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Analytical Modeling of Open- and Closed-Loop Dispersive Molecular Communication Channels with Pulsatile Flow

Theofilos Symeonidis, Fardad Vakilipoor, Timo Jakumeit, Robert Schober, Nunzio Tuccitto, Maximilian Schäfer

arXiv:2610.10102v1cs.ETphysics.flu-dyn

TL;DR

Molecular communication models need to describe both local transport in vessel segments and longer-term circulation, but blood flow is pulsatile. This paper develops and validates a time-variant analytical model, finding close agreement with simulations and identifying conditions for its applicability.

  • Problem

    In-body molecular communication requires models of local diffusion and flow as well as circulation-related repeated arrivals and coupling to vascular-network structure.

    Source: §I · p. 1

  • Method

    The authors derive a time-variant dispersive 1D model for straight ducts, extend it to closed-loop channels, and assess applicability with an Rx-distance indicator and a first-arrival peak-time approximation.

    Source: §I · p. 2, §VI · p. 13

  • Results

    The analytical model closely agrees with 3D PBSs; pulsatility matters most for predominantly advective transport before temporal averaging, and PBS sweeps support S_Rx ≥ 2 as a practical applicability criterion.

    Source: §VI · p. 13

  • Takeaways & Limitations

    Explicitly model pulsatility when temporal averaging is limited; the reduced model is most appropriate for locally uniform vessel segments meeting its rigid-wall and Newtonian-fluid assumptions.

    Source: §VI · p. 13

    The S_Rx criterion alone does not establish physiological applicability; large compliant arteries and microvascular vessels may require additional physiological modeling.

    Source: §VI · p. 13

Abstract

from arXiv · show

Molecular communication (MC) is a communication paradigm in which information is conveyed through the release, propagation, and reception of molecules. Many envisioned healthcare applications of MC are expected to operate inside the human body, where the cardiovascular system (CVS) may serve as the physical propagation environment and molecular transport is governed by diffusion and blood flow. Although blood flow is inherently pulsatile, most analytical MC channel models assume steady flow. In this paper, we develop a time-variant analytical model for dispersive MC channels with pulsatile flow. We derive the straight-duct response as a Normal distribution with time-variant mean and variance, capturing the combined effects of diffusion and pulsatile flow, and extend it to closed-loop channels through a wrapped-Normal representation. The model is validated against three-dimensional (3D) particle-based simulations (PBSs) for synthetic and physiologically motivated velocity waveforms. We further derive a closed-form first-order approximation for the first-arrival peak time and introduce the nondimensional indicator $S_{\mathrm{Rx}}$ for assessing the applicability of the reduced one-dimensional (1D) model. Our results show that pulsatility has the strongest influence when molecular transport is predominantly advective and the temporal flow variations are not averaged out, whereas stronger diffusion and faster pulsations reduce its impact on the received signal. Moreover, a PBS-based parameter sweep supports $S_{\mathrm{Rx}} \ge 2$ as a practical criterion for the applicability of the proposed analytical model. Finally, relating the analytical assumptions to representative blood-vessel classes shows that model applicability depends not only on the transport conditions but also on physiological properties such as vessel rigidity, geometric uniformity, and blood rheology.

I. INTRODUCTION

Most analytical MC channel models assume steady flow, despite pulsatile blood flow and the need to represent both local vessel transport and closed-loop recirculation. The paper develops a time-variant dispersive model and relates its applicability to transport conditions and vessel properties.

I. INTRODUCTION

  • Existing closed-loop models motivate accounting for recirculation, while individual vessel segments can differ in size and flow pulsatility.A single-loop topology is a network-level abstraction, not an assumption of uniform local transport conditions.Source: §I · p. 1
  • Most analytical MC channel models assume steady flow, although blood flow in the cardiovascular system is pulsatile and MC applications are expected inside the body.The paper motivates models that account for diffusion and blood flow in this setting.Source: §I · pp. 1–2
  • The proposed time-variant model derives a straight-channel response and extends it to closed loops, representing both local transport and molecule recirculation.Its stated basis is an effective 1D advection-diffusion equation with time-variant velocity and diffusion coefficient.Source: §I · p. 2
  • The model combines Womersley flow with Aris-Taylor dispersion to capture time-varying mean displacement and effective dispersion from diffusion and pulsatile flow.For impulsive release in a straight duct, its response is a Normal distribution with time-variant mean and variance.Source: §I · p. 2
  • The work also derives a first-order approximation for the direct first-arrival peak time and validates the models against 3D PBSs using synthetic and physiologically motivated waveforms.The study compares time-varying models with steady-flow models to examine pulsatility's effect on the received signal.Source: §I · p. 2

II. PHYSICAL CHANNEL MODEL

  • The physical model starts from 3D advection-diffusion in a cylindrical vessel and reduces axial transport to a 1D dispersive description under a Womersley velocity profile.The accompanying system illustration depicts this progression from a closed-loop channel to an effective 1D model.Source: §II · p. 3, Figure 1 · p. 3, A. 3D Transport Model · p. 3

B. Dispersive Regime and 1D Model Reduction

The 1D reduction relies on dispersive-regime assumptions that make cross-sectional concentration approximately uniform and shear-induced spreading dominant. For Womersley flow, it represents axial transport with time-varying effective velocity and diffusion.

B. Dispersive Regime and 1D Model Reduction

  • In the dispersive regime, axial spreading is dominated by shear-induced dispersion from transverse diffusion and a nonuniform flow profile, rather than axial molecular diffusion alone.The reduced equation uses time-varying coefficients.Source: B. Dispersive Regime and 1D Model… · p. 3
  • The 1D dispersive reduction assumes a long, slender channel, rapid radial diffusion, laminar fully developed radially symmetric flow, and a Newtonian carrier fluid.Under these conditions, concentration is rapidly homogenized across the section and longitudinal dynamics admit an effective 1D equation.Source: B. Dispersive Regime and 1D Model… · p. 3

C. Womersley Velocity Profile

  • For Womersley profiles, the reduced model uses cross-sectionally averaged axial velocity and an effective time-varying diffusion coefficient accounting for molecular diffusion and shear-induced dispersion.The adopted velocity profile is a steady Poiseuille component plus harmonic pulsatile modes; its Womersley number characterizes unsteady inertia relative to viscous diffusion.Source: B. Dispersive Regime and 1D Model… · pp. 3–4, C. Womersley Velocity Profile · p. 4

D. Time-Variant Diffusion Coefficient

The paper specifies a time-varying effective diffusion coefficient for the reduced transport model, then solves that model for straight-duct and closed-loop molecular communication channels.

D. Time-Variant Diffusion Coefficient

  • For the considered Womersley-number ranges, the time-varying effective diffusion coefficient follows an approximation from prior work.The coefficient is determined by the pulsatile velocity described by the Womersley model.Source: D. Time-Variant Diffusion Coefficient · p. 4

III. MC CHANNEL MODEL

  • The channel-model derivation solves the 1D advection-diffusion equation first for a straight duct without recirculation, then extends the solution to closed-loop channels.The straight duct represents local transport within an individual vessel segment.Source: §III · p. 4

A. Straight Duct Channel Model

  • For an impulsive particle release at t = 0 and x = 0, Lemma 1 gives the solution to the 1D equation in an infinitely long straight duct.The supplied passage states the lemma's setup and role, but not the solution's explicit form.Source: A. Straight Duct Channel Model · p. 4

B. Extension to Closed-Loop Channels

The closed-loop solution wraps the straight-duct concentration into periodic images to represent molecular recirculation. In the formal high-frequency limit of the reduced model, pulsatile transport approaches steady-flow mean behavior but has greater effective dispersion.

B. Extension to Closed-Loop Channels

  • The closed-loop concentration is represented by a wrapped-Normal distribution that extends the straight-duct solution to account for molecular recirculation.Periodic images with k > 0 represent forward circulations; k < 0 terms represent spreading across the periodic boundary in the opposite axial direction.Source: B. Extension to Closed-Loop Channels · p. 5

C. High-Frequency Limit of Pulsatile Flow

  • The high-frequency limit is formal: sufficiently large frequencies eventually violate the assumptions underlying the reduced one-dimensional model.The Womersley number scales as the square root of harmonic index times frequency.Source: C. High-Frequency Limit of Pulsatile… · p. 5
  • As pulsation frequency tends to infinity, the mean particle transport converges to that of steady flow and is governed by the temporal mean velocity.Within the reduced model, the oscillatory contributions to mean displacement vanish in this limit.Source: C. High-Frequency Limit of Pulsatile… · p. 5
  • In the high-frequency limit, pulsation increases effective dispersion relative to steady flow, with the additional dispersion depending on squared harmonic amplitudes.The cycle-averaged effective dispersion is also higher for pulsatile flow at a given mean velocity.Source: C. High-Frequency Limit of Pulsatile… · p. 5

D. First-Arrival Peak-Time Approximation

The paper derives a closed-form first-order approximation for the first-arrival peak time from the analytical channel model. The approximation uses a uniform-concentration assumption over the receiver and is most accurate for small-to-moderate pulsatile modulation.

D. First-Arrival Peak-Time Approximation

  • The first-arrival peak-time approximation is derived under a uniform-concentration assumption over the receiver and retains only first-order terms in pulsatile amplitudes.The peak time is defined by the maximum finite-region receiver response; the closed-loop first-arrival response is the k = 0 contribution.Source: D. First-Arrival Peak-Time Approximation · p. 6
  • The peak shift depends on instantaneous modulation at the steady-flow peak time and accumulated displacement from the pulsatile waveform up to that time.These contributions are denoted q0 and Q0, respectively.Source: D. First-Arrival Peak-Time Approximation · p. 6
  • The approximation is most accurate for small-to-moderate modulation, when quadratic and higher-order harmonic terms can be neglected.It describes the first-arrival peak; later recirculating peaks require the full wrapped-Normal solution.Source: D. First-Arrival Peak-Time Approximation · p. 6

IV. SIMULATION RESULTS

The simulation section compares the time-variant analytical responses with three-dimensional particle-based simulations and steady-flow models. Its synthetic tests use sinusoidal and pulsed velocity waveforms, alongside validation scenarios described in the section overview.

IV. SIMULATION RESULTS

  • The validation compares straight-duct and closed-loop receiver responses from the analytical models with 3D particle-based simulations and corresponding steady-flow models.The section also evaluates the first-arrival peak-time approximation.Source: §IV · p. 6

A. Velocity Waveforms

  • The validation considers synthetic and physiologically motivated waveforms, with the 3D flow field constructed to match the 1D waveform's temporal mean and harmonic content.The physiologically motivated waveform is discussed in the following subsection.Source: A. Velocity Waveforms · p. 6

1) Synthetic Velocity Waveforms:

  • The synthetic sinusoidal waveform has oscillation amplitude A = 0.5 and pulsation frequency f.It is represented by a single harmonic with N = 1, M1 = A, and ϕ1 = −π/2.Source: 1) Synthetic Velocity Waveforms · pp. 6–7
  • The synthetic pulsed waveform has duty cycle d = 0.2 and is implemented as a finite-harmonic approximation of an ideal rectangular pulse.The Fourier-series truncation retains enough harmonics to capture the pulse shape adequately; the period is T = 1/f.Source: 1) Synthetic Velocity Waveforms · p. 7

2) Realistic Velocity Waveform:

The physiologically motivated velocity waveform is fitted with a 12-harmonic representation and normalized to preserve its shape across mean velocities. The particle-based validation setup uses a cylindrical channel and parameters selected for the dispersive, low-Womersley regime.

2) Realistic Velocity Waveform:

  • The physiologically motivated waveform is fitted from using 12 harmonics, with its fundamental frequency selected by minimizing mean-squared fitting error.The fitted waveform is normalized by its temporal mean so its shape is independent of the selected mean velocity; Fig. 2 shows it alongside synthetic waveforms.Source: 2) Realistic Velocity Waveform · p. 7, Figure 2 · p. 7
  • The velocity model is described as flexible enough to represent realistic pulsatile flows and synthetic waveforms used in controlled or microfluidic setups.The plotted waveforms are shown for mean velocity ¯u = 1 × 10−4 m/s.Source: 2) Realistic Velocity Waveform · p. 7, Figure 2 · p. 7

B. Particle-Based Simulation Setup

  • The particle-based simulation uses a 3D cylindrical straight channel with reflecting walls and estimates the receiver response by counting particles in a finite cylindrical receiver region.The simulation releases 5 × 10^5 particles uniformly across the inlet cross-section.Source: B. Particle-Based Simulation Setup · p. 7
  • The default setup uses D = 5 × 10−9 m2/s, R = 50 µm, L = 1 mm, xRx = 0.3 mm, and ∆xRx = 0.1 mm, with parameters chosen for the dispersive regime and reduced-model assumptions.Blood-like fluid properties give Womersley numbers below unity for all considered harmonics, supporting use of the analytical model in (13).Source: B. Particle-Based Simulation Setup · pp. 7–8

C. Results for Synthetic Velocity Waveforms

Across synthetic and physiologically motivated pulsatile-flow scenarios, the analytical model agrees closely with particle-based simulations, while pulsatility’s effect depends on transport conditions. The first-order approximation also closely tracks finite-region first-arrival peak times.

C. Results for Synthetic Velocity Waveforms

  • As pulsation frequency increases, the synthetic-waveform response converges toward the corresponding steady-flow response because particles experience multiple velocity oscillations during propagation.The analytical model agrees with PBS across the considered frequencies and both waveforms.Source: C. Results for Synthetic Velocity… · p. 8
  • For varying mean velocities, the analytical model agrees closely with PBS for both synthetic waveforms, while higher velocities make peaks earlier and sharper and pulsatility more pronounced.Lower mean velocities make the response more like steady flow as diffusion smooths flow variations.Source: Figure 4 · p. 8, C. Results for Synthetic Velocity… · p. 8

D. Results for Realistic Velocity Waveform

  • For physiologically motivated flow, the analytical model agrees closely with PBS across the considered scenarios, while pulsatility weakens at larger Rx distances, lower mean velocities, and larger diffusion coefficients.The weaker influence corresponds to a reduced relative contribution of advection compared with diffusion; pulsatility can also shift received-signal peaks.Source: D. Results for Realistic Velocity… · pp. 8–9
  • In the closed-loop case, analytical and PBS responses agree closely, and the response shows delayed peaks from recirculating molecules with diminishing pulsatility over successive circulations.Increasing temporal spreading smooths velocity-induced variations over propagation time.Source: D. Results for Realistic Velocity… · p. 9, Figure 6 · p. 9

E. Accuracy of the First-Arrival Peak-Time Approximation

  • The first-order approximation closely follows the reference first-arrival peak time across sweeps of mean velocity, diffusion coefficient, and Rx distance, with RMSE values from 0.064 s to 0.095 s.Peak time decreases with higher mean velocity, depends weakly on diffusion over the considered range, and increases with Rx distance; the approximation avoids numerical peak searching.Source: E. Accuracy of the First-Arrival… · pp. 9–10, Figure 7 · p. 10

V. REGIME ANALYSIS

The regime analysis uses a conservative nondimensional indicator to summarize whether the reduced 1D model’s assumptions hold over a Tx–Rx distance. A PBS parameter sweep supports S_Rx ≥ 2 as a practical criterion for accurate received-signal approximation.

V. REGIME ANALYSIS

  • The analysis also relates the indicator-based applicability assessment to representative blood-vessel classes.The paper describes this class-level compatibility qualitatively.Source: §V · p. 10

A. PBS-Based Model Applicability Analysis

  • The sweep varies vessel radius, mean velocity, diffusion coefficient, and Tx–Rx distance, and the indicator is used to assess applicability beyond the combinations directly validated by PBS.PBS comparison is computationally demanding at larger propagation distances, whereas S_Rx can be evaluated directly; its component measures increase with Rx distance when the other parameters are fixed.Source: A. PBS-Based Model Applicability… · pp. 10–11, §V · p. 10
  • The reduced 1D model requires three conditions, so S_Rx is defined conservatively by the least-satisfied condition; larger values indicate that the combined assumptions are more strongly satisfied.The measures concern domain slenderness, radial-diffusion versus axial-advection timescales, and axial advection versus axial molecular diffusion.Source: A. PBS-Based Model Applicability… · pp. 10–11
  • A PBS-based parameter sweep supports S_Rx ≥ 2 as a practical criterion for expecting the analytical model to reproduce received-signal dynamics accurately.In the illustrated sweep, radii of 10 µm and 20 µm meet the criterion and show excellent agreement, while larger radii fall below it and agreement deteriorates.Source: A. PBS-Based Model Applicability… · p. 11

B. Applicability to Representative Blood-Vessel Classes

The reduced 1D model is expected to be reliable when both the vessel’s physical assumptions and the SRx ≥ 2 criterion are satisfied. Its fit varies by vessel class, while pulsatility matters most when temporal variations are not averaged out.

B. Applicability to Representative Blood-Vessel Classes

  • The vessel-class assessment concerns local Tx–Rx transport before recirculation and qualitatively weighs geometry, wall rigidity, fluid behavior, and expected pulsatility.Tapering is a secondary consideration; pronounced wall deformation, non-Newtonian rheology, and very small vessel dimensions can impose more fundamental limits.Source: B. Applicability to Representative… · p. 11, Table I · p. 12
  • Large and pulmonary arterioles are candidate local-segment settings when SRx ≥ 2, although their Newtonian-fluid assumption is marginal.Their walls are approximately rigid, tapering is negligible, and pulsatility is moderate.Source: B. Applicability to Representative… · p. 12
  • Small veins and venules are compatible with several assumptions, but weak pulsatility means pronounced pulsatile effects are not expected.Medium and large veins can also be compatible over sufficiently uniform segments when SRx ≥ 2, despite marginal wall rigidity and, in large veins, local geometric uniformity.Source: B. Applicability to Representative… · p. 12
  • Femoral, renal, and medium-sized arteries are conditionally compatible as local segments when wall deformation and radius variation are sufficiently small and SRx ≥ 2.Their wall rigidity and local uniformity may be marginal, while Newtonian-fluid behavior and moderate-to-strong pulsatility generally fit the assessment.Source: B. Applicability to Representative… · p. 12
  • The aorta and pulmonary arteries are least compatible because their compliant walls and complex hemodynamics exceed the model’s assumptions.The model is limited to local-segment approximation there; a fuller treatment would include wall compliance, Windkessel-type buffering, and pulse-wave propagation.Source: B. Applicability to Representative… · p. 12
  • Terminal arterioles and capillaries are less compatible because cell-scale diameters can make cell partitioning, near-wall interactions, and non-Newtonian rheology important.These vessel classes require more specialized microvascular transport models.Source: B. Applicability to Representative… · p. 12

VI. CONCLUSION

  • The analytical model’s straight-duct response uses a time-varying Normal distribution and its closed-loop response a wrapped-Normal representation; both agreed closely with 3D PBSs in the considered regimes.Pulsatility matters most for predominantly advective transport when flow variations are not averaged out; stronger diffusion, longer propagation times, and faster oscillations reduce sensitivity.Source: §VI · p. 13
  • SRx ≥ 2 is a practical applicability criterion, but physiological suitability also requires locally uniform segments with sufficiently rigid walls and Newtonian-fluid behavior.Large compliant arteries and microvascular vessels may require additional physiological modeling.Source: §VI · p. 13, B. Applicability to Representative… · p. 13

APPENDIX A PROOF OF LEMMA 1

The straight-duct solution is obtained by transforming away advection and normalizing time so the diffusion equation has constant coefficients. The closed-loop solution then imposes periodicity, equivalently wrapping the straight-duct response around the loop.

APPENDIX A PROOF OF LEMMA 1

  • A moving spatial coordinate represents cumulative axial displacement, and transforming concentration cancels the advection terms in the transport equation.Source: Appendix A · p. 13
  • A transformed time variable normalizes the time axis so the diffusion coefficient becomes constant, yielding a constant-coefficient diffusion equation.For an impulsive release, the transformed initial condition is a Dirac delta; solving on an infinitely long duct and returning to original variables gives the straight-duct solution.Source: Appendix A · p. 13

APPENDIX B PROOF OF THEOREM 1

  • The closed-loop solution imposes periodic boundary conditions on [0, L], or equivalently sums shifted straight-duct solutions to wrap them around a loop of circumference L.Source: Appendix B · p. 14

APPENDIX C DERIVATION OF THE FIRST-PEAK APPROXIMATION

The derivation starts from the steady-flow first-arrival peak condition, then treats pulsatile modulation as a first-order perturbation. Linearizing the peak condition converts that perturbation into an approximate shift in peak time.

APPENDIX C DERIVATION OF THE FIRST-PEAK APPROXIMATION

  • The first-arrival peak is defined by the zero derivative of the log local concentration, with the steady-flow condition yielding the baseline peak time t0.Source: Appendix C · p. 14
  • Pulsatile modulation perturbs the steady-flow system, and retaining first-order terms approximates changes to the mean displacement and variance.The accumulated modulation and the local-concentration approximation supply the perturbation terms.Source: Appendix C · p. 14
  • A first-order Taylor expansion of the peak condition converts the perturbation into a peak-time shift, with the steady-flow peak condition’s local sensitivity setting the correction.Source: Appendix C · p. 14
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