Pharmacokinetic/Pharmacodynamic Modeling,Evolutionary Algorithms, and Optimal Control Theory: a numerical-based approach

Pharmacokinetic/Pharmacodynamic Modeling,Evolutionary Algorithms, and Optimal Control

Theory: anumerical-based approach

UnpublisheUnpublished Workd Work

https://www.youtube.com/watch?v=s6gCvhSh-qc






Pharmacokinetic/Pharmacodynamic Modeling,Evolutionary Algorithms, and Optimal Control Theory: a

numerical-based approach

J G Pires, August, L’Aquila, 2015.

Abstract.




Abstract.

This paper is the formal mixture of three different areas of knowledge, two from applied math and engineering, and one from applied medicine and pharmacology.

The aforementioned areas are: optimal control theory, evolutionary algorithms, and finally pharmacokinetic/pharmacodynamic modeling, PK/PD for short.

We design and test a computer program for solving problems in optimal control theory based on general assumption and apply to a problem in PD/PK modeling.

Although the code is not as fast as the traditional counterpart, which cannot be used to solve the problem presented, the code shows interesting results. We make an analysis of variance of the code.

Keywords. Pharmacokinetic/pharmacodynamic modeling. optimal control theory. Forward-Backward Sweep Method. Pontryagin Maximum Principle. Evolutionary Algorithms.




Abstract.









Abstract.









Abstract.









Abstract.









Introduction.




IntroductionThis paper is the formal mixture of three different areas of knowledge, two from

applied math and engineering, and one from applied medicine and pharmacology.

The aforementioned areas are: optimal control theory, evolutionary algorithms, and finally pharmacokinetic/pharmacodynamic modeling, PK/PD for short. Three books were used mainly to back up the paper.

The first reference is from applied optimal control, that is to say [1]; other references that had influenced the authors from previous endeavors are [2-3].

The second reference is from pharmacokinetic/pharmacodynamic modeling, which is [4]; a second reference that influenced the author from past researches is [5]. The last but not least is [6], which is a book about evolutionary computing.




























Short literature review





Evolutionary Algorithms for Optimal Control in Fed-Batch Fermentation Processes [13].

In this work, evolutionary algorithms are used to achieve optimal feedforward control in a recombinant bacterial fed-batch fermentation process, that aims at producing a bio-pharmaceutical product.

What is interesting mentioning about [13] is that a smooth solution is achieved. It was a surprise for the author, given the output of the reported algorithm. One explanation for the difference can be done metaphorically.

Since [13] just optimize the differences between the final and initial states, a linear relation, it is like pulling a string in the extremes, it is probably to find a nice curve.





On our case, it is similar to pulling the same string, but with several points within, that can move in any direction, certainly we cannot obtain a nice curve.

Another point is that the authors from [13] had optimize with both ends free, final and initials, it is possible to be done also with Hamiltonian, but the challenge could be how to handle the fact that all conditions on the adjoint equations disappears, no initial or final conditions, but we need them for integrating.

[1] gives a potential way to solve the problem that works for "simple" problems, the issue now is how to extend to our case; we believe that it is possible.




Literature Deficiency.

Systems that can benefit from optimal control theory includes: ordinary differential equations, partial differential equations, discrete equations, stochastic differential equations, integro-difference equations, and combinations [1].

From this list, two are quite important ones, but from the short review done, which includes papers not reported herein, shows a deficiency: stochastic differential equations and partial differential equations.

The former is interesting for the current project of the authors, namely, stochastic models in life sciences and medicines, but the latter is a general interest. The two areas changed a lot in the last decades, it is even possible that most of the techniques widely used at the present time were not developed yet when the cornerstones of optimal control theory were laid down.





The mathematical model




The mathematical model The model

Fig. 1 presents the mathematical model under investigation, this formulation of optimal control problem is known as the Bolza formulation, an alternative formulation is known as Mayer formulation.

This formulation was preferred because: it does not require to solve the integral, the Mayer formulation does not require as well, but it requires the introduction of an extra state variable, then it solves the integral indirectly; it leaves clearly what we are doing, the cost function is showed separated from the dynamical system; it takes advantage of the Hamiltonian formulation, escape from solving the integral.

See [14], [7] for some discussions. Unfortunately we must skip the details about the model. Besides both the model and the method to solve the model are novel, herein we just look into the method to solve the model, the model parameterization is kept the same all over the simulations, future works is to play with the model as well.




The mathematical model




The Evolutionary Algorithm




1.Individual representation

The Evolutionary Algorithm




Results




Results

0.000.010.020.030.040.050.060.070.080.090.10

12 20 30 40 50 60 70 80 90 100 110 200 210 220 250

Number of individuals

rela

tive

impr

ovem

ent




Results




Results




Results




Conclusions, future works, and final remarksOn this paper we have reported an algorithm designed to solve the problem of

optimum drug regimen in cancer therapy; the case studied is not the state of the art, but can be the starting point for something more well-elaborated.

The advantage of the algorithm is that once a more elaborated method is chosen, the limitations of the classical approaches is expected not to be a problem, once we have applied evolutionary computing. The code suffers from speed, but the results sound plausible, comparing with its linear counterpart.

The algorithm can be called the Evolutionary Forward-Backward Sweep Method, in contrast to its deterministic counterpart. Further, the code can been as a prototype for a more advanced programmed, initially called Evolutionary Program for Optimal Control Nonlinear Simulator (EPOP-NonLin Sim).




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Pharmacokinetic/Pharmacodynamic Modeling,Evolutionary Algorithms, and Optimal Control Theory: a numerical-based approach