Oscillations and Multiple Equilibria in Microvascular Blood Flow


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.

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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.

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Correspondence to Nathaniel J. Karst.



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:

$$\begin{aligned} \tilde{Q}_B&= -\tilde{Q}_A \end{aligned}$$
$$\begin{aligned} \tilde{Q}_D&= \tilde{Q}_A + \tilde{Q}_C \end{aligned}$$
$$\begin{aligned} \tilde{Q}_E&= -\tilde{Q}_A - \tilde{Q}_C. \end{aligned}$$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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


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

Applying this result to Eq. 56 gives

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

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

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

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.

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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

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