knows theparameters. In our case, these parameters are the location iomarkers (A)MoleculesATargetNanosensorskFC ab Fig. 1. Schematic diagram of the considered system with K = sensors. of the target and the biomarker secretion rate.3) Finally, we develop a composite hypothesis testing frame-work for the FC, where the location of the possible targetand the secretion rate of the biomarkers are unknown.We then derive a genie-aided detector (GAD), whichprovides a performance upper bound for any realizabledetector at the FC. Furthermore, we propose two sub-optimal detectors for practical detection. The performanceof the proposed detectors is evaluated via Monte Carlosimulation.The rest of this paper is organized as follows. Section IIintroduces the system model. In Section III, we derive thePMF of the number of activated receptors and the optimaldetector for the NSs. In Section IV, we introduce the GADand propose two sub-optimal detectors for the FC. We evaluatethe performance of the proposed detectors via simulations inSection V, and conclude the paper in Section VI.II. S YSTEM M ODEL A ND P RELIMINARIES
A. System Model
We consider an unbounded three-dimensional (3-D) envi-ronment with constant temperature and viscosity, a possibletarget located at position x T , K identical spherical reactive NSswith radius a located at positions x k , k ∈ K , { , , ..., K } , anda spherical FC located at x F with radius b , as depicted in Fig.1. When the target is present in the surveillance area, weassume that it continuously secretes biomarkers, which aredenoted as type A molecules, at position x T into the envi-ronment with secretion rate µ [biomarkers · s − ]. We denotethe presence (abnormality) and absence (normality) of thetarget by hypotheses H and H , respectively. The secretedbiomarkers independently diffuse in the environment withconstant diffusion coefficient D and may reach an NS. Wefurthermore assume that the secreted biomarkers can degradeat a rate of k d [s − ] via a first-order degradation reaction ofthe form A k d → ⊘ , (1)where ⊘ is a species of molecules that is not recognized bythe NSs nor the FC.We assume that each NS has M receptor proteins on itssurface, which we refer to as B molecules and are modeled asdisks with radius r d . Biomarkers that come in contact with anNS may reversibly react with the B molecules on the surface of the NS and activate them via a second-order reversible reactionas follows A + B k f ⇄ k b C , (2)where k f and k b are forward and backward reaction ratesin molecule − m s − and s − , respectively, and one generatedmolecule of type C represents one activated receptor. Wedefine P ( t ; x T , x k ) as the probability that a given biomarkersecreted by the target at time t = and at location x T activates a receptor of the k -th NS centered at location x k at time t . We also assume that there are other sources oftype A molecules. Theses additional molecules are regarded asenvironmental noise. For the number of activated receptors atthe k -th NS, which constitutes the received signal, we definerandom variable (RV) Y k and its realization y k . Based on y k , the k -th NS makes a local hard decision denoted by c k regarding the presence of the target. If the NS decides thatthe target is present, it relays this information to the FC by aone-shot instantaneous emission of N secondary molecules oftype A k at time T with diffusion coefficient D k ; otherwise,the NS does not emit any molecule. For simplicity, we assumethat the emission time T for all NSs is identical.The molecules released by the NSs may reach the FC,which can perform more complex operations than the NSs.Furthermore, the FC may be connected to an outside computerwhich may perform computationally expensive processingtasks if needed. We assume that the FC is reactive with respectto all molecules of type A k , k ∈ K , and has M ′ receptors oftype B k , k ∈ K , on its surface, where each receptor is modeledby a disk with radius r F . For the secondary molecules, wealso assume first-order degradation reactions in the channeland second-order reversible reactions at the FC, i.e., A k k d , k → ⊘ , A k + B k k f , k ⇄ k b , k C k , k ∈ K , (3)where k f , k , k b , k , and k d , k are the corresponding forward,backward, and degradation rates, respectively. We also define P ( T − T ; x k , x F ) as the probability that a given moleculeof type A k , emitted at time T at x k , activates a receptorof type B k at time T at the surface of the FC centered at x F . In addition, we assume that the FC is able to count thenumbers of molecules of type C k , k ∈ K , at time T , whichcorresponds to the signal received from the k -th NS and ismodeled by RV Z k and its realization z k . We assume a sourceof environmental noise for all type C k molecules. At the FC,for the tractability of the analysis and to reduce the complexityof the decision rule, we assume that first a hard decision,denoted by d k , is made regarding the relayed message fromthe k -th NS by comparing z k with a specific threshold. Then,based on all K hard decisions, which are collected in vector d = [ d , d , · · · , d K ] T , the FC decides on the presence of thetarget. B. Preliminaries
In this subsection, we briefly review the reactive receivermodel in [13], as we adopted this model for both the NSsnd the FC. In particular, we derive the receptor activationprobability at the k -th NS, P ( t ; x T , x k ) , as a function of theparameters of the channel between the target and the k -th NS.Using [13] and assuming that the NSs are placed sufficientlyfar from each other such that their received signals do notinfluence each other, the received signals of different NSscan be assumed to be independent. Therefore, we obtain thefollowing expression for P ( t ; x k , x T ) [13, Eq. (29)] P ( t ; x T , x k ) = k f exp (− k d t ) π √ Da k x T − x k k ( α W (cid:16) k x T − x k k− a √ Dt , α √ t (cid:17) ( γ − α )( α − β ) + β W (cid:16) k x T − x k k− a √ Dt , β √ t (cid:17) ( β − γ )( α − β ) + γ W (cid:16) k x T − x k k− a √ Dt , γ √ t (cid:17) ( β − γ )( γ − α ) ) , (4)where k·k is the ℓ norm, W ( n , m ) = exp ( nm + m ) erfc ( n + m ) , erfc (·) denotes the complementary error function, and α , β ,and γ are the solutions of the following system of non-linearequations α + β + γ = (cid:16) + k ⋆ f π aD (cid:17) √ Da ,αγ + βγ + αβ = k b − k d ,αβγ = k b √ Da − k d (cid:16) + k ⋆ f π aD (cid:17) √ Da . (5)We note that α, β, and γ can be complex numbers. In (5), k ⋆ f is given by k ⋆ f = π Dk f ϕ k f a ( − ϕ ) + π D , (6)where ϕ is the same for all NSs and is given by ϕ = Mr d ( k f a + π D ) a ( − λ )( π r d k f + π D ) + Mr d ( k f a + π D ) , (7)and λ = M π r d /( π a ) .Similarly, the receptor activation probability for the k -threceptor type at the FC, P ( T − T ; x k , x F ) , can be obtainedfrom (4) by substituting M , r d , a , D , k f , k b , and k d with M ′ , r F , b , D k , k f , k , k b , k , and k d , k , respectively, and also solving(5) with the parameters of the channel between the k -th NSand the FC.III. D ETECTOR D ESIGN AT N ANOSENSORS
In this section, we first derive the steady-state PMF of thereceived signal at the k -th NS. Then, based on the derivedPMF, we derive the optimal local decision rule at the k -thNS. A. PMF of the Received Signal at the NSs
In this subsection, first, we derive the average value of thereceived signal, i.e., the mean number of activated receptorsat the k -th NS. Then, given this mean, we derive the PMF ofthe received signal. By using (4) and considering the fact thatthe target is secreting biomarkers at a constant rate of µ , theaverage value of the received signal at the k -th NS at time t can be obtained by integrating P ( t ; x T , x k ) over time, i.e., ∫ t µ P ( τ ; x T , x k ) d τ. (8) The asymptotic mean value of the received signal at the k -thNS, denoted by m k , is obtained for t → ∞ , and is given inLemma 1. Lemma 1.
The steady-state average value of the receivedsignal at the k-th NS is given bym k = µ g ( x T , x k ) , µ k f exp (cid:18) (−k x T − x k k + a ) q k d D (cid:19) π √ D a k x T − x k k× ( α ( γ − α )( α − β )( α √ k d + k d ) + β ( β − γ )( α − β )( β √ k d + k d ) + γ ( β − γ )( γ − α )( γ √ k d + k d ) ) . (9) Proof:
Due to the space limitation, we only provide asketch of the proof. In particular, by substituting (4) in (8),taking the limit t → ∞ , and using the following integral [15,Eq. (4.3.34)] ∫ ∞ x = erfc (cid:18) ax + bx (cid:19) exp (cid:16) − c x (cid:17) x d x = ( a + c ) − × (cid:16) a + √ a + c (cid:17) − exp (cid:16) − b ( a + √ a + c ) (cid:17) , (10)which holds if R( b ) > and R( a + c ) > , where R(·) isthe real part operator, we arrive at (9).In Section V, we show that for a finite t , the average valueof the received signal closely approaches the asymptotic value.In the following, we calculate the PMF of the receivedsignal at the NSs, which we use for the subsequent analysis.When the target continuously secretes biomarkers, since therelease time instances of the biomarkers are different, thePMF of the received signal at the k -th NS follows a general Poisson binomial distribution. Although the Poisson binomialdistribution is cumbersome to work with, it can often beapproximated by a Poisson distribution when the number oftrials (secreted biomarkers) is large and the success probability P ( t ; x T , x k ) is small, cf. [16]. Since these conditions are metin typical MC environments, we approximate the receivedsignal at the k -th NS by a Poisson RV. The accuracy of thisapproximation is evaluated in Section V. Furthermore, we alsomodel the additive and independent environmental noise by aPoisson distribution [17], [18] with mean ζ , which is presentunder both hypotheses H and H . Hence, the received signalat the k -th NS is modeled by Y k ∼ (cid:26) Poisson ( ζ ) , under H , Poisson ( ζ + µ g ( x T , x k )) , under H , (11)where Poisson ( x ) denotes the Poisson distribution with mean x and g (· , ·) is defined in (9). Since µ and x T are not known,we obtain the following composite hypothesis testing problem (cid:26) H : if µ = , H : if µ > , x T ( nuisance ) . (12)We note that the discriminator parameter between the twohypotheses is µ ; while x T is a nuisance parameter that onlyexists for hypothesis H . . Optimal Local Detector at the NSs In this subsection, our goal is to design the optimal localdetector based on the received signal y k at the k -th NS thatmaximizes the local detection probability subject to a pre-assigned upper bound ω on the local false alarm probabilityat the NS, i.e., max local detectors P T-Sd , k , subject to P T-Sfa ≤ ω , (13)where P T-Sfa and P
T-Sd , k are respectively the local false alarm andthe detection probabilities for the link between the target andthe k -th NS. We note that P T-Sfa only depends on the decisionthreshold and ζ , while P T-Sd , k depends on the decsion threshold, ζ , and the unknown parameters µ and x T . If the location of thetarget and its biomarker secretion rate were known at the NSs,the optimal detector for (13) would be the Neyman-Pearsondetector [14], which compares the local log-likelihood ratio(LLR) with the maximum threshold that ensures P T-Sfa ≤ ω .Denoting the local LLR of the k -th NS as λ k , we obtain λ k , log P (cid:0) y k (cid:12)(cid:12) H (cid:1) P (cid:0) y k (cid:12)(cid:12) H (cid:1) ! = log (cid:18) µ g ( x T , x k ) + ζ ζ (cid:19) y k − µ g ( x T , x k ) , (14)where P (·) denotes probability. From (14), we obtain thatindependent of the value of the unknown parameters µ > and x T , the local LLR λ k is a monotonic increasing functionof y k . Therefore, by using the Karlin-Rubin theorem [19], wearrive at the following UMP test:c k = (cid:26) , if y k ≤ τ , , if y k > τ , (15)where τ is the decision threshold at the NSs and is the samefor all NSs. Clearly, since P T-Sfa in (15) is a decreasing functionof τ , we obtain τ = { max τ : P T-Sfa ( τ ) − ω ≤ } . Therefore,instead of comparing λ k , which we cannot evaluate due tothe unknown parameters µ and x T , we can directly compareobservation y k with τ , which yields the same performanceas the optimal Neyman-Pearson detector that employs (14)as decision variable and compares it with a correspondingmaximum threshold such that P T-Sfa ≤ ω . Given the proposedoptimal local detector in (15), we can evaluate the local falsealarm probability as followsP T-Sfa = P (cid:0) c k = (cid:12)(cid:12) H (cid:1) = P (cid:0) y k > τ (cid:12)(cid:12) H (cid:1) = ∞ Õ i = τ + exp (− ζ )( ζ ) i i ! , H ( τ , ζ ) , (16)which is a decreasing function of τ . Similarly, we can evaluatethe local detection probability of the k -th NS asP T-Sd , k = P (cid:0) c k = (cid:12)(cid:12) H (cid:1) = H ( τ , ζ + µ g ( x T , x k )) . (17)IV. D ETECTOR D ESIGN AT THE
FCAs mentioned in Section II, the FC is a reactive receiversensitive to all A k , k ∈ K , molecules. Employing the sameapproach as in Section III-B for deriving the local decision rule of the NSs, we arrive at the following initial hard decisionrule at the FC for detection of the signal sent by the k -th NSd k = (cid:26) , if z k ≤ τ , , if z k > τ , (18)where z k is the received signal from the k -th NS, i.e., thenumber of type C k molecules produced at time T at the FC,and τ is the threshold that the FC employs to detect the signalreceived from the NSs. For simplicity, τ is assumed to beidentical for all NSs. Since each NS secretes the secondarymolecules instantaneously at time T , Z k is distributed as Z k ∼ (cid:26) Poisson ( ζ k ) , if c k = , Poisson ( ζ k + N P ( T − T ; x F , x k )) , if c k = , (19)where ζ k , k ∈ K , is the average number of environmentalnoise molecules of type C k , k ∈ K , and N is the numberof molecules of type A k released by the k -th NS if c k = .Similar to (16) and (17), we can derive the false alarm anddetection probabilities for the hard decision rule in (18) forthe link between the k -th NS and the FC, which we denote byP S-Ffa , k and P S-Fd , k , respectively. We note that since the locationsof the NSs are assumed to be known at the FC, both P S-Ffa , k andP S-Fd , k are known by the FC for all k ∈ K .To model the hypothesis test at the FC, we express d k in(18) in terms of the hypotheses H and H . To this end, byconsidering (15) and (18) we arrive at a binary non-symmetricchannel between the target and the FC with the followingtransition probabilities P ( d k = ) = (cid:26) P S-Fd , k P T-Sfa + P S-Ffa , k ( − P T-Sfa ) , ρ , k , under H , P S-Fd , k P T-Sd , k + P S-Ffa , k ( − P T-Sd , k ) , ρ , k , under H , (20)where ρ , k is a function of the unknown parameters µ and x T ,i.e., we can also write ρ , k ( x T , µ ) .At the FC, the goal is to design the optimal detector (basedon hard decision vector d ) that maximizes the global detectionprobability denoted by P d subject to a pre-assigned upperbound ω on the global false alarm probability denoted byP fa , and given τ and τ . That is, max detectors P d , subject to P fa ≤ ω , and given τ and τ . (21)Now, similar to Section III-B, if x T and µ were known atthe FC, the optimal detector for (21) would compare the totalLLR of d with the maximum threshold such that P fa ≤ ω .The total LLR can be obtained as follows LLR = log P (cid:0) d (cid:12)(cid:12) H (cid:1) P (cid:0) d (cid:12)(cid:12) H (cid:1) ! = K Õ k = LLR k (22) , K Õ k = (cid:26) log (cid:18) ρ , k ρ , k (cid:19) d k + log (cid:18) − ρ , k − ρ , k (cid:19) ( − d k ) (cid:27) . Since ρ , k is a function of unknown parameters µ and x T ,we have a similar composite hypothesis testing problem asin (12), and thus cannot directly employ (22). Nevertheless,as a benchmark scheme, we use (22) and assume x T and µ are known. This detector is referred to as GAD and yields anpper bound on the achievable performance of any practicaldetector at the FC that does not know x T and µ .Unlike (14), (22) may not be a monotonic function of d k .Therefore, we cannot use the Karlin-Rubin theorem to changethe structure of the detector in (22) such that it does notrequire knowledge of µ and x T . In addition, since the nuisanceparameter x T apprears only under hypothesis H , we cannotdirectly use many of the detectors proposed in the literaturefor composite hypothesis testing, such as the locally optimumdetector (LOD) or Rao and Wald tests [14]. Instead, in thefollowing subsections, we derive two (generally) sub-optimaldecision rules for the FC. A. Generalized-Likelihood Ratio Test
A common approach for the composite hypothesis testingis the generalized-likelihood ratio test (G-LRT). The G-LRTdecision variable can be expressed as [14]: T G-LRT = K Õ k = ( log (cid:18) ρ , k (cid:0)b x T , b µ (cid:1) ρ , k (cid:19) d k + log (cid:18) − ρ , k (cid:0)b x T , b µ (cid:1) − ρ , k (cid:19) ( − d k ) ) , (23)where b x T , and b µ denote the ML estimates of x T and µ underhypothesis H , i.e., (cid:0)b x T , b µ (cid:1) = argmax x T ,µ P (cid:0) d (cid:12)(cid:12) H , µ, x T (cid:1) = argmax x T ,µ (cid:26) log (cid:0) ρ , k (cid:0)b x T , b µ (cid:1) (cid:1) d k + log (cid:0) − ρ , k (cid:0)b x T , b µ (cid:1)(cid:1) ( − d k ) (cid:27) . (24)To perform the G-LRT, the decision variable in (23) is com-pared with the maximum threshold such that P fa ≤ ω , whichwe denote by τ . Since we cannot analytically solve (24) for b µ and b x T , we find b µ and b x T numerically, e.g. via a grid search,cf. Section V. As a result, the complexity of the G-LRT is highsince the search has to be performed with respect to both x T and µ . Hence, to reduce the complexity of the final decisionrule at the FC, in the following subsection, we derive anotherdetector which is less complex. B. Generalized-Locally Optimum Detector
A different approach for the case that the unknown param-eters are only present under hypothesis H is the detectorproposed in [20]. In the following, we refer to this detector as generalized-locally optimum detector (G-LOD), to underlinethe use of an LOD in the decision rule. The decision variableof the G-LOD can be written as T G-LOD = max x T ∂ log (cid:16) P (cid:16) d (cid:12)(cid:12) H ,µ, x T (cid:17)(cid:17) ∂µ p I ( x T , µ = ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) µ = , (25) Following [14], it can be proven that the performance of the LOD is closeto the Neyman-Pearson detector if µ is very small. where I ( x T , µ ) denotes the Fisher information, i.e., I ( x T , µ ) = E ∂ log (cid:0) P (cid:0) d (cid:12)(cid:12) H , µ, x T (cid:1) (cid:1) ∂ µ ! , (26)and E (·) is the expectation operator. In the following, first wederive the derivative in the numerator of (25) before we useit to evaluate (26). The derivative can be written as ∂ log (cid:0) P (cid:0) d (cid:12)(cid:12) H , µ, x T (cid:1)(cid:1) ∂ µ = K Õ k = ∂ρ , k ∂µ ρ , k d k − ∂ρ , k ∂µ − ρ , k ( − d k ) . (27)To proceed, we need ( ∂ ρ , k )/( ∂ µ ) which is given in Lemma2. Lemma 2.
The derivative of the PMF of decision vector d with respect to µ under hypothesis H is given by ∂ ρ , k ∂ µ = exp (− µ g ( x T , x k ) − ζ ) ( µ g ( x T , x k ) + ζ ) τ τ ! × g ( x T , x k ) (cid:16) P S-Fd , k − P S-Ffa , k (cid:17) . (28) Proof:
By replacing (16) and (17) in (27), we obtain ∂ ρ , k ∂ µ = ∂∂ µ ( H ( τ , µ g ( x T , x k ) + ζ )) (cid:16) P S-Fd , k − P S-Ffa , k (cid:17) , (29)where ∂∂ µ ( H ( τ , µ g ( x T , x k ) + ζ )) = ∞ Õ k = τ + ∂∂ µ exp (− µ g ( x T , x k ) − ζ ) ( µ g ( x T , x k ) + ζ ) k k ! ! = ∞ Õ k = τ + k ! (cid:26) − g ( x T , x k ) exp (− µ g ( x T , x k ) − ζ )× ( µ g ( x T , x k ) + ζ ) k (cid:27) + ∞ Õ k = τ + k ! (cid:26) k g ( x T , x k )× exp (− µ g ( x T , x k ) − ζ )( µ g ( x T , x k ) + ζ ) k − (cid:27) = exp (− µ g ( x T , x k ) − ζ ) ( µ g ( x T , x k ) + ζ ) τ τ ! g ( x T , x k ) . (30)By substituting (30) in (29), we arrive at (28). This completesthe proof.Using similar algebraic calculations, we obtain the follow-ing expression for the Fisher information in (26) I ( x T , µ ) = (cid:18) exp (− µ g ( x T , x k ) − ζ ) ( µ g ( x T , x k ) + ζ ) τ τ ! (cid:19) × (cid:16) g ( x T , x k ) (cid:16) P S-Fd , k − P S-Ffa , k (cid:17) (cid:17) ρ , k ( − ρ , k ) . (31)Now, by plugging (27), (28), and (31) into (25), we obtain thefollowing decision variable for the G-LOD T G-LOD = max x T Í Kk = ϑ k (cid:16) d k ρ , k − − d k − ρ , k (cid:17)rÍ Kk = ϑ k ρ , k ( − ρ , k ) , (32) ABLE IL
IST OF I MPORTANT S IMULATION P ARAMETERS F OR THE I NDIVIDUAL N ETWORK C OMPONENTS [13]
AND THE N ETWORK T OPOLOGY .Param. Value Param. Value D , D k × µ m s − µ s − k d − s − k d , k × − s − k f . × µ m s − mol − k f , k . × µ m s − mol − k b . × − s − k b , k × − s − M , M ′ . × { a , b } { . , } µ m r d × − µ m r F . × − µ m ζ ζ k τ τ x T ( , , ) µ m x F (− , − , ) µ m T ms T ms where ϑ k , g ( x T , x k ) exp (− ζ ) ( ζ ) τ τ ! (cid:16) P S-Fd , k − P S-Ffa , k (cid:17) . (33)The G-LOD makes the final decision on the presence ofthe target by comparing (32) with threshold τ . Thereby, thecomplexity of the G-LOD is lower than that of the G-LRT,since the maximization in (32) is only with respect to x T .V. N UMERICAL R ESULTS
In this section, first, we validate the results derived for themean and the distribution of the number of activated receptors(received signal) at an NS via the particle-based simulatordeveloped in [13]. Then, we consider a sample networkconsisting of several NSs and evaluate the performance ofthe proposed detectors by plotting the global probability ofmissed detection P m = ( − P d ) and the global probability offalse alarm ( P fa ) . Table I summarizes the system parametersthat were used for all simulations, unless stated otherwise.Here, “mol” is used for the abbreviation of “molecule”.Fig. 2 depicts the average received signal at an NS versustime for system parameters x k = ( , , ) µ m, x T = ( , , ) µ m,and µ = . The particle-based simulation results in Figs. 2and 3 were averaged over × independent realizations ofthe channel with a simulation step size of × − µ s. Sincethere are three reaction rates k f , k b , and k d , due to the spacelimit, we only present results for the case when k b is altered.In Fig. 2, we show three sets of curves, where the analyticalcurves are obtained by numerically evaluating (8) for each t and the asymptotic curves are obtained from (9). We observethat the analytical and simulation results are in excellentagreement. In addition, for all considered values of k b , theaverage received signal reaches its asymptotic value before µ s. We also observe that when k b increases, the asymptoticmean number of activated receptors decreases. This is due tothe fact that as k b increases, the rate of the backward reactionincreases which reduces the number of activated receptors ata given time.Fig. 3 shows the histograms for the received signal at an NSfor the same parameters as in Fig. 2 at time t = T = ms.In Fig. 3, we also show the Poisson PMF approximationsfor the received signal at the NS in (11) for ζ = . As t ( µ s) A v e r age r e c e i v ed s i gna l k b =1.5 × −4 k b =5 × −5 k b =5 × −4 Particle−based simulationAnalysisAsymptotic (9)
Fig. 2. Average received signal at an NS centered at ( , , ) µ m as a functionof time for different k b . Value P r obab ili t y Particle−based simulationPoisson Approximation k b =1.5 × −4 k b =5 × −5 k b =5 × −4 Fig. 3. Poisson approximation and histogram, obtained from particle-basedsimulation for the received signal at an NS centered at ( , , ) µ m for different k b . can be observed, the histogram of the received signal at theNS is very well approximated by the Poisson PMF for allconsidered scenarios. Hence, this result confirms the accuracyof the proposed approximation of the received signal at the NSby a Poisson PMF with the mean in (9). Similar observationshave been made for the signals received from the NSs at theFC.In the following, in Figs. 4 and 5, we compare the per-formance of all proposed detectors in terms of the globalfalse alarm and missed detection probabilities by averagingover × independent Monte Carlo simulations, wherethe distributions of the received signals at the NSs and theFC are obtained based on the expressions given in (11) and(19). The locations of the target and the FC are given inTable I. For the NSs, in Fig. 4, for each simulation, weuniformly distribute the centers of K = NSs in a 2-D squaresurveillance area with an edge length of µ m and centeredat the origin. To evaluate the performance of the sub-optimaldetectors, we need to find the ML estimates of x T and µ , whichmay have no analytical solutions. Therefore, as mentionedin Section IV-A, we propose to perform a grid search as anapproximation method to obtain b x T and b µ . To determine b x T and b µ , we use the grid points defined by sets {( x j , y j ′ , ) : x j = − . + j / , y j ′ = − . + j ′ / j , j ′ = , , ..., } and { l µ / , l = , , ..., } , respectively, where x j , y j ′ are in µ m. Therefore, for each realization, for the G-LRT in (23) andthe G-LOD in (32), the maximization is reduced to a searchamong × and candidates, respectively. Note that theFC can be a complex node, e.g., a processor that can performthe grid search or a node that is connected to a computer thatcan perform the grid search.In Fig. 4, we compare the performance of the proposeddetectors by plotting the global probability of missed detection, -4 -3 -2 -1 P fa -4 -3 -2 -1 P m G-LOD, µ = 6 × G-LRT, µ = 6 × GAD, µ = 6 × G-LOD, µ = 4 × G-LRT, µ = 4 × GAD, µ = 6 × Fig. 4. Global probability of missed detection versus global probability offalse alarm for N = , and µ ∈ { × , × } .
30 40 50 60 70 80 90 100 110 120 K -4 -3 -2 -1 P m G-LODG-LRTGAD
Fig. 5. Global probability of missed detection versus the number of NSs K for a given false alarm probability of P fa = − , N = , and µ = × . P m , versus the global probability of false alarm, P fa , for N = and µ ∈ { × , × } . Fig. 4 shows that for µ = × ,the performance of the G-LOD is better than that of the G-LRT, while for µ = × , the G-LRT outperforms the G-LOD. The difference between the performance of the G-LODand the G-LRT can be justified as follows. Since we have useda fixed number of grid points for finding b µ , the accuracy of b µ is worse for larger µ . Therefore, we can expect that for smaller µ the G-LRT outperforms the G-LOD, as in Fig. 4, where theG-LRT performs better than the G-LOD for µ = × .In Fig. 5, we study the impact of the number of NSs K on the performance of the G-LOD and the G-LRT. Tothis end, we show the global probability of missed detectionversus K , given a fixed global probability of false alarm ofP fa = − , N = , and µ = × . Fig. 5 reveals thatas the number of NSs increases, the performance of both G-LOD and G-LRT improves, since both detectors are able toexploit the independent signals received from the different NSsfor performance improvement. Finally, we observe that thereis a considerable gap between the (idealistic) GAD and theproposed sub-optimal detectors which suggests that the designof improved decision rules for the FC is a promising topic forfuture research.VI. C ONCLUSIONS AND F UTURE W ORK
In this paper, we studied the problem of target detectionin MC systems by developing a composite hypothesis testingframework, where we assumed that the location of the targetand the biomarker secretion rate were unknown at the NSs andthe FC. We derived a closed-form expression for the PMF ofthe received signal at each NS. We then proposed a simpledetector for the NSs and showed that it is UMP. Finally, wederived two sub-optimal detectors for the FC to obtain the final decision regarding the presence of a target and evaluatedtheir performance via simulations.A promising topic for future work is the investigation ofdetection schemes for further relaxed assumptions regardingthe available a priori knowledge. In particular, the case wherethe FC also does not know the locations of the NSs is relevant.R
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