Oscillations and Multiple Equilibria in Microvascular Blood Flow

Abstract

We investigate the existence of oscillatory dynamics and multiple steady-state flow rates in a network with a simple topology and in vivo microvascular blood flow constitutive laws. Unlike many previous analytic studies, we employ the most biologically relevant models of the physical properties of whole blood. Through a combination of analytic and numeric techniques, we predict in a series of two-parameter bifurcation diagrams a range of dynamical behaviors, including multiple equilibria flow configurations, simple oscillations in volumetric flow rate, and multiple coexistent limit cycles at physically realizable parameters. We show that complexity in network topology is not necessary for complex behaviors to arise and that nonlinear rheology, in particular the plasma skimming effect, is sufficient to support oscillatory dynamics similar to those observed in vivo.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

References

  1. Arciero JC, Secomb TW (2011) Spontaneous oscillations in a model for active control of microvessel diameters. Math Med Biol. doi:10.1093/imammb/dqr005

  2. di Bernardo M, Budd CJ, Champneys AR, Kowalczyk P, Nordmark AB, Tost GO, Piiroinen PT (2008) Bifurcations in nonsmooth dynamical systems. SIAM Rev 50(4):629–701. doi:10.1137/050625060

    MATH  MathSciNet  Article  Google Scholar 

  3. Carr RT, Lacoin M (2000) Nonlinear dynamics of microvascular blood flow. Ann Biomed Eng 28(6):641–652. doi:10.1114/1.1306346

    Article  Google Scholar 

  4. Carr RT, Geddes JB, Wu F (2005) Oscillations in a simple microvascular network. Ann Biomed Eng 33(6):764–771. doi:10.1007/s10439-005-2345-2

    Article  Google Scholar 

  5. Casey R, Jong Hd, Gouzé JL (2005) Piecewise-linear models of genetic regulatory networks: equilibria and their stability. J Math Biol 52(1):27–56. doi:10.1007/s00285-005-0338-2

    Article  Google Scholar 

  6. Chien S, Tvetenstrand CD, Epstein MA, Schmid-Schönbein GW (1985) Model studies on distributions of blood cells at microvascular bifurcations. Am J Physiol 248(4 Pt 2):H568–H576

    Google Scholar 

  7. Coombes S, Doole SH (2010) Neuronal population dynamics with post inhibitory rebound: a reduction to piecewise linear discontinuous circle maps. 11(3):193–217. doi:10.1080/02681119608806224

  8. Damiano ER (1998) The effect of the endothelial-cell glycocalyx on the motion of red blood cells through capillaries. Microvasc Res 55(1):77–91. doi:10.1006/mvre.1997.2052

    Article  Google Scholar 

  9. Davis JM, Pozrikidis C (2010) Numerical simulation of unsteady blood flow through capillary networks. Bull Math Biol 73(8):1857–1880. doi:10.1007/s11538-010-9595-3

    MathSciNet  Article  Google Scholar 

  10. Davis JM, Pozrikidis C (2014) Self-sustained oscillations in blood flow through a honeycomb capillary network. Bull Math Biol 76(9):2217–2237. doi:10.1007/s11538-014-0002-3

    MATH  MathSciNet  Article  Google Scholar 

  11. Dellimore JW, Dunlop MJ, Canham PB (1983) Ratio of cells and plasma in blood flowing past branches in small plastic channels. Am J Physiol 244(5):H635–H643

    Google Scholar 

  12. Dercole F, Gragnani A, Rinaldi S (2007) Bifurcation analysis of piecewise smooth ecological models. Theor Popul Biol 72(2):197–213. doi:10.1016/j.tpb.2007.06.003

    MATH  Article  Google Scholar 

  13. Fåhræus R (1929) Suspension stability of blood. Physiol Rev 9:241–274

    Google Scholar 

  14. Fåhræus R, Lindqvist T (1931) The viscosity of blood in narrow capillary tubes. J Physiol 96:562–568

    Google Scholar 

  15. Fenton BM, Carr RT, Cokelet GR (1985) Nonuniform red cell distribution in 20 to 100 \(\mu \)m bifurcations. Microvasc Res 29(1):103–126. doi:10.1016/0026-2862(85)90010-X

    Article  Google Scholar 

  16. Forouzan O, Yang X, Sosa JM, Burns JM, Shevkoplyas SS (2012) Spontaneous oscillations of capillary blood flow in artificial microvascular networks. Microvasc Res 84(2):123–132. doi:10.1016/j.mvr.2012.06.006

    Article  Google Scholar 

  17. Geddes JB, Carr RT, Karst N, Wu F (2007) The onset of oscillations in microvascular blood flow. SIAM J Appl Dyn Syst 6(4):694–727. doi:10.1137/060670699

    MATH  MathSciNet  Article  Google Scholar 

  18. Geddes JB, Carr RT, Wu F, Lao Y, Maher M (2010) Blood flow in microvascular networks: a study in nonlinear biology. Chaos Interdiscip J Nonlinear Sci 20(4):045,123. doi:10.1063/1.3530122

    MathSciNet  Article  Google Scholar 

  19. Harris AG, Skalak TC (1993) Effects of leukocyte activation on capillary hemodynamics in skeletal muscle. Am J Physiol 264(3 Pt 2):H909–H916

    Google Scholar 

  20. Janssen BJ, Oosting J, Slaaf DW, Persson PB, Struijker-Boudier HA (1995) Hemodynamic basis of oscillations in systemic arterial pressure in conscious rats. Am J Physiol 269(1 Pt 2):H62–H71

    Google Scholar 

  21. Jeffrey MR, Dankowicz H (2014) Discontinuity-induced bifurcation cascades in flows and maps with application to models of the yeast cell cycle. Phys D Nonlinear Phenom 271:32–47. doi:10.1016/j.physd.2013.12.011

    MATH  MathSciNet  Article  Google Scholar 

  22. Karst CM, Storey BD, Geddes JB (2013) Laminar flow of two miscible fluids in a simple network. Phys Fluids 25(3):033,601. doi:10.1063/1.4794726

    Article  Google Scholar 

  23. Karst NJ, Storey BD, Geddes JB (2014) Spontaneous oscillations in simple fluid networks. SIAM J Appl Dyn Syst 13(1):157–180. doi:10.1137/130926304

    MATH  MathSciNet  Article  Google Scholar 

  24. Kiani MF, Pries AR, Hsu LL, Sarelius IH, Cokelet GR (1994) Fluctuations in microvascular blood flow parameters caused by hemodynamic mechanisms. Am J Physiol 266(5 Pt 2):H1822–H1828

    Google Scholar 

  25. Klitzman B, Johnson PC (1982) Capillary network geometry and red cell distribution in hamster cremaster muscle. Am J Physiol 242(2):H211–H219

    Google Scholar 

  26. Krogh A (1921) Studies on the physiology of capillaries: II. The reactions to local stimuli of the blood-vessels in the skin and web of the frog. J Physiol (Lond) 55(5–6):412–422

    Article  Google Scholar 

  27. Kuznetsov Y (2004) Elements of applied bifurcation theory. Springer, New York. doi:10.1007/978-1-4757-3978-7

    MATH  Book  Google Scholar 

  28. Parthimos D, Osterloh K, Pries AR, Griffith TM (2004) Deterministic nonlinear characteristics of in vivoblood flow velocity and arteriolar diameter fluctuations. Phys Med Biol 49(9):1789–1802. doi:10.1088/0031-9155/49/9/014

    Article  Google Scholar 

  29. Pop SR, Richardson G, Waters SL, Jensen OE (2007) Shock formation and non-linear dispersion in a microvascular capillary network. Math Med Biol 24(4):379–400. doi:10.1093/imammb/dqm007

    MATH  Article  Google Scholar 

  30. Pries AR, Ley K, Claassen M, Gaehtgens P (1989) Red cell distribution at microvascular bifurcations. Microvasc Res 38(1):81–101. doi:10.1016/0026-2862(89)90018-6

    Article  Google Scholar 

  31. Pries AR, Secomb TW, Gaehtgens P, Gross JF (1990) Blood flow in microvascular networks. Experiments and simulation. Circ Res 67(4):826–834

    Article  Google Scholar 

  32. Pries AR, Secomb TW, Gessner T, Sperandio MB, Gross JF, Gaehtgens P (1994) Resistance to blood flow in microvessels in vivo. Circ Res 75(5):904–915. doi:10.1161/01.RES.75.5.904

    Article  Google Scholar 

  33. Rodgers GP, Schechter AN, Noguchi CT, Klein HG, Niehuis QW, Bonner RF (1984) Periodic microcirculatory flow in patients with sickle cell disease. N Engl J Med 311:1534–1538. doi:10.1056/NEJM198412133112403

    Article  Google Scholar 

  34. Secomb TW, Hsu R (1996) Motion of red blood cells in capillaries with variable cross-sections. J Biomech Eng 118(4):538–544. doi:10.1115/1.2796041

    Article  Google Scholar 

  35. Shevkoplyas SS, Gifford SC, Yoshida T, Bitensky MW (2003) Prototype of an in vitro model of the microcirculation. Microvas Res 65(2):132–136. doi:10.1016/S0026-2862(02)00034-1

    Article  Google Scholar 

  36. Simpson DJW, Kompala DS, Meiss JD (2009) Discontinuity induced bifurcations in a model of Saccharomyces cerevisiae. Math Biosci 218(1):40–49. doi:10.1016/j.mbs.2008.12.005

    MATH  MathSciNet  Article  Google Scholar 

  37. Storey BD, Hellen DV, Karst NJ, Geddes JB (2015) Observations of spontaneous oscillations in simple two-fluid networks. Phys Rev E 91:023,004. doi:10.1103/PhysRevE.91.023004

    Article  Google Scholar 

  38. Tawfik Y, Owens RG (2013) A mathematical and numerical investigation of the hemodynamical origins of oscillations in microvascular networks. Bull Math Biol 75(4):676–707. doi:10.1007/s11538-013-9825-6

    MATH  MathSciNet  Article  Google Scholar 

Download references

Acknowledgments

This work was supported by the Babson Faculty Research Fund (N. J. K.) and the National Science Foundation under contract DMS-1211640 (B. D. S. and J. B. G.). We thank the two reviewers of our original submission for their careful attention and helpful suggestions.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Nathaniel J. Karst.

Appendix

Appendix

We assume perturbations as given in Eqs. 3336. The perturbations on \(Q_A\) and \(Q_C\) induce perturbations on the flow rates in the remaining branches:

$$\begin{aligned} \tilde{Q}_B&= -\tilde{Q}_A \end{aligned}$$
(42)
$$\begin{aligned} \tilde{Q}_D&= \tilde{Q}_A + \tilde{Q}_C \end{aligned}$$
(43)
$$\begin{aligned} \tilde{Q}_E&= -\tilde{Q}_A - \tilde{Q}_C. \end{aligned}$$
(44)

The boundary condition on vessel A is given by plasma skimming:

$$\begin{aligned} H_A(0,t) = f_A(Q_A(t), H_1) H_1. \end{aligned}$$
(45)

As we assume \(H_1\) to be constant, only terms related to \(\tilde{Q}_A\) and \(\tilde{H}_A\) are retained after linearization

$$\begin{aligned} h_A&= H_1 \left. {\partial f_A \over \partial Q_A}\right| _* q_A, \end{aligned}$$
(46)

where \((\cdot )|_*\) indicates evaluation at the equilibrium values \((Q_A^*,Q_C^*)\). A similar result exists for the boundary condition on vessel B:

$$\begin{aligned} h_B&= H_1 \left. {\partial g_A \over \partial Q_A}\right| _* q_A. \end{aligned}$$
(47)

The derivative of the plasma skimming function g can always be computed through conservation of mass.

$$\begin{aligned} H_F&= (1-Q_A) g_A H_F + Q_A f_A H_F \end{aligned}$$
(48)
$$\begin{aligned} \partial g_A \over \partial Q_A&= \left( g_A - f_A - Q_A {\partial f_A \over \partial Q_A}\right) (1 - Q_A)^{-1}. \end{aligned}$$
(49)

We assume that \(Q_C(t) > 0\), so that branch B leads to a diverging node. The boundary condition on branch C is

$$\begin{aligned} H_C(0,t)&= f_C\left( {Q_C(t) \over Q_B(t)}, H_B(1,t)\right) H_B(1,t). \end{aligned}$$
(50)

Since the feed hematocrit \(H_B(1,t)\) is not constant, we must account in branch C for the arrival of perturbations originating upstream in vessel B. After linearization, we retain

$$\begin{aligned} h_C = e^{-\lambda \tau _B^*} \left. \left( f_C + H_B {\partial f_C \over \partial H_B}\right) \right| _* h_B + {H_B^* Q_C^* \over (Q_B^*)^2}\left. {\partial f_C \over \partial Q}\right| _* q_A + {H_B^* \over Q_B^*}\left. {\partial f_C \over \partial Q} \right| _* q_C. \end{aligned}$$
(51)

Here \(\partial f_C / \partial Q\) represents differentiation with respect to the function’s flow rate argument \(Q = Q_C / Q_B\). Similarly, from the linearization of boundary condition on branch E, we retain

$$\begin{aligned} h_E&= e^{-\lambda \tau _B^*} \left. \left( g_E + H_B {\partial g_E \over \partial H_B}\right) \right| _* h_B + {H_B^* Q_C^* \over (Q_B^*)^2} \left. {\partial g_E \over \partial Q} \right| _*q_A + {H_B^* \over Q_B^*} \left. {\partial g_E \over \partial Q} \right| _* q_C. \end{aligned}$$
(52)

The boundary condition for vessel D is given by conservation of mass rather than by plasma skimming:

$$\begin{aligned} Q_D(t) H_D(0,t)&= Q_A(t) H_A(1,t) + Q_C(t) H_C(1,t). \end{aligned}$$
(53)

Here, too, we must account for the arrival of perturbations originating upstream, namely in vessels A and C.

$$\begin{aligned} Q_D^* h_D&= -e^{-\lambda \tau _A^*} Q_A^* h_A - e^{-\lambda \tau _C^*} Q_C^* h_C + (H_D^* - H_A^*) q_A + (H_D^* - H_C^*) q_C \end{aligned}$$
(54)

The final two constraints on the perturbations are obtained by linearizing Eqs.  26, 27. In the case of Eq. 26, we have

$$\begin{aligned} Q_A(t) R_A(t)&= Q_B(t) R_B(t) \!+\! Q_C(t) R_C(t) \end{aligned}$$
(55)
$$\begin{aligned} \Rightarrow \tilde{Q}_A(t) R^*_A \!+\! Q^*_A \tilde{R}_A(t)&= \tilde{Q}_B(t) R^*_B + Q^*_B \tilde{R}_B(t) \!+\! \tilde{Q}_C(t) R^*_C \!+\! Q^*_C \tilde{R}_C(t). \quad \end{aligned}$$
(56)

The perturbation on the resistance in branch j can be derived by first expanding the resistance function in Eq. 28:

$$\begin{aligned} R_j(t)&= {128 {L}_j \over \pi d_j^4} \int _0^1 \mu _j(H_j^* + \tilde{H}(x_j,t)) {\text {d}}x_j \end{aligned}$$
(57)
$$\begin{aligned}&= {128 {L}_j \over \pi d_j^4} \int _0^1 \mu _j (H_j^*) + \left. {\partial \mu _j \over \partial H_j}\right| _* \tilde{H}(x_j,t) {\text {d}}x_j. \end{aligned}$$
(58)

Substituting the perturbation on \(H_j\) given in Eq. 33 and discarding the constant terms give

$$\begin{aligned} \tilde{R}_j(t)&= h_j {128 {L}_j \over \pi d_j^4} \left. {\partial \mu _j \over \partial H_j}\right| _* \int _0^1 e^{\lambda (t - \tau _j x_j)}{\text {d}}x_j \end{aligned}$$
(59)
$$\begin{aligned}&= h_j u_j e^{\lambda t}, \end{aligned}$$
(60)

where

$$\begin{aligned} u_j = R_j^* \left. {\partial \ln (\mu _j) \over \partial H_j}\right| _* {1 - e^{-\lambda \tau _j^*} \over \lambda \tau _j^*}. \end{aligned}$$
(61)

Applying this result to Eq. 56 gives

$$\begin{aligned} Q^*_A u_A h_A - Q^*_B u_B h_B - Q^*_C u_Ch_C + q_A (R^*_A + R_B^*) - R^*_C q_C = 0. \end{aligned}$$
(62)

A similar procedure applied to the linearization of Eq. 27 would give

$$\begin{aligned}&Q_A^* u_A h_A - Q_B^* u_B h_B + Q_D^* u_D h_D - Q_E^* u_E h_E\nonumber \\&\quad +\,\left( R_A^* + R_B^* + R_D^* + R_E^*\right) q_A + (R_D^* + R_E^*) q_C = 0. \end{aligned}$$
(63)

Combining Eqs. 46, 47, 51, 52, 54, and 5662 gives a homogeneous system of seven equations which are linear in the seven perturbation amplitudes \(h_A, \ldots , h_E, q_A, q_C\) as claimed in Equation 38. A similar system can be derived if we assume \(Q_C(t) < 0\). In this case, Eqs. 46, 47, and 5663 would remain the same, while Eqs. 51, 52, and 54 would now reflect the fact that node 3 is now converging and node 2 diverging.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Karst, N.J., Storey, B.D. & Geddes, J.B. Oscillations and Multiple Equilibria in Microvascular Blood Flow. Bull Math Biol 77, 1377–1400 (2015). https://doi.org/10.1007/s11538-015-0089-1

Download citation

Keywords

  • Microvascular blood flow
  • Fluid dynamics
  • Nonlinear dynamics
  • Discontinuity-induced bifurcations