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Table 1 (abstract P-175). Equations of balances and kinetics of an employed process model including X v viable cell density, X t total cell density, μ cell-specific growth rate, μ d cell-specific death rate, t time, K S and k Monod kinetic constant and Monod constant for uptake, K Lys cell lysis constant, q cell-specific uptake rate or production rate, respectively, Y kinetic production constant, c concentration, Glc glucose, Gln glutamine, Lac lactate, Amm ammonia, F feed rate, V volume

From: Abstracts from the 25th European Society for Animal Cell Technology Meeting: Cell Technologies for Innovative Therapies

 

Balances with fed-batch terms

 

Kinetics

Biophase

\( \frac{{\boldsymbol{d}\boldsymbol{X}}_{\boldsymbol{v}}}{\boldsymbol{d}\boldsymbol{t}}=\left(\boldsymbol{\mu} -{\boldsymbol{\mu}}_{\boldsymbol{d}}\right)\bullet {\boldsymbol{X}}_{\boldsymbol{v}} \) \( -\frac{{\boldsymbol{F}}_{\boldsymbol{Glc}}+{\boldsymbol{F}}_{\boldsymbol{Gln}}}{\boldsymbol{V}}\bullet {\boldsymbol{X}}_{\boldsymbol{v}} \)

\( \frac{{\boldsymbol{dX}}_{\boldsymbol{t}}}{\boldsymbol{dt}}=\boldsymbol{\mu} \bullet {\boldsymbol{X}}_{\boldsymbol{v}}-{\boldsymbol{K}}_{\boldsymbol{Lys}}\bullet \left({\boldsymbol{X}}_{\boldsymbol{t}}-{\boldsymbol{X}}_{\boldsymbol{v}}\right) \) \( -\frac{{\boldsymbol{F}}_{\boldsymbol{Glc}}+{\boldsymbol{F}}_{\boldsymbol{Gln}}}{\boldsymbol{V}}\bullet {\boldsymbol{X}}_{\boldsymbol{t}} \)

Cell growth & death

\( \boldsymbol{\mu} ={\boldsymbol{\mu}}_{\boldsymbol{max}}\bullet \frac{{\boldsymbol{c}}_{\boldsymbol{Glc}}}{{\boldsymbol{c}}_{\boldsymbol{Glc}}+{\boldsymbol{K}}_{\boldsymbol{S},\boldsymbol{Glc}}}\bullet \frac{{\boldsymbol{c}}_{\boldsymbol{Gln}}}{{\boldsymbol{c}}_{\boldsymbol{Gln}}+{\boldsymbol{K}}_{\boldsymbol{S},\boldsymbol{Gln}}} \)

\( {\boldsymbol{\mu}}_{\boldsymbol{d}}={\boldsymbol{\mu}}_{\boldsymbol{d},\boldsymbol{\min}}+{\boldsymbol{\mu}}_{\boldsymbol{d},\boldsymbol{\max}}\bullet \frac{{\boldsymbol{K}}_{\boldsymbol{S},\boldsymbol{Glc}}}{{\boldsymbol{K}}_{\boldsymbol{S},\boldsymbol{Glc}}+{\boldsymbol{c}}_{\boldsymbol{Glc}}} \)

Liquid phase

\( \frac{{\boldsymbol{dc}}_{\boldsymbol{Glc}}}{\boldsymbol{dt}}=-{\boldsymbol{q}}_{\boldsymbol{Glc}}\bullet {\boldsymbol{X}}_{\boldsymbol{v}} \) \( +\frac{{\boldsymbol{F}}_{\boldsymbol{Glc}}}{\boldsymbol{V}}\bullet {\boldsymbol{c}}_{\boldsymbol{Glc},\boldsymbol{F}}-\frac{{\boldsymbol{F}}_{\boldsymbol{Glc}}+{\boldsymbol{F}}_{\boldsymbol{Gln}}}{\boldsymbol{V}}\bullet {\boldsymbol{c}}_{\boldsymbol{Glc}} \)

\( \frac{{\boldsymbol{dc}}_{\boldsymbol{Gln}}}{\boldsymbol{dt}}=-{\boldsymbol{q}}_{\boldsymbol{Gln}}\bullet {\boldsymbol{X}}_{\boldsymbol{v}} \) \( +\frac{{\boldsymbol{F}}_{\boldsymbol{Gln}}}{\boldsymbol{V}}\bullet {\boldsymbol{c}}_{\boldsymbol{Gln},\boldsymbol{F}}-\frac{{\boldsymbol{F}}_{\boldsymbol{Glc}}+{\boldsymbol{F}}_{\boldsymbol{Gln}}}{\boldsymbol{V}}\bullet {\boldsymbol{c}}_{\boldsymbol{Gln}} \)

\( \frac{{\boldsymbol{dc}}_{\boldsymbol{Lac}}}{\boldsymbol{dt}}={\boldsymbol{q}}_{\boldsymbol{Lac}}\bullet {\boldsymbol{X}}_{\boldsymbol{v}} \) \( -\frac{{\boldsymbol{F}}_{\boldsymbol{Glc}}+{\boldsymbol{F}}_{\boldsymbol{Gln}}}{\boldsymbol{V}}\bullet {\boldsymbol{c}}_{\boldsymbol{Lac}} \)

\( \frac{{\boldsymbol{dc}}_{\boldsymbol{Amm}}}{\boldsymbol{dt}}={\boldsymbol{q}}_{\boldsymbol{Amm}}\bullet {\boldsymbol{X}}_{\boldsymbol{v}} \) \( -\frac{{\boldsymbol{F}}_{\boldsymbol{Glc}}+{\boldsymbol{F}}_{\boldsymbol{Gln}}}{\boldsymbol{V}}\bullet {\boldsymbol{c}}_{\boldsymbol{Amm}} \)

Substrate uptake & metabolite production

\( {\boldsymbol{q}}_{\boldsymbol{Glc}}={\boldsymbol{q}}_{\boldsymbol{Glc},\boldsymbol{\max}}\bullet \frac{{\boldsymbol{c}}_{\boldsymbol{Glc}}}{{\boldsymbol{c}}_{\boldsymbol{Glc}}+{\boldsymbol{k}}_{\boldsymbol{Glc}}}\bullet \left(\frac{\boldsymbol{\mu}}{\boldsymbol{\mu} +{\boldsymbol{\mu}}_{\boldsymbol{max}}}+\mathbf{0.5}\right) \)

\( {\boldsymbol{q}}_{\boldsymbol{Gln}}={\boldsymbol{q}}_{\boldsymbol{Gln},\boldsymbol{\max}}\bullet \frac{{\boldsymbol{c}}_{\boldsymbol{Gln}}}{{\boldsymbol{c}}_{\boldsymbol{Gln}}+{\boldsymbol{k}}_{\boldsymbol{Gln}}} \)

\( {\boldsymbol{q}}_{\boldsymbol{Lac}}={\boldsymbol{Y}}_{\boldsymbol{Lac}/\boldsymbol{Glc}}\bullet \frac{{\boldsymbol{c}}_{\boldsymbol{Glc}}}{{\boldsymbol{c}}_{\boldsymbol{Lac}}}\bullet {\boldsymbol{q}}_{\boldsymbol{Glc}}-{\boldsymbol{q}}_{\boldsymbol{Lac},\boldsymbol{uptake}} \)

if c Glc  ≥ 0.5 mM : q Lac,uptake  = 0

if c Glc  < 0.5 mM : q Lac,uptake  = q Lac,uptake,max

q Amm  = YAmm/Gln ∙ q Gln