Experimental System for Molecular Communication in Pipe Flow With Magnetic Nanoparticles
Wayan Wicke, Harald Unterweger, Jens Kirchner, Lukas Brand, Arman Ahmadzadeh, Doaa Ahmed, Vahid Jamali, Christoph Alexiou, Georg Fischer, Robert Schober
11 Experimental System for MolecularCommunication in Pipe Flow With MagneticNanoparticles
Wayan Wicke, Harald Unterweger, Jens Kirchner, Lukas Brand, ArmanAhmadzadeh, Doaa Ahmed, Vahid Jamali, Christoph Alexiou, Georg Fischer, andRobert Schober
Abstract
In the emerging field of molecular communication (MC), testbeds are needed to validate theoreticalconcepts, motivate applications, and guide further modeling efforts. To this end, this paper presentsa flexible and extendable in-vessel MC testbed based on superparamagnetic iron oxide nanoparticles(SPIONs) dispersed in an aqueous suspension as they are also used for drug targeting in biotechnology.The transmitter is realized by an electronic pump for injection of the SPIONs stored in a syringevia a Y-tubing-connector. A second pump generates a background flow for signal propagation in themain tube, e.g., modeling a part of a chemical reactor or a blood vessel. For signal reception, weemploy a commercial susceptometer, an electronic device including a coil, through which the magneticparticles move and non-intrusively generate an electrical signal. We identify the physical mechanismsgoverning transmission, propagation, and reception of SPIONs as signal carriers and propose a simple two-parameter mathematical model for the system’s channel impulse response (CIR). Reliable communicationis demonstrated for model-agnostic and model-based detection methods for experiments with 400 randomsymbols transmitted via on-off keying modulation with a symbol interval. Moreover, the proposedCIR model is shown to consistently capture the experimentally observed distance-dependent impulseresponse peak heights and peak decays for transmission distances from to
40 cm . I. I
NTRODUCTION
Molecular communication (MC) employs molecules as information carriers necessitatingnew models and experimental tools compared to conventional electromagnetic wave based
This work was presented in part at IEEE SPAWC 2018 [1].
January 7, 2021 DRAFT a r X i v : . [ c s . ET ] J a n communication [2]. The growing interest in this research area is due to its revolutionaryapplications in environments unsuited for electromagnetic waves such as biological environments,e.g., in the human body or within bacterial cultures [3], environments unfavorable due topropagation losses, e.g., liquid-filled pipes, and environments with explosive gases, e.g., fuel pipes[2], [4], [5]. Motivated by these applications, a significant body of theoretical work has beendeveloped, see [6]–[8] for surveys of the current literature. Moreover, for practical demonstrationand to gain more insight regarding relevant physical phenomena, several MC testbeds have beenproposed, see [7, Section V] for an overview of recent artificial and biological MC testbeds.These experimental systems can be categorized as either air-based [9]–[11] or liquid-based[12]–[17] depending on the physical communication medium. Air-based systems have beendeveloped for open space transmission [12] as well as for closed air ducts [10], [11], whichoffer a directed information transfer at the expense of the required infrastructure. Liquid-basedexperimental MC systems usually require vessels and exist in different size scales ranging frommicrofluidics [16], [17], to small pipes [12], [14], to larger ducts [13], [15].In this paper, we study a liquid-based MC system composed of small pipes, i.e., the environmentis bounded, flow-driven, and fluid, like in blood vessels. Experimental systems studying suchenvironments have been reported in [12], [14]. The system in [12] is based on either injectingan acid or a base into water and the detection of the medium’s pH level. The system in [14] issimilar to the one in [12] in that it is based on in-vessel chemical reactions but it employs opticaldetection. However, those systems inherently rely on chemical reactions which complicates theiranalysis (see e.g., [18]) and limits their applicability because many applications require passivesignaling to avoid possible interference with biological processes. Moreover, for detection, thesystem in [12] requires direct access to the liquid and the system in [14] requires an opticallytransparent tubing.In this paper, we present a new testbed with the focus on studying flow-driven transportin simple cylindrical tube systems. To this end, it is crucial to select appropriate signalingmolecules or particles [19], [20]. These signaling particles should ideally possess the followingproperties which the information carriers used in [12]–[14] do not have: 1) The particles shouldbe chemically stable for safe and long-term use, i.e., they should not agglomerate and notinteract with other components of the testbed, such as the respective medium. 2) A sensitiveand non-intrusive detection mechanism is required since, depending on the application, physicalaccess to the tubing may not be feasible or practical. 3) The particles should ideally be tunable January 7, 2021 DRAFT for different application needs, e.g., in their size, and have an established production mechanismfor cost-efficiency.One type of artificial particle that satisfies all of these requirements and is already well-established in biotechnology are biocompatible magnetic nanoparticles [21]. These particles canbe tailored to a particular application by engineering of their size, composition, and coating [22].Moreover, magnetic nanoparticles can be attracted by a magnet and externally visualized [21],which can help detection and supervision. Applications of magnetic nanoparticles include tissueengineering [23], biosensing [24], imaging [25], remotely stimulating cells [26], waste-watertreatment [27], and drug delivery [28].In the context of MC, the use of magnetic nanoparticles as information carriers has beenconsidered in [29]–[32] and [33], where the benefits of attracting them towards a receiver aretheoretically evaluated and the design of a wearable device for detecting them is proposed,respectively. However, an experimental MC system with magnetic nanoparticles as informationcarriers has only been presented in the conference version of this work [1]. Furthermore, for thistestbed, the design and characterization of receiver [34]–[36] and transmitter devices [37], [38]has also been investigated.Similar to [12]–[14], in this paper, we present a testbed for in-vessel MC. Our setup differsin that it uses specifically designed superparamagnetic iron oxide nanoparticles (SPIONs) asinformation carriers, which are biocompatible, clinically safe, and do not interfere with otherchemical processes and thus might be attractive for applications such as monitoring of chemicalreactors where particles stored in a reservoir could be released upon an event like the detectionof a defect which is then communicated to a central control station for further processing. Inthe proposed testbed, SPIONs are injected and transported along a propagation tube by fluidflow which is established by a peristaltic pump. The propagation tube runs through the receiverwhere the magnetic susceptibility of the mixture of water and SPIONs within a section of thetube can be non-intrusively determined. The magnetic susceptibility measured in the tube sectionis proportional to the concentration of the particles within the section. This proportionality ismore amenable to mathematical analysis compared to observing the pH in [3], [12], [18], whichnon-linearly depends on the underlying proton concentration.The contributions of this paper can be summarized as follows: • We present an experimental system for MC based on the flow-driven transport of SPIONsin a tube. All components of the system are described in detail which was not possible in
January 7, 2021 DRAFT [1] due to space constraints. • Extending [1], we provide a physical characterization of the system regarding the relevanteffects for particle injection, propagation by flow in the tube, and reception by the suscep-tometer. In MC terms, we motivate the model of a transparent receiver and characterize thepulse shaping by the injection via an initial volume distribution [39]. • Motivated by the physical characterization, we develop a parametric model for the system’schannel impulse response (CIR) providing insight into the flow-driven transport and thereceiver’s physical properties. We validate the applicability of our model by fitting its param-eters to measurement data of the CIR showing a good agreement despite the simplificationsneeded for analytical tractability. • To demonstrate successful information transmission, we show that reliable detection of on-offkeying (OOK) is possible for transmission distances of up to
40 cm and a symbol durationof . To this end, we propose and evaluate symbol sequence estimation by applying 1)an optimal detection rule assuming a linear pulse-amplitude modulation (PAM) model and2) a model-agnostic heuristic detection rule based on the signal increases and decreasesfollowing symbol intervals with injections and idle times, respectively.The rest of this paper is organized as follows. In Section II, we explain the testbed and itscomponents as well as physical preliminaries. In Section III, we propose a simple CIR model. InSection IV, we describe the employed signal processing and detection algorithms. In Section V,we present experimental data. Finally, in Section VI we conclude the paper and provide directionsfor future work. II. M
AGNETIC N ANOPARTICLE -B ASED T ESTBED
In the following, each component of the proposed MC system and its parameters are charac-terized. A representative photograph of the entire system is shown in Fig. 1a and the systemparameters are summarized in Table I.
A. Testbed Components
In this subsection, we describe the physical setup of the testbed components, including theinformation-carrying particles, the transmitter, the propagation channel, and the receiver.
January 7, 2021 DRAFTwater reservoir background flow pump susceptometerinjection pump syringeY-connectortube to waste container (a) particle tube propagation tubebackground flow tube (b)
Iron oxidecore diametercore-cluster diameterSPION diameter Coating(i.e. small molecules, polymer, etc.) (c)Fig. 1. SPION testbed. (a) Photograph of the testbed showing the water reservoir, the background flow pump, the susceptometer,the pump used for injection, the syringe holding the suspension of SPIONs, and flexible plastic tubes connecting the components.The waste container below the table is not shown. (b) Photograph of the Y-connector with elongated SPION suspension rightafter injection for a slow background flow of 𝑄 b = / min . The injected SPION suspension is elongated by the flow profile.(c) Schematic SPION composition consisting of iron oxide cores forming a core-cluster stabilized by a polymer.
1) Carrier:
We use a specific type of magnetic nanoparticles as information carriers which arereferred to as SPIONs. There are a multitude of possibilities for producing SPIONs, includingthermal, sol-gel, electrochemical, and precipitation approaches. One of the fastest, simplest, andmost efficient methods to synthesize SPIONs is the coprecipitation technique in alkaline mediaas it was first proposed by the authors of [40], [41] in the early 1980s. The first and most crucialstep of this synthesis is the precipitation of magnetite ( Fe O ) from ferric ( Fe + ) and ferrous( Fe + ) salts with a stoichiometric ratio of 2:1 ( Fe + / Fe + ) in an inert atmosphere at a basic pH: + + Fe + + − → Fe O + O , consuming hydroxide ( OH − ) with water ( H O ) as byproduct. January 7, 2021 DRAFT
TABLE IS
YSTEM P ARAMETERS (a) Particle PropertiesParameter ValueHydrodynamic particle radius . Suspension iron stock concentration .
89 mg / mL Suspension magnetic susceptibility × − (SI units)Particle mass . × − kg (b) Testbed SettingsParameter Range/ValueTube radius particle injection .
40 mm
Tube radius background flow 𝑎 .
75 mm
Flow rate particle injection 𝑄 p .
26 mL / min Flow rate background flow 𝑄 b / min Volume particle injection 𝑉 i . µ L Symbol duration 𝑇 Propagation distance 𝑑 to
40 cm
For our synthesis, we use ammonia to start the formation of the particles. Since nanoparticlesin general and our SPIONs in particular are featured with a small size, they possess a largesurface to volume ratio. The resulting high surface energy renders the particles thermodynamicallyunstable and is responsible for their tendency to minimize their energy by agglomeration. In orderto avoid this behavior, a suitable stabilization mechanism is required. Generally, stabilizationcan be achieved by small molecules, polymers, and proteins. It is common to all of them toproduce repulsion either by electrostatic, by steric, or by electrosteric means. Most colloidaldispersions possess an electric surface charge which, depending on the material and the dispersionmedium, gives rise to electrostatic stabilization. However, certain tradeoffs apply for the coatingof SPIONs in MC. First, the synthesis and coating has to be designed to make the particles aslarge as possible, in order to be able to generate a large signal for detection. However, the largerthe particles, the more they are prone to sedimentation. In addition the coating material shouldprovide the particles not only with stability against agglomeration but also render them inertagainst the components of the testbed. For these reasons, we used SPIONs with lauric acid as astabilizing agent [42] (see Fig. 1c).The particles are dispersed in an aqueous suspension and stored in a syringe, which is connectedto a tube with an inner radius of . . The particles have a hydrodynamic radius of . ,an iron stock concentration of .
89 mg / mL , a susceptibility of × − (dimensionless in SIunits), and a concentration of approximately × particles / mL in aqueous suspension. January 7, 2021 DRAFT
2) Transmitter:
The movement of the particles through the tube is established with a computercontrolled peristaltic pump (Ismatec Reglo Digital, Germany), which can provide discrete pumpingactions at a flow rate of .
26 mL / min (maximal speed), injecting a dosage volume of 𝑉 i = . µ L of SPION suspension. Injection speed and volume have been manually chosen so as to minimizethe injection duration for avoiding intersymbol interference (ISI) and achieving a strong signal.The end of the tube with the particles is joined via a Y-connector with another tube of radius .
75 mm providing a background flow, see Fig. 1b. The constant background flow of water has aflow rate of / min and is maintained by a second pump (Ismatec IPC, Germany).Discrete pumping actions with a minimum adjustable separation of are realized by a customLabVIEW (National Instruments, Austin, Texas, USA) graphical user interface (GUI) that triggersthe particle pump.
3) Propagation Channel:
The flow rate in the tube channel is the sum of the rates ofthe background flow and the particle injection. It is hence time-dependent and amounts to .
26 mL / min during particle injection and / min in the remaining time. When particles arepumped into the channel by the transmitter, then the resulting particle cloud is entrained by theflow and simultaneously diluted and elongated, see Fig. 1b.The length of the propagation channel is variable but was set to ,
10 cm ,
20 cm , and
40 cm for the results shown in Section V.
4) Receiver:
At the end of the propagation channel, the tube runs through the air core of anMS2G Bartington susceptometer coil (inner diameter:
10 mm , length:
20 mm , specified sensitivelength: ). When the magnetic particles are within the detection range of the susceptometer,an electrical signal 𝜒 ( 𝑡 ) is generated. This signal is proportional to the number of SPIONs thatare within the detection range at a specific time instance. After the particles have passed throughthe receiver, they are collected in a waste bin together with the water from the backgroundflow. Water has a small negative magnetic susceptibility of about − . × − (SI units) [43,Appendix E]. Hence, the magnitude of the magnetic susceptibility of water is much smaller thanthat of the considered SPION suspension 𝜒 ref = × − (SI units).The susceptibility changes measured at the receiver were recorded by use of the software Bartsoft4.2.1.1 (Bartington Instruments, Witney, UK) provided by the manufacturer of the susceptometer.The susceptometer is a reliable and convenient commercial device for characterizing the magneticsusceptibility of SPION suspensions in the laboratory. Nevertheless, we note that our current usecase of measuring time signals with a short sampling period in the order of . is outside of the January 7, 2021 DRAFT usual mode of operation of this device which has been designed for one-time measurements of bulkprobes under stationary conditions, see [36] for the evaluation of a custom susceptometer design.Hence, care has to be taken when interpreting the measured signal as magnetic susceptibilitysince we operate the susceptometer outside of its specification by evaluating the output signalfor a spatially varying SPION concentration over time.
B. General Considerations
In this subsection, we provide some background on the relevant physical effects affectingthe measurement signals. These considerations will guide our mathematical modeling efforts inSection III.
1) Turbulent or Laminar Flow:
Fluid flow can be categorized as either laminar or turbulent.This categorization determines the appropriate mathematical model to be used. While laminarflow is prevalent in microfluidic applications, turbulent flow is encountered in macroscale ductsin the size range of several centimeter. The relevant parameter which predicts a transition fromlaminar to turbulent flow is the Reynolds number Re ∈ R + which is defined as [44, Chapter 3] Re = 𝑎 · 𝑢 𝜈 , (1)where 𝑎 is the tube radius, 𝑢 is the maximum flow velocity in the tube and can be computed as 𝑢 = 𝑢 eff with the area-averaged velocity in the tube 𝑢 eff = 𝑄 b /( 𝜋𝑎 ) , 𝜈 is the kinematic viscosityof the liquid, and 𝑄 b is the background flow rate, i.e., the total flow rate after injection. For acircular duct, a value of Re ≈ is critical for the transition from laminar to turbulent flow,see [44, Chapter 3]. For the testbed parameters in Table I, we find 𝑢 eff = . / s . Thus, usingthe kinematic viscosity of water 𝜈 = − m / s [44, Chapter 1], we obtain Re = . < and hence expect fully laminar flow. By the reasoning above, we would expect a transition toturbulent flow for an increase of the Reynolds number in (1) by a factor of . For example, allother parameters held equal, we would expect a transition to turbulent flow at an effective flowspeed of 𝑢 eff = − ( 𝑄 b =
150 mL min − ) or for a tube radius of . . We note thatadditional turbulence could also be caused by obstacles in the tube.
2) Diffusion:
In general, the particle motion is governed by both diffusion and transportby fluid flow, assuming a fully dilute aqueous SPION suspension. The relative importance of
January 7, 2021 DRAFT diffusion compared to the transport by fluid flow over a distance of 𝑑 can be quantified by adispersion factor 𝛼 D ∈ R + which is defined as [7, Section II.B] 𝛼 D = 𝑑𝐷𝑎 𝑢 (2)where 𝐷 is the diffusion coefficient of the SPIONs, 𝑎 c is a characteristic distance over whichthe flow velocity varies, here chosen as 𝑎 c = 𝑎 / , and 𝑢 is an effective velocity, here chosenas 𝑢 = 𝑢 eff . When 𝛼 D (cid:28) and 𝛼 D (cid:29) , over a distance of 𝑑 , flow and diffusion dominate thetransport, respectively.The diffusion coefficient can be estimated as [7, Section II] 𝐷 = 𝑘 B 𝑇 m 𝜁 , (3)where 𝑘 B 𝑇 m = . × − J is a characteristic energy with Boltzmann constant 𝑘 B and thetemperature of the liquid medium 𝑇 m =
298 K , and 𝜁 = 𝜋𝜂𝑅 p = . × − kg s − is thefriction coefficient. Here, 𝜂 = × − Pa s is the dynamic viscosity of the liquid medium (water)and 𝑅 p = . is the SPION radius. From (3), we estimate 𝐷 = − m / s for the consideredSPIONs. Hence, for the testbed parameters in Table I, we obtain 𝛼 D = × − (for 𝑑 =
40 cm ).This value is several orders of magnitude smaller than and therefore the impact of diffusion isassumed to be negligible over the considered distances 𝑑 ≤
40 cm . By the reasoning above andtaking 𝛼 D = as critical value, we would expect diffusion to have a noticeable impact for anincrease of the dispersion factor by a factor of . All other parameters held equal, this would bethe case for a decrease of the effective flow velocity to 𝑢 eff = .
56 mm s − ( 𝑄 b = .
06 mL min − ),an increase of the distance to 𝑑 =
30 m , a smaller tube radius of 𝑎 = .
082 mm , or a muchlarger diffusion coefficient of 𝐷 = × − m / s . We note that the diffusion coefficient couldeffectively increase by particle-particle interactions or turbulence [44].
3) Injection:
The injected SPION suspension is in aqueous phase. Hence, after injection, aone-phase flow is expected, rather than a two-phase flow, as would be the case for, e.g., an oilysuspension in water. During injection, we have two joining flows. For an injection flow rate of 𝑄 p = .
26 mL min − and a background flow rate of 𝑄 b = − , we have a net flow rateof 𝑄 b + 𝑄 p = .
26 mL min − during injection and a flow rate of 𝑄 b = − when notinjecting. This corresponds to variations of the effective flow velocity between . − and . − . Following an injection, the increased flow velocity occurs for the injection durationof
197 ms and in principle affects the signal generated by all previous injections. By considering
January 7, 2021 DRAFT0 the flow rates, we can estimate the injection depth from the top to the bottom of the pipe by 𝑎 · 𝑄 p /( 𝑄 p + 𝑄 b ) [44], e.g., for 𝑄 p = 𝑄 b we would have an injection depth of half the pipediameter and for 𝑄 p (cid:28) 𝑄 b the injection depth would be close to 0. For the testbed parametersgiven in Table I, we can determine the injection depth as .
77 mm , i.e., we expect the injectedSPION suspension to reside in the upper half of the tube of radius 𝑎 = .
75 mm . In fact, theparticle volume distribution after injection determines the received signal as the SPION transportis expected to be deterministic driven by the flow.
4) Gravity:
Another relevant effect for the transport of particles is gravity. The gravitationalforce on an individual SPION can be determined as 𝐹 = 𝑚𝑔 = . × − N where 𝑚 = . × − kg is the particle mass, see Table I and 𝑔 = .
81 m / s is the gravitational constant[44]. The resulting drift velocity due to gravity can then be determined as 𝑢 = 𝐹 / 𝜁 = . − [7] where 𝜁 is the friction coefficient in Section II-B2. When we consider a vertical displacementby gravity of 𝑎 / = µ m to be relevantly large, then we obtain a time duration after injectionof 𝑎 / / 𝑢 = . where gravity begins to matter. Since the considered CIRs have a duration ofless than , we expect that the effect of gravity on an individual SPION is negligible. Onthe other hand, considering a time duration of as critical, gravity would begin to matterfor a particle mass of . × − kg which would correspond to a radius larger than
176 nm assuming the same mass density, i.e., assuming the SPION mass is proportional to 𝑅 .
5) Magnetic Susceptibility:
Finally, we consider the receiver device and the measured receivedsignal. The magnetic susceptometer used as receiver is a device comprising an electromagneticcoil with an air-core used as measuring space in an electric resonance circuit. The susceptometeris designed for measuring the magnetic susceptibility of bulk material probes large enough tofill the coil (bulk magnetic susceptibility). This bulk magnetic susceptibility is proportional tothe change of inductance resulting from inserting the bulk material into the coil which canbe measured, e.g., by examining the resonance frequency of the coil [36]. For example, theSPION suspension used in this testbed has a bulk susceptibility of 𝜒 ref and a bulk mixture of theSPION suspension with water at a ratio of 𝑐 ∈ [ , ] can be expected to yield a susceptibility of 𝜒 m = 𝜒 ref · 𝑐 . However, for locally varying concentrations as is the case for our testbed wherethe SPION suspension is being transported and dispersed in the water by the fluid flow, thesusceptometer output does only reflect an average susceptibility.Motivated by the above analysis, in the following section, a mathematical model is establishedaccounting for the transport by laminar flow where the received signal is characterized by the January 7, 2021 DRAFT1TX RX 𝑧𝑦 − 𝑑 − 𝑙 𝑧 / 𝑙 𝑧 / 𝑎 Fig. 2. Sketch of the simplified system model consisting of a straight tube with radius 𝑎 , shown for a ( 𝑧, 𝑦 ) -cut with transmitternode TX concentrated at axial coordinate 𝑧 = − 𝑑 and transparent receiver node RX weighting the SPION concentration. Thelaminar profile in (4) is schematically shown by velocity vectors. injection and a spatially weighted integral of the local susceptibility.III. M ATHEMATICAL S IGNAL M ODEL
In this section, first the modeling assumptions for the channel, transmitter, and receiver aredescribed and then the CIR, i.e., the expected received signal for one single injection, is derived.For quick reference, a sketch of the assumed abstract system model is shown in Fig. 2.
A. Modeling Assumptions
First, we will describe the flow-driven propagation in the tube channel, then we characterizetransmitter and receiver. In the following, we will use cylindrical coordinates for position 𝒙 = ( 𝜌, 𝜙, 𝑧 ) , where 𝜌 , 𝜙 , and 𝑧 are the axial distance, azimuth, and axial coordinates, respectively.
1) Channel:
In general, the flow at the injection site is complicated and time-variant as alludedto in Section II-A3. Moreover, in our testbed, the tube is not perfectly straight and is not infinitebut inherently also includes the Y-connector used for injection. This leads to a complicatedflow profile in general even without injections. Nevertheless, to simplify the analysis and asthe deviations over distances on the order of the inner tube diameter are small, we will assumelaminar flow in a straight tube of circular cross-section. In this case, the non-uniform flow velocityprofile is well known to be [44] 𝑢 ( 𝜌 ) = 𝑢 · (cid:18) − 𝜌 𝑎 (cid:19) , (4)where 𝜌 is the radial distance from the central axis of the tube.Then, the concentration develops over time and space based on the following model forflow-driven transport [7] 𝑐 ( 𝒙 , 𝑡 ) = 𝑐 i ( 𝒙 − 𝑢 ( 𝜌 ) 𝑡 · 𝒆 𝑧 ) , (5) January 7, 2021 DRAFT2 where 𝑐 i ( 𝒙 ) is the assumed initial spatial particle distribution and 𝒆 𝑧 is the unit vector in 𝑧 direction. The concentration satisfies ∭ 𝑐 ( 𝒙 , 𝑡 ) d 𝑉 = 𝑉 i , ∀ 𝑡 where 𝑉 is the volume of the infinitecylinder.
2) Transmitter:
The transmitter is physically realized by the injection pump and the Y-shapedtube connector, see Section II-A2. However, for modeling, we will focus on the initial particledistribution within an infinite cylinder which is created by the injection process, i.e., the volumedistribution of SPIONs right after injection.As first order approximation, we will assume that the initial distribution can be modeled asbeing axially concentrated at the site of injection . With this assumption, the initial concentrationin space can be mathematically expressed as 𝑐 i ( 𝒙 ) = 𝑉 i · 𝑓 ( 𝜌, 𝜙 ) · 𝛿 ( 𝑧 + 𝑑 ) , (6)where 𝑓 ( 𝜌, 𝜙 ) is the distribution in the cross-section of the tube and 𝛿 ( 𝑧 ) is the Dirac deltafunction. The distribution in the cross-section is normalized to ∬ 𝑓 ( 𝜌, 𝜙 ) d 𝐴 = , where 𝐴 denotes the area of the tube cross-section. The transmitter is at 𝑧 = − 𝑑 , see Fig. 2, and theinjection volume is assumed to be concentrated at position 𝑧 = − 𝑑 . We note that in general theinitial distribution in (6) depends on the angular coordinate 𝜙 . However, the received signal willnot depend on the distribution over 𝜙 because of the geometrically symmetric arrangement ofthe receiver surrounding the tube. Therefore, in the following, we will introduce some definitionsto describe the particle distribution over the radial coordinate. a) Definitions: The radial distribution is given by 𝑓 𝜌 ( 𝜌 ) = ∫ 𝜋 𝑓 ( 𝜌, 𝜙 ) 𝜌 d 𝜙 (7)and its cumulative distribution function is given by 𝐹 𝜌 ( 𝜌 ) = ∫ 𝜌 −∞ 𝑓 𝜌 ( ˜ 𝜌 ) d ˜ 𝜌 . For convenience, wealso define an auxiliary distribution as 𝑓 𝑠 ( 𝑠 ) = 𝑎 √ 𝑠 · 𝑓 𝜌 ( 𝑎 √ 𝑠 ) (8)which satisfies ∫ 𝑓 𝑠 ( 𝑠 ) d 𝑠 = and where 𝑠 = 𝜌 / 𝑎 . The corresponding cumulative distributionfunction satisfies 𝐹 𝑠 ( 𝑠 ) = 𝐹 𝜌 ( 𝑎 √ 𝑠 ) according to (8). We note that a more accurate model could be obtained, for example, by numerical simulation of the injection process andevaluating the obtained initial volume distribution [45]. However, as this numerical simulation does not directly give theoreticalinsight, in this paper, we focus on a simple parametric model for the initial distribution.
January 7, 2021 DRAFT3 b) Special Cases:
Let’s consider two common models for injection as important specialcases [19]. First, for a uniform particle distribution, 𝑓 ( 𝜌, 𝜙 ) = /( 𝜋𝑎 ) , we obtain 𝑓 𝜌 ( 𝜌 ) = 𝜌 / 𝑎 , ≤ 𝜌 ≤ 𝑎 and 𝑓 𝑠 ( 𝑠 ) = , ≤ 𝑠 ≤ . Second, for a particle distribution proportional to theflow profile in (4), 𝑓 ( 𝜌, 𝜙 ) = 𝜋𝑎 ( − 𝜌 𝑎 ) , we obtain 𝑓 𝜌 ( 𝜌 ) = 𝑎 𝜌 · ( − 𝜌 𝑎 ) and 𝑓 𝑠 ( 𝑠 ) = ( − 𝑠 ) . c) Parametric Initial Distribution: To generalize from these two important special cases,we propose the following distribution in 𝑠𝑓 𝑠 ( 𝑠 ) = 𝐵 ( 𝛼, 𝛽 ) · 𝑠 𝛼 − · ( − 𝑠 ) 𝛽 − (9)which is the Beta distribution [46] with ≤ 𝑠 ≤ . The parameters shaping the Beta distributionare 𝛼 ≥ and 𝛽 ≥ and the normalization is given by 𝐵 ( 𝛼, 𝛽 ) = Γ ( 𝛼 ) Γ ( 𝛽 ) Γ ( 𝛼 + 𝛽 ) , where Γ (·) is theGamma function.The Beta distribution is limited to the range [ , ] which makes it suitable for modeling therange of parameter 𝑠 . We can also recover the uniform distribution for ( 𝛼 = , 𝛽 = ) and thedistribution proportional to the flow profile for ( 𝛼 = , 𝛽 = ) , where 𝐵 ( , 𝛽 ) = / 𝛽 . Moreover,for 𝛼 ≥ and 𝛽 ≥ , 𝑓 𝑠 ( 𝑠 ) is unimodal with adjustable peak position and peak width as isneeded for our testbed where the SPION mass is concentrated around a certain radial positiondue to the injection. These properties make the Beta distribution a good candidate for modelingthe initial distribution following injection for this testbed.For future reference, by using (8), for 𝑓 𝑠 ( 𝑠 ) in (9), we can write the corresponding radialdistribution as 𝑓 𝜌 ( 𝜌 ) = 𝑎𝐵 ( 𝛼, 𝛽 ) · (cid:16) 𝜌𝑎 (cid:17) 𝛼 − · (cid:18) − 𝜌 𝑎 (cid:19) 𝛽 − . (10)
3) Receiver:
The physical detection device is given by the susceptometer. Motivated by thephysical reception mechanism described in Section II-B5, we assume the following receivedsignal model 𝜒 ( 𝑡 ) = 𝜒 ref · ∭ R 𝑤 ( 𝒙 ) · 𝑐 ( 𝒙 , 𝑡 ) d 𝑉 (11)where 𝑤 ( 𝒙 ) is a weighting function which can be interpreted as a receiver window and isnormalized as ∭ R 𝑤 ( 𝒙 ) d 𝑉 = . For example, for 𝑐 ( 𝒙 , 𝑡 ) = , ∀ 𝒙 , 𝑡 we have 𝜒 ( 𝑡 ) = 𝜒 ref . Thereceiver mechanism underlying (11) can be understood as a transparent receiver [39].In the following, for simplicity, we will assume that the receiver weighting function is arectangular window only dependent on 𝑧 , leaving a three-dimensional characterization of the January 7, 2021 DRAFT4 weighting function for future work. Hence, for our modeling, the rectangular receiver weightingfunction is given as 𝑤 ( 𝒙 ) = 𝜋𝑎 𝑙 𝑧 · rect (cid:18) 𝑧𝑙 𝑧 (cid:19) (12)where 𝑙 𝑧 is the window length and rect ( 𝑧 ) = for | 𝑧 | ≤ . , and rect ( 𝑧 ) = otherwise, i.e., thereceiver is centered at axial position 𝑧 = , see Fig. 2.This concludes our list of modeling assumptions. As will be shown, with these assumptions,we can capture the main characteristics of the measurement signals. For example, when due to theinjection more particles are concentrated close to the central axis than close to the boundary ofthe tube, a faster decay of the received signal over time and a larger peak can be expected due tothe laminar flow profile in (4). The general CIR behavior under the given modeling assumptionsis investigated in the following. B. Channel Impulse Response
In this subsection, we use the modeling assumptions summarized in the previous subsectionto derive a compact closed-form CIR expression usable for fitting measurement data. To thisend, we assume the abstract system model shown in Fig. 2, where the injection is simplifiedto a release concentrated at 𝑧 = − 𝑑 and the receiver applies a spatially-weighted integral of theparticle concentration resulting from the flow-driven transport similar to [47].
1) General Case:
Using (5) and (6) in (11), as shown in the Appendix, we can express theCIR as follows ℎ ( 𝑡 ) = 𝑐 𝑑 · 𝑑𝑙 𝑧 · (cid:18) 𝐹 𝑠 (cid:18) − 𝑑 − 𝑙 𝑧 / 𝑢 𝑡 (cid:19) − 𝐹 𝑠 (cid:18) − 𝑑 + 𝑙 𝑧 / 𝑢 𝑡 (cid:19) (cid:19) , (13)where 𝑐 𝑑 = 𝜒 ref 𝑉 i 𝜋𝑎 𝑑 is a distance-dependent dimensionless scaling factor.We note that with 𝑓 𝑠 ( 𝑠 ) in (9), (13) is a generalization of the solution provided in [1]. Inparticular, (13) reduces to [1, Eq. (5)] for 𝛼 = , 𝛽 ← 𝛽 + and 𝑙 𝑧 = 𝑐 𝑧 . A numerical evaluationof (13) for different initial distributions is presented in Section V-A.
2) CIR Analysis:
For the following analysis, for simplicity, we consider the case where 𝑑 (cid:29) 𝑙 𝑧 ,i.e., the case when the transmitter-receiver distance is much larger than the receiver width.Mathematically, we can employ the limit 𝑙 𝑧 / 𝑑 → in (13). Then, we obtain ℎ ( 𝑡 ) = 𝑐 𝑑 · 𝑑𝑢 𝑡 · 𝑓 𝑠 (cid:18) − 𝑑𝑢 𝑡 (cid:19) , (14)with 𝑓 𝑠 ( 𝑠 ) in (9). January 7, 2021 DRAFT5
Since (14) provides a simple closed-form solution for the CIR independent of the receiverlength, we will use it as our modeling function for fitting measurement results in Section V.For convenience, we plug (9) into (14) and arrive at ℎ ( 𝑡 ) = 𝑐 𝑑 · 𝐵 ( 𝛼, 𝛽 ) · (cid:18) − 𝑑𝑢 𝑡 (cid:19) 𝛼 − · (cid:18) 𝑑𝑢 𝑡 (cid:19) 𝛽 (15)for 𝑡 ≥ 𝑑 / 𝑢 and ℎ ( 𝑡 ) = otherwise. Interestingly, for large 𝑡 , ℎ ( 𝑡 ) decays as / 𝑡 𝛽 . This is relatedto the particle fraction at 𝜌 → 𝑎 which according to (10) depends on 𝛽 but not on 𝛼 .The initial delay of the received signal, 𝑡 = 𝑑𝑢 , (16)predicted by (15), is expected, as this is the time needed for particles initially placed in thecenter of the tube to travel distance 𝑑 , i.e., a measure for the theoretical time offset until whichno signaling particle has reached the receiver yet. Moreover, we note from (15) that ℎ ( 𝑡 ) = for 𝛼 ≠ and ℎ ( 𝑡 ) > for 𝛼 = , i.e., there is a jump at 𝑡 = 𝑡 .Finally, let us consider the position and the height of the peak (maximum) of the derived CIR.For 𝛼 > , the position of the peak of the CIR in (15) can be found at 𝑡 peak = 𝑡 · (cid:18) + 𝛼 − 𝛽 (cid:19) , (17)by equating the time derivative of ℎ ( 𝑡 ) in (15) with zero and solving for 𝑡 . Interestingly, it canbe observed via (16) that for any 𝛼 and 𝛽 , 𝑡 peak is proportional to 𝑑 . By substituting (17) in (15),the height of the peak follows as ℎ peak = 𝑐 𝑑 · 𝐵 ( 𝛼, 𝛽 ) · ( 𝛼 − ) 𝛼 − 𝛽 𝛽 ( 𝛼 + 𝛽 − ) 𝛼 + 𝛽 − , (18)which is inversely proportional to distance 𝑑 for any combination of 𝛼 and 𝛽 .As a summary of the CIR analysis, we conclude that for any choice of 𝛼 and 𝛽 , the peakheight ℎ peak decays proportional to / 𝑑 and ℎ ( 𝑡 ) , for large 𝑡 , decays as / 𝑡 𝛽 .
3) Special Cases:
Let us consider again, the two special cases of a uniform particle distributionand that of a particle distribution proportional to the flow profile described earlier. For these twocases, we expect a decay over time proportional to / 𝑡 and / 𝑡 , respectively, see [19, Chapter 15].This behavior is indeed recovered for ( 𝛼 = , 𝛽 = ) and ( 𝛼 = , 𝛽 = ) in (15).IV. M ODULATION , C
HANNEL E STIMATION , AND D ETECTION
In this section, we discuss the communication and signal processing aspects of our testbed,i.e., the preprocessing of the raw data, channel estimation, and detection algorithms.
January 7, 2021 DRAFT6
A. Data Preprocessing
Preprocessing of the data provided by the susceptometer is needed for a consistent postprocessingsuch as comparing with the developed CIR model and for detection. By manual examination,it turns out that the employed susceptometer delivers samples at sampling times which are notperfectly regular and the absolute time is not synchronized with the injection. For consistency,we employ a resampling by linear interpolation to 10 samples per second which corresponds tothe average sampling times of the measurement data as provided by the susceptometer.In the following, we denote the preprocessed susceptibility signal by 𝜒 [ 𝑖 ] = 𝜒 ( 𝑖 Δ 𝑡 ) where Δ 𝑡 = . is the sampling interval and 𝑖 = , , , . . . . Thereby, 𝜒 ( 𝑡 ) is the underlying butinaccessible preprocessed time-continuous signal.For time synchronization, we look for the start of the first occurrence of 10 consecutive samplessurpassing a threshold chosen as one hundredth of the maximal observed signal amplitude inthat measurement. This time index is labeled as 𝑖 start and the received signal 𝑟 [ 𝑖 ] = 𝜒 [ 𝑖 − 𝑖 start ] is then used for further processing. For future reference, the vector of received signal values isdenoted by 𝒓 . B. Channel Estimation
The information to be detected is represented as follows. We assume that information isrepresented by a sequence 𝒂 ∈ { , } 𝐾 of OOK symbols with 𝑎 [ 𝑘 ] ∈ { , } , 𝑘 ∈ { , , . . . , 𝐾 − } ,where 𝐾 is the number of transmitted symbols. This series of amplitude coefficients is thenmodulated on pumping pulses as described in Section II-A2. For PAM detection, we assume thefollowing basic pulse amplitude modulation model for the noise-free received signal 𝑠 [ 𝑖 ; 𝒂 , 𝒉 ] = 𝐾 − ∑︁ 𝑘 = 𝑎 [ 𝑘 ] · ℎ [ 𝑖 − 𝑘 𝐼 ] (19)where 𝑖 is the sampling index, and 𝐼 = 𝑇 / Δ 𝑡 = is the oversampling factor corresponding tosymbol interval 𝑇 . For convenience, the vector of noise-free received signal values is denoted by 𝒔 ( 𝒂 , 𝒉 ) . In particular, 𝒉 ∈ R 𝑁 𝐼 with entries ℎ [ 𝑖 ] , 𝑖 = , , . . . , 𝑁 𝐼 − , are the samples of the CIRwhich can be obtained by estimation using training data, as described in the following, and 𝑁 isthe memory length measured in numbers of symbols.For sequence estimation, we need to know the overall CIR ℎ [ 𝑖 ] . For our numerical results, weobtain this CIR by estimation based on training data sent at the start of transmission. To thisend, we denote the sequence of training symbols as 𝒂 t ∈ { , } 𝐾 t and the training samples of the January 7, 2021 DRAFT7 received signal as 𝒓 t ∈ R 𝐾 t 𝐼 , where 𝐾 t is the length of the training sequence. In a similar manner, 𝒔 ( 𝒂 t , 𝒉 ) denotes the vector of model transmit training signal samples. In the following, wedescribe two channel estimation schemes, one based on the CIR model developed in Section III-Band another one which directly estimates all samples of the CIR.
1) Model-based CIR Estimation:
For estimating the model parameters, we consider thefollowing optimization problem ˆ 𝛼, ˆ 𝛽, ˆ 𝛾 = arg min 𝛼,𝛽,𝛾> (cid:107) 𝒓 t − 𝒔 ( 𝒂 t , 𝛾 · 𝒉 ( 𝛼, 𝛽 )) (cid:107) , (20)where the samples of the CIR are given as 𝒉 ( 𝛼, 𝛽 ) = ℎ ( 𝑖 Δ 𝑡 + 𝑡 ; 𝛼, 𝛽 ) , 𝑖 = , , . . . , 𝑁 𝐼 − . Thisis a non-linear least-squares optimization problem with three parameters (scaling parameter 𝛾 and model parameters 𝛼 and 𝛽 ). The estimated CIR is then given by ˆ 𝒉 = ˆ 𝛾 · 𝒉 ( ˆ 𝛼, ˆ 𝛽 ) .
2) Direct Estimation of CIR:
Directly estimating the samples of the CIR is a common strategyfor channel estimation [48]. In particular, we use a least-squares scheme solving for 𝒉 : ˆ 𝒉 = arg min 𝒉 ∈ R 𝑁 𝐼 (cid:107) 𝒓 t − 𝒔 ( 𝒂 t , 𝒉 ) (cid:107) . (21)This is a linear least-squares optimization problem with 𝑁 𝐼 variables. Typically,
𝑁 𝐼 > , i.e., alarger number of variables has to be estimated compared to the parametric approach in (20). Inthe numerical results in Section V, we compare the performance of both methods. C. Detection Algorithms
For detection, i.e., estimation of the transmitted bit sequence from the received signal, weconsider both sequence estimation assuming the PAM structure in (19) as well as a model-agnosticheuristic detection scheme, namely increase detection.
1) Sequence Estimation:
For sequence estimation, we employ the Viterbi algorithm whichsolves the following optimization problem [49, Chapter 10] ˆ 𝒂 i = arg min 𝒂 i ∈{ , } 𝐾 𝑖 (cid:107) 𝒓 − 𝒔 ( [ 𝒂 t 𝒂 i ] , ˆ 𝒉 ) (cid:107) , (22)where 𝒂 i is the sequence of information symbols of length 𝐾 i and ˆ 𝒉 can be either estimatedbased on our proposed model via (20) or directly via (21). We note that this criterion is optimalwith respect to the error rate in case of impairment by additive white Gaussian noise [49] but isnot necessarily optimal for the unknown distortions in our testbed. The performance in terms ofthe number of decision errors achievable with sequence estimation is evaluated in Section V. January 7, 2021 DRAFT8
2) Increase Detection:
In the following, as a baseline for the sequence estimation describedbefore, we consider a simple version of the detection scheme described in [50]. In particular, thisdetection method does not rely on the PAM model introduced above. Instead, it is a heuristicattempt to exploit the observed characteristics of the received signal. In particular, in a givensymbol interval, for a binary “1” the received signal exhibits an increase in the current symbolinterval (after an appropriate delay) whereas for a binary “0” the received signal is non-increasingon average. This appears to be a convenient signal characteristic to exploit for detection when theexact channel distortions are unknown. Hence, one heuristic approach for detection is, for eachsymbol interval and despite the ISI, to check whether the signal is significantly increasing or not.In particular, we employ for the estimated symbol sequence ˆ 𝒂 the following detection rule ˆ 𝑎 [ 𝑘 ] = , if 𝑟 [ 𝑖 [ 𝑘 ]] − 𝑟 [ 𝑖 [ 𝑘 ]] > 𝜉 , otherwise (23)where 𝑖 [ 𝑘 ] = 𝑘 𝐼, 𝑘 ∈ [ , 𝐾 ) is the starting time of the 𝑘 th symbol interval and 𝑖 [ 𝑘 ] = 𝑖 [ 𝑘 ] + 𝐼 off ,where 𝐼 off ∈ [ , 𝐼 ) is a sampling offset. Moreover, 𝜉 is the detection threshold which needs tobe carefully selected to balance sensitivity to noise (if 𝜉 is too small) and a bias for detectingbinary “0” (if 𝜉 is too large).For choosing the detection parameters 𝐼 off and 𝜉 , there are different options. In this paper, weobtain these parameters based on peak position 𝑡 peak in (17) and peak height ℎ peak in (18) of theproposed CIR equation in (15) where, for simplicity, we assume that 𝛼 = 𝛽 = yields a reasonablecharacterization of the CIR. In particular, we choose 𝜉 = ℎ peak / and 𝐼 off = min { 𝐼 − , 𝑖 peak } ,where 𝑖 peak = (cid:100)( 𝑡 peak − 𝑡 )/ Δ 𝑡 (cid:99) with (cid:100)·(cid:99) denoting rounding to the closest integer value, i.e., wedetermine the index of the expected peak position without accounting for ISI and not exceedingthe symbol interval length. We note that this detection rule can be seen as a generalization to theone employed in [9] where 𝑖 [ 𝑘 ] is fixed to the middle of the 𝑘 th symbol interval and 𝑖 [ 𝑘 ] isfixed to the end of the 𝑘 th symbol interval.V. E XPERIMENTAL R ESULTS
In this section, we evaluate the analytical model equations in Section III and fit the parametersof the analytical model to experimental measurement data for the CIR. Then, we evaluate theperformance of the proposed detection schemes. In the following, the parameter values providedin Table I apply unless indicated otherwise.
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Time (s) S ca l e d S u s ce p ti b ilit y Impulse response /a a f() Initial distribution =1=2 =1=1=3=3 =3=3=1=2=1=1
Fig. 3. Dependence of the CIR on different initial distributions. (Left) CIR for distance 𝑑 =
10 cm scaled with 𝑐 𝑑 in (13)for different Beta initial distribution parameters and receiver weighting function lengths. (Right) Corresponding model initialdistribution 𝑓 𝜌 ( 𝜌 ) . CIRs strongly depend on the particle distribution, e.g., higher CIR peaks and faster decays are observed whenrelatively more particles are concentrated in the center of the tube rather than at the boundary ( 𝜌 = 𝑎 ). A. CIR Model Evaluation
To better illustrate the properties of the proposed CIR model, we investigate the dependenceof the CIR model equation (13) on the initial particle distribution as well as the impact of theweighting function lengths. To this end, in Fig. 3, we show (left) the numerical evaluation of theCIR and (right) the corresponding initial particle distributions in terms of the radial distribution 𝑓 𝜌 ( 𝜌 ) in (8). For each initial distribution, CIRs are shown for a receiver length of 𝑙 𝑧 =
20 mm corresponding to the length of the susceptometer housing (see Fig. 1a), 𝑙 𝑧 = correspondingto the sensitive region specified in the manual of the susceptometer, and 𝑙 𝑧 = which can beseen as an approximation and for which the closed-form expression is given in (15). For the Betainitial distribution, we consider the parameter pairs ( 𝛼 = , 𝛽 = ) corresponding to a uniformdistribution, ( 𝛼 = , 𝛽 = ) corresponding to a distribution proportional to the flow profile asintroduced in Section III-A2, and ( 𝛼 = , 𝛽 = ) which is chosen arbitrarily. For all shownCIRs, we observe an initial delay of about which is in good agreement with the starting time 𝑡 = .
09 s in (16). More accurately, the signals start at time ( 𝑑 − 𝑙 𝑧 / )/ 𝑢 because the receiverweighting function is centered at 𝑧 = and extends to 𝑧 = − 𝑙 𝑧 / , see Fig. 2.Overall, the observed CIR shapes depend strongly on the initial particle distribution but lesson the receiver weighting function length. Nevertheless, the CIR shapes seem more affected bythe choice of different weighting function lengths in case of the uniform distribution and thedistribution proportional to the flow profile. This is in particular the case for the peak valuewhich, in this case ( 𝛼 = ), coincides with the signal starting time (see (17)) and can be attributed January 7, 2021 DRAFT0 to the significant portion of particles around 𝜌 = , see Fig. 3 (right). This portion of particlestravels in a relatively concentrated manner due to the flat flow profile around 𝜌 = , see Fig. 2.Hence, the integral over space in (11) depends more strongly on the window length. This is incontrast to the CIR for ( 𝛼 = , 𝛽 = ) where the CIR depends on the window function lengthless strongly. In that case, the initial CIR increase is more smoothly and the approximate peaktime occurs at 𝑡 = .
82 s via (17).In summary, a variety of different CIR shapes can be realized by considering different initialparticle distributions. However, only minor variations in the CIR shape are observed for theconsidered different weighting function lengths. This is especially true for the distributions withdiminishing mass at 𝜌 = that are expected for the presented testbed, see also Fig. 1b where mostof the visible particle cloud resides within the upper half of the tube. Hence, in the following,the approximation of zero window length in (15) is used for fitting of the measurement data dueto its mathematical simplicity. Nevertheless, the more general CIR expression in (13) might beconvenient when investigating the effect of different coil lengths for a custom susceptometer. B. Conducted Experiments
To test the applicability of our analytical model, we make the following two experiments. a) Pulse Train:
For this experiment, we transmit a fixed sequence of 15 binary “1” viaOOK with symbol durations 𝑇 =
20 s ,
40 s ,
60 s , and
60 s for distances of 𝑑 = ,
10 cm ,
20 cm , and
40 cm such that by visual inspection no ISI is present, i.e., we interpret the observedconsecutive pulses as realizations of the CIR. From this data, we can then obtain a measuredaverage CIR and also evaluate variations of the CIR. b) Data Transmission:
For this experiment, we transmit a fixed sequence of 400 randomly(for time synchronization, the first symbol is fixed to be a “1”) chosen binary OOK symbols withsymbol duration 𝑇 = for distances 𝑑 = ,
10 cm ,
20 cm , and
40 cm , i.e., we take ISI intoaccount. From this data, we obtain estimates of the CIR using both (20) and (21) and performdetection using both sequence estimation and increase detection.
C. CIR Estimation
In this subsection, we evaluate the channel estimation scheme as described in Section IV-Bvisually and in terms of the root mean square error (RMSE).
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Time (s) S ca l e d S u s ce p ti b ilit y Impulse response /a a f() Initial distribution
Fig. 4. Fitting of CIR model. (Left) 15 overlayed CIR realizations (gray curves) and estimated CIR model (blue, orange, green,and red curves) for distances of 𝑑 = ,
10 cm ,
20 cm , and
40 cm . The CIRs are scaled by 𝑐 𝑑 and shifted to time 𝑡 . Thereis a good fit between model and measurement. (Right) Corresponding fitted initial distributions. The fitting parameters are ( 𝛼 = . , 𝛽 = . , 𝛾 = . ) , ( 𝛼 = . , 𝛽 = . , 𝛾 = . ) , ( 𝛼 = . , 𝛽 = . , 𝛾 = . ) , ( 𝛼 = . , 𝛽 = . , 𝛾 = . ) . Thefitted initial distributions are consistent across all considered distances. In Fig. 4, we evaluate the channel estimation based on our proposed CIR model in (20) forthe pulse train experiments described in Section V-B. Thereby, we consider CIR lengths of , , , and symbol durations (excluding the initial delay) for distances 𝑑 = ,
10 cm ,
20 cm , and
40 cm . On the left hand side, for each transmission distance, we show an overlay of15 CIR measured pulses (gray curves) as well as the estimated CIR with fitted parameters (blue,orange, green, and red curves) according to (20). For illustration, the synchronized CIRs areshifted to start consistently at time 𝑡 = 𝑑 / 𝑢 and all CIRs are scaled by 𝑐 𝑑 . On the right handside, we show the corresponding fitted initial particle distributions.From the CIR data (left), we can observe that the measured CIRs do not show much variationacross the considered 15 realizations. Furthermore, the peak height for the scaled CIRs is similarfor all considered transmission distances 𝑑 which means that the peak heights of the unscaledCIRs scale approximately as / 𝑑 . Moreover, the CIRs become significantly broader for increasingdistance which can be associated with increasing levels of ISI. From the fitted initial particledistributions (right), we can observe a similarity for all considered distances which is consistentwith our model since the initial distribution is assumed to depend on the injection but not on thetransmission distance. The initial radial particle distributions exhibit a peak around 𝜌 = . 𝑎 and diminishing mass at 𝜌 = and 𝜌 = 𝑎 .In summary, the derived CIR model can fit measurement data remarkably well despite theunderlying simplifying assumptions and is consistent in terms of peak decay over distance and January 7, 2021 DRAFT2
10 20 30 40 50 60 70 80 90 100
Training symbols -1 R M S E Model fit (20)Sample fit (21)
Fig. 5. Quality of channel estimation. The average RMSE per sample for the whole received signal is shown as a function ofthe number of training symbols used for channel estimation for distances 𝑑 = ,
10 cm ,
20 cm , and
40 cm . The receivedsignals are scaled by 𝑐 𝑑 . Channel estimation by fitting of the proposed model and by fitting of all CIR samples are compared.The CIR estimate using the proposed model is reasonably accurate for all considered numbers of training symbols. stable in the initial particle distribution for different distances. Moreover, the measured individualCIRs are not significantly affected by noise or other distortions, i.e., the randomness of the CIRis limited for the considered operation of the testbed.In the following, we investigate how the proposed CIR model generalizes to the data transmissionexperiments described in Section V-B. To this end, we perform CIR estimation on the first 𝐾 t symbols and then evaluate the root mean square error (RMSE) for all 400 symbols 𝒂 . The RMSEis normalized per sample and can be computed as RMSE = √︃ | 𝒓 − 𝒔 ( 𝒂 , 𝛾 · 𝒉 ) | / / 𝑐 𝑑 . Thereby,for obtaining 𝒉 both the model-based estimate (20) as well as the sample-based estimate (21) inSection IV-B are evaluated.The corresponding results for distances 𝑑 = ,
10 cm ,
20 cm , and
40 cm are shown inFig. 5, where the RMSE for the information symbols is shown as a function of the numberof training symbols used for CIR estimation. For increasing distance, we generally have largererrors and the RMSE generally decreases for more training symbols, i.e., the estimates are moreaccurate if the training is longer but generally worse for more ISI. This mismatch could becaused by several physical effects related to the injection or reception and is worthwhile to studyin future work. The model-based estimate works reasonably well for all considered numbers oftraining symbols, i.e., the estimate generalizes well even for small numbers of training symbols.The sample-based estimate strongly depends on the number of training symbols and improves asthe number of training symbols increases. Thereby, the model-based estimate outperforms the
January 7, 2021 DRAFT3
TABLE IIN
UMBER OF ERRORS FOR THE LAST
DATA SYMBOLS
Scheme
Increase Detection 2 0 15 17Sequence Estimation (model) 0 0 11 35Sequence Estimation (sample) 0 0 0 31 sample-based estimate for smaller numbers of training symbols while the sample-based estimatecan be slightly better for long training.
D. Detection
To investigate the performance of the proposed detection schemes, we apply increase detectionand sequence estimation as described in Section IV for detection of the 400 OOK symbolsequence transmitted in our data transmission experiments described in Section V-B. Thereby,based on the RMSE results in Fig. 5, we choose the training as follows. On the one hand,for CIR estimation using (20), the first 𝐾 t = , , , and symbols are used as trainingsymbols, as the RMSE does not significantly decrease for longer training sequences. On theother hand, for CIR estimation using (21), the first 𝐾 t = , , , and symbols are usedas training symbols, as the RMSE decreases significantly for larger numbers of training symbols.The remaining 𝐾 i = − 𝐾 t symbols are used for evaluating the proposed detection algorithmsbut not for channel estimation.The corresponding decision error results are summarized in Table II where for comparisononly errors for the last 300 data symbols are reported. For distances of up to
10 cm , no or onlya small number of decision errors are observed for all considered detection schemes. For adistance of
20 cm , some symbol errors are observed for both increase detection and sequenceestimation with model-based CIR estimation whereas sequence estimation with sample-basedCIR estimation still shows no errors. In this scenario, because of the long training sequence,the sample-based CIR estimate is accurate and hence detection benefits from the more complexsequence estimation algorithm. For a distance of
40 cm , all considered detection schemes causedecision errors whereby increase detection results in fewer errors than sequence estimation. Inthis case, the worse performance of sequence estimation can be attributed to the relatively large
January 7, 2021 DRAFT4
354 356 358 360 362 364 366 368
Time (s) S ca l e d S u s ce p ti b ilit y
354 356 358 360 362 364 366 368
Time (s) S ca l e d S u s ce p ti b ilit y
354 356 358 360 362 364 366 368
Time (s) S ca l e d S u s ce p ti b ilit y
354 356 358 360 362 364 366 368
Time (s) S ca l e d S u s ce p ti b ilit y MeasuredModel (true)Model (est)Increase
Fig. 6. Frames of the received signal towards the end of transmission. Symbol intervals are separated by gray vertical linesand the transmitted bit sequence is shown as black text. Detection errors are indicated by red, blue, and green background forincrease detection, sequence estimation, and both, respectively. For increase detection, the highlighted black signal points arecompared with the signal value at the beginning of each symbol interval. For sequence estimation, the model PAM signal usingthe detected symbol sequence and the transmitted symbol sequence are shown by black solid and dashed lines, respectively. Themore severe ISI for increasing distances limits the detection performance for larger distances.
CIR estimation error for larger distances, see Fig. 5. Increase detection does not rely on CIRestimation, and thus exhibits a similar number of decision errors as for a distance of
20 cm .In summary, with any of the presented detection schemes, reliable communication with onlyfew decision errors is possible for distances of at least up to
40 cm . Nevertheless, non-coherentdetection schemes like the proposed increase detection might cope better with unknown distortionsand non-linearities in cases of severe ISI as is the case for larger distances. Coherent detectionschemes like the proposed sequence estimation are expected to perform well with enough trainingdata where less training is required for the proposed model CIR. However, we note that thepresented results correspond to just a single realization of the received signal for the transmissionof 400 symbols and more experiments are necessary to thoroughly evaluate different detectionschemes.To visualize the model mismatch and to illustrate the detection algorithms and the errorevents we show excerpts of the received signal in Fig. 6 for transmission distances of 𝑑 = , January 7, 2021 DRAFT5
10 cm ,
20 cm , and
40 cm . In particular, for all considered distances, we show the measuredreceived signal scaled by 𝑐 𝑑 for a 15 symbol time frame from
354 s to
369 s towards the endof transmission. Increase detection is visualized by the symbol interval starting times and thesignal point used for detection within the interval. Sequence estimation with the model CIR isvisualized by the hypothetical PAM signal using both the estimated symbol sequence as well asthe true one.Generally, as expected from the CIRs in Fig. 4, we observe increasingly less pronounced peaksand more ISI for increasing distances. The model PAM signal appears to follow the measuredreceived signal reasonably well but the mismatch between measurement and model becomesmore pronounced for larger distances (note that the y-axes are scaled differently).The highlighted decision errors can be explained as follows. For a transmission distance of 𝑑 = , for the symbol interval starting at
354 s , a “1” to “0” error is observed for increasedetection which is due to the second signal sample being not significantly larger than the signalsample at the beginning despite there being a significant peak. This can be explained by a timesynchronization error, e.g., shifting the two sample points a little bit to the right the bit “1” couldbe correctly detected. However, this shift cannot be generally applied because all other symbolswould be affected as well, e.g., for the symbol interval starting at
362 s the timing is fine. Fortransmission distance
20 cm , for the symbol interval starting at time
363 s , a “0” to “1” errori,s observed for both considered detection schemes. Thereby, for increase detection the error iscaused by the second sample being larger than the first sample which might again be caused bysmall errors introduced by the susceptometer software or noise. For sequence estimation, theerror can be interpreted as a "lift up" of the signal to better follow the measurement at later times,e.g., compare dashed "true" line with solid "detected" line at time
367 s . The other observeddecision errors can be explained in a similar manner.In summary, the linear PAM model with the estimated CIR is in good agreement with themeasurement results. However, for larger distances, here for
40 cm , strong and potentially non-linear ISI is present. Moreover, time synchronization and improved receiver concepts constituteinteresting topics for future work.
January 7, 2021 DRAFT6
VI. C
ONCLUSIONS
A. Summary
In this paper, we have presented a new testbed for the investigation of flow-driven MC systemsas encountered in the cardiovascular system and chemical reactors. To this end, we demonstratedthe applicability of SPIONs for signaling which are particles engineered to be chemically stable,to avoid agglomeration, and to not engage in reactions with surrounding molecules. Moreover,by their magnetic property, SPIONs are detectable without direct access to the tube channel.After a review of the relevant physical effects, we proposed a simple mathematical modelbased on laminar flow-driven particle transport, a parametric initial SPION distribution with twoparameters, and a transparent receiver. Channel estimation for several measurements with andwithout ISI confirmed the applicability of the proposed CIR model for different transmissiondistances and training sequence lengths. Moreover, symbol detection schemes with and withoutusing the CIR model were shown to enable reliable communication for example measurements.Potential applications of SPION based MC include reporting sensing results and carrying controlinformation in industrial, microfluidic or biomedical settings, especially at locations where otherforms of communication can not be employed.
B. Outlook
We highlight the following directions for future theoretical and experimental work. There areseveral interesting preprocessing and detection schemes that could be evaluated with this testbedincluding matched filtering [51], optimal coherent and non-coherent [52] as well as adaptive,learned [53], and feature-based heuristic detection schemes [54]. To this end, it will also beuseful to develop further mathematical models for the received signal, including a statisticalcharacterization of noise and other distortions, e.g., by diffusion, turbulent flow, the injection,the properties of the employed fluid, an overall non-linearity, and time-variant flow [7]. Furthercomprehensive measurements will help in validating these models and algorithms. These modelswill also help in developing novel channel estimation [48] and synchronization [55] schemeswhich again can be model-based to different degrees. In addition to detection, also differentmodulation schemes and transmission from a single transmitter to multiple receivers as well asfrom multiple transmitters to a single receiver could be investigated as suitable extensions ofthe presented point-to-point link. A better theoretical understanding will also help guiding the
January 7, 2021 DRAFT7 hardware development. This includes the optimization of the receiver device [34], employingdifferent pumps for better control of the injection, changing the injection mechanism, e.g.,replacing the Y-connector by a needle, and testing different types of particles. Moreover, thetestbed could potentially be expanded by implementing a network of ducts, changing the carrierliquid, employing magnets for particle movement control, and scaling of its size. Furthermore,particles could be additionally tagged with other chemicals. These extensions could also facilitatethe use of higher-order modulation, e.g., by using different particle types, combining optical andmagnetic measurements of the particles, or using different forms of injection.A
PPENDIX D ERIVATION OF F LOW -D RIVEN M ODEL I MPULSE R ESPONSE
In this appendix, we derive the model CIR in (13). For the following derivation, we assumecylindrical coordinates 𝒙 = ( 𝜌, 𝜙, 𝑧 ) .In general, from (11) and (5), the received signal due to a single release at time 𝑡 = can bewritten as 𝜒 ( 𝑡 ) = 𝜒 ref · ∭ R 𝑤 ( 𝒙 ) · 𝑐 i ( 𝒙 − 𝑢 ( 𝜌 ) · 𝑡 · 𝒆 𝑧 ) d 𝑉 . (24)Now, using (12) and (6) and (7), we arrive at ℎ ( 𝑡 ) = 𝜒 ref 𝑉 i 𝜋𝑎 𝑙 𝑧 ∫ 𝑎 ∫ ∞−∞ rect ( 𝑧 / 𝑙 𝑧 ) · 𝑓 𝜌 ( 𝜌 ) · 𝛿 ( 𝑧 − 𝑢 ( 𝜌 ) 𝑡 + 𝑑 ) d 𝑧 d 𝜌. (25)For convenience, we substitute 𝜌 with 𝑠 = ( 𝜌 / 𝑎 ) and use 𝑓 𝑠 ( 𝑠 ) in (8). Then, we obtain ℎ ( 𝑡 ) = 𝜒 ref 𝑉 i 𝜋𝑎 𝑙 𝑧 ∫ ∫ ∞−∞ rect ( 𝑧 / 𝑙 𝑧 ) · 𝑓 𝑠 ( 𝑠 ) · 𝛿 ( 𝜑 ( 𝑠 )) d 𝑧 d 𝑠, (26)where 𝜑 ( 𝑠 ) = 𝑧 − ˜ 𝑢 ( 𝑠 ) · 𝑡 + 𝑑 and ˜ 𝑢 ( 𝑠 ) = 𝑢 · ( − 𝑠 ) which is simply obtained from (4) bysubstituting 𝜌 with 𝑠 . Now, we use the properties of the Dirac delta function to simplify the term 𝛿 ( 𝜑 ( 𝑠 )) . To this end, we note that 𝜑 ( 𝑠 ) = for 𝑠 = − ( 𝑧 + 𝑑 )/( 𝑢 𝑡 ) provided 𝑧 + 𝑑 < ˜ 𝑢 ( 𝑠 ) · 𝑡 .Thus, we can rewrite the delta function as [56] 𝛿 ( 𝜑 ( 𝑠 )) = | 𝜑 (cid:48) ( 𝑠 ) | 𝛿 ( 𝑠 − 𝑠 ) , (27)where 𝜑 (cid:48) ( 𝑠 ) = 𝑢 𝑡 . Then, using the sifting property of the Dirac delta function [56], we arrive at ℎ ( 𝑡 ) = 𝜒 ref 𝑉 i 𝜋𝑎 𝑙 𝑧 · 𝑢 𝑡 ∫ ∞−∞ rect ( 𝑧 / 𝑙 𝑧 ) · 𝑓 𝑠 (cid:18) − 𝑧 + 𝑑𝑢 𝑡 (cid:19) d 𝑧. (28)Finally, by straightforward integration and using the definition of 𝐹 𝑠 ( 𝑠 ) , we arrive at (13). Thisconcludes the proof. January 7, 2021 DRAFT8 R EFERENCES [1] H. Unterweger, J. Kirchner, W. Wicke, A. Ahmadzadeh, D. Ahmed, V. Jamali, C. Alexiou, G. Fischer, and R. Schober,“Experimental molecular communication testbed based on magnetic nanoparticles in duct flow,” in
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