Compartmental OED

Compartmental OED
State dimension:	1
Differential states:	12
Discrete control functions:	1

The Compartmental OED problem looks for an optimal measurement strategy to determine a single parameter in a one-dimensional ODE model, where we can directly measure the single state. The following description is taken from [1].

Here we consider the open one-compartment model with first-order absorption input fitted by Button (unpublished Ph.D. thesis, Texas A&M University, 1979) to the results of an experiment in which six horses each received 15 mg/kg of theophylline as aminophylline by intragastric administration.

The optimal integer control functions shows bang bang behavior.

Mathematical formulation

For the three-dimensional parameter $p=({\theta }_1,{\theta }_2,{\theta }_3)$ the original initial value problem is given by $\dot{x}(t)=:f(t,p)={\theta }_3\cdot (-{\theta }_1\cdot \exp (-{\theta }_1\cdot t)+{\theta }_2\cdot \exp (-{\theta }_2\cdot t)),x(0)=0.$

We assume both $x_0$ and $t_f$ to be fixed and are only interested in when to measure, with an upper bound $M$ on the measuring time. We can measure the state directly, i.e. $h(x(t))=x(t)$.

Now we formulate the OED problem:

$\begin{array}{lll}\underset{y,G,F,z,w}{\min }& & \text{trace}(F^{-1}(t_f))\\ \text{subject to}\\ \dot{y}(t)& =& f(t,p)\\ \dot{G}(t)& =& f_p(y(t),p)\\ \dot{F}(t)& =& w(t)(h_y(y(t))G(t){)}^T(h_y(y(t))G(t))\\ \dot{z}(t)& =& w(t),\\ y(0)& =& y_0\\ G(0)& =& 0\\ F(0)& =& 0,\\ z(0)& =& 0\\ w(t)& \in & {𝒲}\\ z(t_f)& \le & M\end{array}$

Parameters

These fixed values are used within the model:

$t_f=40;{𝒲}=[0,1];M=5;p=(0.05884,4.298,21.80)$

Reference Solutions

Here is one local solution to the above control problem.

• Reference solution plots
• 
  Sensitivities and measurement control for $\theta =(0.05884,4.298,21.80)$.

References

[1] Optimum Experimental Designs for Properties of a Compartmental Model, A. Atkinson, K. Chaloner, A. Herzbert, https://www.jstor.org/stable/pdf/2532547.pdf