Ex Vivo Lymphatic Perfusion System for Independently Controlling Pressure Gradient and Transmural Pressure in Isolated Vessels

Abstract

In addition to external forces, collecting lymphatic vessels intrinsically contract to transport lymph from the extremities to the venous circulation. As a result, the lymphatic endothelium is routinely exposed to a wide range of dynamic mechanical forces, primarily fluid shear stress and circumferential stress, which have both been shown to affect lymphatic pumping activity. Although various ex vivo perfusion systems exist to study this innate pumping activity in response to mechanical stimuli, none are capable of independently controlling the two primary mechanical forces affecting lymphatic contractility: transaxial pressure gradient, $\Delta P$, which governs fluid shear stress; and average transmural pressure, $P_{\text {avg}}$, which governs circumferential stress. Hence, the authors describe a novel ex vivo lymphatic perfusion system (ELPS) capable of independently controlling these two outputs using a linear, explicit model predictive control (MPC) algorithm. The ELPS is capable of reproducing arbitrary waveforms within the frequency range observed in the lymphatics in vivo, including a time-varying $\Delta P$ with a constant $P_{\text {avg}}$, time-varying $\Delta P$ and $P_{\text {avg}}$, and a constant $\Delta P$ with a time-varying $P_{\text {avg}}$. In addition, due to its implementation of syringes to actuate the working fluid, a post-hoc method of estimating both the flow rate through the vessel and fluid wall shear stress over multiple, long (5 s) time windows is also described.

Appendix: Time Window Length Calculation

In order to determine what constitutes a long $\Delta t$, we must first estimate the transient dynamics relating the instantaneous syringe velocity (averaged between the two syringes), $v_{\text {avg}}$, to the transaxial pressure gradient, $\Delta P$. The instantaneous syringe velocity averaged between the two syringes is defined as follows:

$$\begin{aligned} v_{\text {avg}}(k) = \frac{\Delta x_{1,k}\delta _k - \Delta x_{2,k}\delta _k}{2 T_s} \end{aligned}$$

(A.1)

where $T_s$ is the sampling time of the identification (see “Post-Experiment Shear Stress Estimation” section for additional nomenclature). Thus, using the data from the identification experiment shown in Fig. 2, one may reconstruct another identification of a single-input, single-output (SISO) system with $v_{\text {avg}}$ as the input and $\Delta P$ as the output. The validation data for this identification is shown in Fig. A1a (model order, $n=4$, with good agreement), while the corresponding dynamic characteristics of this model is shown in Fig. A1b.

Figure A1

(a) Validation data (from the experiment in Fig. 2) for the SISO dynamic model of the ELPS with average instantaneous syringe velocity between the two syringes as the input and transaxial pressure gradient as the output. (b) Bode plot of this SISO model of the ELPS showing the frequency response

Of course, this model does not take into account the dynamics between $\Delta P$ and the flow rate through the vessel, which certainly contain some fluid compliance and inertance. However, assuming these effects are on the same order of magnitude as between $v_{\text {avg}}$ and $\Delta P$ (or smaller), a value of $\Delta t$ much larger than these dynamics should suffice. To quantify the speed of these dynamics, the the 2% settling time, $T_{\text {set}}$, is found from simulating the model in Fig. A1 in response to a step input. For this model, $T_{\text {set}}$ is approximately 0.3 s; thus, to ensure $\Delta t$ is long ($> 10\,T_{\text {set}}$), the authors define:

$$\begin{aligned} \Delta t&\gg T_{\text {set}}\nonumber \\&= 5 \text { sec} \end{aligned}$$

(A.2)