Fig 1.
Physiological systems used as case studies.
The four circuits shown here represent deterministic dynamic models described by ordinary differential equations (ODEs) [6]. (A) Linear circuit, integral feedback. (B) Linear circuit, proportional-integral feedback. (C) Nonlinear circuit of hormonal reactions. (D) Glucose homeostasis circuit, also known as βIG model. Parameters p and s (or si) represent gain constants of the feedback loops present in the circuits.
Fig 2.
Illustration of the phenomenon of dynamical compensation in a physiological circuit.
The upper row shows the normal behaviour of the βIG model for a given value of the si parameter. The second row shows the evolution of the steady state after a change in the value of si. After a long adaptation period, which can take months, a new steady state is reached (note that in the plots in this row no external inputs are applied, for clarity of visualization; if they were, periodic peaks similar to the ones in the first and third rows would appear superimposed on the plotted curves). Then, as shown in the third row, the response of the glucose concentration for the new parameter value is the same as the initial one (this does not happen for insulin and β-cell mass).
Fig 3.
Dynamical compensation and structural unidentifiability in the βIG model.
The first row reproduces the last row of Fig 2 and illustrates the phenomenon of dynamical compensation: after the system has adapted to the new value of si, the time-evolution of the glucose concentration (G) for the new value of (si/2) is the same as it was with the old value before adaptation (si). The second row illustrates the phenomenon of structural unidentifiability: without the need for any adaptation, the time-evolution of the glucose concentration (G) is the same for any value of the parameter si, as long as any deviations from the original value are compensated by changes in the parameter p. Note that, since the upper and lower plots of G are identical, if glucose is the only measured quantity both phenomena cannot be distinguished. However, the behaviour of the other state variables (I, β) can be very different, as can be noticed from the third and fourth columns.
Table 1.
Structurally unidentifiable parameters for different configurations of the βIG model.
Each table entry in the four rightmost columns shows the structurally unidentifiable parameters for a given choice of measured outputs (different rows) and parameters considered unknown (different columns). Four representative choices of parameters are studied: (i) with all the model parameters {α, γ, c, p, si} considered unknown, (ii) with the two parameters {p, si} that may exhibit dynamical compensation considered unknown, (iii) with all but p unknown, and (iv) with only one parameter, si, considered unknown.
Fig 4.
Unidentifiability of si from measurements of glucose off steady state.
The upper plot shows the time-course of plasma glucose concentration in the βIG model in a regime of periodic inputs (pulses) of external glucose from meals, similarly as in Figs 2 and 3. At time 100000 there is a sudden change in the insulin sensitivity parameter, si, which is halved as a result of external perturbations. This produces an abrupt change in the glucose level, which then undergoes a long period of adaptation until its baseline returns to the original level. The lower plots show in more detail the behaviour of glucose and the other two state variables, insulin and β-cell mass, in the hours immediately before and after the change in si. These plots illustrate that it is not possible to infer the value of the si parameter in the βIG model by measuring only glucose, even if measurements off steady state are available and p is known. This is indicated by the fact that the dynamic time-course of glucose concentration (G) is identical for and
(left plot in lower row), as long as the initial conditions of the two unmeasured states, insulin (middle plot) and β-cell mass (right plot), are multiplied by the same factor k. The figure shows results for k = 3 and
.