Introduction to dynamical systems analysis in quantitative systems pharmacology: basic concepts and applications

Abstract

Quantitative systems pharmacology (QSP) can be regarded as a hybrid of pharmacometrics and systems biology. Here, we introduce the basic concepts related to dynamical systems theory that are fundamental to the analysis of systems biology models. Determination of the fixed points and their local stabilities constitute the most important step. Illustration of a phase portrait further helps investigate multistability and bifurcation behavior. As a motivating example, we examine a cell circuit model that deals with tissue inflammation and fibrosis. We show how increasing the severity and duration of inflammatory stimuli divert the system trajectories towards pathological fibrosis. Simulations that involve different parameter values offer important insights into the potential bifurcations and the development of efficient therapeutic strategies. We expect that this tutorial serves as a good starting point for pharmacometricians striving to widen their scope to QSP and physiologically-oriented modeling.

Keywords: Bifurcation; Dynamical Systems Theory; Multistability; Quantitative Systems Pharmacology; Systems Biology.

Figure 1. Phase portrait of the tumor-normal cell competition model. The fixed points are (X, Y) = (0, 0), (0, 1), and (1, 0). The fixed point at (0, 0) represents the state of no cells, which is unstable. So long as there are no tumor cells, the trajectory starting from (0, 0) converges to (0, 1), consisting of only normal cells. With the emergence of a single tumor cell, however, the state shifts towards (1, 0), which is the only stable fixed point characterized by 100% tumor cells.

Figure 2. Phase portrait of the predator-prey equation. The 2 nullclines meet at (0, 0) and (1, 1), respectively. The direction field suggests that the trajectories do not converge to (1, 1) but rather circle around it with a constant radius.

Figure 3. Phase portraits illustrating direction fields, nullclines, and three different trajectories under four different scenarios of (A) no inflammation, (B) I = 0.1, Dur = 1, (C) I = 0.1, Dur = 5, and (D) I = 0.5, Dur = 1.

Figure 4. The effect of reducing the autocrine secretion rate of platelet derived growth factor by mF (β₃). The curvature of mF-nullcline decreases, expanding the basin of attraction towards the healing state.

Figure 5. The effect of increasing platelet derived growth factor's rate of endocytosis by mF (α₂), which is similar to that of decreasing β₃.

Figure 6. Decreasing the growth rate (λ₁) or increasing the death rate of mF (μ₁) lead to similar effects on the phase portrait.