Quantifying uncertainty, variability and likelihood for ordinary differential equation models

<p>Abstract</p> <p>Background</p> <p>In many applications, ordinary differential equation (ODE) models are subject to uncertainty or variability in initial conditions and parameters. Both, uncertainty and variability can be quantified in terms of a probability density function on the state and parameter space.</p> <p>Results</p> <p>The partial differential equation that describes the evolution of this probability density function has a form that is particularly amenable to application of the well-known method of characteristics. The value of the density at some point in time is directly accessible by the solution of the original ODE extended by a single extra dimension (for the value of the density). This leads to simple methods for studying uncertainty, variability and likelihood, with significant advantages over more traditional Monte Carlo and related approaches especially when studying regions with low probability.</p> <p>Conclusions</p> <p>While such approaches based on the method of characteristics are common practice in other disciplines, their advantages for the study of biological systems have so far remained unrecognized. Several examples illustrate performance and accuracy of the approach and its limitations.</p

Singular Value Analysis and Balanced Realizations for Nonlinear Systems

For linear control systems minimal realization theory and the related model reduction methods play a crucial role in understanding and handling the system. These methods are well established and have proved to be very successful. In particular the method called balanced truncation gives a good reduced order model with respect to the input-output behavior. This method relies on the relation with the system Hankel operator, which plays a central role in minimal realization theory. Specifically, the Hankel operator supplies a set of similarity invariants, the so called Hankel singular values, which can be used to quantify the importance of each state in the corresponding input-output system. The Hankel operator can also be factored into a composition of observability and controllability operators, from which Gramian matrices can be defined and the notion of balanced realization follows. This linear theory is rather complete and the relations between and interpretations in the state-space and input-output settings are fully understood. This paper gives an overview of the series of research on balanced realization and the related model order reduction method based on nonlinear singular value analysis. Section 2 explains the taken point of view on singular value analysis for nonlinear operators. Section 3 briefly reviews the linear balancing method and balanced truncation in order to show the way of thinking for the nonlinear case. Section 4 treats the state-space balancing method. Then, in Section 5 we continue with balanced realizations based on the singular value analysis of the nonlinear Hankel operator. Furthermore, in Section 6 balanced truncation based on the method of Section 5 is presented. Finally, in Section 7 a numerical simulation illustrates how the proposed model order reduction method works for real-world systems.

Comparing BRIN-BD11 culture producing insulin using different type of microcarriers

This research was conducted to examine the growth profile, growth kinetics, and insulin-secretory responsiveness of BRIN-BD11 cells grown in optimized medium on different types of microcarriers (MCs). Comparisons were made on modified polystyrene (Hillex® II) and crosslinked polystyrene Plastic Plus (PP) from Solohill Engineering. The cell line producing insulin was cultured in a 25 cm2 T-flask as control while MCs based culture was implemented in a stirred tank bioreactor with 1 L working volume. For each culture type, the viable cell number, glucose, lactate, glutamate, and insulin concentrations were measured and compared. Maximum viable cell number was obtained at 1.47 × 105 cell/mL for PP microcarrier (PPMCs) culture, 1.35 × 105 cell/mL Hillex® II (HIIMCs) culture and 0.95 × 105 cell/mL for T-flask culture, respectively. The highest insulin concentration has been produced in PPMCs culture (5.31 mg/L) compared to HIIMCs culture (2.01 mg/L) and T-flask culture (1.99 mg/L). Therefore overall observation suggested that PPMCs was likely preferred to be used for BRIN-BD11 cell culture as compared with Hillex® II MCs