Costly materials, limited resources, and lengthy experiments are constraints that hinder our ability to acquire quantifiable measurements from biological systems. Experimental design approaches are computational techniques for extracting the most useful information from experiments yet to be performed [

1]. These techniques are needed for the study of biological systems to conserve the limited resources that are allocated for performing experiments. Application of these techniques to biological systems has introduced novel mathematical algorithms and models to life sciences, while also requiring the development of new mathematical theories and programming tools [

2]. An important aspect of experimental design for biological systems is model calibration, which requires the estimation of parameters such as kinetic and diffusivity constants [

3]. The development of accurate biological models is constrained by the financial costs and time required to perform biological experiments, often leading to a collection of sparse datasets with which to estimate the parameters of proposed model structures. Experimental design provides a method to yield the best estimates from data given the limitations in data collection, component observability and limited system excitability.

The development and application of experimental design has a rich history spread across a wide range of fields. An excellent review article by Pronzato has condensed the underlying concepts behind the most widely used techniques of experimental design for nonparametric and parametric models [

1]. The reader is referred to the review article and the works cited therein for a thorough understanding of statistical methods for experimental design.

Typically, parameter estimation problems begin by claiming that observations

are perturbed from ideal model outputs

**g**(

**x**,

*θ**) by an error

*ε*, such that

where

**x**_{i }are the model states at

*k *different times or experimental conditions,

*θ** are the true parameter values and the errors, ε

_{i}, are statistically independent with zero mean and variance

. It is assumed that the errors can be defined by probability density functions, often assumed to be independent and identically distributed Gaussian random variables with zero mean and variance σ

^{2 }for mathematical convenience. The unknown parameter vector can then be determined by the maximum likelihood estimate

. As

*k *→ ∞ the difference between

and

*θ** can be described by a normal distribution with zero mean and covariance matrix, Σ, which is bounded from below by the inverse of the Fisher Information Matrix (FIM) according to the Cramér-Rao inequality [

1].

Experimental design aims to maximize information, or minimize uncertainty, about unknown model parameters by exploring experimental configurations such as the sampling times where new measurements should be acquired, the desired number of measurements to add, which system components should be measured, etc. The criteria used to evaluate the information of a design are derived from scalar functions of the FIM [

1]. A-optimal design, for example, minimizes trace(FIM

^{-1}), or equivalently minimizes of the sum of squared lengths of the axes of asymptotic confidence ellipsoids for

*θ*. E-optimality refers to designs where the longest axis of asymptotic confidence ellipsoids for

*θ *is minimized, which is equivalent to maximizing the minimum eigenvalue of the FIM. D-optimal design maximizes det(FIM) and corresponds to minimizing the volume of asymptotic confidence ellipsoids for

*θ*.

Although there is a large body of work dedicated to experimental design using statistical methods [

1], several problems arise when using these approaches for the modeling of biological systems [

4]. Kreutz and Timmer state several of the difficulties in using experimental design for biological systems: i) models are often large and the number of measurements is very limited, ii) relative noise levels of 10% or more are standard for biochemical data, iii) little prior knowledge exists. These considerations make it difficult to correctly characterize the distribution of uncertainty in the model, which is the primary pillar upon which FIM approaches for experimental design are based. Even if the correct distribution is obtained, accurate parameter estimations using the FIM are usually valid only when a large number of data points are available, which is not often the case for biological systems [

5]. Rather, the finite range of values that system component concentrations can take on at a given time more appropriately characterizes the uncertainty in biological systems. This bound can be inferred based on the experimental technology, the characteristics of limited replicates, and/or first principles knowledge. Therefore, a set membership framework is more appropriate for the development of experimental design for many biological systems, where the error is bounded with no other hypothesis given regarding its distribution [

6].

A key aspect of experimental design for bounded-error models is how to characterize the set of parameter values that are consistent with all data measurements. Initial methods for constructing this set use conservative bounding approaches based on ellipsoids to characterize the parameter sets. More precise parameter set estimations can be obtained using interval analysis [

7,

8],

*but these interval techniques have not previously been applied to experimental design approaches*. Apart from the method used to bound the parameter set, proper experimental design metrics are important because they provide a logical link between physical resources and mathematical constructs. Traditional experimental design criteria for bounded-error models minimize the volume of parameter sets that are consistent with the data [

6,

9-

11]. However, the information provided by this metric may not be useful to a biologist. Other metrics that are related directly to the uncertainty of specific parameters or the effects on unmeasurable model states may be of more interest. Such biologically relevant information can be obtained from simple criteria functions previously not used in experimental design for bounded-error models. Set membership experimental design methods have recently regained attention. Hasenauer et al. have developed a set-based experimental design method using semidefinite programming with V-optimality as the only design metric [

12]. The expected information content from additional measurements is determined using a Monte-Carlo approach to simulate different parameters, input sequences and measurement errors. While this method demonstrates the usefulness of bounded-error techniques, there is a lack of connection between the design metric and biological interpretation. Additionally, the use of a Monte-Carlo approach to simulate the effect of additional measurements requires a large number of simulations and can be very time consuming. Bounded approaches, such as the one we outline in this paper, allow for the impact of uncertainty to be assessed without needing to perform Monte-Carlo simulations.

In this work, we develop an experimental design framework that utilizes interval analysis to generate the set of parameters and state bounds consistent with all data measurements. This approach leverages many of the recent advances in bounded-error parameter and state estimation methods [

7,

8], including increased accuracy through the use of interval analysis instead of bounded ellipsoids, as the base of our experimental design framework. Our novel framework uses parameter and state estimations based on initial data measurements, which may provide data for only a subset of the model states, to estimate measurement bounds at candidate time points of interest to the experimenter (times when measurements have not been taken). We then use these estimated measurements at the candidate time points to evaluate which candidate measurements furthest reduce model uncertainty. We propose a method for combining candidate time points to determine the effect of adding multiple measurements. We present biologically relevant design metrics to evaluate candidate designs in order to address issues associated with making a better connection between modelers and experimentalists. These contributions comprise a bounded-error experimental design framework that can be applied to nonlinear continuous-time systems when few data measurements are available. This framework can be used to balance the availability of resources with the addition of one or more measurement points to improve the predictability of resulting models.