Arrhenius rate integrals can play an important role in thermal codes for biomedical applications and we have added the capability to TDiff and HeatWave. The purpose is find spatial regions where significant chemical changes have occured in applications like RF tumor ablation. The changes induced in tissues by heating are not a simple function of the maximum temperature. Biological damage depends on both temperature and the time over which heating is applied. The reaction rate for any endothermic chemical reaction can be approximated by the Arrhenius expression:
dn/dt = -A exp(-ΔE/RT) n.
In the equation, n is the number of entities (molecules, living cells, ...) that have not yet reacted, R is the universal gas constant (8.315 J/mol-degK) and T is the temperature in °K. The two reaction parameters A and ΔE have units of 1/s and J/mol respectively. The exponential form reflects the fact that endothermic reactions involves a quantum tunneling process.
The equation has the solution:
n(t)/n0 = exp[-Ω(t)]
where
Ω(t) = A ∫ dt' exp[-ΔE/RT(t')].
The intergral is taken from t' = 0.0 to t' = t. In biomedical applications the quantity Ω is often called the Arrhenius damage integral. A value Ω = 1 indicates that about 63% of the cells have been modified by the reaction.
During dynamic thermal solutions, TDiff and HeatWave determine the damage integral in all elements for which reaction-rate parameters have been defined. The postprocessors can plot spatial variations of Ω and can also determine surfaces of fixed Ω. One problem in implementing the capability is finding and interpreting values of A and ΔE. The first and second columns of table below list some values for mammalian tissue reported in the literature. Note that values of A vary by more than 200 orders of magnitude, and values of ΔE by a factor ~40. I have developed alternate forms of the reaction parameters for the thermal programs that have two advantages:
- They have values that are in a reasonable range (≤ 1000).
- They have easily-understood physical meanings.
We can rewrite the equation for the Arrhenius damage integral in the form
Ω(t) = ∫ dt' exp{Λ [1 - Tc/T(t')]}.
The two new parameters are related to the original ones by
Λ = ln(A)
and
Tc = ΔE/RΛ
These quantities are listed in columns 3 and 4 of the table. Note the values of the critical temperature Tc. Even though there are huge variations of A and ΔE between tissues, the critical temperature varies by less than 5%. Similarly, the temperature-range parameter Λ varies by only about a factor of 40.
We can understand the physical meanings of Tc and Λ by considering a system with fixed temperature T0. In this case, the equation for Ω has the form:
Ω = Δt exp[Λ (1 - Tc/T0)].
The equation shows that when T0 = Tc, the tissue reaches Ω = 1 in a time interval Δt = 1.0 s. In other words, at the critical temperature 63% of the cells are deactivated within 1 second. In general, higher temperatures are required to alter tissues with higher values of Tc.
To understand the meaning of Λ, suppose that we limit attention to a relatively small range of temperature near Tc such that
T0 = Tc - ΔT,
where ΔT ≪ Tc. In this case, we can solve for the time interval in the equation for Ω:
Δt ≅ Ω exp (Λ*ΔT/Tc).
For given values of Ω and the fractional temperature difference ΔT/Tc, the required heating time increases with higher values of Λ. As an illustration, consider calculation of the required heating time to reach Ω = 1 for liver tissue at 50 °C (T0 = 323.15 °K). Inserting parameters from the table into the above equation, we find that Δt = 62.4 s.
| Tissue |
A (1/s) |
ΔE (J/mol) |
Λ |
Tc (degK) |
| Liver |
7.39E39 |
2.58E5 |
91.8 |
337.7 |
| Bulk skin |
1.80E51 |
3.27E5 |
118.0 |
333.1 |
| Tendon (rat tail) |
6.66E79 |
5.21E5 |
183.8 |
341.1 |
| Tendon (rabbit patella) |
1.14E86 |
5.623E5 |
198.1 |
341.2 |
| Cell death |
2.98E80 |
5.06E5 |
185.3 |
328.7 |
| Microvascular blood flow |
1.98E106 |
6.67E5 |
244.8 |
327.7 |
| Protein coagulation |
7.39E37 |
2.58E5 |
87.2 |
355.4 |
| Epidermis |
3.10E98 |
6.27E5 |
226.8 |
328.1 |
| Porcine epidermis |
4.32E64 |
4.16E5 |
148.8 |
336.2 |
| Chordae tendinae |
1.30E53 |
3.57E5 |
122.3 |
351.0 |
| Porcine epidermis |
4.11E53 |
3.39E5 |
123.5 |
330.1 |
| Rat skin collagen |
1.61E45 |
3.06E5 |
104.1 |
353.5 |
| Rabbit muscle |
3.12E20 |
1.28E5 |
47.2 |
326.2 |
| Human aorta |
5.60E63 |
4.30E5 |
146.8 |
352.3 |
| Kangaroo tendon |
3.01E89 |
5.89E5 |
206.0 |
343.9 |
| Lens capsule |
3.85E137 |
8.60E5 |
316.8 |
326.5 |
| RIT |
6.66E79 |
5.21E5 |
183.8 |
340.9 |
| RIT (acetic acid) |
3.81E218 |
1.31E6 |
503.3 |
313.0 |
| RIT |
1.90E54 |
3.70E5 |
125.0 |
356.0 |
| Porcine cornea |
2.07E15 |
1.06E5 |
35.3 |
361.4 |
| Joint capsule |
4.00E5 |
3.40E4 |
12.9 |
317.0 |
| Joint capsule |
1.85E32 |
2.34E5 |
74.3 |
378.8 |
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