How to implement Reference tracking and control for level of hypnosis in Patients through a pharmacokinetic and pharmacodynamic model.

The goal is to design the control U as an anesthetic drug infusion rate such that the level of hypnosis (in our case Bis index) converges to a desired value. After researching I saw a pharmackinetic plus a pharmacodynamic model can be used to model the dynamics.

But I have some problems trying to derive the closed loop transfer function.

So I am making use of a pharmacokinetic model to describe the distribution of the anesthetic drug in the body. I use a three compartmental model illustrated as follows, where the constants represent the frequency at which the drug is redistributed from compartment i to compartment j.

The input of this model is U which is the drug's infusion rate while its output is the drug concetration of the primary compartment C1.

I then use a pharmacodynamic model to model the drugs effect, mapping the primary compartment concentration C1 to a level of hypnosis through 2 steps.

Step 1 : Map to

represents the drug concentration of the site where the drug actually causes its effect on the body.

Step 2 : Map to Bis index

The last step is to then map to an actual level of hypnosis to measure the patients consciousness. I used a hill sigmoid function for this.

where a andγrepresents given parameters of the patient.

This last step introduces non linearity to the model which I attempted solving with feedback linarization in the following way.

In the laplace domain we have the following , rewriting Bis as

hence the feedback linearization can be applied as

The problem is simulating this model and trying to make the Bis(t) converge to a certain value setting v as a pid control.

The goal isto find the control such that the bis index from an initial value of 1 decreases to a desired bis index in the range of 0.5.

My questions are :

1) Given the feedback linearization I proposed, How do I calculate the closed loop transfer function of this system (the problem is the non linear hill block), I tried feedback(bis*u,1) in matlab but I am not sure.

2) given the error as e(t)= (Bisdesired - Bis), which functions or commands in matlab can I apply to I make e(t) converge to zero with a pid controller through feedback linearization.

Answer 1

[Star Strider]

Unless I missed something in the problem statement, the initial goal is to estimate the parameters. A linear model with identified parameters can then be used with Control System Toolbox functions to model it as a control system. (I do not have the Model Predictive Control Toolbox, although that would appear to be applicable for this research.)

I posted a link to relevant Answers posts in the related Question How of get Initial values of a pharmacokinetic and pharmacodynamic model..

[Mohammed Azeez]

Hi, Thank you very much but the goal isnt to estimate the parameters of the model. The goal is deriving the closed loop transfer function of the model and controlling it. The difficulty lies in the presence of the non linearity of the hill function for which I adopted feedback linearization. But still I am uncertain of the closed loop transfer function.

For example if the model was completely linear,

T the closed loop transfer function of the following system would be = feedback(G*C,1)

Now immagine block G was non linear as in my case, What would be closed loop transfer function ?

[Star Strider]

My pleasure!

This appears to me at the outset to be a model predictive control problem (although I’ve not yet finished reading the papers). I did something similar to see if I could devise an ‘artificial beta cell’ (I’m a fellowship-trained endocrinologist) a couple decades ago, however the big problem with that is that the system (Type 1 diabetic human) is both unobservable and uncontrollable (too many hidden states), so I gave up on it. This is a more tractable problem although definitely not trivial, and since I couldn’t find my MPC textbooks (from a rigorous course a quarter century ago), I found and ordered a couple recent replacements. This is best done in state space, since that is simply easier, although the mathematics are a bit more complicated.