Model Theory

Overview

perfusio implements the self-driving bioprocess methodology of Gadiyar et al. (2026). The core model is a hybrid state-space model that combines:

  1. A mechanistic skeleton (ODEs for CHO cell culture kinetics), and

  2. A Gaussian Process residual layer (Stepwise-GP, SW-GP) that corrects for model misspecification.

Mechanistic Model

The CHO perfusion model tracks \(n = 9\) state variables:

\[ \mathbf{x}(t) \in \mathbb{R}^{9} \]

Index

Symbol

Units

0

\(X_v\) (VCD)

\(10^6\) cells mL\(^{-1}\)

1

\(X_d\) (Viability)

fraction

2

\(S_{\text{glc}}\) (Glucose)

g L\(^{-1}\)

3

\(S_{\text{gln}}\) (Glutamine)

mmol L\(^{-1}\)

4

\(S_{\text{glu}}\) (Glutamate)

mmol L\(^{-1}\)

5

\(S_{\text{lac}}\) (Lactate)

mmol L\(^{-1}\)

6

\(S_{\text{amm}}\) (Ammonia)

mmol L\(^{-1}\)

7

\(S_{\text{pyr}}\) (Pyruvate)

mmol L\(^{-1}\)

8

\(P\) (Titer)

mg L\(^{-1}\)

Growth Kinetics

Specific growth rate (dual Monod with inhibition):

\[ \mu = \mu_{\max} \cdot \frac{S_{\text{glc}}}{K_S + S_{\text{glc}}} \cdot \frac{S_{\text{gln}}}{K_N + S_{\text{gln}}} \cdot \left(1 - \frac{X_v}{X_{\max}}\right) \]

with \(\mu_{\max} = 0.040\;\mathrm{h}^{-1}\), \(K_S = 0.15\), \(K_N = 0.04\).

Glucose Consumption

Pirt maintenance + growth-coupled:

\[ q_S = \frac{\mu}{Y_{XS}} + m_S \]

with Warburg switch when \(S_{\text{lac}} > L_{\text{thresh}}\).

Product Formation

Luedeking–Piret kinetics:

\[ q_P = \alpha \mu + \beta \]

Mass Balances (continuous perfusion)

\[ \frac{dX_v}{dt} = (\mu - \mu_d - D_b)\,X_v \]
\[ \frac{dS}{dt} = D_f\,(S_f - S) - D_h\,S - q_S\,X_v \]

where \(D_f\) is the perfusion (feed) dilution rate and \(D_h = D_f - D_b\) the harvest dilution rate.

Stepwise-GP Residual Layer

The SW-GP predicts the next absolute state \(\mathbf{c}_{t+1}\) directly (not a rate residual), trained on one-step pairs \((\mathbf{c}_t, \mathbf{u}_t, t) \to \mathbf{c}_{t+1}\):

\[ \mathbf{c}_{t+1} = f_{\text{GP}}(\mathbf{c}_t,\;\mathbf{u}_t,\;t) \]

The hybrid model decomposes predictions as:

\[ \mathbf{c}_{t+1} = \underbrace{\mathbf{c}_t + \Delta t\,\mathbf{r}_{\text{mech}}}_{\text{mechanistic Euler}} + \underbrace{(\hat{\mathbf{c}}_{t+1} - \mathbf{c}_{\text{mech}})}_{\text{GP residual}} \]

where \(\hat{\mathbf{c}}_{t+1}\) is the GP posterior mean and \(\mathbf{c}_{\text{mech}}\) is the mechanistic Euler prediction. This form ensures the mechanistic prior anchors extrapolation while the GP corrects in-distribution errors.

Rollout is performed either via moment-matching (fast, propagates mean and variance analytically) or Monte Carlo (unbiased, default \(S = 100\) paths).

References

  • Gadiyar, C. J., et al. (2026). Biotechnology and Bioengineering, 123(2), 391–405.

  • Hutter, S., et al. (2021). Computers & Chemical Engineering, 151, 107373.

  • Cruz-Bournazou, M. N., et al. (2022). Digital Chemical Engineering, 1, 100005.