# (254d) Parameter Set Selection for Signal Transduction Models With Significant Uncertainties

- Conference: AIChE Annual Meeting
- Year: 2013
- Proceeding: 2013 AIChE Annual Meeting
- Group: Food, Pharmaceutical & Bioengineering Division
- Session:
- Time:
Tuesday, November 5, 2013 - 9:24am-9:42am

Mathematical models composed of ODEs are widely used to describe the dynamic behavior of biological/biomedical systems. The accuracy of these models not only depends on the structure of the model which is determined by the physics, chemistry, and biology of the system, but also relies on adjustable parameters in the model, many of which are either taken from the literature or estimated using experimental data. A number of studies have investigated various aspects of parameter estimation of dynamic system [1-3]. However, before any parameter estimation is performed, it is important to determine if all parameters are numerically identifiable and, if not, then what subset of parameters can be accurately estimated [4]. Examples of particular interest for this work are models of signal transduction pathways as these models tend to contain a large number of parameters, many of which contain significant uncertainties and which cannot be estimated from available measurements.

Sensitivity analysis is a powerful tool to study how model parameter variations can qualitatively or quantitatively influence model behavior. As such, a variety of methods for parameter set selection based on sensitivity analysis have been proposed in the literatures. These methods include, but are not limited to, genetic algorithms [5], collinearity index methods [6], column pivoting methods [7], Gram-Schmidt orthogonalization methods [8], and clustering methods [9]. A systematic scheme for parameter set selection is based on optimality criteria computed from the Fisher information matrix which is closely related to the parameter-output sensitivity matrix [10].

It is important to note that all of the above mentioned methods for parameter set selection utilize local sensitivity analysis. The main drawback of local sensitivity analysis is that the sensitivity vectors are dependent on the parameter values which are not precisely known prior to parameter estimation. This may result in identification of parameter subsets that are suboptimal which can have significant impact on the model’s prediction accuracy if the parameter uncertainty is large [11]. One alternative to local sensitivity analysis is global sensitivity analysis which simultaneously varies multiple parameters, often over a large range, so that the uncertainty description of parameters is incorporated into the sensitivity analysis procedure. Unfortunately, it is difficult to interpret the result from global sensitivity analysis, e.g. the Morris method [12], sampling-based method [13], and a variance-based method [14], for experiment design or parameter set selection [11].

This work addresses the challenges mentioned above for parameter set selection for dynamic systems under uncertainty. This is achieved by combing a hierarchical clustering method [9] and dynamic optimization techniques [15] allowing simultaneous variation of all parameters over a large range to quantify the effect of the uncertainty in the parameter space on the sensitivity vectors. Unlike the local approach where parameters are clustered according to their sensitivity vectors [9], the presented approach clusters the parameters based upon the sensitivity cones. Important challenges that are dealt with in this work arise for the computation of the sensitivity cones, not only because of the uncertainty description over all parameter, but also because not all cones have the same angle as the sensitivity vectors of some parameters are significantly more affected by uncertainty than those corresponding to other parameters.

One of the key contributions of this work is to determine the angle of each sensitivity cone and make use of this information to determine threshold cutoff values for grouping parameter via hierarchical clustering. The angel of the sensitivity cone is calculated by solving a dynamic optimization problem. The problem is formulated by discretizing the ODE system with the 3-point Radau collocation method in AMPL [16] and is solved with an interior-point nonlinear solver IPOPT [17] integrated with MA86 [18].

The proposed systematic scheme for parameter set selection for dynamic systems under uncertainty is applied to two examples, including a model of a signal transduction pathway. The examples are presented in detail and comparisons of the prediction accuracy for optimal and non-optimal parameter sets are discussed.

**Reference**

[1] Poyton A, Varziri MS, McAuley KB, McLellan P, & Ramsay JO. Parameter estimation in continuous-time dynamic models using principal differential analysis. *Computers & Chemical Engineering*2006;30:698-708.

[2] Ramsay JO, Hooker G, Campbell D, & Cao J. Parameter estimation for differential equations: a generalized smoothing approach. *Journal of the Royal Statistical Society: Series B (Statistical Methodology) *2007;69:741-796.

[3] Lin Y, & Stadtherr MA. Deterministic global optimization for parameter estimation of dynamic systems. *Industrial & Engineering Chemistry Research *2006;45:8438-8448.

[4] Chu Y, & Hahn J. Parameter set selection for estimation of nonlinear dynamic systems. *AIChE Journal*2007;53:2858-2870.

[5] Chu Y, & Hahn J. Integrating parameter selection with experimental design under uncertainty for nonlinear dynamic systems. *AIChE Journal*2008;54:2310-2320.

[6] Brun R, Reichert P, & Künsch HR. Practical identifiability analysis of large environmental simulation models. *Water Resources Research *2001;37:1015-1030.

[7] Velez-Reyes M, & Verghese G. Subset selection in identification, and application to speed and parameter estimation for induction machines. Control Applications, 1995, *Proceedings of the 4th IEEE Conference on IEEE*. 1995. p. 991-997.

[8] Lund BF, & Foss BA. Parameter ranking by orthogonalization—Applied to nonlinear mechanistic models. *Automatica*2008;44:278-281.

[9] Chu Y, & Hahn J. Parameter set selection via clustering of parameters into pairwise indistinguishable groups of parameters. *Industrial & Engineering Chemistry Research *2008;48:6000-6009.

[10] Walter E, & Pronzato L. Qualitative and quantitative experiment design for phenomenological models—a survey. *Automatica *1990;26:195-213.

[11] Chu Y, & Hahn J. Quantitative optimal experimental design using global sensitivity analysis via quasi-linearization. *Industrial & Engineering Chemistry Research *2010;49:7782-7794.

[12] Morris MD. Factorial sampling plans for preliminary computational experiments. *Technometrics *1991;33:161-174.

[13] Hornberger GM, & Spear R. Approach to the preliminary analysis of environmental systems. *Journal of Environmental Management *1981;12.

[14] Atherton R, Schainker R, & Ducot E. On the statistical sensitivity analysis of models for chemical kinetics. *AIChE Journal *1975;21:441-448.

[15] Srinivasan B, Palanki S, & Bonvin D. Dynamic optimization of batch processes: I. Characterization of the nominal solution. *Computers & Chemical Engineering *2003;27:1-26.

[16] Fourer R, Gay D, & Kernighan B. AMPL: a modeling language for mathematical programming. 2002. *Duxbury Press*.

[17] Wächter A, & Biegler LT. On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming. *Mathematical Programming*2006;106:25-57.

[18] HSL(2013). collection of Fortran codes for large-scale scientific computation. *http://www.hslrl.ac.uk*.

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