Short-Term Regulatory System

The arterial baroreflex arc is implemented according to the feedback system illustrated in Figure 4. This system is aimed at tracking a setpoint ($sp$) pressure through the following sequence of events. The baroreceptors sense $P_a(t)$ and relay this pressure to the autonomic nervous system (ANS). The ANS compares the deviation between the sensed pressure and $P_a^{sp}$ with zero and then responds by adjusting four parameters of the pulsatile heart and circulation in order to keep the ensuing $P_a(t)$ near $P_a^{sp}$. The four adjustable parameters are $F(t)$, $C_{l,r}(t)$ at end-systole ( $C_{l,r}^{es}(t)$), $Q_v^0(t)$, and $R_a(t)$. The ANS controls these parameters based on the history of $P_a(t)-P_a^{sp}$ specifically according to the following nonlinear, dynamical mapping:

 

$$ap(t)=\int (G_pp(\tau)+G_ss(\tau))S\,{\rm arctan} \left(\frac{P_a(t-\tau)-P_a^{sp}}{S}\right)d\tau+ap^{sp},$$ (1)

where $ap$ may represent any of the four adjustable parameters, the arctan function (which is parametrized by the constant $S$) imposes arterial baroreflex saturation, $p(t)$ and $s(t)$ are unit-area effector mechanisms which respectively represent the fast, parasympathetic limb of the ANS and the slower, sympathetic limb (both $\alpha$- and $\beta$-sympathetic sublimbs; see Figure 5), and $G_p$ and $G_s$ reflect the respective static gain values of the effector mechanisms. Note that in order to map $F(t)$ to the times of onset of ventricular contraction (which amounts to re-initiating the variable, ventricular compliance time evolution), an ``integrate and fire'' model of the sinoatrial node is incorporated in the model.

  

Figure 4: Block diagram of the feedback system depicting the arterial baroreflex arc.

The cardiopulmonary baroreflex arc is also implemented according to a feedback diagram analogous to Figure 4. However, the sensed pressure here is defined to be the effective right atrial transmural pressure ( $P_{\text{\lq\lq $ra$''}}^{tr}(t)=P_{\text{\lq\lq $ra$''}}(t)-P_{th}(t)$) of the pulsatile heart and circulation model.

  

Figure 5: Unit-area effector mechanisms representing (a) the fast, parasympathetic limb $p(t)$ and (b) the slower, sympathetic limb $s(t)$. These effector mechanisms characterize the dynamical properties of the block labelled ANS in Figure 4.

The direct neural coupling mechanism between respiration and heart rate is characterized by a linear, time-invariant impulse response which maps fluctuations in instantaneous lung volume ($Q_{lu}(t)$; see Section 2.3) to fluctuations in $F(t)$. The impulse response is defined here by a linear combination of $s(t)$ and $p(t)$, each of which are advanced in time by 1.5 s in order to account for the noncausality of this mechanism [6,9].