Solitonic Windkessel Model for Intracranial Aneurysm

Figures

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  Figure 1. (A) Doubly composed intracranial aneurysm consisting of mother and daughter aneurysms (also called dumbbell-shaped aneurysm in the preceding works). The blood flow F is the amount of blood flow pumping from the heart, F1 is the blood flow in the mother aneurysm and F2 is the blood flow in the daughter aneurysm, where the circulative flows with the opposite direction are assumed in the aneurysms. (B) Solitonic Windkessel model for the doubly composed intracranial aneurysm. The corresponding electric circuit is shown, where C1 and C2 corresponds to the size/dimension of mother and daughter aneurysms, respectively. R determines the branching ratio F1 = F of the blood flow. The constants L1 and L2 are related to the elasticity of blood vessel wall. P0, P1 and P2 denote the local pressures, and F, F1, and F2 are blood flows.

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  Figure 2. Blood flow F(t) and blood pressure P(t) in the main vessel are given in the left panel (A) and the right panel (B), respectively. Environmental variables are determined by the heartbeat: blood pressure P(t) and blood flow F(t) are plotted for 0–20 s. These are regularly provided by the heart. The time periodic oscillation is fixed by the typical beat 70 bpm (Table 1), the blood pressure is assumed to range from 80 to 120 mmHg and the blood flow to be 40 to 60 mL/min.

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  Figure 3. (A) The branching of blood flow from F(t) to F1(t) and F2(t) at a given setting (R = 2.2, C1 = 0.100, C2 = 0.005 and L1 = 10, L2 = 20). Here, we take R = 2.2 for the local blood flow in the main vessel F–F1 not to change too much (less than 10% in all the calculated cases) compared to F. It simulates the typical case of doubly composed aneurysm. Calculation shows that the blood flow in the artery F is almost equal to the blood flow in the parent artery F–F1, and therefore the blood flows in the aneurysms are kept to be small. Depending on the choice of four parameters (C1, C2, L1, and L2), flows F1 and F2 can be either positive, zero or negative. (B) In the same parameter setting, the local pressures P1(t) and P2(t) are calculated, which note that the blood pressure at the parent artery is equal to P(t) regardless of the choice of parameter. In this parameter setting, the amplitude of local pressure P1(t) is almost equal to P(t), while another local pressure P2(t) oscillates with irregular beat and small amplitude.

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  Figure 4. Maximum value of local blood pressure P2 depending on the size of daughter aneurysms. By fixing the size parameter C1 of mother aneurysm, the size ratio between mother and daughter aneurysms is changed as C2/C1 = 0.20, 0.30, … 1.00 for a fixed C1 = 0.10. C2/C1 = 1.00 corresponds to the ideal dumbbell shape. Note that the blood flow and blood pressure in the artery are almost unchanged (yellow and red ones in Figure 3), while local flows F1 and F2 and local pressures P1 and P2 easily change their maximum amplitude depending on the parameter. For the growth of daughter aneurysm, an ordinary scenario with L1 = 2L2 is demonstrated in the left panel (A), and a solitonic scenario with L1 = L2 is demonstrated in the right panel (B) (cf. Section 2.2). In all the presented cases, P1 is no more than 170 mmHg.

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  Figure 5. Maximum value of the local blood pressure P1 depending on the aneurysmal compliance L1. The daughter aneurysms are not assumed to be formed well (C2 = 0.1, C2 = 0.001, L2 = 0). By changing the compliance parameter L1 of mother aneurysm, the local pressure P1 of mother aneurysm increases linearly. Note that the blood flow F–F1 and blood pressure P in the parent artery is nothing different from those given by Figure 3.

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  Figure 6. Blood flow (A) and blood pressure (B) at a given setting (R = 2.2, C1 = 0.100, C2 = 0.100 and L1 = 30, L2 = 60). Although the amplitude of resonating pressure |P2| becomes large, the resonating flow F2 is kept to be small enough to be less than 10 mL/min.