Simulation of a chain of collapsible contracting lymphangions with progressive valve closure

The aim of this investigation was to achieve the first step toward a comprehensive model of the lymphatic system. A numerical model has been constructed of a lymphatic vessel, consisting of a short series chain of contractile segments (lymphangions) and of intersegmental valves. The changing diameter of a segment governs the difference between the flows through inlet and outlet valves and is itself governed by a balance between transmural pressure and passive and active wall properties. The compliance of segments is maximal at intermediate diameters and decreases when the segments are subject to greatly positive or negative transmural pressure. Fluid flow is the result of time-varying active contraction causing diameter to reduce and is limited by segmental viscous and valvular resistance. The valves effect a smooth transition from low forward-flow resistance to high backflow resistance. Contraction occurs sequentially in successive lymphangions in the forward-flow direction. The behavior of chains of one to five lymphangions was investigated by means of pump function curves, with variation of valve opening parameters, maximum contractility, lymphangion size gradation, number of lymphangions, and phase delay between adjacent lymphangion contractions. The model was reasonably robust numerically, with mean flow-rate generally reducing as adverse pressure was increased. Sequential contraction was found to be much more efficient than synchronized contraction. At the highest adverse pressures, pumping failed by one of two mechanisms, depending on parameter settings: either mean leakback flow exceeded forward pumping or contraction failed to open the lymphangion outlet valve. Maximum pressure and maximum flow-rate were both sensitive to the contractile state; maximum pressure was also determined by the number of lymphangions in series. Maximum flow-rate was highly sensitive to the transmural pressure experienced by the most upstream lymphangions, suggesting that many feeding lymphatics would be needed to supply one downstream lymphangion chain pumping at optimal transmural pressure.     

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

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.     

Numerical simulation of cardiovascular dynamics with different types of VAD assistance

A variety of methods by which mechanical circulatory support (MCS) can be provided have been described. However, the haemodynamic benefits of the different methods have not been adequately quantified. The aim of this paper is to compare the haemodynamic effects of six forms of MCS by numerical simulation. Three types of ventricular assist device (VAD) are studied: positive displacement; impeller and a novel reciprocating-valve design. Similarly, three pumping modes are modelled: constant flow; counterpulsation and copulsation. The cardiovascular system is modelled using an approach developed previously, using the concentrated parameter method by considering flow resistance, vessel elasticity and inertial effects of blood in individual conduit segments. The dynamic modelling of displacement and impeller pumps is represented by VAD inlet/outlet flow-rate changes. The dynamics of the reciprocating-valve pump is modelled with a specified displacement profile. Results show that in each simulation, the physiological variables of mean arterial pressure and systemic flow are adequately maintained. Modulation of the impeller pump flow profile produces a small (5 mmHg) oscillatory component to arterial pressure, whereas the displacement and reciprocating-valve pumps generate substantial arterial pressure and flow pulsatility. The impeller pump requires the least power input, the reciprocating valve pump slightly more, and the displacement pump the most. The in parallel configuration of the impeller and displacement pump designs with respect to the left ventricle provides near complete unloading and can cause the aortic valve to remain closed throughout the entire cardiac cycle with the attendant risk of aortic valve leaflet fusion following prolonged support. The in series configuration of the reciprocating-valve pump avoids this shortcoming but activation must be carefully synchronized to the cardiac cycle to allow adequate coronary perfusion. The reciprocating-valve pump is associated with haemodynamic advantages and a favourable power consumption.     

Active lymphatic pumping and sheep lung lymph flow

Active (intrinsic) lymphatic pumping may be an important factor determining lymph flow from the lungs. Unfortunately, in most experiments, it is very difficult to determine the influence of active pumping vs. passive factors on lymph flow. However, 1) the pumping activity (stroke volume and frequency) of isolated lymphatic segments varies nonlinearly with transmural pressure, and 2) the lung lymph flow from awake sheep varies nonlinearly with lymphatic outflow pressure. Accordingly, if lymphatic pumping significantly influences lung lymph flow, then it should be possible to describe the sheep lung lymph flow vs. outflow pressure data with the pumping activity data. To test this, we used published lymphatic pumping activity data to develop a mathematical model of the lymphatic pump for a segment of lymphatic vessel. Flow vs. outflow pressure relationships obtained from simulations with this model were very similar to the data from sheep. Our results indicate that both passive factors and active lymphatic pumping contribute to lymph flow, and our model may allow investigators to distinguish the effects of active pumping vs. passive factors in the regulation of lymph flow.     

A multiscale sliding filament model of lymphatic muscle pumping

The lymphatics maintain fluid balance by returning interstitial fluid to veins via contraction/compression of vessel segments with check valves. Disruption of lymphatic pumping can result in a condition called lymphedema with interstitial fluid accumulation. Lymphedema treatments are often ineffective, which is partially attributable to insufficient understanding of specialized lymphatic muscle lining the vessels. This muscle exhibits cardiac-like phasic contractions and smooth muscle-like tonic contractions to generate and regulate flow. To understand the relationship between this sub-cellular contractile machinery and organ-level pumping, we have developed a multiscale computational model of phasic and tonic contractions in lymphatic muscle and coupled it to a lymphangion pumping model. Our model uses the sliding filament model (Huxley in Prog Biophys Biophys Chem 7:255–318, 1957) and its adaptation for smooth muscle (Mijailovich in Biophys J 79(5):2667–2681, 2000). Multiple structural arrangements of contractile components and viscoelastic elements were trialed but only one provided physiologic results. We then coupled this model with our previous lumped parameter model of the lymphangion to relate results to experiments. We show that the model produces similar pressure, diameter, and flow tracings to experiments on rat mesenteric lymphatics. This model provides the first estimates of lymphatic muscle contraction energetics and the ability to assess the potential effects of sub-cellular level phenomena such as calcium oscillations on lymphangion outflow. The maximum efficiency value predicted (40%) is at the upper end of estimates for other muscle types. Spontaneous calcium oscillations during diastole were found to increase outflow up to approximately 50% in the range of frequencies and amplitudes tested.

     

Determinants of valve gating in collecting lymphatic vessels from rat mesentery

Secondary lymphatic valves are essential for minimizing backflow of lymph and are presumed to gate passively according to the instantaneous trans-valve pressure gradient. We hypothesized that valve gating is also modulated by vessel distention, which could alter leaflet stiffness and coaptation. To test this hypothesis, we devised protocols to measure the small pressure gradients required to open or close lymphatic valves and determine if the gradients varied as a function of vessel diameter. Lymphatic vessels were isolated from rat mesentery, cannulated, and pressurized using a servo-control system. Detection of valve leaflet position simultaneously with diameter and intraluminal pressure changes in two-valve segments revealed the detailed temporal relationships between these parameters during the lymphatic contraction cycle. The timing of valve movements was similar to that of cardiac valves, but only when lymphatic vessel afterload was elevated. The pressure gradients required to open or close a valve were determined in one-valve segments during slow, ramp-wise pressure elevation, either from the input or output side of the valve. Tests were conducted over a wide range of baseline pressures (and thus diameters) in passive vessels as well as in vessels with two levels of imposed tone. Surprisingly, the pressure gradient required for valve closure varied >20-fold (0.1-2.2 cmH(2)O) as a passive vessel progressively distended. Similarly, the pressure gradient required for valve opening varied sixfold with vessel distention. Finally, our functional evidence supports the concept that lymphatic muscle tone exerts an indirect effect on valve gating.     

Intrinsic pump-conduit behavior of lymphangions

Lymphangions, segments of lymphatic vessels bounded by valves, have characteristics of both ventricles and arteries. They can act primarily like pumps when actively transporting lymph against a pressure gradient. They also can act as conduit vessels when passively transporting lymph down a pressure gradient. This duality has implications for clinical treatment of several types of edema, since the strategy to optimize lymph flow may depend on whether it is most beneficial for lymphangions to act as pumps or conduits. To address this duality, we employed a simple computational model of a contracting lymphangion, predicted the flows at both positive and negative axial pressure gradients, and validated the results with in vitro experiments on bovine mesenteric vessels. This model illustrates that contraction increases flow for normal axial pressure gradients. With edema, limb elevation, or external compression, however, the pressure gradient might reverse, and lymph may flow passively down a pressure gradient. In such cases, the valves may be forced open during the entire contraction cycle. The vessel thus acts as a conduit, and contraction has the effect of increasing resistance to passive flow, thus inhibiting flow rather than promoting it. This analysis may explain a possible physiological benefit of the observed flow-mediated inhibition of the lymphatic pump at high flow rates.     

Frequency-dependent power output and skeletal muscle design

Cyclically contracting muscles provide power for a variety of processes including locomotion, pumping blood, respiration, and sound production. In the current study, we apply a computational model derived from force–velocity relationships to explore how sustained power output is systematically affected by shortening velocity, operational frequency, and strain amplitude. Our results demonstrate that patterns of frequency dependent power output are based on a precise balance between a muscle's intrinsic shortening velocity and strain amplitude. We discuss the implications of this constraint for skeletal muscle design, and then explore implications for physiological processes based on cyclical muscle contraction. One such process is animal locomotion, where musculoskeletal systems make use of resonant properties to reduce the amount of metabolic energy used for running, swimming, or flying. We propose that skeletal muscle phenotype is tuned to this operational frequency, since each muscle has a limited range of frequencies at which power can be produced efficiently. This principle also has important implications for our understanding muscle plasticity, because skeletal muscles are capable of altering their active contractile properties in response to a number of different stimuli. We discuss the possibility that muscles are dynamically tuned to match the resonant properties of the entire musculoskeletal system.     

Innervation and control of tension in abdominal muscles of Blaberus discoidalis and Gromphadorhina portentosa

In Blaberus discoidalis and Gromphadorhina portentosa, the distribution of motor axons to the muscles which control movements of the spiracular valves at both respiratory and non-respiratory spiracles is identical. Both fast and slowly contracting heads of the opener muscles are innervated by an excitatory motor axon. Physiological properties of the opener excitor axon correlate with valve function. The slowly contracting head of the opener muscle is, in addition, innervated by a common inhibitor which also occasionally innervates closer muscle fibers. Activation of the common inhibitor terminates contraction of slowly contracting opener muscle fibres and initiates a rapid relaxation of these fibres.     

Contraction of collecting lymphatics: organization of pressure-dependent rate for multiple lymphangions

The paper describes the extension of a previously developed model of pressure-dependent contraction rate to the case of multiple lymphangions. Mechanical factors are key modulators of active lymphatic pumping. As part of the evolution of our lumped-parameter model to match experimental findings, we have designed an algorithm whereby the time until the next contraction depends on lymphangion transmural pressure in the contraction just completed. The functional dependence of frequency on pressure is quantitatively matched to isobaric contraction experiments on isolated lymphatic segments. When each of several lymphangions is given this ability, a scheme for their coordination must be instituted to match the observed synchronization. Accordingly, and in line with an experiment on an isolated lymphatic vessel segment in which we measured contraction sequence and conduction delay, we took the fundamental principle to be that local timing can be overridden by signals to initiate contraction that start in adjacent lymphangions, conducted with a short delay. The scheme leads to retrograde conduction when the lymphangion chain is pumping against an adverse pressure difference, but antegrade conduction when contractions occur with no or a favourable pressure difference. Abolition of these conducted signals leads to chaotic variation of cycle-mean flow-rate from the chain, diastolic duration in each lymphangion, and inter-lymphangion delays. Chaotic rhythm is also seen under other circumstances. Because the model responds to increasing adverse pressure difference by increasing the repetition rate of contractions, it maintains time-average output flow-rate better than one with fixed repetition rate.     

Immediate exercise hyperemia: contributions of the muscle pump vs. rapid vasodilation.

A striking characteristic of the blood flow adaptation at exercise onset is the immediate and substantial increase in the first few (0-5 s) seconds of exercise. The purpose of this mini-review is to put into context the present evidence regarding mechanisms responsible for this phase of exercise hyperemia. One potential mechanism that has received much attention is the mechanical effect of muscle contraction (the muscle pump). The rapid vasodilatory mechanism(s) is another possible mechanism that has recently been shown to exist. This review will provide the reader with 1) an understanding of the basic physics of blood flow and the theories of muscle pump function, 2) a critical examination of evidence both for and against the contribution of the muscle pump or rapid vasodilatory mechanisms, and 3) an awareness of the limitations and impact of experimental models and exercise modes on the contribution of each of these mechanisms to the immediate exercise hyperemia. The inability to measure microvenular pressure continues to limit investigators to indirect assessments of the muscle pump vs. vasodilatory mechanism contributions to immediate exercise hyperemia in vivo. Future research directions should include examination of muscle-contraction-induced resistance vessel distortion as a trigger for rapid smooth muscle relaxation and further investigation into the exercise mode dependency of muscle pump vs. rapid vasodilatory contributions to immediate exercise hyperemia.     

Modeling flow in collecting lymphatic vessels: one-dimensional flow through a series of contractile elements

The lymphatic system comprises a series of elements, lymphangions, separated by valves and possessed of active, contractile walls to pump interstitial fluid from its collection in the terminal lymphatics back to the main circulation. Despite its importance, there is a dearth of information on the fluid dynamics of the lymphatic system. In this article, we describe linked experimental and computational work aimed at elucidating the biomechanical properties of the individual lymphangions. We measure the static and dynamic mechanical properties of excised bovine collecting lymphatics and develop a one-dimensional computational model of the coupled fluid flow/wall motion. The computational model is able to reproduce the pumping behavior of the real vessel using a simple contraction function producing fast contraction pulses traveling in the retrograde direction to the flow.     

Lymphatic vessels transition to state of summation above a critical contraction frequency

Although behavior of lymphatic vessels is analogous to that of ventricles, which completely relax between contractions, and blood vessels, which maintain a tonic constriction, the mixture of contractile properties can yield behavior unique to lymphatic vessels. In particular, because of their limited refractory period and slow rate of relaxation, lymphatic vessels lack the contractile properties that minimize summation in ventricles. We, therefore, hypothesized that lymphatic vessels transition to a state of summation when lymphatic vessel contraction frequency exceeds a critical value. We used an isovolumic, controlled-flow preparation to compare the time required for full relaxation with the time available to relax during diastole. We measured transmural pressure and diameter on segments of spontaneously contracting bovine mesenteric lymphatic vessels during 10 isovolumic volume steps. We found that beat-to-beat period (frequency(-1)) decreased with increases in diameter and that total contraction time was constant or slightly increased with diameter. We further found that the convergence of beat-to-beat period and contraction cycle duration predicted a critical transition value, beyond which the vessel does not have time to fully relax. This incomplete relaxation and resulting mechanical summation significantly increase active tension in diastole. Because this transition occurs within a physiological range, contraction summation may represent a fundamental feature of lymphatic vessel function.     

Blood flow augmentation by intrinsic venular contraction in vivo

Venomotion, spontaneous cyclic contractions of venules, was first observed in the bat wing 160 years ago. Of all the functional roles proposed since then, propulsion of blood by venomotion remains the most controversial. Common animal models that require anesthesia and surgery have failed to provide evidence for venular pumping of blood. To determine whether venomotion actively pumps blood in a minimally invasive, unanesthetized animal model, we reintroduced the batwing model. We evaluated the temporal and functional relationship between the venous contraction cycle and blood flow and luminal pressure. Furthermore, we determined the effect of inhibiting venomotion on blood flow. We found that the active venous contractions produced an increase in the blood flow and exhibited temporal vessel diameter-blood velocity and pressure relationships characteristic of a peristaltic pump. The presence of valves, a characteristic of reciprocating pumps, enhances the efficiency of the venular peristaltic pump by preventing retrograde flow. Instead of increasing blood flow by decreasing passive resistance, venular dilation with locally applied sodium nitroprusside decreased blood flow. Taken together, these observations provide evidence for active venular pumping of blood. Although strong venomotion may be unique to bats, venomotion has also been inferred from venous pressure oscillations in other animal models. The conventional paradigm of microvascular pressure and flow regulation assumes venules only act as passive resistors, a proposition that must be reevaluated in the presence of significant venomotion.     

Active and passive properties of carotid arteries]

A mathematical model of carotid arteries is constructed from the known experimental data. Passive properties of the vascular wall are characterized by an alternating module of elasticity, the active ones by the specific power of muscle contraction. Its maximum value (0,023 n/m-3) is shown to be reached with intravascular pressure 190 mm Hg. The dependence of inner radius on the power of muscle contraction is studied at different values of intravascular pressure. It is shown that theactive properties of carotid arteries are essentially determined by their passive properties and depend on the stretching of the vessel wall.     

Relaxation of diaphragm muscle.

Relaxation is the process by which, after contraction, the muscle actively returns to its initial conditions of length and load. In rhythmically active muscles such as diaphragm, relaxation is of physiological importance because diaphragm must return to a relatively constant resting position at the end of each contraction-relaxation cycle. Rapid and complete relaxation of the diaphragm is likely to play an important role in adaptation to changes in respiratory load and breathing frequency. Regulation of diaphragm relaxation at the molecular and cellular levels involves Ca(2+) removal from the myofilaments, active Ca(2+) pumping by the sarcoplasmic reticulum (SR), and decrease in the number of working cross bridges. The relative contribution of these mechanisms mainly depends on sarcomere length, muscle tension, and the intrinsic contractile function. Increased capacity of SR to take up Ca(2+) can arise from increased density of active SR pumping sites or in slow-twitch fibers from phosphorylation of phospholamban, whereas impaired coupling between ATP hydrolysis and Ca(2+) transport into the SR or intracellular acidosis reduces SR Ca(2+) pump activity. In experimental conditions of decreased contractile performance, slowed, enhanced, or unchanged relaxation rates have been reported in vitro. In vivo, a slowing in the rate of decline of the respiratory pressure is generally considered an early reliable index of respiratory muscle fatigue. Impaired relaxation rate may, in turn, favor mismatch between blood flow and metabolic demand, especially at high breathing frequencies.     

Effects of muscle contraction on pulsatile pressure-flow relations in femoral bed

Femoral arterial pressure-flow relations and vascular impedance were studied during isometric contraction of the gastrocnemius-plantaris muscle group in anesthetized dogs. Contractions were synchronized with the electrocardiogram to occur in the first or second half of the cardiac cycle and included twitches as well as low-, intermediate-, and high-frequency tetanuses. The effects of fatigue and recovery were also documented. Marked changes in pressure and flow waveforms and corresponding femoral arterial input impedance spectra were seen for all contraction modes. Impedance moduli and estimated characteristic impedance were elevated regardless of contraction mode and were associated with fluctuations in impedance phase. All tetanuses placed in the first half of the cardiac cycle produced a striking and consistent reversal of impedance phase for the fundamental harmonic from negative to positive values which decreased with progressive fatigue. During recovery, impedance spectra were unchanged from control spectra. We have demonstrated marked alterations in pressure and flow waveforms and impedance spectral patterns during isometric contraction in the canine hindlimb. These changes may be explained by 1) markedly increased wave reflection as a result of muscle contraction and/or 2) the generation of a retrograde pulse by contracting muscle that fuses with the antegrade pulse of cardiac origin.     

Modeling of the mechanical function of the human gastroesophageal junction using an anatomically realistic three-dimensional model

The aim of this study was to combine the anatomy and physiology of the human gastroesophageal junction (the junction between the esophagus and the stomach) into a unified computer model. A three-dimensional (3D) computer model of the gastroesophageal junction was created using cross-sectional images from a human cadaver. The governing equations of finite deformation elasticity were incorporated into the 3D model. The model was used to predict the intraluminal pressure values (pressure inside the junction) due to the muscle contraction of the gastroesophageal junction and the effects of the surrounding structures. The intraluminal pressure results obtained from the 3D model were consistent with experimental values available in the literature. The model was also used to examine the independent roles of each muscle layer (circular and longitudinal) of the gastroesophageal junction by contracting them separately. Results showed that the intraluminal pressure values predicted by the model were primarily due to the contraction of the circular muscle layer. If the circular muscle layer was quiescent, the contraction of the longitudinal muscle layer resulted in an expansion of the junction.In conclusion, the model provided reliable predictions of the intraluminal pressure values during the contraction of a normal gastroesophageal junction. The model also provided a framework to examine the role of each muscle layer during the contraction of the gastroesophageal junction.     

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

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(L)Q(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.     

Incorporating measured valve properties into a numerical model of a lymphatic vessel

An existing lumped-parameter model of multiple lymphangions (lymphatic vascular segments) in series is adapted for the incorporation of recent physiological measurements of lymphatic vascular properties. The new data show very marked nonlinearity of the passive pressure–diameter relation during distension, relative to comparable blood vessels, and complex valve behaviour. Since lymph is transported as a result of either the active contraction or the passive squeezing of vascular segments situated between two one-way valves, the performance of these valves is of primary importance. The valves display hysteresis (the opening and closing pressure drop thresholds differ), a bias to staying open (both state changes occur when the trans-valve pressure drop is adverse) and pressure-drop threshold dependence on transmural pressure. These properties, in combination with the strong nonlinearity that valve operation represents, have in turn caused intriguing numerical problems in the model, and we describe numerical stratagems by which we have overcome the problems. The principal problem is also generalised into a relatively simple mathematical example, for which solution detail is provided using two different solvers.