Optimal postnodal lymphatic network structure that maximizes active propulsion of lymph

Abstract

The lymphatic system acts to return lower-pressured interstitial fluid to the higher-pressured veins by a complex network of vessels spanning more than three orders of magnitude in size. Lymphatic vessels consist of lymphangions, segments of vessels between two unidirectional valves, which contain smooth muscle that cyclically pumps lymph against a pressure gradient. Whereas the principles governing the optimal structure of arterial networks have been identified by variations of Murray's law, the principles governing the optimal structure of the lymphatic system have yet to be elucidated, although lymph flow can be identified as a critical parameter. The reason for this deficiency can be identified. Until recently, there has been no algebraic formula, such as Poiseuille's law, that relates lymphangion structure to its function. We therefore employed a recently developed mathematical model, based on the time-varying elastance model conventionally used to describe ventricular function, that was validated by data collected from postnodal bovine mesenteric lymphangions. From this lymphangion model, we developed a model to determine the structure of a lymphatic network that optimizes lymph flow. The model predicted that there is a lymphangion length that optimizes lymph flow and that symmetrical networks optimize lymph flow when the lymphangions downstream of a bifurcation are 1.26 times the length of the lymphangions immediately upstream. Measured lymphangion lengths (1.14 +/- 0.5 cm, n = 74) were consistent with the range of predicted optimal lengths (0.1-2.1 cm). This modeling approach was possible, because it allowed a structural parameter, such as length, to be treated as a variable.

Fig. 1.

Pressure-volume relationship of a lymphatic segment. Stroke volume, the difference between end-diastolic volume (V_ed) and end-systolic volume (V_es), is determined using inlet pressure (P_in), outlet pressure (P_out), slope of end-systolic pressure-volume relationship (E_max), slope of end-diastolic pressure volume relationship (E_min), and dead volume [i.e., volume at 0 pressure (V₀)].

Fig. 2.

Symmetrical lymphatic vessel structure at a confluence. Flow (Q̇) is governed by conservation of mass, with Q̇₁ = Q̇₂ + Q̇₃. Ratio of lymphangion length (l₁/l₂) and lymphangion radius at end diastole (r₁/r₂) is 1.26, with the assumption of a symmetrical network with constant lymphangion endothelial shear stress and ejection fractions.

Fig. 3.

Validation of 1st-order (analytic) approximation of lymph flow in a lymphangion (Eq. 4, solid line) (Eq. 2). Experimental data from 4 vessels were used to validate the model first reported by Venugopal et al. (31) for comparison purposes (▴, ⧫, •, and ○). Dashed line, linear regression of pooled data; dashed curves, 95% confidence intervals. Scatter in data represents variation among vessels.

Fig. 4.

Flow from a lymphangion is maximal at optimal length (l_opt). Flow is normalized by its maximum value.

Fig. 5.

Variation of lengths obtained from postnodal bovine mesenteric lymphatic vessel (1.14 ± 0.5 cm, n = 74). Equation 5 was used to predict that optimal lengths can range from 0.1 to 2.1 cm (dashed lines), consistent with data.