SME
Tissue Engineering
Bioreactors
Use this for “C∗ = 0.21 mM, kLa = 5 h⁻¹, and 2 × 10⁶ cells/mL each use 0.2 pmol/h. What is the dissolved oxygen, and how many cells can the vessel feed?”
Diagram
OUR = qX = 0.2 × 2 × 10⁶ × 10⁻⁶ = 0.4 mM/h.
Supply
equals
uptake:
C
=
C∗
−
OUR
kLa
=
0.21
−
0.4
5
=
0.13
mM.
Xₘₐₓ
=
kLa·C∗
÷
q
=
5.25 ×
10
6
cells/mL.
This is an example. Type your own values to start a new problem.
OUR (mM/h)
=
q (pmol/(cell·h))
×
X (cells/mL)
C (mM)
=
C∗ (mM)
−
OUR (mM/h)
÷
kLa (h⁻¹)
Xₘₐₓ (cells/mL)
=
kLa (h⁻¹)
×
C∗ (mM)
÷
q (pmol/(cell·h))
Formulas
OUR
=
q
×
X
10
6
0.4
=
0.2
×
2 ×
10
6
10
6
C
=
C
∗
−
OUR
kLa
0.13
=
0.21
−
0.4
5
Xₘₐₓ
=
kLa
×
C
∗
×
10
6
q
5.25 ×
10
6
=
5
×
0.21 ×
10
6
0.2
Assumptions
•
Steady state: the gas dissolves oxygen at kLa(C∗ − C), just as fast as the cells use it.
•
The medium is well mixed, so the dissolved oxygen C is the same everywhere in the vessel.
•
Every cell takes the same q, so the uptake rate is OUR = qX.
•
1 pmol per cell per hour times cells per mL, divided by 10⁶, is mM per hour.
Step-by-step
WE KNOW
C∗ = 0.21 mM, kLa = 5 h⁻¹, q = 0.2 pmol/(cell·h), X = 2 × 10⁶ cells/mL
FIND
oxygen uptake rate (OUR), dissolved oxygen (C), highest density fed (Xₘₐₓ)
Step 1 · Find oxygen uptake rate (OUR)
Use OUR = q × X ÷ 10⁶
Each
cell
takes
q
;
X
cells
in
each
mL
take
q
×
X
pmol
per
mL
each
hour.
OUR
=
0.2
×
2 ×
10
6
10
6
OUR
=
400,000
10
6
OUR = 0.4 mM/h
Step 2 · Find dissolved oxygen (C)
Use C = C∗ − OUR ÷ kLa
Supply
equals
uptake:
kLa(
C
∗
−
C
)
=
OUR,
so
the
oxygen
sits
OUR
kLa
below
saturation.
C
=
0.21
−
0.4
5
C
=
0.21
−
0.08
C
=
0.13
mM
Step 3 · Find highest density fed (Xₘₐₓ)
Use Xₘₐₓ = kLa × C∗ × 10⁶ ÷ q
C
falls
to
0
when
the
uptake
reaches
the
most
the
gas
can
supply,
kLa·
C
∗.
Xₘₐₓ
=
5
×
0.21 ×
10
6
0.2
Xₘₐₓ
=
1,050,000
0.2
Xₘₐₓ
=
5.25 ×
10
6
cells/mL
Check
Put the numbers back in. Both sides should match.
0.4
=
0.2
×
2 ×
10
6
10
6
✓
0.13
=
0.21
−
0.4
5
✓
5.25 ×
10
6
=
5
×
0.21 ×
10
6
0.2
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.