Outflow Boundary Conditions for Arterial Networks with Multiple Outlets

Abstract

Simulation of blood flow in three-dimensional geometrically complex arterial networks involves many inlets and outlets and requires large-scale parallel computing. It should be based on physiologically correct boundary conditions, which are accurate, robust, and simple to implement in the parallel framework. While a secondary closure problem can be solved to provide approximate outflow conditions, it is preferable, when possible, to impose the clinically measured flow rates. We have developed a new method to incorporate such measurements at multiple outlets, based on a time-dependent resistance boundary condition for the pressure in conjunction with a Neumann boundary condition for the velocity. Convergence of the numerical solution for the specified outlet flow rates is achieved very fast at a computational complexity comparable to the widely used Resistance or Windkessel boundary conditions. The method is verified using a patient-specific cranial vascular network involving 20 arteries and 10 outlets.

Appendix A

In the following we derive the resistance–flow rate relationship in a network of five vessels, shown in Fig. 23.

Figure 23

Sketch of 1D model of five arteries: one inlet and five outlets

The Q _1,2/Q _1,3 ratio is given by

$$ \frac{Q_{1,2}}{Q_{1,3}} = \frac{R_3-L_{1,3}K_{1,3}}{R_2-L_{1,2}K_{1,2}}. $$

(12)

Thus,

$$ R_3 = L_{1,3}K_{1,3}+\frac{Q_{1,2}}{Q_{1,3}} (R_2-L_{1,2}K_{1,2}). $$

(13)

Now we apply (6) to segments 3,5 to obtain

$$ P_5-P_3 = L_{3,5}K_{3,5}Q_{3,5} $$

(14)

and using the resistance boundary condition P _j  = R _j Q _j , we get

$$ R_5Q_{3,5}-R_3Q_{1,3} = L_{3,5}K_{3,5}Q_{3,5}. $$

(15)

Next, we substitute R ₃ in (15) by formula (13)

$$ R_5Q_{3,5}-Q_{1,3} \left(L_{1,3}K_{1,3}+\frac{Q_{1,2}}{Q_{1,3}} (R_2-L_{1,2}K_{1,2})\right) = L_{3,5}K_{3,5}Q_{3,5} $$

(16)

and, assuming that $|L_jK_j| \ll 1$ and $|L_jK_j| \ll R_j$, we obtain

$$ R_5Q_{3,5} - R_2 Q_{1,2} = L_{3,5}K_{3,5}Q_{3,5} \approx 0. $$

(17)

From Eq. (17) we obtain

$$ \frac{Q_{3,5}}{Q_{1,2}} \approx \frac{R_2}{R_5}. $$

Similarly, we can obtain

$$ \frac{Q_{3,4}}{Q_{1,2}} \approx \frac{R_2}{R_4}. $$

So far we derived the Q _i /Q _j relations for a steady flow but similarly to the previous case, we can derive the $\hat{Q}_{i,k}/\hat{Q}_{j,k}$ relations for unsteady flow and show that

$$ \frac{\hat{Q}_{i,k}}{\hat{Q}_{j,k}} \approx \frac{R_j}{R_i} $$

and therefore

$$ \frac{Q_{i}(t)}{Q_{j}(t)} \approx \frac{R_j}{R_i}. $$

By neglecting friction we can extend the R–Q relation to the network with an arbitrary number of segments. For example, using the model provided in Fig. 23 we obtain:

$$ R_1Q_{\rm in} \approx R_2 Q_{1,2} \approx R_3 Q_{1,3} \approx R_5 Q_{3,5} \approx R_4 Q_{3,4} \ldots $$