SME
Biotransport · Problem type
How deep oxygen reaches
Use this for “How thick can an engineered tissue be before its center runs out of oxygen?”
Chart
Vertex (200, 0), axis of symmetry x = 200
This is an example. Type your own values to start a new problem.
L (μm)
=
2
×
D (cm²/s)
×
C₀ (mM)
Q (mol/(L·s))
Formulas
L
=
10,000
×
√(2
×
D
×
C₀
1,000 ×
Q
)
200
=
10,000
×
√(2
×
(2
×
10
-5
)
×
0.2
1,000 × (2 ×
10
-5
)
)
Assumptions
•
A flat layer is fed oxygen from one face only; none leaves through the far side.
•
Steady state: the cells use oxygen at the same rate Q everywhere, and nothing changes with time.
•
The profile obeys D d²C/dx² = Q, so it is the parabola C = C₀(1 − x ÷ L)², falling to zero at L.
•
Tissue thicker than L has a core with no oxygen; thinner tissue has some left at its far side.
Step-by-step
WE KNOW
C₀ = 0.2 mM, D = 2 × 10⁻⁵ cm²/s, Q = 2 × 10⁻⁵ mol/(L·s)
FIND
critical depth (L)
Step 1 · Find critical depth (L)
Use L = 10,000 × √(2 × D × C₀ ÷ (1,000 × Q))
Oxygen fed in at the face, 2DC₀, just covers what the layer uses, QL². The 1000 turns mM into mol per liter; × 10,000 turns cm into μm.
L
=
10,000
×
√(2
×
(2
×
10
-5
)
×
0.2
1,000 × (2 ×
10
-5
)
)
L
=
10,000
×
√(2
×
(2
×
10
-5
)
×
0.2
0.02
)
L
=
10,000
×
√((4
×
10
-5
)
×
0.2
0.02
)
L
=
10,000
×
√(
8 ×
10
-6
0.02
)
L
=
10,000
×
√
0.0004
L
=
10,000
×
0.02
L
=
200
μm
Check
Put the numbers back in. Both sides should match.
200
=
10,000
×
√(2
×
(2
×
10
-5
)
×
0.2
1,000 × (2 ×
10
-5
)
)
✓
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