Damaged Tissue
Add a Thermal Damage subnode under the Biological Tissue node to calculate tissue damage.
In hyperthermia and cryogenic processes, tissue necrosis (permanent damage or death of living tissue) occurs when one of the two following phenomena happens:
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An excessive thermal energy is absorbed (hyperthermia process) or released (cryogenic process)
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A critical high (hyperthermia process) or low (cryogenic process) temperature is exceeded
Correspondingly, COMSOL Multiphysics has two ways to model energy absorption — computing the period of time the tissue remained in the necrotic temperature interval and direct time integration of the energy.
Temperature Threshold
In the first form of damage integral, tissue necrosis occurs in four cases:
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When the temperature exceeds the hyperthermia damage temperature Td, h for more than a certain time period td, h,
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When the temperature falls below the cryogenic damage temperature Td, c for more than a certain time period td, c,
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Instantly after the temperature exceeds the hyperthermia necrosis temperature Tn, h,
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Instantly after the temperature falls below the cryogenic necrosis temperature Tn, c.
For the first two cases, the damaged tissue indicator, α, defined either by
for hyperthermia analysis, or by
for cryogenic analysis, with
is the ratio of the period of time when T > Td, h to the time limit td, h, or the ratio of the period of time when T < Td, c to the time limit td, c. It gives an indication of damage state of the tissue. When it reaches 1, the tissue is necrotic. The fraction of necrotic tissue corresponds to the quantity min(α, 1).
For the last two cases, the necrosis time indicator, αnecr, defined either by
for hyperthermia analysis, or by
for cryogenic analysis, with
evaluates the period of time when T > Tn, h or the period of time when T < Tn, c. If αnecr > 0, the tissue is necrotic because it already reached the necrosis temperatures Tn, h or Tn, c at some time step of the simulation. Hence, the fraction of necrotic tissue due to immediate necrosis is equal to 1 if αnecr > 0 and 0 otherwise.
Combining all cases, the overall fraction of necrotic tissue, θd, is equal to:
(4-38)
Arrhenius Kinetics
The second form of damage integral is applicable only for hyperthermia processes and provides the degree of tissue injury, α, based on the polynomial Arrhenius equation:
Here, A is the frequency factor (SI unit: 1/s), and ΔE is the activation energy for the irreversible damage reaction (SI unit: J/mol). The parameters A and ΔE are dependent on the type of tissue and have been characterized for liver tissues by Jacques et others (Ref. 7) to be A = 7.39 ⋅ 1039 s–1 and ΔE = 2.577 ⋅ 105 J/mol. See Ref. 8, Ref. 9, and Ref. 10 for the characterization of these parameters for prostate, skin, and fat. See also Ref. 11 and Ref. 12 for more references on biological tissues material properties.
The fraction of necrotic tissue is then expressed by:
(4-39)
Thermal Properties
The material properties of the damaged tissue are redefined to take into account the influence of tissue injury. If ρd, Cp, d, and kd denote the density, heat capacity at constant pressure, and thermal conductivity of the necrotic tissue, respectively, then two effective quantities are defined:
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The effective thermal conductivity, keff = θdkd + (1 − θd)k
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The effective heat capacity at constant pressure, (ρCp)eff = θdρdCp, d + (1 − θd)ρCp
In these equalities, θd takes one of the two definitions given above in Equation 4-38 or Equation 4-39 according to the integral form chosen.
Heat Source
A cooling or heating source is associated with the reaction leading to damage of tissue. Depending on the damage integral model, this source is expressed as follows:
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Temperature threshold for hyperthermia analysis:
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Temperature threshold for cryogenic analysis:
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Energy absorption: