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A bootstrap-aggregated hybrid semi-parametric modeling framework for bioprocess development

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Abstract

Hybrid semi-parametric modeling, combining mechanistic and machine-learning methods, has proven to be a powerful method for process development. This paper proposes bootstrap aggregation to increase the predictive power of hybrid semi-parametric models when the process data are obtained by statistical design of experiments. A fed-batch Escherichia coli optimization problem is addressed, in which three factors (biomass growth setpoint, temperature, and biomass concentration at induction) were designed statistically to identify optimal cell growth and recombinant protein expression conditions. Synthetic data sets were generated applying three distinct design methods, namely, Box–Behnken, central composite, and Doehlert design. Bootstrap-aggregated hybrid models were developed for the three designs and compared against the respective non-aggregated versions. It is shown that bootstrap aggregation significantly decreases the prediction mean squared error of new batch experiments for all three designs. The number of (best) models to aggregate is a key calibration parameter that needs to be fine-tuned in each problem. The Doehlert design was slightly better than the other designs in the identification of the process optimum. Finally, the availability of several predictions allowed computing error bounds for the different parts of the model, which provides an additional insight into the variation of predictions within the model components.

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Correspondence to Moritz von Stosch.

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Appendix

Appendix

A: E. coli simulation fed-batch model

The model describes the production of viral capsid protein by a recombinant E. coli strain in a fed-batch bioreactor. This model has been proposed by [21], which is an adaptation of the model by [27]. The model comprises the material balances for biomass, substrate, and product concentration as well as the overall mass balance in a stirred tank bioreactor:

$$\frac{{{\text{d}}X}}{{{\text{d}}t}} = \mu \cdot X - D \cdot X,$$
(7)
$$\frac{{{\text{d}}S}}{{{\text{d}}t}} = - v_{S} \cdot X - D \cdot \left( {S - S_{f} } \right),$$
(8)
$$\frac{{{\text{d}}P}}{{{\text{d}}t}} = v_{P} \cdot X - D \cdot P,$$
(9)
$$\frac{{{\text{d}}W}}{{{\text{d}}t}} = u_{F} ,$$
(10)

with \(\mu\), \(v_{S}\), and \(v_{P}\) the specific rates of biomass growth (1/h), substrate uptake (1/h), and product formation (U/g/h), \(X\), \(S\), and \(P\) the biomass (g/kg), substrate (g/kg), and product concentrations (U/kg), \(D = u_{F} /W\) (1/h) the dilution rate, and \(u_{F}\) the feeding rate (kg/h).

The specific biomass growth rate was modeled using the expression:

$$\mu = \mu_{\text{max} } \cdot \frac{S}{{S + K_{S} }} \cdot \frac{{K_{i} }}{{S + K_{i} }} \cdot \exp \left( {\alpha \cdot \left( {T - T_{\text{ref}} } \right)} \right),$$
(11)

with \(\mu_{ \text{max} }\) = 0.737 (1/h), \(K_{S}\) = 0.00333 (g/kg), \(K_{i} = 93.8\) (g/kg), \(\alpha = 0.0495\) (1/C), \(T_{\text{ref}} = 37\) (°C), and \(T\) (°C) the temperature of the culture broth.

The specific substrate uptake rate is modeled via:

$$v_{S} = \frac{1}{{Y_{XS} }} \cdot \mu + m,$$
(12)

with \(Y_{XS} = 0.46\) (g/g) and \(m = 0.0242\) (g/g/h).

The specific product formation rate is modeled by

$$v_{P} = \frac{{I_{D} }}{{T_{\text{PX}} }} \cdot \left( {\frac{{v_{P,\text{max} ,T} \cdot \mu \cdot k_{m} }}{{k_{\mu } + \mu + \mu^{2} /k_{i\mu } }} - p_{X} } \right),$$
(13)

with

$$v_{P,\text{max} ,T} = \frac{{5 \cdot 10^{10} \cdot \exp \left( {\frac{{ - A_{\text{eng}} }}{{R \cdot \left( {T + 273.15} \right)}}} \right)}}{{1 + 3 \cdot 10^{93} \cdot \exp \left( {\frac{{ - R_{\text{eng}} }}{{R \cdot \left( {T + 273.15} \right)}}} \right)}},$$
(14)

with \(A_{\text{eng}} = 62\) (kJ/mol), \(R_{\text{eng}} = 551\) (kJ/mol), \(R = 8.3144e - 3\) (kJ/mol/K), \(T_{PX} = 1.495\)(h),\(p_{X} = 50\)(U/g),\(k_{\mu } = 0.61\)(1/h), \(k_{m} = 751\)(U/g), \(k_{i\mu } = 0.0174\) (1/h), and the induction parameter \(I_{D} = 0\) before induction and \(I_{D} = 1\) afterwards.

For the feeding rate, an exponential profile was adopted to match a desired constant specific biomass growth, \(\mu_{\text{set}}\), that is

$$u_{F} = \frac{1}{{S_{f} \cdot Y_{XS} }} \cdot \mu_{\text{set}} \cdot X_{0} \cdot W_{0} \cdot \exp \left( {\mu_{\text{set}} \cdot \left( {t - t_{0} } \right)} \right),$$
(15)

where \(X_{0} = X\left( {t_{0} } \right)\) (g/kg) is the initial biomass concentration and \(W_{0} = W\left( {t_{0} } \right)\) (kg) is the initial weight of the culture broth.

The process was divided into two phases, a growth and a production phase. During the growth phase, \(\mu_{\text{set}} = 0.3\) (h−1) and \(T = 34\) (C). The duration of the growth phase was adapted to yield the initial biomass concentration, Xind, set out by the DoEs. The substrate concentration in the feeding solution was set to \(S_{f} = 300\) (g/kg). Data for online variables were logged every 6 min. The biomass and product concentrations (offline variables) were measured 20 times during each fermentation. The data were corrupted with 5% Gaussian (white) noise.

B: E. coli hybrid semi-parametric model

The parametric part of the hybrid model is based on the material balance equations of biomass and product, that is

$$\frac{{{\text{d}}X}}{{{\text{d}}t}} = \mu \cdot X - D \cdot X,$$
(16)
$$\frac{{{\text{d}}P}}{{{\text{d}}t}} = v_{p} \cdot X - D \cdot P,$$
(17)

where \(D\) is the dilution rate, \(X\) and \(P\) are the biomass and product concentrations (to note that the hybrid model does not consider substrate dynamics), with specific reaction rates \(\mu\) and \(v_{p}\). Thus, the volumetric rate Eq. (2) simplifies as follows for the present problem:

$$r\left( {c,x} \right) = \left[ {\mu ,v_{p} } \right]^{\text{T}} \cdot X.$$
(18)

The specific rates \(\mu\) and \(v_{p}\) are much more difficult to establish; thus, they were modeled by a simple feedforward neural network with three layers only:

$$g = \left[ {\mu ,v_{p} } \right]^{\text{T}} = w^{2,1} \cdot \tanh \left( {w^{1,1} \cdot f\left( {c,x} \right) + w^{1,2} } \right) + w^{2,2} ,$$
(19)

with \(w = \{ w^{1,1} ,w^{1,2} ,w^{2,2} ,w^{2,2} \}\). The network has only three inputs, namely, biomass, \(X\), the feeding rate, \(F\), and cultivation temperature, \(T\). Thus, the pre-processing function (Eq. (3)) reduces to the following form:

$$f\left( {c,x} \right) = \left[ {X,F,T} \right]^{\text{T}} .$$
(20)

Preliminary tests have shown that five neurons in the hidden layer are optimal for the present case study data set used, which corresponds to \(\dim \left( w \right) = 4 \times 5 + 6 \times 2 = 32\) parameters to be identified in each run. The number of hidden nodes of the neural network was thus selected to be five in all studies performed.

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Pinto, J., de Azevedo, C.R., Oliveira, R. et al. A bootstrap-aggregated hybrid semi-parametric modeling framework for bioprocess development. Bioprocess Biosyst Eng 42, 1853–1865 (2019). https://doi.org/10.1007/s00449-019-02181-y

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Keywords

  • Hybrid semi-parametric modeling
  • Hybrid modeling
  • Bagging
  • Design of experiments
  • Sampling error
  • Data portioning
  • Ensemble methods