# A one-dimensional mathematical model of collecting lymphatics coupled with an electro-fluid-mechanical contraction model and valve dynamics

- 72 Downloads

## Abstract

We propose a one-dimensional model for collecting lymphatics coupled with a novel Electro-Fluid-Mechanical Contraction (EFMC) model for dynamical contractions, based on a modified FitzHugh–Nagumo model for action potentials. The one-dimensional model for a deformable lymphatic vessel is a nonlinear system of hyperbolic Partial Differential Equations (PDEs). The EFMC model combines the electrical activity of lymphangions (action potentials) with fluid-mechanical feedback (circumferential stretch of the lymphatic wall and wall shear stress) and lymphatic vessel wall contractions. The EFMC model is governed by four Ordinary Differential Equations (ODEs) and phenomenologically relies on: (1) environmental calcium influx, (2) stretch-activated calcium influx, and (3) contraction inhibitions induced by wall shear stresses. We carried out a stability analysis of the stationary state of the EFMC model. Contractions turn out to be triggered by the instability of the stationary state. Overall, the EFMC model allows emulating the influence of pressure and wall shear stress on the frequency of contractions observed experimentally. Lymphatic valves are modelled by extending an existing lumped-parameter model for blood vessels. Modern numerical methods are employed for the one-dimensional model (PDEs), for the EFMC model and valve dynamics (ODEs). Adopting the geometrical structure of collecting lymphatics from rat mesentery, we apply the full mathematical model to a carefully selected suite of test problems inspired by experiments. We analysed several indices of a single lymphangion for a wide range of upstream and downstream pressure combinations which included both favourable and adverse pressure gradients. The most influential model parameters were identified by performing two sensitivity analyses for favourable and adverse pressure gradients.

## Keywords

One-dimensional model for lymphatics FitzHugh–Nagumo Collecting lymphatics Lymphangions Lymphatic action potential## Notes

### Acknowledgements

The authors gratefully acknowledge the suggestions given by Prof. Christian Vergara from the Department of Mathematics, Politecnico di Milano, Italy. The authors also acknowledge the excellent work done by anonymous referees that greatly contributed to improve this paper.

### Compliance with ethical standards

### Conflict of Interest

The authors declare that they have no conflict of interests.

## Supplementary material

## References

- Alastruey A.J (2006) Numerical modelling of pulse wave propagation in the cardiovascular system: development, validation and clinical applications. Ph.D. thesis, University of LondonGoogle Scholar
- Alastruey J, Parker KH, Peiró J, Sherwin SJ (2008) Lumped parameter outflow models for 1-D blood flow simulations: effect on pulse waves and parameter estimation. Commun Comput Phys 4(2):317–336MathSciNetzbMATHGoogle Scholar
- Baish JW, Kunert C, Padera TP, Munn LL (2016) Synchronization and random triggering of lymphatic vessel contractions. PLoS Comput Biol 12(12):e1005,231CrossRefGoogle Scholar
- Bertram C, Macaskill C, Moore J (2016) Pump function curve shape for a model lymphatic vessel. Med Eng Phys 38(7):656–663CrossRefGoogle Scholar
- Bertram CD, Macaskill C, Davis MJ, Moore JE (2014) Development of a model of a multi-lymphangion lymphatic vessel incorporating realistic and measured parameter values. Biomech Model Mechanobiol 13(2):401–416CrossRefGoogle Scholar
- Bertram C.D., Macaskill C, Davis M.J., Moore J.E (2016) Consequences of intravascular lymphatic valve properties: a study of contraction timing in a multi-lymphangion model. American Journal of Physiology: Heart and Circulatory Physiology 310(7), ajpheart.00,669.2015Google Scholar
- Bertram CD, Macaskill C, Davis MJ, Moore JE (2017) Valve-related modes of pump failure in collecting lymphatics: numerical and experimental investigation. Biomech Model Mechanobiol 16(6):1987–2003CrossRefGoogle Scholar
- Bertram CD, Macaskill C, Moore JE (2011) Simulation of a chain of collapsible contracting lymphangions with progressive valve closure. J Biomech Eng 133(1):011,008CrossRefGoogle Scholar
- Bertram CD, Macaskill C, Moore JE (2014) Incorporating measured valve properties into a numerical model of a lymphatic vessel. Comput Methods Biomech Biomed Eng 17(14):1519–1534CrossRefGoogle Scholar
- Borsche R, Kall J (2016) High order numerical methods for networks of hyperbolic conservation laws coupled with ODEs and lumped parameter models. J Comput Phys 327:678–699MathSciNetCrossRefzbMATHGoogle Scholar
- Breslin JW (2014) Mechanical forces and lymphatic transport. Microvasc Res 96:46–54CrossRefGoogle Scholar
- Caulk AW, Dixon JB, Gleason RL (2016) A lumped parameter model of mechanically mediated acute and long-term adaptations of contractility and geometry in lymphatics for characterization of lymphedema. Biomech Model Mechanobiol 15(6):1601–1618CrossRefGoogle Scholar
- Caulk AW, Nepiyushchikh ZV, Shaw,R, Dixon JB, Gleason RL (2015) Quantification of the passive and active biaxial mechanical behaviour and microstructural organization of rat thoracic ducts. J R Soc Interface 12(108):20150,280CrossRefGoogle Scholar
- Contarino C, Toro EF, Montecinos GI, Borsche R, Kall J (2016) Junction-generalized Riemann problem for stiff hyperbolic balance laws in networks: an implicit solver and ADER schemes. J Comput Phys 315:409–433MathSciNetCrossRefzbMATHGoogle Scholar
- Davis MJ (2015) Is nitric oxide important for the diastolic phase of the lymphatic contraction/relaxation cycle? Proc Nat Acad Sci 113(2):E105–E105CrossRefGoogle Scholar
- Davis MJ, Rahbar E, Gashev AA, Zawieja DC, Moore JE (2011) Determinants of valve gating in collecting lymphatic vessels from rat mesentery. Am J Physiol Heart Circ Physiol 301(1):H48–H60CrossRefGoogle Scholar
- Davis MJ, Scallan JP, Wolpers JH, Muthuchamy M, Gashev AA, Zawieja DC (2012) Intrinsic increase in lymphangion muscle contractility in response to elevated afterload. Am J Physiol Heart Circ Physiol 303(7):H795–H808CrossRefGoogle Scholar
- Formaggia L, Quarteroni A, Veneziani A (2009) Cardiovascular Mathematics. Modeling and simulation of the circulatory system. Springer, BerlinzbMATHGoogle Scholar
- Franzone PC, Pavarino LF, Scacchi S (2014) Mathematical cardiac electrophysiology. Springer, BerlinCrossRefzbMATHGoogle Scholar
- Gajani G.S, Boschetti F, Negrini D, Martellaccio R, Milanese G, Bizzarri F, Brambilla A (2015) A lumped model of lymphatic systems suitable for large scale simulations. In: 2015 European conference on circuit theory and design (ECCTD). Institute of Electrical & Electronics Engineers (IEEE)Google Scholar
- Gashev AA (2002) Inhibition of the active lymph pump by flow in rat mesenteric lymphatics and thoracic duct. J Physiol 540(3):1023–1037CrossRefGoogle Scholar
- Gashev AA, Davis MM, Delp MD, Zawieja DC (2004) Regional variations of contractile activity in isolated rat lymphatics. Microcirculation 11(6):477–492CrossRefGoogle Scholar
- van Griensven A, Meixner T, Grunwald S, Bishop T, Diluzio M, Srinivasan R (2006) A global sensitivity analysis tool for the parameters of multi-variable catchment models. J Hydrol 324(1–4):10–23CrossRefGoogle Scholar
- Hodgkin AL, Huxley AF (1952) A quantitative description of membrane current and its application to conduction and excitation in nerve. J Physiol 117(4):500–544CrossRefGoogle Scholar
- Jamalian S, Davis MJ, Zawieja DC, Moore JE (2016) Network scale modeling of lymph transport and its effective pumping parameters. PLoS ONE 11(2):e0148,384CrossRefGoogle Scholar
- Kunert C, Baish JW, Liao S, Padera TP, Munn LL (2015) Mechanobiological oscillators control lymph flow. Proc Nat Acad Sci 112(35):10938–10943CrossRefGoogle Scholar
- LeFloch PG (2002) Hyperbolic systems of conservation laws. Springer, BerlinCrossRefzbMATHGoogle Scholar
- Liang F, Takagi S, Himeno R, Liu H (2009) Biomechanical characterization of ventricular–arterial coupling during aging: A multi-scale model study. J Biomech 42(6):692–704CrossRefGoogle Scholar
- MacDonald AJ, Arkill KP, Tabor GR, McHale NG, Winlove CP (2008) Modeling flow in collecting lymphatic vessels: one-dimensional flow through a series of contractile elements. Am J Physiol Heart Circ Physiol 295(1):H305–H313CrossRefGoogle Scholar
- Margaris K, Black RA (2012) Modelling the lymphatic system: challenges and opportunities. J R Soc Interface 9(69):601–612CrossRefGoogle Scholar
- McHale NG, Roddie IC (1976) The effect of transmural pressure on pumping activity in isolated bovine lymphatic vessels. J Physiol 261(2):255–269CrossRefGoogle Scholar
- Müller LO, Parés C, Toro EF (2013) Well-balanced high-order numerical schemes for one-dimensional blood flow in vessels with varying mechanical properties. J Comput Phys 242:53–85MathSciNetCrossRefzbMATHGoogle Scholar
- Müller LO, Toro EF (2014) Enhanced global mathematical model for studying cerebral venous blood flow. J Biomech 47(13):3361–3372CrossRefGoogle Scholar
- Munn LL (2015) Mechanobiology of lymphatic contractions. Semin Cell Dev Biol 38:67–74CrossRefGoogle Scholar
- Mynard JP, Davidson MR, Penny DJ, Smolich JJ (2012) A simple, versatile valve model for use in lumped parameter and one-dimensional cardiovascular models. Int J Numer Methods Biomed Eng 28(6–7):626–641MathSciNetCrossRefGoogle Scholar
- Nagumo J, Arimoto S, Yoshizawa S (1962) An active pulse transmission line simulating nerve axon. Proc IRE 50(10):2061–2070CrossRefGoogle Scholar
- Ohhashi T, Azuma T, Sakaguchi M (1980) Active and passive mechanical characteristics of bovine mesenteric lymphatics. Am J Physiol 239(1):H88–95Google Scholar
- Quarteroni A, Lassila T, Rossi S, Ruiz-Baier R (2017) Integrated heart—coupling multiscale and multiphysics models for the simulation of the cardiac function. Comput Methods Appl Mech Eng 314:345–407MathSciNetCrossRefGoogle Scholar
- Quarteroni A, Veneziani A, Vergara C (2016) Geometric multiscale modeling of the cardiovascular system, between theory and practice. Comput Methods Appl Mech Eng 302:193–252MathSciNetCrossRefGoogle Scholar
- Rahbar E, Moore JE (2011) A model of a radially expanding and contracting lymphangion. J Biomech 44(6):1001–1007CrossRefGoogle Scholar
- Rahbar E, Weimer J, Gibbs H, Yeh AT, Bertram CD, Davis MJ, Hill MA, Zawieja DC, Moore JE (2012) Passive pressure-diameter relationship and structural composition of rat mesenteric lymphangions. Lymphat Res Biol 10(4):152–163CrossRefGoogle Scholar
- Reddy N.P (1974) A discrete model of the lymphatic system. Ph.D. thesis, Texas A&M UniversityGoogle Scholar
- Scallan JP, Wolpers JH, Muthuchamy M, Zawieja DC, Gashev AA, Davis MJ (2012) Independent and interactive effects of preload and afterload on the pump function of the isolated lymphangion. Am J Physiol Heart Circ Physiol 303(7):H809–H824CrossRefGoogle Scholar
- Strocchi M, Contarino C, Zhang Q, Bonmassari R, Toro E (2017) A global mathematical model for the simulation of stenoses and bypass placement in the human arterial system. Appl Math Comput 300:21–39MathSciNetGoogle Scholar
- Sun Y, Sjoberg BJ, Ask P, Loyd D, Wranne B (1995) Mathematical model that characterizes transmitral and pulmonary venous flow velocity patterns. Am J Physiol Heart Circ Physiol 268(1):H476–H489CrossRefGoogle Scholar
- Swartz M (2001) The physiology of the lymphatic system. Adv Drug Deliv Rev 50(1–2):3–20CrossRefGoogle Scholar
- Telinius N, Majgaard J, Kim S, Katballe N, Pahle E, Nielsen J, Hjortdal V, Aalkjaer C, Boedtkjer DB (2015) Voltage-gated sodium channels contribute to action potentials and spontaneous contractility in isolated human lymphatic vessels. J Physiol 593(14):3109–3122CrossRefGoogle Scholar
- Toro EF (2009) Riemann solvers and numerical methods for fluid dynamics, 3rd edn. Springer, BerlinCrossRefzbMATHGoogle Scholar
- Toro EF (2016) Brain venous haemodynamics, neurological diseases and mathematical modelling. A Review. Appl Math Comput 272:542–579MathSciNetGoogle Scholar
- Toro EF, Billett SJ (2000) Centred TVD schemes for hyperbolic conservation laws. IMA J Numer Anal 20:47–79MathSciNetCrossRefzbMATHGoogle Scholar
- Toro EF, Müller L, Cristini M, Menegatti E, Zamboni P (2015) Impact of jugular vein valve function on cerebral venous haemodynamics. Curr Neurovascular Res 12(4):384–397CrossRefGoogle Scholar
- Toro EF, Siviglia A (2013) Flow in collapsible tubes with discontinuous mechanical properties: mathematical model and exact solutions. Commun Comput Phys 13(02):361–385MathSciNetCrossRefzbMATHGoogle Scholar
- Venugopal AM, Stewart RH, Laine GA, Dongaonkar RM, Quick CM (2007) Lymphangion coordination minimally affects mean flow in lymphatic vessels. Am J Physiol Heart Circ Physiol 293(2):H1183–H1189CrossRefGoogle Scholar
- Wilson JT, van Loon R, Wang W, Zawieja DC, Moore JE (2015) Determining the combined effect of the lymphatic valve leaflets and sinus on resistance to forward flow. J Biomech 48(13):3584–3590CrossRefGoogle Scholar
- Zawieja DC (2009) Contractile physiology of lymphatics. Lymphat Res Biol 7(2):87–96CrossRefGoogle Scholar
- Zawieja DC, Davis KL, Schuster R, Hinds WM, Granger HJ (1993) Distribution, propagation, and coordination of contractile activity in lymphatics. Am J Physiol Heart Circ Physiol 264(4):H1283–H1291CrossRefGoogle Scholar