Development of a model of a multi-lymphangion lymphatic vessel incorporating realistic and measured parameter values

Abstract

Our published model of a lymphatic vessel consisting of multiple actively contracting segments between non-return valves has been further developed by the incorporation of properties derived from observations and measurements of rat mesenteric vessels. These included (1) a refractory period between contractions, (2) a highly nonlinear form for the passive part of the pressure–diameter relationship, (3) hysteretic and transmural-pressure-dependent valve opening and closing pressure thresholds and (4) dependence of active tension on muscle length as reflected in local diameter. Experimentally, lymphatic valves are known to be biased to stay open. In consequence, in the improved model, vessel pumping of fluid suffers losses by regurgitation, and valve closure is dependent on backflow first causing an adverse valve pressure drop sufficient to reach the closure threshold. The assumed resistance of an open valve therefore becomes a critical parameter, and experiments to measure this quantity are reported here. However, incorporating this parameter value, along with other parameter values based on existing measurements, led to ineffective pumping. It is argued that the published measurements of valve-closing pressure threshold overestimate this quantity owing to neglect of micro-pipette resistance. An estimate is made of the extent of the possible resulting error. Correcting by this amount, the pumping performance is improved, but still very inefficient unless the open-valve resistance is also increased beyond the measured level. Arguments are given as to why this is justified, and other areas where experimental data are lacking are identified. The model is capable of future adaptation as new experimental data appear.

Appendix

The passive constitutive relation is a nonlinear function $\Delta p_{\mathrm{tm}} = f_{\mathrm{p}}(D)$, where

$$\begin{aligned} f_p \left( D\right)&= P_\mathrm{d} \left[ c_1 \left( {\frac{D}{c_9}-c_2} \right) ^{2}+c_3 \exp \left( {c_4\left( {\frac{D}{c_9}-c_5}\right) } \right) \right. \\&\left. +\,c_6 +c_7 \left( {\frac{D}{c_9}-c_8}\right) +c_{10} \left( {\frac{c_9}{D}}\right) ^{3}\right] , \\ \text{ with } c_9&= \frac{D_\mathrm{d}}{c_{11}}. \end{aligned}$$

When $P_{\mathrm{d}}$ defines the slope $4P_{\mathrm{d}}/D_{\mathrm{d}}$ of the function at $\Delta p_{\mathrm{tm}} = 0$, the constants have the values $c_{1} = -2.34457751, c_{2} = 1.1262924, c_{3} = 3.76013762, c_{4} = 79.991135, c_{5} = 1.0028029, c_{6} = 1.59133174, c_{7} = 3.69692633, c_{8} = 0.20699868, c_{10} = -0.0180867408$ and $c_{11}$ = 0.32538081. This system is convenient in the numerical model; however, a different scaling system is preferred in experiments. The distension-limiting region of the curve to the right of the knee is reached by $\Delta p_{\mathrm{tm}} = 5\,\text{ cm }\;\text{ H }_{2}$O; the corresponding diameter $c_{9}$ is used to normalise the measurements of $D\;(D_{\mathrm{d}}$ then takes the value $c_{11})$. $P_{\mathrm{d}}$ can be used to refer to this point if so desired (for the defining curve under consideration, it then takes the value 4,905 = 5 $\times$ 981 dyn $\text{ cm }^{-2}$), in which case the values of $c_{1}, c_{3}, c_{6}, c_{7}$ and $c_{10}$ must be multiplied by 0.7464031.

The active length–tension relation for contraction is a nonlinear function $M_{\mathrm{d}}(D)$, where

$$\begin{aligned} M_{\mathrm{d}} \left( D\right)&= \frac{1}{1+\exp \left( {-s_{\mathrm{d}} \left( {D-D_{\mathrm{a}}}\right) }\right) }+\frac{1}{1+\exp \left( {s_{\mathrm{d}}\left( {D-D_{\mathrm{b}}}\right) }\right) }-1, \end{aligned}$$

with $D_{\mathrm{a}} = 0.85c_{9}, D_{\mathrm{b}} = \text{2c }_{9} \text{ and } s_{\mathrm{d}} = 3.25/D_\mathrm{d}$.