Field Precision title

Arrhenius rate integrals in finite-element thermal solutions

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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