What is a State-Observer and why is it useful?

A state-observer is an algorithm used to estimate the current state of a system. It is a model-based approach, hence it needs an accurate process model to describe the dynamics of the given system. This model can be entirely data-driven, such as neural networks; mechanistic, e.g., derived from differential equations or a hybrid model (data-driven and mechanistic parts). Essential for the estimation of the systems state however is in all cases, that knowledge about the systems inputs and outputs is available in form of measurements.

The question is, why do I need to put work into deriving or training a model and then still need to feed my measurements into the algorithm. Often, especially in systems biology or related fields, it is very difficult to have necessary measurement systems available. State-estimation not only allows to predict the state of otherwise unmeasurable states, but gives the opportunity to reduce unnecessary measurements and therefore reduce acquisition costs as well as operating costs. Though, this assumption is only valid if enough states are measurable such that the process is entirely observable. Additionally, a state-observer can be used as a fault detection system when observing large deviations between prediction and measurements or it is used as the basis for model-based control approaches such as model predictive control.

What are Particle Filters?

Prominent state-observer algorithms used are different Kalman filters and particle filters. Contrary to Kalman filters the particle filter is able handle even strong non-linearities in the process model and measurement functions. It is monte carlo based and therefore relies on many simulations and consequently increased computational power. The algorithm consists of one initialization step at the beginning followed by four distinct operations which repeat with each filter iteration. The filter is initialized by distributing many sampling points (particles) either uniformly in the complete design space of the process or as gaussian distribution around a known initial condition. Each particle has an associated weight which is equally distributed during the initialization stage. The repeating operations during each iteration of the filter are as follows:

  1. Prediction: The movement of each particle in the design space is predicted based on the underlying model.
  2. Update: The weight of each particle gets scaled based on the error of the predicted position and the measured values.
  3. Resampling: If the amount of significantly contributing particles (high enough weight) is too low, particles with low weight get replaced by duplicates of high weight particles.
  4. Estimation: The weighted mean and covariance of the particles distribution is calculated.

A simulation of a particle filter for a refolding process visualizes the folding of solubilized protein S to either native protein N or aggregated protein A (Figure 1).

Figure 1: Circulating steps of a particle filter for a simulated refolding process from solubilized protein S to either native protein N or aggregated protein A. Particles in green, scaled with their associated weight; measurement in black; calculated mean in red.

Biopharmaceutical Application Examples

As mentioned in the beginning, the application of particle filters poses a great opportunity for the biopharmaceutical field, where sometimes few or even no measurements are available for certain process states. Two examples are presented here spanning both the field of upstream and downstream processes.

The first example is the state estimation of a eukaryotic fed-batch fermentation process with Penicillium chrysogenum. The particle filter is used to estimate concentrations of biomass, precursor and product (Penicillin) as well as the cell specific growth rate and maximal product conversion rate. Figure 2 shows the adaptation of the maximal product conversion rate, which is adapted at each update step to describe the measured values of the product concentration with higher accuracy [1].

Figure 2: Particle filter with adapting maximal product conversion rate to accurately describe the Penicillin production. Top: Parameter (line) with confidence interval (dashed lines); bottom: estimation of the concentration of Penicillin (line) and offline measurements (circles).

The second example describes the estimation of a simulated refolding process using the fed-batch dilution method. Solubilized protein is fed into a refolding buffer over a period of five hours, followed by a batch phase, where remaining solubilized protein can fold (Figure 3). The approach for state estimation and subsequent control of refolding processes is reviewed in [2].

Figure 3: Particle filter for the estimation of a protein refolding process. Measured values (circles) are generated from the simulations (lines) by adding random noise at distinct time steps. The predictions from the particle filter after each evaluation are depicted as crosses.

Benefits and Potential

The particle filter as tool for state estimation has many benefits. It leverages its knowledge conservation in form of process models and can handle even strongly non-linear systems. In combination with model-based control approaches it has the potential to revolutionize protein refolding processes by transferring them from inferior empirical applications with low yields towards knowledge-based processes with higher yields and space-time yields. Additionally, process models make these processes easier to transfer to new products, therefore posing a solution as a generic tool to optimize refolding processes, reduce time-to-market and simplify process development by model-based experimental design.

[1] Kager, J. et al., 2018. State estimation for a penicillin fed-batch process combining particle filtering methods with online and time delayed offline measurements. Chem. Eng. Sci. 177, 234–244.

[2] Pauk, J.N. et al., 2021. Advances in monitoring and control of refolding kinetics combining PAT and modeling. Appl Microbiol Biotechnol 105, 2243–2260.