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# Thermal Therapy in Urologic Systems: A Comparison of Arrhenius and Thermal Isoeffective Dose Models in Predicting Hyperthermic Injury

[+] Author and Article Information
Xiaoming He

Department of Mechanical Engineering, and Biomedical Engineering Program, University of South Carolina, Columbia, SC 29208xmhe@sc.edu

Sankha Bhowmick

Department of Mechanical Engineering, and Biomedical Engineering and Biotechnology Program, University of Massachusetts Dartmouth, North Dartmouth, MA 02747

John C. Bischof

Department of Mechanical Engineering, Department of Urologic Surgery, and Department of Biomedical Engineering, University of Minnesota, Minneapolis, MN 55455bischof@umn.edu

The term ‘kinetics parameters’ rather than ‘kinetic parameters’ is used in this paper in accordance with one of the reviewer’s suggestions. In the thermal injury literature, however, ‘kinetic parameters’ is more commonly used than ‘kinetics parameters’.

For example, a survival of 30% was used to determine injury rate and linear fitting to the survival versus heating time was performed in Ref. 26. A survival of 90% was used to determine injury rate and, instead of Eq. 3, a linear fitting to the survival versus heating time was performed in Ref. 27. A survival of 10% and 1% was used to determine injury rate in Refs. 29,42, respectively.

After carefully examining the original data of survival versus exposure time, only data below $55°C$ were used in this study because the cell survival above $55°C$ for all exposure times studied was minimal $(<10%)$, indicating injury saturation. In other words, cell survival is so low that the injury measurement is not sensitive enough to tell the difference at different exposure times when the temperature is above $55°C$. The injury saturation is actually suggested by the time-temperature relationships predicted using the previously reported kinetics parameters which shows that the exposure time required for complete cell death does not change much even though the temperature increases from $50°C$ to $70°C$ (see Fig. 2).

Note that the use of different numbers of significant decimal points for temperature conversion from Celsius to Kelvin (i.e., 273 versus 273.15) and for the universal gas constant $Rg$ (i.e., 1.98 versus $1.986 cal mole−1 K−1$) may greatly affect the prediction of thermal injury as well. These constants should be kept consistent with those used for extracting the kinetics parameters to estimate thermal injury using the Arrhenius model.

Of note, a correction factor of 0.5 is used in the formula for the calculated $R$ values in Table 2 based on the observation that an average of $R$ calculated using Eq. 8 and $R45$ might be able to give a better prediction of thermal threshold than either $R$ value alone, as discussed in the previous paragraph.

J Biomech Eng 131(7), 074507 (Jun 05, 2009) (12 pages) doi:10.1115/1.3128671 History: Received November 09, 2008; Revised January 23, 2009; Published June 05, 2009

## Abstract

The Arrhenius and thermal isoeffective dose (TID) models are the two most commonly used models for predicting hyperthermic injury. The TID model is essentially derived from the Arrhenius model, but due to a variety of assumptions and simplifications now leads to different predictions, particularly at temperatures higher than $50°C$. In the present study, the two models are compared and their appropriateness tested for predicting hyperthermic injury in both the traditional hyperthermia (usually, $43–50°C$) and thermal surgery (or thermal therapy/thermal ablation, usually, $>50°C$) regime. The kinetic parameters of thermal injury in both models were obtained from the literature (or literature data), tabulated, and analyzed for various prostate and kidney systems. It was found that the kinetic parameters vary widely, and were particularly dependent on the cell or tissue type, injury assay used, and the time when the injury assessment was performed. In order to compare the capability of the two models for thermal injury prediction, thermal thresholds for complete killing (i.e., 99% cell or tissue injury) were predicted using the models in two important urologic systems, viz., the benign prostatic hyperplasia tissue and the normal porcine kidney tissue. The predictions of the two models matched well at temperatures below $50°C$. At higher temperatures, however, the thermal thresholds predicted using the TID model with a constant $R$ value of 0.5, the value commonly used in the traditional hyperthermia literature, are much lower than those predicted using the Arrhenius model. This suggests that traditional use of the TID model (i.e., $R=0.5$) is inappropriate for predicting hyperthermic injury in the thermal surgery regime $(>50°C)$. Finally, the time-temperature relationships for complete killing (i.e., 99% injury) were calculated and analyzed using the Arrhenius model for the various prostate and kidney systems.

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## Figures

Figure 1

An example showing how to extract kinetics parameters in the Arrhenius model using the isothermal method: (a) determining the cell injury rate k at various temperatures by fitting the injury model (i.e., Eq. 3) to cell survival data, and (b) determining the kinetics parameters by linear fitting using Eq. 4. The activation energy and frequency factor can be determined from the slope and the intercept of the line, respectively.

Figure 2

A comparison of the time-temperature relationship for complete killing at 3 h post-thermal therapy (assuming 99% injury as the criterion) in thermal therapy of in vitro AT-1 prostate cancer tissue predicted using Eq. 3 and the kinetics parameters determined in this study (circle), those reported previously in Ref. 27 (triangle), and those corrected from the reported using Eq. 6 (asteroid)

Figure 3

A comparison of the predictions of thermal injury in human benign prostatic hyperplasia (BPH) tissue assayed by histology and chronic necrosis of normal porcine kidney (PK) tissue assayed by histology (see Table 2) using the Arrhenius model and thermal isoeffective dose (TID) model with the R parameter evaluated in three different ways: 0.5, R45, and Eq. 8

Figure 4

A comparison of the R values in the TID model for thermal injury in three different model systems: attached AT-1 cells assayed by clonogenics, BPH assayed by histology, and chronic necrosis of normal porcine kidney tissue assayed by histology (see Table 2)

Figure 5

Relationship between the activation energy and the natural logarithm of the frequency factor in the Arrhenius model: the kinetics parameters in Table 2 match well with the master line reported in Ref. 5

Figure 6

The exposure time-temperature relationship for complete killing (assuming 99% injury as the criterion) of various prostate systems (i.e., protein, cell, and tissue) predicted by the Arrhenius model (Eq. 3) using the kinetics parameters given in Table 2: (a) overall comparison of all the available prostate systems; (b) comparison between in situ protein and suspended AT-1 cells; (c) comparison between AT-1 tissue and suspended and attached AT-1 cells; (d) comparison between in vitro and in vivo AT-1 tissue; and (e) comparison between in vitro AT-1 and in vitro human BPH tissue. The dotted lines with different symbols are for protein, the dashed and dotdashed lines with different symbols are for cells, and the solid lines with different symbols are for tissue. Histology, dye uptake, and clonogenics assays are represented by the colors red (lines 1 and 9), blue (lines 2–8), and black (lines 10–12), respectively. The numbers in the figure represent different prostate systems, time point of injury assessment, and injury assay: 1, human BPH tissue: in vitro, 3 day culture, histology; 2, human BPH tissue: in vitro, 3 day culture, EthD; 3, protein in AT-1 cells: ≤59°C in Ref. 25; 4, AT-1 cells (attached): 3 h culture, PI; 5, AT-1 cells (suspended): 3 h culture, PI; 6, AT-1 tissue: in vitro, 1 day culture, EthD; 7, AT-1 tissue: in vitro, 3 h culture, EthD; 8, AT-1 tissue: in vitro, 3 day culture, EthD; 9, AT-1 tissue: in vivo, 3 days in host, histology; 10, AT-1 cells (attached): clonogenics; 11, protein in AT-1 cells: ≤52°C in Ref. 25; and 12, AT-1 cells (suspended): clonogenics.

Figure 7

The exposure time-temperature relationship for complete killing (assuming 99% injury as the criterion) of various kidney systems (i.e., cell and tissue) predicted using the Arrhenius model (Eq. 3) with the kinetics parameters given in Table 2: (a) an overall comparison of the available kidney systems, (b) comparison between various single cell systems, and (c) comparison of in vitro versus in vivo tissue injury in normal porcine kidneys. The dashed lines with different symbols are for cells and the solid lines with different symbols are for tissue. Histology, dye uptake, and clonogenics assays are represented by the color red (lines 2, 3 and 5), blue (lines 1, 4, 6, and 7), and black (lines 8–9), respectively. The numbers in the figure represent different kidney systems, time point of injury assessment, and injury assay: 1, human RCC A498 cells (attached), 1 day culture, EthD; 2, porcine kidney (PK) tissue: thermal fixation; 3, porcine kidney tissue: 2 day culture, acute necrosis; 4, human RCC SN12 cells (attached): 3 h culture, PI; 5; porcine kidney tissue: 7 days in host, chronic necrosis; 6, human RCC SN12 cells (suspended): 3 h culture, PI; 7, VX9 cells: 1 day culture, EthD; 8, porcine kidney cells (attached): clonogenics; and 9, baby hamster kidney (BHK) cells (attached): clonogenics.

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