The shortcomings of accurate rate estimations in cultivation processes and a solution for precise and robust process modeling

Abstract

The accurate estimation of cell growth or the substrate consumption rate is crucial for the understanding of the current state of a bioprocess. Rates unveil the actual cell status, making them valuable for quality-by-design concepts. However, in bioprocesses, the real rates are commonly not accessible due to analytical errors. We simulated Escherichia coli fed-batch fermentations, sampled at four different intervals and added five levels of noise to mimic analytical inaccuracy. We computed stepwise integral estimations with and without using moving average estimations, and smoothing spline interpolations to compare the accuracy and precision of each method to calculate the rates. We demonstrate that stepwise integration results in low accuracy and precision, especially at higher sampling frequencies. Contrary, a simple smoothing spline function displayed both the highest accuracy and precision regardless of the chosen sampling interval. Based on this, we tested three different options for substrate uptake rate estimations.

Keywords: Bioprocess development; Cubic smoothing spline; Fed-batch fermentation; Growth rate; Substrate uptake rate.

Fig. 1

Simulated a Monod and b non-competitive model process parameters and biomass concentration variation due to random sampling error at 12.5%, 7.5% and 2.5% CV for the Monod model (c) with a sampling interval of 0.5 h and the non-competitive model (d) with a sampling interval of 1 h are presented. For c, d the number of simulated fed-batch processes n = 100

Fig. 2

a, b The estimated growth rates at different sampling intervals and their respective standard deviations (depicted by the area) at a biomass determination precision of 2.5% coefficient of variation (CV). c, d The resulting CV of the growth rate µ as a function of the sampling interval and at different biomass determination precisions for Monod model (a, c) and the non-competitive model (b, d) The number of simulated processes n = 100. Data above 100% are not depicted

Fig. 3

a Spline fittings with p 0 and 1 of noisy biomass data (12.5% CV of biomass determination). b RMSE as a function of the sampling interval, the CV of biomass determination and the fitting parameter p of the spline function. c RMSE at a p of 0.4 at different sampling intervals. The coefficient of variation (CV) of the growth rate for the Monod model (d) and the non-competitive model (e) as a function of the sampling interval and CV of biomass determination for a fitting parameter p of 0.4. For b–e the number of simulated processes n = 100

Fig. 4

Comparing the RMSE values of the stepwise integral estimations (a) and stepwise integral estimations using a moving average (n = 4) as a function of the sampling interval and CV of biomass determination. b The timely deviation (%) from the time point when the simulated µ changed 15% (non-competitive model) derived from utilizing moving average with a window size of 3 and 4. The number of simulated processes n = 100

Fig. 5

Specific substrate uptake rate estimation via option 1 (a) 2 (b) and 3 (c) over the time course of a fed-batch (n = 100) for a sampling interval of 1 h and precision of 12.5% CV for the biomass determination are presented. The averaged values and their respective standard deviations of the three different options over the time course of the process (d), the resulting RMSE values for each option and sampling point (e), and MAPE for all three options (f) are displayed. The number of simulated processes n = 100