16.6 Quantitative Analysis The equilibrium value of pressure is given by Eq. (16.6), when _u is equal to the zero vector. When considering the parameter values of Table 16.1, we have Peq ¼ PA,eq PB,eq " #¼ 7:041kPa 2:255kPa " # ð 16:13Þ The phase portraits presented in this section are ones that are created with the model parameter values set to those presented in Table 16.1. The behavior we observe in the phase space is indicative of two different time scales acting simultaneously. Figures 16.5 and 16.4 illustrate this behavior. In response to an initial condition with nonzero position yet zero velocity, the pressure PB exponentially increases to a value on the order of microseconds (Fig. 16.4). Figure 16.5 is a zoomed-out version of Fig. 16.4. Here we see that PB is observed to start returning to its equilibrium value on the order of seconds. Figures 16.6 and 16.7 share initial conditions where at t ¼0, uA ¼uB ¼0 andwhere vA and vB are nonzero. The initial values of vA and vB for each solution path are shown on the border of Fig. 16.7. It is apparent that, at a very fast time scale, the dynamics quickly converge onto a manifold of lower dimension. Fig. 16.2 Solution trajectories projected onto the uA uB plane. All velocity initial conditions are set to zero Fig. 16.3 Family of intracranial pressure solutions with respect to time 16 A Mechano-Hydraulic Model of Intracranial Pressure Dynamics 115
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