A General Analytical Approximation to Impulse Response of 3-D Microfluidic Channels in Molecular Communication

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1 A General Analytical Approximation to Impulse Response of 3- Microfluiic Channels in Molecular Communication Fatih inç, Stuent Member, IEEE, Bayram Cevet Akeniz, Stuent Member, IEEE, Ali Emre Pusane, Member, IEEE an Tuna Tugcu, Member, IEEE, arxiv:84.7v [cs.et] 4 Jul 8 Abstract In this paper, the impulse response for a 3- microfluiic channel in the presence of Poiseuille flow is obtaine by solving the iffusion equation in raial coorinates. Using the raial istribution, the axial istribution is then approximate accoringly. Since Poiseuille flow velocity changes with raial position, molecules have ifferent axial properties for ifferent raial istributions. We, therefore, present a piecewise function for the axial istribution of the molecules in the channel consiering this raial istribution. Finally, we lay evience for our theoretical erivations for impulse response of the microfluiic channel an raial istribution of molecules through comparing them using various Monte Carlo s. Inex Terms 3- microfluiic channel with flow, non-uniform iffusion, Poiseuille flow, impulse response of the microfluiic channel, molecular communication I. INTROUCTION Molecular communication via iffusion MCv) is one of the most promising areas for nanonetworking ue to its biocompatability. It is base on encoing information symbols by releasing messenger molecules into a fluiic environment. Release molecules iffuse through the environment uner Brownian motion an the receiver makes a ecision on the transmitte symbols by observing or absorbing the release molecules. There are several ifferent channel moels propose in the molecular communication literature. An extensive survey that involves the compilation of these channel moels is presente in []. espite the large number of ifferent iffusion channels in the literature, they all have one common problem ue to the nature of the iffusion: inter symbol interference ISI). Since the movement of molecules are slow an ranom in Brownian motion, some of the release molecules may not reach to the estination until the esire time. This possibly leas to an averse effect on ecoing. There are many moulation an equalization methos to eliminate the molecules that cause ISI [], [3], [4], [5], [6]. In aition to these methos, channel moels that iminish ISI have been propose by consiering the reasons that cause ISI. It is clear that, ISI occurs ue to the ispersion an slow movement of the molecules. Especially in unboune environments, the molecules are uniformly isperse in the space an hence, the number of molecules reaching the receiver ecreases. Therefore, using barriers in the channel can be a reasonable approach to keep the molecules closer to the estination an is a more realistic channel type consiering biomeical applications. As propose in [7], vessel-like structures are goo caniates for long range molecular communication since they preserve release molecules in a guie range. Another beneficial factor in molecular communication channel for reucing ISI is flow which increases the spee of the molecules []. Therefore, using microfluiic channels assiste by flow oes not only iminish ISI but also increases the ata rate. There are various works relate to microfluiic or vessellike channels. In [8], a ecoing scheme for molecular communications in bloo vessels has been propose. In [9], a rectangular microfluiic channel with flow has been moelle an analyze. In [], numerical capacity analysis of the vessel like molecular channel with flow is examine. In [], a partially covering receiver in a vessel-like channel without flow is examine, an the channel characteristics are analyze. Although the channel impulse responses for unboune channels are erive, the channel impulse response for microfluiic channel with Poiseuille flow has not been erive yet in the molecular communication literature except for some special cases. In [], the flow moels of microfluiic channels with ifferent cross-section area are presente an the impulse response is erive by solving a - iffusionavenction equation, which is only vali for some specific cases. In [3], channel impulse responses of point an planar transmitter in a microfluiic channel have been erive for ispersion an flow ominate cases. For the first case, raial istribution of the molecules is assume to be uniform an the system is reuce to - to solve the channel response. For the latter case, only the flow is consiere by neglecting the effect of iffusion an the channel impulse response is obtaine accoringly. Although that paper inclues an elegant an extensive work for microfluiic channels with Poiseuille flow, it is only vali for the uniform raial istribution or flow ominate regions. This assumptions occurs, if Peclet number, a unitless number that compares the effects of flow an iffusion, is much higher or much lower than the ratio of the raius of the channel an the istance between the transmitter an receiver. Therefore, for other cases, erivation of the channel impulse response still remains as an open problem. In this paper, the channel impulse response of microfluiic channel that involves Poiseuille flow is erive when an arbitrarily place point transmitter an a planar observing receiver that fully covers the cross-section of the channel. In orer to etermine this function, firstly the

2 h Fig.. General channel moel for a point transmitter an a planar observing receiver. formula of the raial istribution of the molecules in time is erive. Using this istribution, the average velocity, isplacement of a molecule an finally the probability of observation of a molecule by the receiver which can also be regare as channel impulse response) are obtaine. All these functions enable us to examine a microfluiic channel without any real time. Furthermore, using these functions, some channel properties like require time to reach uniform raial istribution in channel is extracte. The main contributions of this paper to the literature can be liste as follows: erivation of the raial istribution of the release molecules in a vessel-like molecular communication channel uner the effects of Poiseuille flow an iffusion. A piecewise approximation of the axial istribution of the release molecules for all cases. II. SYSTEM MOEL The consiere system moel is epicte in Fig.. For the sake of simplicity, the coorinates in the channel are efine using cylinrical coorinates r, x), where r = y + z [, ] an x [, ]. In this figure, a point transmitter place at an arbitrary axial x irection) an raial r irection) position enote by, an a circular planar observing receiver is place x r away from the axial position of the point transmitter. The bounaries of the microfluiic channel are reflecting an there is a Poiseuille flow that changes with the axial position as vr) = v m r, ) where v m is the maximum velocity that occurs at the center of the microfluiic channel. The planar observer receiver observes the number of molecules passing through its surface an makes a ecision base on its observations without removing the messenger molecules from the environment. In aition to the flow, iffusion also exists in the channel an the iffusion process is moelle using the iffusion coefficient, which is relate to the variance of the Brownian motion. Since the flow is only available on the x axis, for each t secons, the isplacement of a molecule at a raial position r can be moelle in Cartesian coorinates as x = tvr) + N, t), y = N, t), z = N, t). ) From Eq. ), one can easily observe that the istribution of y an z hence, raial isplacement r) are not epenent on x, but istribution of x is epenent on r. In other wors, the movement of the molecules along the raial axis is purely iffusive while movement of the molecules along the axial axis is a combination of the iffusion an flow, whose velocity is etermine by raial position. In orer to compare the effect of flow an iffusion, Peclet number P e) is a useful imensionless metric that can be obtaine for the microfluiic channel as P e = v m. 3) Note that for pure iffusion P e =, an for pure flow i.e., avection) P e approaches. III. CHANNEL IMPULSE RESPONSE In orer to erive the impulse response of the channel, we nee to erive the joint raial, axial, an time istribution of the molecules, p x, r, t,, x r ), in the microfluiic channel. As can be seen in Eq. ), the raial istribution is inepenent of the axial istribution, but the axial istribution is epenent on the raial istribution. Consiering this fact, we can rewrite p x, r, t,, x r ) as

3 3 p x, r, t,, x r ) = p r, t,, x r ) p x, t r,,, x r ). 4) Therefore, our aim is fining the raial istribution p r, t,, x r ) first, an then using this istribution to obtain the axial istribution p x, t r,,, x r ). Once p x, r, t,, x r ) an p r, t,, x r ) are erive, the channel impulse response of the circular observer is obtaine as n hit t,, x r ) = t x r p x, t r,,, x r ) x. 5) Having foun n hit t,, x r ), one can easily fin the fraction of the observe molecules by the receiver until time t, t,, x r ), by integrating n hit t,, x r ) with respect to time as t,, x r ) = t n hit τ,, x r ) τ. 6) A. erivation of the raial istribution p r, t,, x r ) The raial istribution can be obtaine by simply solving the iffusion equation or by rawing comparison to the heat flow, as iscusse in [4]. Here, we shall fin the raial istribution of molecules using the former metho. To escribe the iffusion of the molecule insie the raial region, a solution to the Fick s Law, satisfying the necessary bounary conitions is neee. The equation is given as P r, t,, x r ) = P r, t,, x r ), 7) t where P r, t,, x r ) is the probability ensity of the molecule. The bounaries are reflecting, meaning that the probability current normal to the bounaries shoul be zero. Furthermore, the molecule is assume to be situate at a istance away from the origin for t =, which results in the following two bounary conitions: P r, t,, x r ) =, 8a) r r= P r, t,, x r ) = δr ), 8b) t= πr where we recall that, uner Neumann bounary conitions, the Laplacian operator ) is guarantee to have a unique solution up to an aition of a constant, which can be regare as the normalization constant. In orer to solve P r, t,, x r ) seperation of variable anatsz is use as which leas to the equation P r, t,, x r ) = φr, θ)t t), 9) φr, θ) = T t) φr, θ) T t) = µ, from which one can easily euce that: T t) = Ce µt, ) an arrive at the Neumann-eigenvalue problem for the Laplacian operator: φr, θ) = µ φr, θ). ) At this point, it is important to recall some key properties of Laplacian operator [5] [7]. The eigenvalues µ / are nonnegative an real, as well as the eigenvectors corresponing to istinct eigenvalues being orthogonal an forming a basis for all possible solutions. The non-negativity of the eigenvalues ensures that T t) oes not ten to infinity as t, whereas the orthogonal basis guarantees a unique solution. Here, we invoke the iea of angular symmetry SO) symmetry) in the system. ue to SO) symmetry, the position-epenent part of the ansatz epens only on the istance from the origin an not the angle-θ, i.e., φr, θ) = φr). This choice eliminates certain eigenvalues an corresponing eigenvectors, from the solution. Nonetheless, the coefficients corresponing to the non-symmetrical eigenvectors are zero ue to the symmetry of the system, removing our buren for further calculations. Rewriting the eigenvalue equation in polar coorinates, we obtain: r φ r) + rφ r) + µ r φr) =, where φ r) enotes the erivative of φr) with respect to r. The most general solution is µ µ φr) = J r + cy r, where J n an Y n are the Bessel s function of the first an secon kin, respectively an c is a constant to be etermine by the bounary conitions. Here, the coefficient of J is chosen arbitrarily as the overall coefficient C of the solution is lumpe into the time epenent part in ). The solution can now be shape accoring to the bounary conitions given in Eq. 8). One important observation is that, for t >, the probability ensity function P r, t,, x r ) oes not iverge for r =, ), resulting in c =. The most general solution is then of the form: µn P r, t,, x r ) = C n J r e µ n t, n= where C n an µ n are to be specifie by the bounary conitions. We arbitrarily efine the starting inex as n =. Once the bounary conition given in Eq. 8a) is invoke, we arrive at P r, t,, x r ) r µn = = J r= ) =. The first orer Bessel function J r) has infinitely many zeros. These zeros, efine as β n = µn, then constitute infinitely many terms for the solution. With the notation β =, the solution is a sum of infinitely many terms given as P r, t,, x r ) = r ) C n J β n e µ n t. n=

4 4 Before imposing the final conition given in Eq. 8b), it is useful to note the following normalization ientity for Bessel functions [8], which is given as xj β n x)j β m x) x =.5J β n ) δ nm, where δ nm is the Kronecker elta function. Using this ientity for the initial conition given in Eq. 8b), we conclue that C n = J βn π J β n), from which we fin the final solution to be J βn ) P r, t,, x r ) = π J β n) J βn r n= ) e β n t, ) where the first term β = ) correspons to the uniform istribution P r, t,, x r ) /π, as t. Practically, it takes much shorter time to reach the uniform istribution. Consiering P r, t,, x r ) in Eq. ), while n = correspons to the uniform istribution, n = is the ominating term that makes P r, t,, x r ) non-uniform. Therefore, if this term is arbitrarily small that can be assume to be effectively zero, we can fin the require time that raial istribution becomes uniform as ke β t < ɛ, 3) where we efine the boun ) J β r) k an Jβ J β) ɛ/π as the allowe eviation in the raial istribution from the uniformity. Here, we note that π is not lumpe into the parameter ɛ to keep the measures unitless, since as t, P r, t,, x r ) /π an ɛ can be interprete as the fractional error. Therefore, P r, t,, x r ) becomes uniform if the following conition is satisfie: t k log. 4) ɛ β We note that this boun is in line with the finings of [9]. Taking ɛ =, we obtain t β log) + logk)). 5) Since β 3.83 an k is on the orer of, a boun on time for uniform raial istribution can be approximate as t t 3. 6) Therefore, for t t, we can expect raial homogeneity. Having foun the probability ensity function P r, t,, x r ), we can fin the raial istribution as p r, t,, x r ) = πrp r, t,, x r ). 7) B. erivation of the axial istribution p x, t r,,, x r ) The axial istribution of the molecules in the microfluiic channel has ifferent characteristics for ifferent p r, t,, x r ). In particular, as iscusse in [], if p r, t,, x r ) is uniformly istribute, p x, t r,,, x r ) can be easily ientifie as p x, t r,,, x r ) = 4πe t exp x v mt 4 e t ), 8) tvm which is equivalent to N, et an can be consiere as a - Brownian) motion with effective iffusion coefficient e = + P e shifte with average axial isplacement 48 for any time t. The fact that Eq. 8) is vali for uniformly istribute p r, t,, x r ) implies it is vali when following conition is satisfie, P e P c = 4x r, 9) as explaine in []. This implication makes sense since in orer to obtain a uniform istribution in raial space, the channel shoul have either small raius ) or high iffusion coefficient ) so that the molecules isperse in the channel rapily to reach the uniform state. On the other han, as P e increases, it takes some time for p r, t,, x r ) to become uniform. Hence, uring this perio, Eq.8) cannot be use. We therefore propose two ifferent regions, namely uniform an non-uniform raial regions, an using these regions a partial function for the erivation of p x r, t r,,, x r ) is obtaine: ) p x r, t r,,, x r ) for uniform raial istribution range: As state before, p x, t r,,, x r ) can be written using Eq. 8), if raial istribution is uniform. We have alreay shown that for t > t the raial istribution is uniform. Therefore for this perio we can present the axial istribution using Eq. 8). On the other han since the average isplacement is ifferent for t > t ue to non-uniform raial istribution, we shoul evaluate the average isplacement an plug it to Eq. 8) instea of v mt. For this aim, the average axial isplacement of a molecule x exp t) for t secons can be evaluate using p r, t,, x r ) as x exp t) = t p r, τ,, x r ) vr) r τ. ) Once x exp t) is obtaine, p x r, t r,,, x r ) is obtaine using Eq.8) as

5 timems) a) = 5µm, =, P e = 5, P c = 4 b) = 5µm, =, P e = 75, P c = 333, t = 3/4s c) = µm, =, P e =, P c =, t = 4/3s timems) ) = µm, =, P e = 5, P c =. 5 5 timems) e) = 5µm, =.5 P e = 75, P c = timems) f) = 4µm, =.5 P e =, P c = 5 Fig.. Simulation an cumulative axial molecule istribution, t,, x r) for = µm/s, x r = 5mm, v m = mm/s an ifferent values which inicate ifferent P e regions. raial istribution pr,t) x r m) x 5 a) = 5 6 m, t = ms raial istribution pr,t) 5 x r m) x 5 b) = 5 6 m, t = s raial istribution pr,t) 5 x r m) x c) = 4 6 m, t = s raial istribution pr,t) 6 x r m) x ) = 4 6 m, t = 8s Fig. 3. Simulation an raial molecule istribution, p r, t,, x r) for = m/s, x r = 5mm, v m = mm/s an ifferent values an time t. Noting that, p r, t,, x r) = πrp r, t,, x r), linearity on p r, t,, x r) implies uniformity on P r, t,, x r). p x r, t r,,, x r ) = exp x ) r x exp t)) 4 e t = f Xt x r ). 4πe t ) where f Xt x) N x exp t), e t). ) p x r, t r,,, x r ) for non-uniform raial istribution range: When P e is comparable to or greater than P c, the release molecules nee some time to reach a uniform raial istribution. Until that time, ispersion of molecules is limite; hence, a new axial istribution moel shoul be efine. Let vt) be the average velocity an v a t) be the require average velocity to traverse a istance x r at time t, respectively. Then, one can easily efine the following relation: v a t) = x r t. ) Therefore, any molecule whose average velocity is higher than v a t) until time t passes through the receiver. Consiering this fact, the probability of exceeing x r istance until time t can be written as P rob xt) x r ) = P rob vt) v a t)). 3) where xt) is a ranom variable that efines the axial istribution of a molecule an this istribution is not known. Alternatively, one can calculate the probability of exceeing the average require velocity v a t) for a given time t. Note

6 6 that, the require average velocity involves two terms coming from the flow an iffusion as v a t) = v m r t) ) + U t, 4) where r t) is the require raial position to achieve v a t) with U, which is the axial isplacement component coming from the iffusion istribute with f U u) N, t). Using p r, t,, x r ), we can obtain P rob vt) v a t)) consiering the velocity terms coming from the flow an iffusion using p r, t,, x r ) as P rob vt) v a t)) = r t) t t p r, τ,, x r ) τrf U u)u, 5) where the result of the first two integral in 5) gives the average velocity istribution of a molecule ue to flow an the outermost integral evaluates the contribution of the iffusion. Once P rob xt) x r ) is obtaine, the axial istribution p x, t r,,, x r ) can also be obtaine using this probability as p x, t r,,, x r ) = P rob xt) x k )) x k xk =x, 6) where we fin the probability ensity function p x, t r,,, x r ) from the corresponing cumulative istribution function P rob xt) x). Finally, using 5), n hit t,, x r ) can be partially represente as n hit t,, x r ) = t P rob vt) v at)), t t 7) x r f Xt x) x t > t t Accoringly, t,, x r ) can be obtaine as t,, x r ) = { P rob vt) v a t)) t t F Xt x r ) t > t 8) where F Xt x r ) is the cumulative normal istribution whose mean an variance are x exp t) an e t, respectively. Even though the erivations from 5)-8) are mainly heuristic an cumbersome, t,, x r ) is easy to evaluate an to compare with the results, hence is of significant interest for the verification of our finings. IV. SIMULATION RESULTS We have verifie the erive cumulative istribution of molecules observe by the receiver, t,, x r ) an the raial istribution of molecules at the channel p r, t,, x r ) using Monte Carlo s. For both functions, the comparisons are mae for three cases: ) P e P c ) P c P e 3) P e P c, namely P e, is much greater than P c, P c is much greater than P e an P e is comparable with P c, respectively. In particular parameters are liste in Table I. For all s, 5 molecules are release an their positions are upate every t = 3 s using Eq. ). TABLE I PARAMETERS t = 3 s x r v m P e P c i) P e P c 5 µm 5mm mm/s m/s 5 4 iii) P c P e 5µm 5mm mm/s m/s iii) P e P c 4µm 5mm mm/s m/s 5 ii)p c P e µm 5mm mm/s m/s 5 In Fig., the erive expression x r, t,, x r ) is verifie using Monte Carlo s. As can be seen from the figure, the erive formula fits with s for all three regions. Especially for Fig. b) an c), the propose piecewise function, which is separate by t, can be easily ientifie. For the time uration before t, the first equation in 8) is use while the secon equation is use for the time uration after t. On the other han for other s t is either too small or too high so that only one function is use. Furthermore, as hence P e ) increases, the time neee for the molecules to pass through the receiver ecreases an converges to x r /v m. This is expecte, since as the raius of the channel increases, the raial position of the molecules oes not isperse so much in the beginning. Hence, the flow spee affecting these molecules is aroun v m. On the other han, as ecreases, the molecules can reach the bounary of the channel rapily, which reuces the velocity of the flow, an thus, it takes more time to reach the receiver. Furthermore, as inicate in Fig. e) an f), the erive expression is verifie for ifferent initial raial positions. Another interesting observation from these figures can be obtaine by comparing them with Fig. b) an c), respectively. As the initial raial position moves from center to bounary, in other wors as increases, the initial spee of the molecules ecreases. Hence, it takes more time to reach the receiver compare to the = case. Nonetheless, for all cases, all molecules will be observe by the receiver in a short uration compare to the other channel moels in the literature ue to flow. In Fig. 3, the raial istributions for ifferent channel parameters an time are presente both theoretically an numerically. As can be seen from these figures, our propose formula is verifie with the s. Furthermore, as inicate in Figs. 3b) an 3), for higher times, p r, t,, x r ) turns out to be linear; hence, there is a uniform raial istribution. In particular, for the parameters in Fig. 3a) an 3b), at t = ms, the raial istribution is nonuniform while at t=s, this istribution becomes uniform. This is expecte since for these parameters, t = 3.75s, resulting a non-uniform istribution for the time below t. The similar situation can also be observe for Fig. 3c) an 3) since for this case t 5.3s.

7 7 V. CONCLUSION Microfluiic channels with flow is a very goo caniate for molecular communication since the bounaries an flow reuce the ISI an increase the ata rate, which are still consiere as open problems for many channel types. Therefore, the istribution of molecules in the channel an the impulse response are necessary to etermine the characteristics of the channel. Although there are many ifferent attempts for the erivation of the impulse response of microfluiic channels, all consier some specific cases by reucing the avectioniffusion equation to the case for a - channel, which is vali for uniform raial istribution. On the other han, it may take some time to have uniform raial istribution, for smaller iffusion coefficient or higher channel raius. Until that time, the reuce - moel to solve axial istribution, hence that channel impulse response, is not applicable for these cases. We, therefore, firstly erive the raial istribution of the molecules insie the channel with respect to time, an accoringly etermine the require time that the molecules reach the uniform raial state. Then, using the raial istribution an the time that molecules reach the uniform state, we erive the piecewise channel impulse response. Finally we verify the erive formulas of the channel impulse response an raial istribution of molecules with Monte Carlo s. VI. ACKNOWLEGEMENTS This research was partially supporte by the Scientific an Technical Research Council of Turkey TUBITAK) uner Grant number 6E96 an by TETAM: Telecommunications an Informatics Technologies Research Center at Bogazici University uner grant number PT-7K6. Fatih inç woul like to thank Professor Achim Kempf for insightful iscussions on the spectral theory of the Laplacian operator. [9] A. O. Bicen an I. F. 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