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First-order approximation for the pressure-flow relationship of spontaneously contracting lymphangions

Michael E. DeBakey Institute, Texas A&M University, College Station, Texas

Submitted 6 July 2007 ; accepted in final form 3 March 2008

To return lymph to the great veins of the neck, it must be actively pumped against a pressure gradient. Mean lymph flow in a portion of a lymphatic network has been characterized by an empirical relationship (P_in – P_out = –P_p + R_LQ_L), where P_in – P_out is the axial pressure gradient and Q_L is mean lymph flow. R_L and P_p are empirical parameters characterizing the effective lymphatic resistance and pump pressure, respectively. The relation of these global empirical parameters to the properties of lymphangions, the segments of a lymphatic vessel bounded by valves, has been problematic. Lymphangions have a structure like blood vessels but cyclically contract like cardiac ventricles; they are characterized by a contraction frequency (f) and the slopes of the end-diastolic pressure-volume relationship [minimum value of resulting elastance (E_min)] and end-systolic pressure-volume relationship [maximum value of resulting elastance (E_max)]. Poiseuille's law provides a first-order approximation relating the pressure-flow relationship to the fundamental properties of a blood vessel. No analogous formula exists for a pumping lymphangion. We therefore derived an algebraic formula predicting lymphangion flow from fundamental physical principles and known lymphangion properties. Quantitative analysis revealed that lymph inertia and resistance to lymph flow are negligible and that lymphangions act like a series of interconnected ventricles. For a single lymphangion, P_p = P_in (E_max – E_min)/E_min and R_L = E_max/f. The formula was tested against a validated, realistic mathematical model of a lymphangion and found to be accurate. Predicted flows were within the range of flows measured in vitro. The present work therefore provides a general solution that makes it possible to relate fundamental lymphangion properties to lymphatic system function.

mathematical modeling; edema

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