Cell-mass maximization in fed-batch cultures

Abstract

Complete solutions are provided for cell-mass maximization for free and fixed final times and constant and variable yields. The optimal feed rate profile is a concatenation of maximum, minimum and singular feed rates. The exact sequence and duration of each feed rate depends primarily on the initial substrate concentration, and degenerate cases arise due to the magnitude constraint on the feed rate and the length of final time t _f. When the final time is free and not in the performance index, it is infinite for constant yield so that any form of feed rate leads to the same amount of cells, while for variable yield the singular feed rate is exponential and maximizes the yield. For fixed final time the singular feed rate for constant yield is exponential and maximizes the specific growth rate by maintaining the substrate concentration constant, while for variable yield, it is semi-exponential and the substrate concentration starts near the maximum specific growth rate and moves toward the maximum yield. A simple sufficient condition for existence of singular feed rate requires an existence of a region bounded by the maxima of specific growth and cellular yield. Otherwise, the optimal feed rate profile is a bang-bang type and the bioreactor operates in batch mode.

Appendix

Proof that zero substrate concentration at the final time S(t _f) =  0 implies that the final time t _f is infinity

During the filling stage, 0 ≤  t ≤  t _F, F >  0, the substrate concentration cannot be zero. Therefore, we consider the subsequent batch period, t _F <  t ≤ t _f in which F =  0 and V =  V(t _f) =  V _max. The mass balance equations are

$$ \dot{X} = \mu (S)X $$

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$$ \dot{S} = - \sigma (S)X $$

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$$ \dot{V} = 0. $$

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Since the specific rates μ (S)andσ (S) are usually ratios of polynomials in S and must vanish at S =  0, we can rewrite, for example,

$$ \sigma (S) = \frac{{SQ(S)}}{{P(S)}} $$

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where P(S) and Q(S) are polynomials in S and the degree of P(S) is greater than that of SQ(S). Substituting Eq. 4 into 2 and rearranging, we obtain

$$ {\int\limits_{S(t_{{\rm F}})}^{S(t)} {\frac{{P(S)}}{{SQ(S)}}}}{\rm d}S = - {\int\limits_{t_{{\rm F}}}^t {X{\rm d}\tau}}. $$

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Using Heaviside expansion we can expand the left-hand side (LHS) assuming the roots of Q(S), a _i , i = 1,2,3,...,n are non-repeating (case of repeating roots can be similarly handled),

$$ \begin{aligned} {\int\limits_{S(t_{{\rm F}})}^{S(t)} {\frac{{P(S)}}{{SQ(S)}}}}{\rm d}S &= {\int\limits_{S(t_{{\rm F}})}^{S(t)} {{\sum\limits_{i = 1}^n {\frac{{P(a_{i})}}{{a_{i} {Q}^{\prime}(a_{i}) + {Q}^{\prime}(a_{i})}}}}}}\frac{1}{{S - a_{i}}}{\rm d}S \\&= {\int\limits_{S(t_{{\rm F}})}^{S(t)} {\frac{{P(0)}}{{{Q}^{\prime}(0)}}\frac{1}{S}{\rm d}S}} + {\int\limits_{S(t_{{\rm F}})}^{S(t)} {{\sum\limits_{i = 2}^n {\frac{{P(a_{i})}}{{a_{i} {Q}^{\prime}(a_{i}) + {Q}^{\prime}(a_{i})}}}}}}\frac{1}{{S - a_{i}}}{\rm d}S \\&= \frac{{P(0)}}{{{Q}^{\prime}(0)}}\ln \frac{{S(t)}}{{S(t_{{\rm F}})}} + {\sum\limits_{i = 2}^n {\frac{{P(a_{i})}}{{a_{i} {Q}^{\prime}(a_{i}) + {Q}^{\prime}(a_{i})}}}}\ln \left[\frac{{S(t) - a_{i}}}{{S(t_{{\rm F}}) - a_{i}}}\right] \\ \end{aligned} $$

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which at t =  t _f reduces to

$$ \frac{{P(0)}}{{{Q}^{\prime}(0)}}\ln \frac{{S(t_{{\rm f}})}}{{S(t_{{\rm F}})}} + {\sum\limits_{i = 2}^n {\frac{{P(a_{i})}}{{a_{i} {Q}^{\prime}(a_{i}) + {Q}^{\prime}(a_{i})}}}}\ln \left[\frac{{S(t_{{\rm f}}) - a_{i}}}{{S(t_{{\rm F}}) - a_{i}}}\right] = - {\int\limits_{t_{{\rm F}}}^{t_{{\rm f}}} {X{\rm d}t}} = - X(t_{{\rm F}}){\int\limits_{t_{{\rm F}}}^{t_{{\rm f}}} {\exp {\int\limits_{t_{{\rm F}}}^t {\mu {\rm d}\tau}}{\rm d}t}} $$

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where the last equality is obtained by integrating Eq. 1. If S(t _f) =  0, the LHS is negative infinity. Therefore, in order for the RHS to be negative infinity the final time t _f must be infinity since the integrand is finite. In other words, the final time must be infinity, t _f =  ∞, in order for the substrate concentration to reach a zero value, S(t _f) =  0. This conclusion holds true, unless there is a zero-order term in specific rates due to a maintenance term or cell lysis.