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473169 Bioprocess Optimization Under Uncertainty Using Ensemble Modeling

Mathematical modeling has become an indispensible tool in bioprocess design and optimization [1], especially after the implementation of Quality by Design for biopharmaceuticals by the US Food and Drug Administration’s (FDA) office [2]. Since the majority of bioprocesses involve batch or fed-batch culture and are therefore dynamic in nature, kinetic modeling using ordinary differential equation (ODE) is the predominant framework to model these processes. However, the development of first principle dynamic bioprocess models often faces significant challenges due to high degree of process nonlinearity and limited datasets for parameter estimation (model training), the combination of which leads to model identifiability problem. This problem causes uncertainty in the modeling, and as a consequence, there often exist a family of equivalent kinetic models and model parameters, which are consistent with the available data and information on the bioprocess. Such uncertainty, if not addressed properly, could lead to suboptimal bioprocess design and operation.

In this work, we investigated several strategies for bioprocess optimization under uncertainty using ensemble modeling. The bioprocess optimization under uncertainty involved the maximization of mean-standard deviation objective function:

*U*(*c*)* = E _{θ}*(

*f*(

*c, θ*))

*+ α√V*(

_{θ}*f*(

*c, θ*))

where
*c* denotes the vector of operating conditions, *θ* denotes the
vector of model parameters, f(*c,θ*) describes the process objective
function (e.g. yield or product titer), *α* is a weighting factor, and *E _{θ}*
(f(

*c,θ*)) and

*V*(f(

_{θ}*c,θ*)) are the expected and variance of

*f*(

*c,θ*) over the posterior probability density function of the parameters

*P*(

*θ*). Similar objective functions have been proposed previously for bioprocess optimization under uncertainty [3, 4]. Our work differed from the previous studies in the generation of the parameter ensemble, allowing efficient evaluation of the objective function, especially for high dimensional parameters. We previously created an ensemble modeling toolbox for MATLAB, called REDEMPTION (Reduced Dimension Ensemble Modeling and Parameter Estimation) [5], which offered a user-friendly interface for kinetic ODE modeling of bioprocesses using time series process data, including the identification of an ensemble (family) of statistically equivalent model parameters. In particular, we used a combination of an out-of-equilibrium Adaptive Monte Carlo and multiple ellipsoids-based uniform sampling, called HYPERSPACE [6], to obtain the ensemble of parameters whose negative log-likelihood function values were lower than a given threshold. In the bioprocess optimization above, we approximated the expectation function over the posterior distribution of parameters using the following:

*E _{θ}*(

*g*(

*θ*)) = Σ

_{θ}

_{∈}

_{Ω}*g*(

*θ*)

*P*(

*θ*)

where *Ω* denotes the
parameter ensemble.

We
evaluated the performance of the proposed strategy using a case study of
monoclonal antibodies (mAb) production. We employed the kinetic bioprocess
model proposed by Kontoravdi et al. [1, 7] to simulate *in silico* time-series concentration
data, contaminated with Gaussian noise at 50% coefficient of variation.
We subsequently used this dataset to obtain the maximum likelihood (ML)
parameter estimate. We further generated an ensemble of ~400,000 parameter
combinations using a threshold of 1.2 times the negative log-likelihood value
of the ML estimate. The bioprocess optimization involved finding the
optimal starting glutamine concentration that maximizes the mAb concentration
at the end of batch culture, for different initial glucose concentrations. We
implemented two strategies: *α *= 0 (i.e. maximization of expected value
of *f*(*c,θ*)) and *α *= -1.282 (corresponding to 90% lower
confidence bound of *f*(*c,θ*)). We compared our strategy to
maximizing *f*(*c,θ*) directly using the ML parameter estimate. The
final mAb concentration for each optimal glutamine concentration was computed
using the model used to generate the data above. The comparison in Table 1
showed that the optimal glutamine concentrations using the proposed bioprocess
optimization under uncertainty led to higher final mAb concentration than those
using the ML parameter estimate, especially at high initial glucose
concentration. In particular, the strategy using *α *= 0 consistently
outperformed the optimization using the ML model. The result thus
demonstrated the benefit and importance of considering parameter uncertainty in
bioprocess optimization.

Table 1. Final mAb concentrations from different optimization strategies.

| mAb concentration (10 | ||

Initial glucose concentration (mM) | Proposed method ( | Proposed method ( | Optimization using ML model |

25 | 1.3127 | 1.2742 | 1.3020 |

50 | 1.3238 | 1.3105 | 1.1856 |

75 | 1.3116 | 1.3158 | 1.1390 |

100 | 1.3042 | 1.3183 | 1.1347 |

1.
Kiparissides, A., et al., *'Closing the loop' in biological systems modeling
- From the in silico to the in vitro.* Automatica, 2011. **47**(6): p.
1147-1155.

2.
Rathore, A.S. and H. Winkle, *Quality by design for biopharmaceuticals.*
Nature Biotechnology, 2009. **27**(1): p. 26-34.

3.
Mandur, J. and H. Budman, *Robust optimization of chemical processes using
Bayesian description of parametric uncertainty.* Journal of Process Control,
2014. **24**(2): p. 422-430.

4.
Kim, K.K.K. and R.D. Braatz, *Probabilistic Analysis and Control of Uncertain
Dynamic Systems: Generalized Polynomial Chaos Expansion Approaches.*2012
American Control Conference (Acc), 2012: p. 44-49.

5.
Liu, Y., E. Manesso, and R. Gunawan, *REDEMPTION: reduced dimension ensemble
modeling and parameter estimation.* Bioinformatics, 2015. **31**(20): p.
3387-3389.

6.
Zamora-Sillero, E., et al., *Efficient characterization of high-dimensional
parameter spaces for systems biology.* BMC Systems Biology, 2011. **5**:
p. 142.

7.
Tatiraju, S., M. Soroush, and R. Mutharasan, *Multi-rate nonlinear state and
parameter estimation in a bioreactor.* Biotechnology and Bioengineering,
1999. **63**(1): p. 22-32.

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