Feedback Linearization Using Neural Networks Applied to ...

Feedback Linearization Using Neural Networks Applied to Advanced Pharmacodynamic and Pharmacogenomic Systems Alexandru Floares Oncological Institute Cluj-Napoca Str. Republicii, Nr. 34-36, Cluj-Napoca, 400015, Romania Email: alexandru [email protected]

Abstract— At present pharmacological modeling is developing from an empirical, descriptive discipline into a mechanistic science. Also, new and important fields like pharmacogenomics appeared. As a consequence pharmacology is dealing with large nonlinear control systems. The intent of this paper is to show that all this systems being based on a limited array of mechanisms and having some structural peculiarities are good candidate for the application of feedback linearization techniques. With our approach, feedback linearization using neural networks (multilayer perceptrons) nonlinear pharmacodynamic and pharmacogenomic systems can be linearized. The generality of this approach and the implications, due to the fact that unlike Jacobian linearization feedback linearization is not only locally valid, are also shown.

I. I NTRODUCTION In the last years the reductionist paradigm which dominated the last century life sciences becomes gradually balanced with the systemic view of living organisms. In clinical pharmacology, detailed data about the complex molecular mechanisms of the interactions between drug(s) and organism become available. Most notably, the target genes of many drugs are being discovered and the differential genes expression induced by drugs can be investigated by microarray techniques [1]. This data allow conceptual and mathematical models building. Unfortunately, this is a very difficult task because of the inherent complexity of the nonlinear physiological control mechanisms interacting with an external control, which is the drug(s) dosage regimen. Also, the dimensions of this nonlinear control system is very high; drugs usually alter the expression of thousands genes. Building mathematical models is important for understanding this pharmacological systems. At least as important is to be able to adequately control them even with a limited understanding - a black box approach. Both approaches have the same very important goal: optimizing and individualizing medical therapy in the presence of variate degrees of knowledge and uncertainty. Building mathematical models requires detailed knowledge of the mechanism involved and the estimation of numerous parameters; it is a tedious and time consuming process. This probably explains the law impact of this approach in the clinical practice even for simple models. Neural networks (NN), despite their recent rapid growth in the implementation in various fields of applications, have their potential in clinical pharmacology largely unexplored. The essential features of NN, nonlinearity, adaptivity, independence of statistical and other modeling assumptions, fault tolerance, universality, and real time operation, make them suitable for clinical

pharmacology applications. Feedback linearization (FBL) is one of the most important nonlinear control design strategy developed during the last few decades [2]. This approach may results in linearization which are valid for larger practical operating points of the system, as opposed to a local Jacobian linearization about an operating point. Neural adaptive control of feedback linearizable nonlinear systems was first proposed in [3] and extensively analyzed in [4]. They greatly simplify the modeling task and have the potential of a greater clinical impact. In a previous work we obtained the best published result in a cancer chemotherapy problem using neural network feedback linearization (NN FBL) [5]. This motivates this study and the use of multilayer perceptrons, instead of other possible NN like radial basis functions or dynamical neural networks [6]. II. P HARMACOLOGICAL S YSTEMS A. Pharmacogenomics Data For illustrating the proposed methods, the synthetic corticoid (CS) methylprednisolone (MPL) pharmacogenomic (PG) data, studied using gene microarray in rat liver (see [7]), were investigated. The dataset is available online at http://microarray.cnmcresearch.org/ (link Programs in Genomic Applications. Methylprednisolone (MPL), with different dosage regimens, is indicated in various conditions: endocrine disorders, rheumatic disorders, collagen diseases, dermatologic diseases, allergic states and inflammatory processes, hematologic disorders, neoplastic diseases, edematous states, and nervous system diseases. Therefore, optimization and individualization of clinical therapy with corticoids is very important. In the pharmacological experiment, forty-three male rats weighing 225 to 250 g received a single intravenous bolus dose of 50 mg/kg MPL. Rats were sacrificed and liver, an important action site for corticoids, excised at 17 time points over 72 hours. Four untreated rats were sacrificed at 0 hours as controls. RNAs from individual livers were used to investigate 8000 genes with Affymetrix GeneChips. Cluster analysis revealed six temporal pattern consisting of 197 CSresponsive probes representing 143 genes. B. Pharmacological Mathematical Models Pharmacokinetics (PK), the relationship between time and plasma concentration, can be simply described as ”what the body does to the drug”. The clinical interpretation of pharmacokinetic results requires another set of information,

the relationship between plasma concentrations (or dose) and effect, or pharmacodynamics (PD). This can be described as ”what the drug does to the body” 1) Preliminary Remarks and Definitions: A multivariate nonlinear dynamic system with m inputs {u1 , ..., um } and p outputs {y1 , ..., yp } is described in a state space form by the following equations: x˙ = f (x) +

m X

gj (x)uj

(1)

j=1

yi = hi (x), i = 1, ..., p where x = [x1 , ..., xn ]T ∈