Research Papers

Improving Accuracy in Arrhenius Models of Cell Death: Adding a Temperature-Dependent Time Delay

[+] Author and Article Information
John A. Pearce

Temple Foundation Professor (#3)
Department of Electrical and Computer
The University of Texas at Austin,
1616 Guadalupe Street UTA 7.352,
Austin, TX 78701
e-mail: jpearce@mail.utexas.edu

1Corresponding author.

Manuscript received July 20, 2015; final manuscript received October 15, 2015; published online November 2, 2015. Assoc. Editor: Ram Devireddy.

J Biomech Eng 137(12), 121006 (Nov 02, 2015) (7 pages) Paper No: BIO-15-1362; doi: 10.1115/1.4031851 History: Received July 20, 2015; Revised October 15, 2015

The Arrhenius formulation for single-step irreversible unimolecular reactions has been used for many decades to describe the thermal damage and cell death processes. Arrhenius predictions are acceptably accurate for structural proteins, for some cell death assays, and for cell death at higher temperatures in most cell lines, above about 55 °C. However, in many cases—and particularly at hyperthermic temperatures, between about 43 and 55 °C—the particular intrinsic cell death or damage process under study exhibits a significant “shoulder” region that constant-rate Arrhenius models are unable to represent with acceptable accuracy. The primary limitation is that Arrhenius calculations always overestimate the cell death fraction, which leads to severely overoptimistic predictions of heating effectiveness in tumor treatment. Several more sophisticated mathematical model approaches have been suggested and show much-improved performance. But simpler models that have adequate accuracy would provide useful and practical alternatives to intricate biochemical analyses. Typical transient intrinsic cell death processes at hyperthermic temperatures consist of a slowly developing shoulder region followed by an essentially constant-rate region. The shoulder regions have been demonstrated to arise chiefly from complex functional protein signaling cascades that generate delays in the onset of the constant-rate region, but may involve heat shock protein activity as well. This paper shows that acceptably accurate and much-improved predictions in the simpler Arrhenius models can be obtained by adding a temperature-dependent time delay. Kinetic coefficients and the appropriate time delay are obtained from the constant-rate regions of the measured survival curves. The resulting predictions are seen to provide acceptably accurate results while not overestimating cell death. The method can be relatively easily incorporated into numerical models. Additionally, evidence is presented to support the application of compensation law behavior to the cell death processes—that is, the strong correlation between the kinetic coefficients, ln{A} and Ea, is confirmed.

Copyright © 2015 by ASME
Topics: Temperature , Delays , Leakage
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Fig. 5

The Arrhenius fit to PC3 data including a time delay is improved to an acceptable level of accuracy. Temperatures from 44 to 52 °C, as marked.

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Fig. 4

Time delay versus temperature regression line is: td (s) = 2703 − 49.6 T (°C) or 16,252 − 49.6 T (K)

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Fig. 3

Arrhenius plot for the PC3 cell survival curve data. The regression fit is: ln{k} = 77.90–26,721/T with r2 = 0.889.

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Fig. 2

PC3 cell survival curve at 44 °C. Experiment data in the constant-rate region (solid squares) are fit by: C(t) = 2.639 exp{ −1.702 × 10−3t), r2 = 0.996 (solid line). The remaining shoulder region data are shown as open circles.

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Fig. 1

PC3 cell survival curve at 44 °C [9]. Experiment data (solid circles) and single-step irreversible Arrhenius model fit (solid line).

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Fig. 6

The Arrhenius fit to AT1 calcein leakage data with time delay is improved to an acceptable level of accuracy. Temperatures: 45 (solid squares), 50 (solid circles), 55 (solid triangles), 60 (open circles), and 70 °C (open squares). Time delay Arrhenius fit lines as marked, no time delay is required at 70 °C.

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Fig. 9

AT1 clonogenicity data compared to the Arrhenius fits with time delay added according to Eq. (9) for the same experiment data. Temperatures: 45 (solid squares), 46 (solid circles), 47 (solid triangles), 48 (solid nabla), and 50 °C (open squares). The constant-rate region Arrhenius coefficients are: A = 5.948 × 1082 and Ea = 521.04 (kJ mol−1) and are labeled. The time-delay fit line now fairly estimates the measured values with the exception of 47 °C.

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Fig. 10

The Arrhenius fit to CHO loss of clonogenicity data with time delay is improved to an acceptable level of accuracy. Temperatures: 43 (solid squares), 44 (solid circles), 45.5 (solid triangles), and 46.5 °C (open circles). Time delay Arrhenius fit lines as marked.

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Fig. 11

Plot of calculated kinetic coefficients from Table 1 on the He–Bischof line. The hollow symbols are coefficients from the original data, and the solid symbols represent the coefficients including a time delay. The inverted triangles = AT1 calcein leakage data, the left going arrow heads = PC3 data, the open diamond = AT1 PI data, the open triangle = SN12 PI data, the circles = AT1 clonogenicity data, and the squares = CHO clonogenicity data.

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Fig. 7

AT1 PI uptake data do not exhibit an observable time delay. Temperatures: 40 (open squares), 45 (solid circles), 50 (solid triangles), 55 (open circles), and 60 °C (solid squares). The Arrhenius fit lines are based on the published coefficients: A = 2.99 × 1037 and Ea = 244.8 (kJ mol−1) and are labeled.

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Fig. 8

AT1 clonogenicity data compared to the simple Arrhenius fits without time delay. Temperatures: 45 (solid squares), 46 (solid circles), 47 (solid triangles), 48 (solid nabla), and 50 °C (open squares). The Arrhenius fit lines are based on the published coefficients: A = 1.04 × 1084 and Ea = 526.39 (kJ mol−1) and are labeled. In all cases, the fit line grossly overestimates the measured values.




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