HOME HELP FEEDBACK SUBSCRIPTIONS ARCHIVE SEARCH TABLE OF CONTENTS		QUICK SEARCH:   [advanced] Author: Keyword(s): Year:  Vol:  Page:

This Article Abstract Full Text (PDF) All Versions of this Article: 294/5/H2144    most recent 00781.2007v1 Alert me when this article is cited Alert me if a correction is posted Citation Map Services Email this article to a friend Similar articles in this journal Similar articles in PubMed Alert me to new issues of the journal Download to citation manager Citing Articles Citing Articles via HighWire Citing Articles via Web of Science (3) Citing Articles via Google Scholar Google Scholar Articles by Quick, C. M. Articles by Stewart, R. H. Search for Related Content PubMed PubMed Citation Articles by Quick, C. M. Articles by Stewart, R. H.

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

	ABSTRACT

TOP ABSTRACT METHODS RESULTS DISCUSSION APPENDIX GRANTS REFERENCES

 
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

Insight from heart-arterial interaction to mathematically model the lymphatic system. The recurring analogies to both blood vessels and cardiac ventricles were recently leveraged to develop a physics-based mathematical model of a lymphangion (27, 34). With the extension of the transmission line equation description of arteries (26, 35), the resulting lymphangion model included the effects of lymph inertia and viscosity. Similarly, with the extension of the classical time-varying elastance model used to describe cardiac ventricles (31, 32), it also included a description of diastolic tone and contractility. The resulting lymphangion model was more complicated than either arterial or ventricular models, since it included both vessel parameters (radii and length) and ventricular parameters (minimum and maximum elastance). Despite the neglect of nonideal valve behavior such as lymph backflow through valves, model predictions of flow were validated experimentally (27, 34). The complexity of this detailed model allowed an exploration of the behavior of lymphatic vessels consisting of multiple interacting lymphangions (34) and an illustration that lymphangions act like passive conduits when inlet pressure is raised higher than outlet pressure (27). With the requirement of the numerical solution, however, the model results were only valid for the particular values of assumed parameters. Nonetheless, this approach has allowed lymph flow to be predicted from specific lymphangion properties and fundamental physical principles.

Empirical description of lymph flow in a network. Experiments performed on intact lymphatic networks in vivo indicate that increasing pressure in a particular lymphatic vessel increases flow downstream (9) but inhibits upstream flow (12, 20). This fundamental property has been used to explain the interaction between hepatic and intestinal lymph vessels (30) and the sensitivity of lymph flow measurements to cannula height (20). The measured change in lymph flow with a change in outlet pressure is remarkably linear over a wide range of pressures (8–10). To characterize a large number of discrete pressure-flow data points with a simple empirical formula, Drake et al. (9) drew an analogy to an electrical circuit often used to characterize blood flow (Fig. 1A). It consisted of a pressure source [pump pressure (P_p)] and a resistor representing an effective lymphatic resistance (R_L). The effective lymph driving pressure acting to propel lymph is thus the sum of the interstitial driving pressure and P_p. This model is empirical in nature in that P_p and R_L cannot be predicted from known properties of lymphatic vessels and fundamental physical principles but are found from the linear regression of measured data. In fact, the interpretation of R_L and P_p as an actual resistance and generated pressure has been vigorously disputed (1). As of yet, the values of P_p and R_L have not been related to lymphangion properties such as contraction frequency, lymphatic vessel tone, lymphangion contractility, or even anatomical structures such as vessel length or radius. Poiseuille's law provided a simple algebraic formula to relate the blood pressure-flow relationship to fundamental properties of a blood vessel. No analogous formula exists that relates the lymph pressure-flow relationship to fundamental properties of a lymphangion. The purpose of the present work, therefore, is to derive an algebraic formula predicting lymphangion flow from fundamental physical principles and known lymphangion properties, which will be used to interpret the effective resistance (9) of a spontaneously contracting lymphangion.

View larger version (8K): [in this window] [in a new window]   Fig. 1. A: empirical model developed by Drake et al. (Ref. 9) to describe the relationship of lymph flow (QL) and both inlet pressure (Pin) and outlet pressure (Pout). It consists of an effective lymphatic resistance (RL) and pump pressure (Pp). Values of Pp and RL are chosen to fit measured data. B: relationship of QL and Pout. Slope of this relationship is 1/RL, and the pressure axis intercept is –Pp. C: the model of a pumping lymphangion was based on the time-varying elastance model first used by Suga and colleagues (Refs. 31 and 32). Emin and Emax, minimum and maximum values of resulting elastance of the slopes of the end-diastolic pressure-volume and end-systolic pressure-volume relationships, respectively; Ved and Ves, end-diastolic and end-systolic volumes, respectively. D: the relationship of QL and Pout in terms of properties describing lymphangions. f, contraction frequency.

	METHODS

TOP ABSTRACT METHODS RESULTS DISCUSSION APPENDIX GRANTS REFERENCES

(1)

R_L and P_p were parameters without particular physical interpretation and can be found from the slope and intercept of a measured pressure-flow relationship (Fig. 1B).

Model for lymphangion pumping based on the pressure-volume relationship. A very different approach to modeling lymph flow was taken by Venugopal et al. (34) and Quick et al. (27), who related pulsatile lymph flow to the mechanical properties of lymphangions. With a reliance on the similarity of lymphangion pressure-volume relationships to those of cardiac ventricles, the concept of the time-varying elastance model originally developed by Suga and colleagues (31, 32) for cardiac ventricles was applied. First, a measured relationship of instantaneous transmural pressure, P(t), is divided by a measured volume, V(t).

(2)

V₀ is the dead volume, the theoretical volume at zero pressure. The resulting elastance, E(t), varies as a function of time so that it predicts increases in developed pressure with increased filling. Lymphangion contractility is characterized by the maximum value of E(t), E_max, equivalent to the slope of the end-systolic pressure-volume relationship. Lymphangion tone is characterized by the minimum value of E(t), E_min, equivalent to the slope of the end-diastolic pressure-volume relationship (EDPVR).

Model for lymph flow based on classical hydrodynamics. The second component of the model developed by Venugopal et al. (34) and Quick et al. (27) borrowed the extensive theoretical framework conventionally used to characterize blood flow in arteries (26, 35). Described in detail elsewhere (27, 34), the formulation came from linearizing the Navier-Stokes equation. The result is two terms. The first term of Eq. 3 is a pressure drop due to Q_L, resistance of the valve (R_valve) (28), and a Poiseuillean resistance that arises from viscous losses and is a function of lymph viscosity (µ) and lymphangion radius (r). The second term characterizes the pressure drop (i.e., the net force) that accelerates the mass of the lymph. This inertial term is proportional to the rate of change in flow (dQ/dt) and lymph density () and inversely proportional to r². The pressure drop across a small segment of lymphatic vessel (P) is thus the sum of these two terms.

(3)

The value of P thus represents the combined contribution of resistance and inertia that counteracts the pressure developed by lymphangion contraction (Eq. 2). Q_L in Eq. 3 can be calculated from changes in V in Eq. 2. As described in detail elsewhere (34), the time-varying elastance characterized by Eq. 2 describes the storage of lymph and forms a component that is equivalent to a time-varying capacitance. This mathematical model was evaluated numerically, and its predictions were validated by in vitro experiments on lymphangions (27) and lymphatic vessels (34).

Lymphatic vessel model validation. To test the lymphangion model, data previously reported by Venugopal et al. (34) were plotted. Briefly, segments of postnodal bovine mesenteric vessels were studied with an isobaric preparation. The inlet pressure was maintained at 5.0 mmHg, and the outlet pressure was varied from 5.0 to 7.5 mmHg. Mean flow from each vessel was measured for 1 min after every change in pressure gradient. Data from all vessels (n = 4) were pooled, and mean flow was plotted as a function of axial pressure gradient. The results of a linear regression and associated 95% confidence intervals were also plotted.

	RESULTS

TOP ABSTRACT METHODS RESULTS DISCUSSION APPENDIX GRANTS REFERENCES

 
Approximation of mean lymph flow assuming lymphangions pump against a pressure gradient. Equation 2 can be used to approximate mean lymph flow by following the derivation first developed by Sunagawa et al. (33) to approximate cardiac output. First, end-diastolic volume (V_ed) can be approximated by end-diastolic pressure (P_ed) divided by E_min plus a dead volume. Second, end-systolic volume (V_es) can be approximated by end-systolic pressure (P_es) divided by E_max plus a dead volume. Flow is then equal to contraction frequency (f) multiplied by the stroke volume (i.e., V_ed – V_es), represented by the width of the pressure-volume loop in Fig. 1C. If lymph inertia and resistance are neglected, P_ed can then be approximated by P_in, and P_es can be approximated by P_out.

(4)

A similar formulation is derived in the APPENDIX (Eq. A1) to describe the pressure-flow relationship in the special case that the inlet pressure is raised above the outlet pressure. Equations 4 and A1 are thus first-order, linear approximations of the lymph pressure-flow relationship in a lymphangion (e.g., Fig. 1D).

Comparison of algebraic approximation and complex numerical solution to quantify the effects of lymph inertia and viscosity. The algebraic approximation for lymph flow in a lymphangion (which notably lacks viscous and inertial effects; Eq. 4) can be compared with numerical simulation results from the model of Venugopal et al. (34) and Quick et al. (27) (which includes viscous and inertial effects; Eq. 3) to quantify the role of lymph resistance and inertia. Figure 2 illustrates the mean flow calculated by assuming that E_min = 106 mmHg/ml, E_max = 246 mmHg/ml, inlet pressure = 5 mmHg, and outlet pressure was varied. Figure 2 is completely equivalent to the numerical solution in Venugopal et al. (34), which contains the large numbers of details needed to fully characterize the model. The approximation of Eq. 4 results in <2% error in mean flow. Although the deceleration of the blood entering ventricles results in a pressure gradient that tends to close the valve (13), the contribution of inertia is small (29), and for the example given in Fig. 2, the deceleration of lymph causes the pressure gradient across a valve to be very low (0.001% of the total pressure gradient).

View larger version (14K): [in this window] [in a new window]   Fig. 2. Validation of first-order (analytical) approximation of lymph flow in a lymphangion (solid line) (Eq. 4) with a complex hydrodynamic simulation () (Eqs. 2 and 3) that includes the effects of lymph resistance and inertia. Complex hydrodynamic model of lymphangion was previously evaluated experimentally and described in detail in both Venugopal et al. (Ref. 34) and Quick et al. (Ref. 27). Experimental data from 4 vessels were used to validate model reproduced from Venugopal et al. (Ref. 34) for comparison purposes (closed symbols). Linear regression of data and 95% confidence intervals are represented by dashed lines. Scatter in data represents variation among vessels.

Reconciliation of first-order approximation with empirical description. The approximation of mean lymph flow in a lymphangion presented in the present work (Eq. 4) is linear, much like the empirical model of Drake et al. (9) (Eq. 1). With Q_L in Eq. 1 set equal to Q_L in Eq. 4, a physical interpretation can be found for R_L and P_p.

(5A)

(5B)

The resulting interpretation for a single lymphangion is illustrated in Fig. 3. An alternate formulation is derived in the APPENDIX (Eqs. A1–A3) for the special case that the inlet pressure is raised higher than the outlet pressure.

View larger version (6K): [in this window] [in a new window]   Fig. 3. Identification of physiological parameters that affect RL and Pp. The value of Pp is a function of Pin and the relative change in elastance (Emax – Emin)/Emin, which characterize lymphangion tone (Emin) and contractility (Emax). Thus increasing filling pressure or contractility increases the value of Pp. The value of RL is a function of Emax and f. Increasing f thus decreases the effective RL.

	DISCUSSION

TOP ABSTRACT METHODS RESULTS DISCUSSION APPENDIX GRANTS REFERENCES

Reinterpreting the pump-conduit duality of lymphangions. The present work provides insight that has not been available in previous attempts to mathematically model lymphangions (27, 34). The complex mathematical model used in the present work for validation purposes (Fig. 2 and Eqs. 2 and 3) was used previously to illustrate that lymphangions exhibit both pump and conduit behavior (27). However, this complex model assumed that the pressure generated by lymphangion contraction was countered internally by both inertia and resistance (Eq. 3). All three components were believed to play a role unless the normal pressure gradient was reversed and the inlet pressure rose above the outlet pressure with interstitial edema, limb elevation, external compression, or contraction of upstream lymphangions (21, 25, 30), as described in detail by Quick et al. (27). With a sufficient pressure gradient to passively propel flow in diastole, the valves are forced open during the entire contraction cycle, and cyclical contraction no longer acts to propel lymph. Vessels in this special case were shown to be solely conduits. The present work, however, expands this insight. When lymphangions are pumping lymph up a pressure gradient, all three components do not contribute equally to lymph flow. In fact, the pressure generated by lymphangion contraction (Eq. 2) is so much larger than lymph inertia and resistance (Eq. 3) that neither plays a significant role. Thus lymphangions act solely as pumps when they transport lymph against a pressure gradient. When the pressure gradient is reversed, lymphangions act solely as conduits, and the resulting resistance is primarily a function of the radius (see Eq. A2). Equations 5 and A2 elucidate which lymphangion properties affect the slope of the pressure-flow relationships when lymphangions are acting as either pumps or conduits. However, neither approximation adequately describes the transition between the two behaviors when the axial pressure gradient is near zero, when contraction in systole and passive flow in diastole both contribute to lymph flow. In this case, lymph resistance and inertia may play an important role in lymph flow.

Sacrificing model accuracy to arrive at an algebraic formulation. To derive a simple algebraic formula relating lymphangion properties to lymph flow, a number of assumptions were made. First, with the assumption of the time-varying elastance model (Eq. 2), the present model does not include behavior analogous to the Hill effect, where the velocity of contraction limits the force generated by lymphatic muscle (6), or to the Bainbridge effect, where filling increases the frequency of contraction (18, 23). Similarly, the slope of the EDPVR (i.e., E_min) is assumed to be constant, whereas it is known to be highly nonlinear and sensitive to loading conditions (2, 22). Each effect could be addressed by modifying E_max or E_min to be a function of pressure, since Eq. 4 would still hold. In contrast, adding complications such as lymphatic backflow through the valves (7) or changes in upstream lymphangion volume due to valve bulging backward may not be worth the cost, since adding the effect would not allow an algebraic solution. More importantly, the simple model derived in the present work neglected both lymph inertia and the effect of resistance to lymph flow, which had very small effects in larger postnodal lymphatic vessels (Fig. 2). Although the resistance to lymph flow is markedly higher in microlymphatic vessels, the flow is also markedly lower. From Eq. 3, the maximum pressure drop in microlymphatic vessels approximated from parameter values reported by Dixon et al. (7) is 0.046 mmHg, suggesting that inertial and resistance effects are negligible even in microlymphatic vessels. To test our simplified model, we relied on a complex realistic mathematical model that was taken as our gold standard, because it is not possible to quantify the influence of lymph inertia and resistance directly from experimental data. The gold standard, however, does not include fluid behavior such as turbulence that may modify lymph flow. Nonetheless, the gold standard we used to test Eq. 4 was itself extensively validated with experimental data by both Venugopal et al. (34) and Quick et al. (27). Adding complexities to the simple, algebraic model (Eq. 4) may not be worth the cost if the resulting equations have to be solved numerically by computer simulation.

Scope of model embodied in analytical approximation. Mathematical models cannot account for all possible behaviors that can manifest in a physiological system; the process of developing a mathematical model involves critical choices of which behaviors to exclude. Five particular choices limiting the scope of the present work are worthy of note, particularly because there are experimental situations requiring a customization of the model. First, there are no explicit vascular biomechanics. Although lymphangion behavior could be characterized in terms of wall tension (24), this level of detail would not allow the use of simple indexes of lymphangion contractility such as E_max and E_min (27, 34). Second, there is no collapsible vessel behavior. Although lymphatic vessels can be collapsed when they are faced with a negative transmural pressure (1), the model only embodies dynamics of a cylindrical vessel subjected to positive transmural pressures. Collapsible vessel behavior requires a completely different set of equations and can lead to highly nonlinear behavior (4). Third, there is no stretch-induced contraction. Although the model incorporates increase in stroke volume and work with increased filling, there is no explicit threshold that initiates spontaneous contraction as pressure rises. Fourth, there are no extrinsic pumping mechanisms. Although extrinsic mechanisms can play a very large role in lymph flow (19), we only addressed lymph flow due to intrinsic pumping. We did not attempt to characterize the expansion of lymphangions by the interstitium or the compression by arterial pulsation, skeletal muscle contraction, loads on skin, or intestinal peristalsis (1, 3). Each mechanism could be addressed by making specific changes to E_max and E_min to reflect changes in the lymphatic pressure-volume relationship. Finally, there are no gravitational effects. The present work focused on the pumping of lymph in horizontal vessels. Although this assumption may be reasonable for the thoracic duct in most animals studied experimentally, it would notably be inappropriate for a standing human. Appropriate modifications could be made to Eq. 3 if thoracic lymph flow in an upright human was to be studied. Finally, the scope of the present work did not include the reproduction of many excellent experimental studies that manipulated interstitial pressure or lymphatic outlet pressure (8–10, 12, 22, 23). Although it may be conceivable to reproduce such experiments, and also include the simultaneous measurement of all lymphangion pressures, maximum diameters, minimum diameters, and contraction frequencies of all relevant lymphangions, this is beyond the scope of the present work.

Advantages of algebraic approximation. Our simple lymphangion model has a number of advantages over more complicated mathematical models that require simulation by computer. Although complex mathematical models can yield details such as regional endothelial shear stress, they have to be evaluated numerically (7); that is, every parameter has to be assigned a particular value, and the output of the computer simulation is a number or a graphical plot. The simplicity of our algebraic model thus yields five concrete benefits: 1) ease of use: our algebraic model is easy to manipulate without specialized training in solving differential equations; 2) generality of result: although published numerical solutions assume particular parameter values to address a particular research question, our model is a tool that investigators can use by customizing parameter values to answer new questions; 3) conceptual transparency: our model elucidates how three lymphangion properties act synergistically or antagonistically to affect lymph flow (e.g., E_max/f; Eq. 5B), an insight that cannot come from plotting lymph flow as a function of four or more parameters; 4) modeling networks of vessels: our model is amenable to calculating lymph flow in a network of vessels, which is currently not feasible using computational fluid dynamics; and 5) analysis of experimental data: our simple model can be used to infer lymphangion properties from the slopes and intercepts of experimentally measured data. The present work therefore provides a general solution that makes it possible to relate fundamental lymphangion properties to lymphatic system function.

Implications to basic science of interstitial fluid balance and potential to influence clinically relevant research. Studies in intestinal lymphatic vessels have reported that effective resistance (R_L) decreases with volume infusion (20) and portal venous hypertension (11). Similar decreases in R_L are observed in the heart, lung, liver, skeletal muscle, and kidney (8, 20) with inferior vena caval pressure elevation. Interventions used in these studies increase not only the lymphatic transmural pressure but also lymph flow. Increased lymphatic transmural pressure increases the lymphangion contraction frequency (1). With the insight provided by the present work, we can now attribute this decrease in R_L, in part, to increases in lymphatic contraction frequency (Eq. 5B). Because the reported reductions in R_L are greater than expected for the observed increases in contraction frequency, we can infer that contractility (indicated by E_max) may also have declined. This inference is supported by a recent report from in vitro experiments that shows increased luminal flow (and thus presumably endothelial shear stress) inhibits lymphatic contractility (15, 17). Equation 5B provides further insight by suggesting the nonintuitive result that increasing lymphangion contractility (i.e., increasing E_max) can increase effective lymphangion resistance, which may not be overcome by the increase in effective pump pressure (Eq. 5A). This suggests that an appropriate goal for the clinical treatment of edema may not necessarily be to stimulate lymphangion contraction but instead to inhibit it. Furthermore, if the inlet pressure rises above the outlet pressure and lymphangions transition from pumps to conduits (27) as described in the APPENDIX (Eq. A3), increasing contractility could inhibit passive flow. Thus our first-order approximation of the lymphangion pressure-flow relationship not only increases our understanding of the basic science of interstitial fluid balance by relating in vitro studies of lymphangions to in vivo studies of the lymphatic system under edemagenic conditions, but it also suggests a new approach for clinical intervention.

	APPENDIX

TOP ABSTRACT METHODS RESULTS DISCUSSION APPENDIX GRANTS REFERENCES

 
If the inlet pressure to a lymphangion exceeds its outlet pressure, it is possible to have lymph flow during lymphatic diastole (27). In this case, the valves would remain open, and lymphangions would not be able to generate a pressure by contraction. The total resistance to flow in diastole would therefore be equal to the resistance of the upstream valve, R_valve, and the resistance of the lymphangion approximated from Poiseuille's law (first term of Eq. 3). With the assumption of a cylindrical geometry, the total volume of the lymphangion would equal r²L. With a substitution into Eq. 2, and with the assumption that chamber pressure is an average of inlet (P_in) and outlet (P_out) pressures, lymph flow at end diastole (Q_ed) would equal

(A1)

where r₀ is the theoretical radius at zero pressure (corresponding to the dead volume V₀ in Eq. 2). The effective resistance when a lymphangion is acting like a conduit in diastole (R_ed) would be a function of lymphangion tone, characterized by the minimum elastance E_min.

(A2)

However, with the assumption that the lymphangion is still contracting, the radius would decrease to a minimum value at end systole. The resulting resistance (R_es) therefore would be governed by the maximum elastance E_max.

(A3)

Increasing contractility (i.e., increasing E_max) would therefore decrease lymph flow when P_in > P_out.

	GRANTS

TOP ABSTRACT METHODS RESULTS DISCUSSION APPENDIX GRANTS REFERENCES

 
Portions of this work were supported by National Institutes of Health Grants NIH-K25-HL-070608 (to C. M. Quick), CDC-620069 (to G. A. Laine), and CDC-623086 (to G. A. Laine) and American Heart Association Grants AHA-0565116Y (to C. M. Quick) and AHA-0365127Y (to R. H. Stewart).

	FOOTNOTES

 

Address for reprint requests and other correspondence: C. M. Quick, Michael E. DeBakey Inst., TAMU 4466, Texas A&M Univ., College Station, TX 77843-4466 (e-mail: cquick{at}tamu.edu)

The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.

	REFERENCES

TOP ABSTRACT METHODS RESULTS DISCUSSION APPENDIX GRANTS REFERENCES

 

1. Aukland K, Reed RK. Interstitial-lymphatic mechanisms in the control of extracellular fluid volume. Physiol Rev 73: 1–78, 1993.[Abstract/Free Full Text]
2. Benoit JN, Zawieja DC, Goodman AH, Granger HJ. Characterization of intact mesenteric lymphatic pump and its responsiveness to acute edemagenic stress. Am J Physiol Heart Circ Physiol 257: H2059–H2069, 1989.[Abstract/Free Full Text]
3. Bridenbaugh EA, Gashev AA, Zawieja DC. Lymphatic muscle: a review of contractile function. Lymphat Res Biol 1: 147–158, 2003.[CrossRef][Medline]
4. Brower RW, Noordergraaf A. Pressure-flow characteristics of collapsible tubes: a reconciliation of seemingly contradictory results. Ann Biomed Eng 1: 333–355, 1973.[CrossRef][Web of Science][Medline]
5. Davies PF, Spaan JA, Krams R. Shear stress biology of the endothelium. Ann Biomed Eng 33: 1714–1718, 2005.[CrossRef][Web of Science][Medline]
6. Davis MJ, Price A, Gashev AA, Wang WW, Zawieja DC, Zhang R. Force velocity relationships of rat mesenteric lymphatics and arteries. 2005 Experimental Biology meeting (on CD-ROM) (Abstract). FASEB J 19: 123.121, 2005.
7. Dixon BJ, Greiner ST, Gashev AA, Cote GL, Moore JE, Zawieja DC. Lymph flow, shear stress, and lymphocyte velocity in rat mesenteric prenodal lymphatics. Microcirculation 13: 597–610, 2006.[CrossRef][Web of Science][Medline]
8. Drake R, Giesler M, Laine G, Gabel J, Hansen T. Effect of outflow pressure on lung lymph flow in unanesthetized sheep. J Appl Physiol 58: 70–76, 1985.[Abstract/Free Full Text]
9. Drake RE, Allen SJ, Katz J, Gabel JC, Laine GA. Equivalent circuit technique for lymph flow studies. Am J Physiol Heart Circ Physiol 251: H1090–H1094, 1986.[Abstract/Free Full Text]
10. Drake RE, Dhother S, Oppenlander VM, Gabel JC. Lymphatic pump function curves in awake sheep. Am J Physiol Regul Integr Comp Physiol 270: R486–R488, 1996.[Abstract/Free Full Text]
11. Dunbar BS, Elk JR, Drake RE, Laine GA. Intestinal lymphatic flow during portal venous hypertension. Am J Physiol Gastrointest Liver Physiol 257: G94–G98, 1989.[Abstract/Free Full Text]
12. Eisenhoffer J, Elias RM, Johnston MG. Effect of outflow pressure on lymphatic pumping in vitro. Am J Physiol Regul Integr Comp Physiol 265: R97–R102, 1993.[Abstract/Free Full Text]
13. Fung YC. Biomechanics: Motion, Flow Stress, Growth. New York: Springer-Verlag, 1990.
14. Gashev AA, Zawieja DC. Physiology of human lymphatic contractility: a historical perspective. Lymphology 34: 124–134, 2001.[Web of Science][Medline]
15. Gashev AA, Davis MJ, Zawieja DC. Inhibition of the active lymph pump by flow in rat mesenteric lymphatics and thoracic duct. J Physiol 540: 1023–1037, 2002.[Abstract/Free Full Text]
16. Gashev AA, Orlov RS, Borisov AV, Kliuchin'ski T, Andreevskaia MV, Bubnova NA, Borisova RP, Andreev IA, Erofeev NP, Priklonskaia EG. The mechanisms of lymphangion interaction in the process of lymph movement. [In Russian.] Fiziol Zh SSSR Im IM Sechenova 76: 1489–1508, 1990.
17. Gasheva OY, Zawieja DC, Gashev AA. Contraction-initiated NO-dependent lymphatic relaxation: a self-regulatory mechanism in rat thoracic duct. J Physiol 575: 821–832, 2006.[Abstract/Free Full Text]
18. Hargens AR, Zweifach BW. Contractile stimuli in collecting lymph vessels. Am J Physiol Heart Circ Physiol 233: H57–H65, 1977.[Abstract/Free Full Text]
19. Ikomi F, Schmid-Schonbein GW. Lymph pump mechanics in the rabbit hind leg. Am J Physiol Heart Circ Physiol 271: H173–H183, 1996.[Abstract/Free Full Text]
20. Laine GA, Allen SJ, Katz J, Gabel JC, Drake RE. Outflow pressure reduces lymph flow rate from various tissues. Microvasc Res 33: 135–142, 1987.[CrossRef][Web of Science][Medline]
21. Laine GA, Hall JT, Laine SH, Granger J. Transsinusoidal fluid dynamics in canine liver during venous hypertension. Circ Res 45: 317–323, 1979.[Free Full Text]
22. Li B, Silver I, Szalai JP, Johnston MG. Pressure-volume relationships in sheep mesenteric lymphatic vessels in situ: response to hypovolemia. Microvasc Res 56: 127–138, 1998.[CrossRef][Web of Science][Medline]
23. McHale NG, Roddie IC. The effect of transmural pressure on pumping activity in isolated bovine lymphatic vessels. J Physiol 261: 255–269, 1976.[Abstract/Free Full Text]
24. Meisner JK, Stewart RH, Laine GA, Quick CM. Lymphatic vessels transition to state of summation above a critical contraction frequency. Am J Physiol Regul Integr Comp Physiol 293: R200–R208, 2007.[Abstract/Free Full Text]
25. Mortillaro NA, Taylor AE. Interstitial fluid pressure of ileum measured from chronically implanted polyethylene capsules. Am J Physiol Heart Circ Physiol 257: H62–H69, 1989.[Abstract/Free Full Text]
26. Noordergraaf A. Circulatory System Dynamics. New York: Academic, 1978.
27. Quick CM, Venugopal AM, Gashev AA, Zawieja DC, Stewart RH. Intrinsic pump-conduit behavior of lymphatic vessels. Am J Physiol Regul Integr Comp Physiol 292: R1510–R1518, 2007.[Abstract/Free Full Text]
28. Reddy NP, Krouskop TA, Newell PH Jr. A computer model of the lymphatic system. Comput Biol Med 7: 181–197, 1977.[CrossRef][Web of Science][Medline]
29. Schmid-Schonbein GW. Microlymphatics and lymph flow. Physiol Rev 70: 987–1028, 1990.[Abstract/Free Full Text]
30. Stewart RH, Laine GA. Flow in lymphatic networks: interaction between hepatic and intestinal lymph vessels. Microcirculation 8: 221–227, 2001.[CrossRef][Web of Science][Medline]
31. Suga H, Sagawa K. Instantaneous pressure-volume relationships and their ratio in the excised, supported canine left ventricle. Circ Res 35: 117–126, 1974.[Abstract/Free Full Text]
32. Suga H, Sagawa K, Shoukas AA. Load independence of the instantaneous pressure-volume ratio of the canine left ventricle and effects of epinephrine and heart rate on the ratio. Circ Res 32: 314–322, 1973.[Abstract/Free Full Text]
33. Sunagawa K, Maughan WL, Burkhoff D, Sagawa K. Left ventricular interaction with arterial load studied in isolated canine ventricle. Am J Physiol Heart Circ Physiol 245: H773–H780, 1983.[Abstract/Free Full Text]
34. Venugopal AM, Stewart RH, Laine GA, Quick CM. Lymphangion coordination minimally affects flow in lymphatic vessels. Am J Physiol Heart Circ Physiol 293: H1183–H1189, 2007.[Abstract/Free Full Text]
35. Womersely J. An elastic tube theory of pulse transmission and oscillatory flow in mammalian arteries. WADC Tech, 1957.

This article has been cited by other articles:

				R. M. Dongaonkar, G. A. Laine, R. H. Stewart, and C. M. Quick Balance point characterization of interstitial fluid volume regulation Am J Physiol Regulatory Integrative Comp Physiol, July 1, 2009; 297(1): R6 - R16. [Abstract] [Full Text] [PDF]

				C. M. Quick, B. L. Ngo, A. M. Venugopal, and R. H. Stewart Lymphatic pump-conduit duality: contraction of postnodal lymphatic vessels inhibits passive flow Am J Physiol Heart Circ Physiol, March 1, 2009; 296(3): H662 - H668. [Abstract] [Full Text] [PDF]

				A. M. Venugopal, C. M. Quick, G. A. Laine, and R. H. Stewart Optimal postnodal lymphatic network structure that maximizes active propulsion of lymph Am J Physiol Heart Circ Physiol, February 1, 2009; 296(2): H303 - H309. [Abstract] [Full Text] [PDF]

This Article Abstract Full Text (PDF) All Versions of this Article: 294/5/H2144    most recent 00781.2007v1 Alert me when this article is cited Alert me if a correction is posted Citation Map Services Email this article to a friend Similar articles in this journal Similar articles in PubMed Alert me to new issues of the journal Download to citation manager Citing Articles Citing Articles via HighWire Citing Articles via Web of Science (3) Citing Articles via Google Scholar Google Scholar Articles by Quick, C. M. Articles by Stewart, R. H. Search for Related Content PubMed PubMed Citation Articles by Quick, C. M. Articles by Stewart, R. H.