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Mixture of Inverse Gaussians for Hemodynamic Transport (MIGHT) in Vascular Networks

Timo Jakumeit, Bastian Heinlein, Leonie Richter, Sebastian Lotter, Robert Schober, Maximilian Schäfer

TL;DR

The paper tackles the challenge of modeling molecular transport in complex vascular networks for cardiovascular molecular communication. It introduces MIGHT, a closed-form channel model that represents the received molecular flux as a finite sum of weighted inverse Gaussian distributions, with path parameters derived from physical VN properties and flow. The authors validate MIGHT against convolution-based models and COMSOL simulations, demonstrating accuracy across VN with multiple transport paths and under high Péclet numbers, and they show how the model enables structural reduction of VN and estimation of representative networks from observed signals. This yields a practical, scalable tool for analyzing, simplifying, and inferring VN dynamics in molecular communication applications with potential clinical impact.

Abstract

Synthetic molecular communication (MC) in the cardiovascular system (CVS) is a key enabler for many envisioned medical applications in the human body, such as targeted drug delivery, early cancer detection, and continuous health monitoring. The design of MC systems for such applications requires suitable models for the signaling molecule propagation through complex vessel networks (VNs). Existing theoretical models offer limited analytical tractability and lack closed-form solutions, making the analysis of large-scale VNs either infeasible or not insightful. To overcome these limitations, in this paper, we propose a novel closed-form physical model, termed MIGHT, for advection-diffusion-driven transport of signaling molecules through complex VNs. The model represents the received molecule flux as a weighted sum of inverse Gaussian (IG) distributions, parameterized by physical properties of the network. The proposed model is validated by comparison with an existing convolution-based model and finite-element simulations. Further, we show that the model can be applied for the reduction of large VNs to simplified representations preserving the essential transport dynamics and for estimating representative VN based on received signals from unknown VNs.

Mixture of Inverse Gaussians for Hemodynamic Transport (MIGHT) in Vascular Networks

TL;DR

The paper tackles the challenge of modeling molecular transport in complex vascular networks for cardiovascular molecular communication. It introduces MIGHT, a closed-form channel model that represents the received molecular flux as a finite sum of weighted inverse Gaussian distributions, with path parameters derived from physical VN properties and flow. The authors validate MIGHT against convolution-based models and COMSOL simulations, demonstrating accuracy across VN with multiple transport paths and under high Péclet numbers, and they show how the model enables structural reduction of VN and estimation of representative networks from observed signals. This yields a practical, scalable tool for analyzing, simplifying, and inferring VN dynamics in molecular communication applications with potential clinical impact.

Abstract

Synthetic molecular communication (MC) in the cardiovascular system (CVS) is a key enabler for many envisioned medical applications in the human body, such as targeted drug delivery, early cancer detection, and continuous health monitoring. The design of MC systems for such applications requires suitable models for the signaling molecule propagation through complex vessel networks (VNs). Existing theoretical models offer limited analytical tractability and lack closed-form solutions, making the analysis of large-scale VNs either infeasible or not insightful. To overcome these limitations, in this paper, we propose a novel closed-form physical model, termed MIGHT, for advection-diffusion-driven transport of signaling molecules through complex VNs. The model represents the received molecule flux as a weighted sum of inverse Gaussian (IG) distributions, parameterized by physical properties of the network. The proposed model is validated by comparison with an existing convolution-based model and finite-element simulations. Further, we show that the model can be applied for the reduction of large VNs to simplified representations preserving the essential transport dynamics and for estimating representative VN based on received signals from unknown VNs.

Paper Structure

This paper contains 16 sections, 3 theorems, 49 equations, 5 figures.

Table of Contents

  1. Introduction
  2. System Model
  3. Vessel Network Definition
  4. Advective-Diffusive Molecule Transport
  5. Inverse Gaussian Mixture Channel Model
  6. Single Pipe Model
  7. Multiple Pipe Path Model
  8. Vessel Network Model
  9. Model Validation
  10. Applications of Proposed Model
  11. Vessel Network Reduction
  12. Estimating Vessel Networks from Signals
  13. Conclusion
  14. Proof of Theorem 1
  15. Proof of Theorem 2
  16. ...and 1 more sections

Key Result

Theorem 1

The FPT $T_i(z)$ at $x=z$ of a single molecule entering pipe $p_i$ at $t = 0$ is IG-distributed, i.e., with mean, variance, and scale parameter given by The molecule flux $j_i(z,t)$ in $\per s$ at $x = z$ in $p_i$ is the PDF $f_{\mathrm{IG}}(t,z;\mu,\theta)$ of $T_i(z)$ in eq:igd, i.e., Proof: The proof is provided in Appendix sec:proof1.

Figures (5)

  • Figure 1: Network topology notation. Exemplary VN with associated notation for the topology.
  • Figure 2: System and channel model.a) Molecules released at the inlet $n_\mathrm{in}$ propagate via advection and diffusion through a VN comprising vessels, bifurcations, and junctions, reaching the outlet $n_\mathrm{out}$. The VN is represented as a directed graph. b) The VN is decomposed into paths $P_k$ from $n_\mathrm{in}$ to $n_\mathrm{out}$. c) The molecule flux at each path outlet follows an IG, parameterized by lengths $l_i$, radii $r_i$, and flow velocities $\bar{u}_i$ of the vessels in $P_k$. d) The network flux at the outlet is the sum of path fluxes, weighted by path fractions $\gamma_{P_k}$, derived from the flow rates $Q_i$.
  • Figure 3: Model validation.$N_\mathrm{obs}(t)$ simulated in COMSOL, predicted by \ref{['eqn:Nobs']}, the convolution in \ref{['eq:path-conv']}, and the convolution-based model from Jakumeit2024 for different VN. COMSOL velocity magnitudes are color-coded on the left to indicate flow patterns qualitatively.
  • Figure 4: Structural reduction of VN. For each VN, the original topology and the discarded pipes, as well as the resulting $N_\mathrm{obs}(t)$ are shown for various $\alpha$. Pipe lengths are depicted to scale. In a) and b), radii were drawn from a normal distribution with mean $0.5mm$ and standard deviation $0.1mm$. In c), $r_i=0.5mm$.
  • Figure 5: VN estimation.a) Prototype VN. b)–e) Estimated VN from COMSOL data (b), c)) and synthetic signals (d), e)) reproduce the observed signals across different VN sizes. Fitted IG signals and estimated outputs are shown in green and blue, respectively. The numbers of paths in the original and estimated VN are shown in gray and blue in the subplots. Inlet and outlet pipes are highlighted in yellow and rose.

Theorems & Definitions (6)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • proof
  • proof
  • proof