# Monitoring the Recombinant Adeno-Associated Virus Production using Extended Kalman Filter

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Unstructured Mechanistic Kinetic Model Formulation for rAAV Production

#### 2.2. Extended Kalman Filter

**P**), which contains the error covariance of the predicted state variables values.

**P**can be formulated using linear terms of a Taylor expansion in the continuous-time domain by the following differential equation:

**Prediction step:**Using the initial condition, the predicted state variables vector (${\mathit{\psi}}_{k/k-1}$) and predicted error covariance matrix of state (${\mathbf{P}}_{k|k-1}$) were estimated by solving 9 and 11 from ${t}_{k-1}$ to ${t}_{k}$, where a new measurement is obtained at time k. It is noteworthy that the initial conditions in the prediction step are initial estimation error covariance matrix of state ${\mathbf{P}}_{i,i}(t=0)$, and initial state variables vector $\mathit{\psi}(t=0)$ (composed of state variables and parameters of UMKM).

**Correction step:**In this step, the estimates of the prediction step (${\mathit{\psi}}_{k/k-1}$ and ${\mathbf{P}}_{k|k-1}$) were combined with the measured value (${\mathbf{Z}}_{k}$) and Kalman gain (${\mathbf{K}}_{k}$) to provide corrected state variables vector (${\mathit{\psi}}_{k/k}$) and corrected error covariance matrix of state (${\mathbf{P}}_{k|k}$) using the following equations:

**P**and

**K**to be updated with information about the UMKM parameters, since we used a simple UMKM, and

**P**and

**Q**with uncorrelated elements. As we wanted to estimate the other state variables and the parameters from the Xv measurements available in discrete time, the ${\mathbf{H}}_{2}$ used in our case was ${\mathbf{H}}_{2}$ = diag([1 0 0 0 0 0 1 1 1 1 1 1 0]) in cell expansion phase and ${\mathbf{H}}_{2}$ = diag([1 0 0 0 0 0 1 1 1 1 1 1 1]) in the viral vector production phase. ${\mathbf{H}}_{1}$= diag([1 0 0 0 0 0 0 0 0 0 0 0 0]) was used in both phases since we only had Xv measurements.

#### 2.3. ODE Parameter Estimation Approaches

#### 2.3.1. Neural Ordinary Differential Equation

#### 2.3.2. Bayesian Inference Parameters Estimation

#### 2.4. Datasets

#### 2.4.1. Data Description

#### 2.4.2. Shake-Flasks Dataset

_{2}in a shaker incubator (Infors, Basel, Switzerland) at 120 rpm agitation rate and transfected at 36 h post inoculation (hpi) with the viable cell density at around $1\times {10}^{6}$ cells/mL for all runs. The shake-flasks were harvested at 84 hpi (48 h post transfection). Samples were taken every 12 h until harvest.

#### 2.4.3. Bioreactor 1 Dataset

_{2}overlay and sodium bicarbonate addition PID control loop. The temperature was maintained at 37 °C using a heating jacket. Cells were inoculated into a bioreactor with an initial viable cell density of $0.36\times {10}^{6}$ cells/mL in the same medium as the shake-flasks experiment. The cells were transfected at 57 hpi with viable cell density at $1.27\times {10}^{6}$ cells/mL. The bioreactor was harvested at 114 hpi (61 hpt). Samples were taken approximately every 24 h before transfection and approximately 12 h after transfection until harvest.

#### 2.4.4. Bioreactor 2 Dataset

#### 2.4.5. Current Approach: Quantitative Analysis

#### 2.5. Parameters of the UMKM and the Extended Kalman Filter

**R**,

**P**, and

**Q**) are presented in Table 1. We estimated the parameters of the UMKM for the cell expansion and viral vector production phases with two different approaches, NODE and Bayesian inference, using the bioreactor 1 dataset, and the results of these approaches are discussed in Section 3.1.

**R**represents only the variance of Xv, since only Xv is measured, and is therefore one-dimensional.

#### 2.6. Evaluation

#### 2.6.1. Calibration Using Offline Values

#### 2.6.2. EKF Test

## 3. Results

#### 3.1. UMKM Parameter Estimation with NODE and Bayesian Inference

#### 3.2. EKF Calibration

#### 3.3. EKF Test

**R**) used in the calibration process (Table 5 and Table 6). However, the initial condition of state variables vector $\mathit{\psi}(t=0)$ in CEP was composed of state variables of the bioreactor 2 dataset (column 3 of Table 7) and the values of Bayesian UMKM parameters estimation (column 4 of Table 3). Figure 5 presents the results of EKF estimations in CEP. Plot A shows the online Xv measurements with noise (orange line) and EKF estimations following the exponential behavior of Xv (blue line). Plots B, C, D, and E present the final estimated values (last column of Table 7). The parameters estimation (plot F) presented the same behavior obtained by the calibration. It did not present the final values as being significantly different from the initial parameters. This is because the initial conditions of UMKM state variables in bioreactor 1 and 2 datasets are the same in CEP.

## 4. Discussion

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Wang, D.; Tai, P.W.; Gao, G. Adeno-associated virus vector as a platform for gene therapy delivery. Nat. Rev. Drug Discov.
**2019**, 18, 358–378. [Google Scholar] [CrossRef] [PubMed] - Bulcha, J.T.; Wang, Y.; Ma, H.; Tai, P.W.; Gao, G. Viral vector platforms within the gene therapy landscape. Signal Transduct. Target. Ther.
**2021**, 6, 1–24. [Google Scholar] [CrossRef] [PubMed] - Naso, M.F.; Tomkowicz, B.; Perry, W.L.; Strohl, W.R. Adeno-associated virus (AAV) as a vector for gene therapy. BioDrugs
**2017**, 31, 317–334. [Google Scholar] [CrossRef] [Green Version] - Keeler, A.M.; Flotte, T.R. Recombinant adeno-associated virus gene therapy in light of Luxturna (and Zolgensma and Glybera): Where are we, and how did we get here? Annu. Rev. Virol.
**2019**, 6, 601–621. [Google Scholar] [CrossRef] [PubMed] - Srivastava, A.; Mallela, K.M.; Deorkar, N.; Brophy, G. Manufacturing challenges and rational formulation development for AAV viral vectors. J. Pharm. Sci.
**2021**, 110, 2609–2624. [Google Scholar] [CrossRef] [PubMed] - Food and Drug Administration. Guidance for Industry, PAT-A Framework for Innovative Pharmaceutical Development, Manufacturing and Quality Assurance. Available online: https://www.fda.gov/media/71012/download (accessed on 15 July 2022).
- Gimpel, A.L.; Katsikis, G.; Sha, S.; Maloney, A.J.; Hong, M.S.; Nguyen, T.N.; Wolfrum, J.; Springs, S.L.; Sinskey, A.J.; Manalis, S.R.; et al. Analytical methods for process and product characterization of recombinant adeno-associated virus-based gene therapies. Mol.-Ther.-Methods Clin. Dev.
**2021**, 20, 740–754. [Google Scholar] [CrossRef] [PubMed] - Ohadi, K.; Aghamohseni, H.; Legge, R.L.; Budman, H.M. Fluorescence-based soft sensor for at situ monitoring of chinese hamster ovary cell cultures. Biotechnol. Bioeng.
**2014**, 111, 1577–1586. [Google Scholar] [CrossRef] - Claßen, J.; Graf, A.; Aupert, F.; Solle, D.; Höhse, M.; Scheper, T. A novel LED-based 2D-fluorescence spectroscopy system for in-line bioprocess monitoring of Chinese hamster ovary cell cultivations—Part II. Eng. Life Sci.
**2019**, 19, 341–351. [Google Scholar] [CrossRef] [Green Version] - Whelan, J.; Craven, S.; Glennon, B. In situ Raman spectroscopy for simultaneous monitoring of multiple process parameters in mammalian cell culture bioreactors. Biotechnol. Prog.
**2012**, 28, 1355–1362. [Google Scholar] [CrossRef] - Abu-Absi, N.R.; Kenty, B.M.; Cuellar, M.E.; Borys, M.C.; Sakhamuri, S.; Strachan, D.J.; Hausladen, M.C.; Li, Z.J. Real time monitoring of multiple parameters in mammalian cell culture bioreactors using an in-line Raman spectroscopy probe. Biotechnol. Bioeng.
**2011**, 108, 1215–1221. [Google Scholar] [CrossRef] - Berry, B.; Moretto, J.; Matthews, T.; Smelko, J.; Wiltberger, K. Cross-scale predictive modeling of CHO cell culture growth and metabolites using R aman spectroscopy and multivariate analysis. Biotechnol. Prog.
**2015**, 31, 566–577. [Google Scholar] [CrossRef] [PubMed] - Faassen, S.M.; Hitzmann, B. Fluorescence spectroscopy and chemometric modeling for bioprocess monitoring. Sensors
**2015**, 15, 10271–10291. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Tsopanoglou, A.; del Val, I.J. Moving towards an era of hybrid modelling: Advantages and challenges of coupling mechanistic and data-driven models for upstream pharmaceutical bioprocesses. Curr. Opin. Chem. Eng.
**2021**, 32, 100691. [Google Scholar] [CrossRef] - Chhatre, S. Modelling approaches for bio-manufacturing operations. Meas. Monit. Model. Control. Bioprocesses
**2012**, 132, 85–107. [Google Scholar] - Udugama, I.A.; Lopez, P.C.; Gargalo, C.L.; Li, X.; Bayer, C.; Gernaey, K.V. Digital Twin in biomanufacturing: Challenges and opportunities towards its implementation. Syst. Microbiol. Biomanufacturing
**2021**, 1, 257–274. [Google Scholar] [CrossRef] - Luo, Y.; Kurian, V.; Ogunnaike, B.A. Bioprocess systems analysis, modeling, estimation, and control. Curr. Opin. Chem. Eng.
**2021**, 33, 100705. [Google Scholar] [CrossRef] - Reyes, S.J.; Durocher, Y.; Pham, P.L.; Henry, O. Modern Sensor Tools and Techniques for Monitoring, Controlling, and Improving Cell Culture Processes. Processes
**2022**, 10, 189. [Google Scholar] [CrossRef] - Koutinas, M.; Kiparissides, A.; Pistikopoulos, E.N.; Mantalaris, A. Bioprocess systems engineering: Transferring traditional process engineering principles to industrial biotechnology. Comput. Struct. Biotechnol. J.
**2012**, 3, e201210022. [Google Scholar] [CrossRef] [Green Version] - Kotidis, P.; Pappas, I.; Avraamidou, S.; Pistikopoulos, E.N.; Kontoravdi, C.; Papathanasiou, M.M. DigiGlyc: A hybrid tool for reactive scheduling in cell culture systems. Comput. Chem. Eng.
**2021**, 154, 107460. [Google Scholar] [CrossRef] - Narayanan, H.; Sokolov, M.; Morbidelli, M.; Butté, A. A new generation of predictive models: The added value of hybrid models for manufacturing processes of therapeutic proteins. Biotechnol. Bioeng.
**2019**, 116, 2540–2549. [Google Scholar] [CrossRef] - Narayanan, H.; Behle, L.; Luna, M.F.; Sokolov, M.; Guillén-Gosálbez, G.; Morbidelli, M.; Butté, A. Hybrid-EKF: Hybrid model coupled with extended Kalman filter for real-time monitoring and control of mammalian cell culture. Biotechnol. Bioeng.
**2020**, 117, 2703–2714. [Google Scholar] [CrossRef] [PubMed] - Fernandes-Platzgummer, A.; Badenes, S.M.; da Silva, C.L.; Cabral, J.M. Bioreactors for Stem Cell and Mammalian Cell Cultivation. Bioprocess. Technol. Prod. Biopharm. Bioprod.
**2018**, 131–173. [Google Scholar] [CrossRef] - Mears, L.; Stocks, S.M.; Albaek, M.O.; Sin, G.; Gernaey, K.V. Mechanistic fermentation models for process design, monitoring, and control. Trends Biotechnol.
**2017**, 35, 914–924. [Google Scholar] [CrossRef] - Nguyen, T.N.; Sha, S.; Hong, M.S.; Maloney, A.J.; Barone, P.W.; Neufeld, C.; Wolfrum, J.; Springs, S.L.; Sinskey, A.J.; Braatz, R.D. Mechanistic model for production of recombinant adeno-associated virus via triple transfection of HEK293 cells. Mol.-Ther.-Methods Clin. Dev.
**2021**, 21, 642–655. [Google Scholar] [CrossRef] [PubMed] - Zhang, D.; Del Rio-Chanona, E.A.; Petsagkourakis, P.; Wagner, J. Hybrid physics-based and data-driven modeling for bioprocess online simulation and optimization. Biotechnol. Bioeng.
**2019**, 116, 2919–2930. [Google Scholar] [CrossRef] - Kourti, T. 4.11-Multivariate Statistical Process Control and Process Control, Using Latent Variables. 2020. Comprehensive Chemometrics
**2020**, 275–303. [Google Scholar] [CrossRef] - Ohadi, K.; Legge, R.L.; Budman, H.M. Development of a soft-sensor based on multi-wavelength fluorescence spectroscopy and a dynamic metabolic model for monitoring mammalian cell cultures. Biotechnol. Bioeng.
**2015**, 112, 197–208. [Google Scholar] [CrossRef] - Yousefi-Darani, A.; Paquet-Durand, O.; Hitzmann, B. The Kalman Filter for the Supervision of Cultivation Processes. Springer International Publishing.
**2020**, 177, 95–125. Available online: https://link.springer.com/chapter/10.1007/10_2020_145 (accessed on 15 July 2022). - Paquet-Durand, O.; Zettel, V.; Yousefi-Darani, A.; Hitzmann, B. The Supervision of Dough Fermentation Using Image Analysis Complemented by a Continuous Discrete Extended Kalman Filter. Processes
**2020**, 8, 1669. [Google Scholar] [CrossRef] - Selişteanu, D.; Șendrescu, D.; Georgeanu, V.; Roman, M. Mammalian cell culture process for monoclonal antibody production: Nonlinear modelling and parameter estimation. Biomed Res. Int.
**2015**, 2015, 598721. [Google Scholar] [CrossRef] [Green Version] - Chahal, P.S.; Schulze, E.; Tran, R.; Montes, J.; Kamen, A.A. Production of adeno-associated virus (AAV) serotypes by transient transfection of HEK293 cell suspension cultures for gene delivery. J. Virol. Methods
**2014**, 196, 163–173. [Google Scholar] [CrossRef] [PubMed] - Kyriakopoulos, S.; Ang, K.S.; Lakshmanan, M.; Huang, Z.; Yoon, S.; Gunawan, R.; Lee, D.Y. Kinetic modeling of mammalian cell culture bioprocessing: The quest to advance biomanufacturing. Biotechnol. J.
**2018**, 13, 1700229. [Google Scholar] [CrossRef] [PubMed] - Tang, P.; Xu, J.; Louey, A.; Tan, Z.; Yongky, A.; Liang, S.; Li, Z.J.; Weng, Y.; Liu, S. Kinetic modeling of Chinese hamster ovary cell culture: Factors and principles. Crit. Rev. Biotechnol.
**2020**, 40, 265–281. [Google Scholar] [CrossRef] [PubMed] - Xing, Z.; Bishop, N.; Leister, K.; Li, Z.J. Modeling kinetics of a large-scale fed-batch CHO cell culture by Markov chain Monte Carlo method. Biotechnol. Prog.
**2010**, 26, 208–219. [Google Scholar] [CrossRef] - Jin, X.B.; Robert Jeremiah, R.J.; Su, T.L.; Bai, Y.T.; Kong, J.L. The new trend of state estimation: From model-driven to hybrid-driven methods. Sensors
**2021**, 21, 2085. [Google Scholar] [CrossRef] - Ji, Z.; Brown, M. Joint state and parameter estimation for biochemical dynamic pathways with iterative extended Kalman filter: Comparison with dual state and parameter estimation. Open Autom. Control. Syst. J.
**2009**, 2, 69–77. [Google Scholar] [CrossRef] [Green Version] - Brockwell, P. Time series analysis. Encycl. Stat. Behav. Sci. 2005. [CrossRef]
- Dua, V.; Dua, P. A simultaneous approach for parameter estimation of a system of ordinary differential equations, using artificial neural network approximation. Ind. Eng. Chem. Res.
**2012**, 51, 1809–1814. [Google Scholar] [CrossRef] - Lee, K.; Parish, E.J. Parameterized neural ordinary differential equations: Applications to computational physics problems. Proc. R. Soc. A
**2021**, 477, 20210162. [Google Scholar] [CrossRef] - Rackauckas, C.; Innes, M.; Ma, Y.; Bettencourt, J.; White, L.; Dixit, V. Diffeqflux. jl-A julia library for neural differential equations. arXiv
**2019**, arXiv:1902.02376. [Google Scholar] - Xia, H.; Suliafu, V.; Ji, H.; Nguyen, T.; Bertozzi, A.; Osher, S.; Wang, B. Heavy ball neural ordinary differential equations. Adv. Neural Inf. Process. Syst.
**2021**, 34, 18646–18659. [Google Scholar] - Chen, R.T.; Rubanova, Y.; Bettencourt, J.; Duvenaud, D. Neural Ordinary Differential Equations. Advances in Neural Information Processing Systems.
**2018**, 31. Available online: https://arxiv.org/abs/1806.07366 (accessed on 15 July 2022). - Alahmadi, A.A.; Flegg, J.A.; Cochrane, D.G.; Drovandi, C.C.; Keith, J.M. A comparison of approximate versus exact techniques for Bayesian parameter inference in nonlinear ordinary differential equation models. R. Soc. Open Sci.
**2020**, 7, 191315. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Ge, H.; Xu, K.; Ghahramani, Z. Turing: A language for flexible probabilistic inference. In Proceedings of the International Conference on Artificial Intelligence and Statistics, PMLR, Playa Blanca, Spain, 9–11 April 2018; pp. 1682–1690. [Google Scholar]
- Dhadphale, J.M.; Unni, V.R.; Saha, A.; Sujith, R. Neural ODE to model and prognose thermoacoustic instability. Chaos Interdiscip. J. Nonlinear Sci.
**2022**, 32, 013131. [Google Scholar] [CrossRef] [PubMed] - Feldt, R. BlackBoxOptim.jl. Available online: https://github.com/robertfeldt/BlackBoxOptim.jl (accessed on 15 July 2022).
- Shanmugavelayutham, G.J.C. Convergence analysis of differential evolution variants on unconstrained global optimization functions. arXiv
**2011**, arXiv:1105.1901. [Google Scholar] - Joshi, P.R.; Bernier, A.; Moço, P.D.; Schrag, J.; Chahal, P.S.; Kamen, A. Development of a scalable and robust AEX method for enriched rAAV preparations in genome-containing VCs of serotypes 5, 6, 8, and 9. Mol.-Ther.-Methods Clin. Dev.
**2021**, 21, 341–356. [Google Scholar] [CrossRef] [PubMed] - Judd, S.; Judd, C. Chapter 3–Design, operation and maintenance. In The MBR Book: Principles and Applications of Membrane Bioreactors for Water and Wastewater Treatment, 2nd ed.; Butterworth-Heinemann: Oxford, UK, 2011; pp. 55–207. [Google Scholar] [CrossRef]
- O’Sullivan, M.; O’Sullivan, J. Reservoir modeling and simulation for geothermal resource In Book Geothermal Power Generation; Woodhead Publishing: Sawston, UK, 2016; pp. 165–199. [Google Scholar] [CrossRef]
- Kornecki, M.; Strube, J. Accelerating biologics manufacturing by upstream process modelling. Processes
**2019**, 7, 166. [Google Scholar] [CrossRef]

**Figure 1.**EKF for rAAV production monitoring—the EKF performs a continuous estimation of UMKM state variables and parameters. The proposed EKF can use offline (

**b**,

**c**) and online (

**g**) measurements of Xv collected from a bioreactor (

**a**). The Xv measurements (the only measured state variable) are inputs (

**d**) of EKF (

**e**), which uses them to estimate the other state variables during rAAV production (

**f**).

**Figure 2.**Current approach used for rAAV production monitoring: offline and online quantitative analysis. (

**a**) The rAAV production was performed in shake-flasks and bioreactors. The offline measurements (

**b**,

**c**) were performed with the samples collected from these two environments and quantified in three different devices. Quantified state variables are Xv, Glc, Gln, Lac, Amm, and rAAV viral titer (

**d**). The online measurement of Xv was performed in bioreactor 2 (

**e**). The offline quantification of Xv, GLC, GLN, LAC, and AMM takes around 30 min for one set of data point, and the quantification of viral titer in rAAV production is carried out only at the end of production.

**Figure 3.**EKF calibration in CEP with bioreactor 1 dataset: the UMKM predictions were performed with parameters (${\mu}_{{X}_{v}}$, ${\mu}_{Glc}$, ${\mu}_{Gln}$, ${\mu}_{Lac}$, ${\mu}_{Amm}$, ${k}_{deg}$, and ${\mu}_{AAV}$) estimated by Bayesian inference (red lines in plots (

**A**–

**E**)), and EKF estimation (blue lines in plots (

**A**–

**E**)) was performed using these UMKM parameters as initial parameters, and were updated during the process. However, the final UMKM parameters found by EKF are not very different from those used as initial parameters obtained by Bayesian inference; see plot (

**F**). The EKF and UMKM estimations were very similar.

**Figure 4.**EKF calibration in VVPP with bioreactor 1 dataset: the UMKM predictions were performed with parameters (${\mu}_{{X}_{v}}$, ${\mu}_{Glc}$, ${\mu}_{Gln}$, ${\mu}_{Lac}$, ${\mu}_{Amm}$, ${k}_{deg}$, and ${\mu}_{AAV}$) estimated by Bayesian inference (red lines in plots (

**A**–

**F**)). EKF predictions (blue lines in plots (

**A**–

**F**)) were performed using the parameters estimated by Bayesian inference as initial UMKM parameters since they were updated during the process. However, despite some fluctuations, the final parameters found are not very different from those used as initial parameters; see plot (

**G**). The EKF was able to use the Xv measurement (with an outlier) and, similar to UMKM, estimate GLC, LAC, AMM, and rAAV.

**Figure 5.**EKF test with bioreactor 2 dataset for CEP-EKF estimation for the cell expansion phase of the upstream process. The Plot (

**A**) shows the online Xv measurements with noise (orange line) and EKF estimations following the exponential behavior of Xv (blue line). The last values estimated by EKF in CEP are considered as the initial condition of state variables in VVPP (

**B**–

**E**). The parameter estimation had some fluctuations regarding the LAC parameter, but the final parameters found are not very different from those used as initial parameters; see plot (

**F**).

**Figure 6.**EKF test with bioreactor 2 dataset for VVPP: EKF estimation for the viral vector production phase of the upstream process. The UMKM predictions were performed with parameters estimated by Bayesian inference (red lines in plots (

**A**–

**D**)) using bioreactor 1 dataset, and EKF estimation (blue lines in plots (

**A**–

**D**)) was performed using these parameters as initial parameters. They were updated during the process; see Plot (

**E**). The EKF was able to use the Xv measurement and performed estimation near to offline measurements of GLC, LAC, and rAAV.

Shake-Flask Dataset (Runs 1, 2 and 3) | Bioreactor 1 Dataset | Bioreactor 2 Dataset | Bioreactor 2 Dataset | |
---|---|---|---|---|

Offline Measurements | Offline Measurements | Offline Measurements | Online Measurements | |

Xv | Xv|GLC|GLN|LAC| AMM|AAV | Xv (CEP) and GLC|LAC|AAV (VVPP) | Xv (CEP and VVPP) | |

UMKM Parameters Estimation (${\mu}_{{X}_{v}}$, ${\mu}_{Glc}$, ${\mu}_{Gln}$, ${\mu}_{Lac}$, ${\mu}_{Amm}$, ${k}_{deg}$, ${\mu}_{AAV}$) | - | ✓ | - | - |

Initial Estimation Error (IEE) covariance matrix of states (${\mathbf{P}}_{i,i}$) | - | ✓ | - | - |

Error covariance matrix of process model (${\mathbf{Q}}_{i,i}$) | - | ✓ | - | - |

Measurement noise variance R | ✓ | - | - | - |

EKF Calibration | - | ✓ | - | - |

EKF Test | - | - | ✓ | ✓ |

**Table 2.**Initial conditions of state variables in CEP and VVPP for the EKF calibration using bioreactor 1 dataset.

State Variables | Name | Value in CEP | Value in VVPP ^{1} |
---|---|---|---|

Xv | Viable cells | 0.36 × ${10}^{6}$ c/mL | 1.27 × ${10}^{6}$ c/mL |

GLC | Glucose | 32.19 mM | 24.1 mM |

GLN | Glutamine | 5.03 mM | 3.54 mM |

LAC | Lactate | 0.111 mM | 7.88 mM |

AMM | Ammonium | 0.33 mM | 1.46 mM |

AAV | AAV viral titer | 0 VG/mL | 0 VG/mL |

^{1}The VVPP values are different from CEP values because they are different phases of upstream process.

**Table 3.**Estimated parameters for the UMKM in cell expansion phase (CEP) and viral vector production phase (VVPP) by NODE and Bayesian inference.

NODE | Bayesian Inference | |||||
---|---|---|---|---|---|---|

Parameter | CEP | VVPP | CEP (Mean) | CEP (StD) | VVPP (Mean) | VVPP (StD) |

${\mu}_{Xv}$ (h${}^{-1}$) | 0.0295 | 0.0068 | 0.0299 | 0.0004 | 0.0065 | 0.0003 |

${\mu}_{GLC}$ $\left(\mathrm{mmol}\phantom{\rule{3.33333pt}{0ex}}{10}^{-6}\mathrm{c}\phantom{\rule{3.33333pt}{0ex}}{\mathrm{h}}^{-1}\right)$ | 0.1850 | 0.0955 | 0.1895 | 0.0028 | 0.0973 | 0.0050 |

${\mu}_{GLN}$ $\left(\mathrm{mmol}\phantom{\rule{3.33333pt}{0ex}}{10}^{-6}\mathrm{c}\phantom{\rule{3.33333pt}{0ex}}{\mathrm{h}}^{-1}\right)$ | 0.0329 | 0.0172 | 0.0350 | 0.0030 | 0.0213 | 0.0031 |

${\mu}_{LAC}$ $\left(\mathrm{mmol}\phantom{\rule{3.33333pt}{0ex}}{10}^{-6}\mathrm{c}\phantom{\rule{3.33333pt}{0ex}}{\mathrm{h}}^{-1}\right)$ | 0.2500 | 0.0243 | 0.2544 | 0.0045 | 0.0214 | 0.0031 |

${\mu}_{AMM}$ $\left(\mathrm{mmol}\phantom{\rule{3.33333pt}{0ex}}{10}^{-6}\mathrm{c}\phantom{\rule{3.33333pt}{0ex}}{\mathrm{h}}^{-1}\right)$ | ${10}^{-5}$ | 0.0001 | 0.0001 | 0.0001 | 0.0001 | 2.9 × ${10}^{-5}$ |

${k}_{deg}$ (h${}^{-1}$) | 0.0050 | 0.0022 | 0.0049 | 0.0001 | 0.0020 | 0.0003 |

${\mu}_{AAV}$ $({10}^{9}\phantom{\rule{3.33333pt}{0ex}}\mathrm{vg}/\mathrm{mL}\phantom{\rule{3.33333pt}{0ex}}\mathrm{h}\phantom{\rule{3.33333pt}{0ex}}{10}^{6}\mathrm{c})$ | 0 | 0.0622 | 0 | 0 | 0.0644 | 0.0027 |

**Table 4.**RMSE values of the EKF and UMKM estimations regards bioreactor 1 dataset in the calibration.

EKF | EKF | UMKM | UMKM | ||
---|---|---|---|---|---|

State Variables | Name | RMSE Value in CEP | RMSE Value in VVPP | RMSE Value in CEP | RMSE Value in VVPP |

Xv | Viable cells | 0.009 | 0.235 | 0.027 | 0.258 |

GLC | Glucose | 0.425 | 1.529 | 0.456 | 1.47 |

GLN | Glutamine | 0.076 | 0.367 | 0.085 | 0.363 |

LAC | Lactate | 1.021 | 0.655 | 1.082 | 0.635 |

AMM | Ammonium | 0.014 | 0.123 | 0.015 | 0.12 |

AAV | AAV viral titer | - | 0.722 | - | 0.819 |

Parameter | Name | Value in CEP | Value in VVPP |
---|---|---|---|

${\mathrm{P}}_{1,1}\phantom{\rule{3.33333pt}{0ex}}({\mathrm{c}}^{2}/{\mathrm{mL}}^{2})$ | Viable cells IEE | 0.00 | 0.00 |

${\mathrm{P}}_{2,2}\phantom{\rule{3.33333pt}{0ex}}\left({\mathrm{mM}}^{2}\right)$ | Glucose IEE | 0.00 | 0.00 |

${\mathrm{P}}_{3,3}\phantom{\rule{3.33333pt}{0ex}}\left({\mathrm{mM}}^{2}\right)$ | Glutamine IEE | 0.00 | 0.00 |

${\mathrm{P}}_{4,4}\phantom{\rule{3.33333pt}{0ex}}\left({\mathrm{mM}}^{2}\right)$ | Lactate IEE | 0.00 | 0.00 |

${\mathrm{P}}_{5,5}\phantom{\rule{3.33333pt}{0ex}}\left({\mathrm{mM}}^{2}\right)$ | Ammonium IEE | 0.00 | 0.00 |

${\mathrm{P}}_{6,6}\phantom{\rule{3.33333pt}{0ex}}({\mathrm{VG}}^{2}/{\mathrm{mL}}^{2})$ | AAV viral titer IEE | 0.00 | 0.00 |

${\mathrm{P}}_{7,7}\phantom{\rule{3.33333pt}{0ex}}\left({\mathrm{h}}^{-2}\right)$ | ${\mu}_{Xv}$ IEE | 1.71 × ${10}^{-6}$ | 7.92 × ${10}^{-7}$ |

${\mathrm{P}}_{8,8}\phantom{\rule{3.33333pt}{0ex}}\left(\mathrm{mmol}\phantom{\rule{3.33333pt}{0ex}}{10}^{-12}\mathrm{c}\phantom{\rule{3.33333pt}{0ex}}{\mathrm{h}}^{-2}\right)$ | ${\mu}_{GLC}$ IEE | 1.53 × ${10}^{-6}$ | 2.56 × ${10}^{-5}$ |

${\mathrm{P}}_{9,9}\phantom{\rule{3.33333pt}{0ex}}\left(\mathrm{mmol}\phantom{\rule{3.33333pt}{0ex}}{10}^{-12}\mathrm{c}\phantom{\rule{3.33333pt}{0ex}}{\mathrm{h}}^{-2}\right)$ | ${\mu}_{GLN}$ IEE | 1.81 × ${10}^{-6}$ | 1.05 × ${10}^{-5}$ |

${\mathrm{P}}_{10,10}\phantom{\rule{3.33333pt}{0ex}}\left(\mathrm{mmol}\phantom{\rule{3.33333pt}{0ex}}{10}^{-12}\mathrm{c}\phantom{\rule{3.33333pt}{0ex}}{\mathrm{h}}^{-2}\right)$ | ${\mu}_{LAC}$ IEE | 2.55 × ${10}^{-5}$ | 9.59 × ${10}^{-6}$ |

${\mathrm{P}}_{11,11}\phantom{\rule{3.33333pt}{0ex}}\left(\mathrm{mmol}\phantom{\rule{3.33333pt}{0ex}}{10}^{-12}\mathrm{c}\phantom{\rule{3.33333pt}{0ex}}{\mathrm{h}}^{-2}\right)$ | ${\mu}_{AMM}$ IEE | 2.97 × ${10}^{-9}$ | 6.71 × ${10}^{-10}$ |

${\mathrm{P}}_{12,12}\phantom{\rule{3.33333pt}{0ex}}\left({\mathrm{h}}^{-2}\right)$ | ${k}_{deg}$ IEE | 3.37 × ${10}^{-9}$ | 8.71 × ${10}^{-8}$ |

${\mathrm{P}}_{13,13}\phantom{\rule{3.33333pt}{0ex}}(\phantom{\rule{3.33333pt}{0ex}}{\mathrm{vg}}^{2}/{\mathrm{mL}}^{2}\phantom{\rule{3.33333pt}{0ex}}{\mathrm{h}}^{2}\phantom{\rule{3.33333pt}{0ex}}{10}^{12}\mathrm{c})$ | ${\mu}_{AAV}$ IEE | 0 | 4.30 × ${10}^{-6}$ |

**Table 6.**Measurement noise variance

**R**and error covariance matrix of process model ${\mathbf{Q}}_{i,i}$ for the EKF.

Parameter | Name | Value in CEP | Value in VVPP |
---|---|---|---|

$\mathrm{R}\phantom{\rule{3.33333pt}{0ex}}({\mathrm{c}}^{2}/{\mathrm{mL}}^{2})$ | Viable cells MNV ^{1} | 0.006 | 0.006 |

${\mathrm{Q}}_{1,1}\phantom{\rule{3.33333pt}{0ex}}({\mathrm{c}}^{2}/{\mathrm{mL}}^{2})$ | Viable cells PNV ^{2} | 0.0006 | 0.000006 |

${\mathrm{Q}}_{2,2}\phantom{\rule{3.33333pt}{0ex}}\left({\mathrm{mM}}^{2}\right)$ | Glucose PNV | 0.0006 | 0.0006 |

${\mathrm{Q}}_{3,3}\phantom{\rule{3.33333pt}{0ex}}{\mathrm{mM}}^{2}$ | Glutamine PNV | 0.0006 | 0.0006 |

${\mathrm{Q}}_{4,4}\phantom{\rule{3.33333pt}{0ex}}\left({\mathrm{mM}}^{2}\right)$ | Lactate PNV | 0.0006 | 0.0006 |

${\mathrm{Q}}_{5,5}\phantom{\rule{3.33333pt}{0ex}}\left({\mathrm{mM}}^{2}\right)$ | Ammonium PNV | 0.0006 | 0.0006 |

${\mathrm{Q}}_{6,6}\phantom{\rule{3.33333pt}{0ex}}({\mathrm{VG}}^{2}/{\mathrm{mL}}^{2})$ | AAV viral titer PNV | 0.0006 | 0.0006 |

${\mathrm{Q}}_{7,7}\phantom{\rule{3.33333pt}{0ex}}\left({\mathrm{h}}^{-2}\right)$ | ${\mu}_{Xv}$ PNV | 1.71 × ${10}^{-7}$ | 7.92 × ${10}^{-8}$ |

${\mathrm{Q}}_{8,8}\phantom{\rule{3.33333pt}{0ex}}\left(\mathrm{mmol}\phantom{\rule{3.33333pt}{0ex}}{10}^{-12}\mathrm{c}\phantom{\rule{3.33333pt}{0ex}}{\mathrm{h}}^{-2}\right)$ | ${\mu}_{GLC}$ PNV | 1.53 × ${10}^{-5}$ | 1.16 × ${10}^{-5}$ |

${\mathrm{Q}}_{9,9}\phantom{\rule{3.33333pt}{0ex}}\left(\mathrm{mmol}\phantom{\rule{3.33333pt}{0ex}}{10}^{-12}\mathrm{c}\phantom{\rule{3.33333pt}{0ex}}{\mathrm{h}}^{-2}\right)$ | ${\mu}_{GLN}$ PNV | 1.81 × ${10}^{-5}$ | 1.05 × ${10}^{-5}$ |

${\mathrm{Q}}_{10,10}\phantom{\rule{3.33333pt}{0ex}}\left(\mathrm{mmol}\phantom{\rule{3.33333pt}{0ex}}{10}^{-12}\mathrm{c}\phantom{\rule{3.33333pt}{0ex}}{\mathrm{h}}^{-2}\right)$ | ${\mu}_{LAC}$ PNV | 2.55 × ${10}^{-4}$ | 15.59 × ${10}^{-6}$ |

${\mathrm{Q}}_{11,11}\phantom{\rule{3.33333pt}{0ex}}\left(\mathrm{mmol}\phantom{\rule{3.33333pt}{0ex}}{10}^{-12}\mathrm{c}\phantom{\rule{3.33333pt}{0ex}}{\mathrm{h}}^{-2}\right)$ | ${\mu}_{AMM}$ PNV | 2.97 × ${10}^{-9}$ | 0.11 × ${10}^{-8}$ |

${\mathrm{Q}}_{12,12}\phantom{\rule{3.33333pt}{0ex}}\left({\mathrm{h}}^{-2}\right)$ | ${k}_{deg}$ PNV | 3.37 × ${10}^{-9}$ | 0.71 × ${10}^{-8}$ |

${\mathrm{Q}}_{13,13}\phantom{\rule{3.33333pt}{0ex}}({\mathrm{vg}}^{2}/{\mathrm{mL}}^{2}\phantom{\rule{3.33333pt}{0ex}}{\mathrm{h}}^{2}\phantom{\rule{3.33333pt}{0ex}}{10}^{12}\mathrm{c})$ | ${\mu}_{AAV}$ PNV | 0 | 15.30 × ${10}^{-6}$ |

^{1}MNV—measurement noise value;

^{2}PNV—process noise value.

State Variable | Name | Value in CEP | Value in VVPP ^{1} |
---|---|---|---|

Xv | Viable cells | 0.2512 × ${10}^{6}$ c/mL | 1.0011 × ${10}^{6}$ c/mL |

GLC | Glucose | 32.19 mM | 26.7219 mM |

GLN | Glutamine | 5.03 mM | 4.0299 mM |

LAC | Lactate | 0.111 mM | 7.2925 mM |

AMM | Ammonium | 0.33 mM | 1.5469 mM |

AAV | AAV viral titer | 0 VG/mL | 0 VG/mL |

^{1}These values are related to the final EKF estimation for CEP with bioreactor 2 dataset; see Figure 5.

**Table 8.**RMSE values of the EKF and UMKM estimations regarding bioreactor 2 dataset in the EKF test.

State Variable | Name | RMSE of UMKM Estimation with Parameters Estimated by BI | RMSE of EKF Estimation |
---|---|---|---|

GLC | Glucose | 2.931 | 0.778 |

LAC | Lactate | 2.29 | 0.228 |

AAV | AAV viral titer | 2.616 | 0.355 |

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**MDPI and ACS Style**

Iglesias, C.F., Jr.; Xu, X.; Mehta, V.; Akassou, M.; Venereo-Sanchez, A.; Belacel, N.; Kamen, A.; Bolic, M.
Monitoring the Recombinant Adeno-Associated Virus Production using Extended Kalman Filter. *Processes* **2022**, *10*, 2180.
https://doi.org/10.3390/pr10112180

**AMA Style**

Iglesias CF Jr., Xu X, Mehta V, Akassou M, Venereo-Sanchez A, Belacel N, Kamen A, Bolic M.
Monitoring the Recombinant Adeno-Associated Virus Production using Extended Kalman Filter. *Processes*. 2022; 10(11):2180.
https://doi.org/10.3390/pr10112180

**Chicago/Turabian Style**

Iglesias, Cristovão Freitas, Jr., Xingge Xu, Varun Mehta, Mounia Akassou, Alina Venereo-Sanchez, Nabil Belacel, Amine Kamen, and Miodrag Bolic.
2022. "Monitoring the Recombinant Adeno-Associated Virus Production using Extended Kalman Filter" *Processes* 10, no. 11: 2180.
https://doi.org/10.3390/pr10112180