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
Your Session has timed out. Please sign back in to continue.


Henriques, F. C. , 1947, “ Studies of Thermal Injury V. The Predictability and Significance of Thermally Induced Rate Processes Leading to Irreversible Epidermal Injury,” Arch. Pathol., 43(5), pp. 489–502.
Henriques, F. C. , and Moritz, A. R. , 1947, “ Studies of Thermal Injury in the Conduction of Heat to and Through Skin and the Temperatures Attained Therein: A Theoretical and Experimental Investigation,” Am. J. Pathol., 23(4), pp. 530–549. [PubMed]
Moritz, A. R. , 1947, “ Studies of Thermal Injury III. The Pathology and Pathogenesis of Cutaneous Burns: An Experimental Study,” Am. J. Pathol., 23(6), pp. 915–941. [PubMed]
Moritz, A. R. , and Henriques, F. C. , 1947, “ Studies in Thermal Injury II: The Relative Importance of Time and Surface Temperature in the Causation of Cutaneous Burns,” Am. J. Pathol., 23(5), pp. 695–720. [PubMed]
He, X. , Bhowmick, S. , and Bischof, J. C. , 2009, “ Thermal Therapy in Urologic Systems: A Comparison of Arrhenius and Thermal Isoeffective Dose Models in Predicting Hyperthermic Injury,” ASME J. Biomech. Eng., 131(1), p. 745071.
Pearce, J. A. , 2013, “ Comparative Analysis of Mathematical Models of Cell Death and Thermal Damage Processes,” Int. J. Hyperthermia, 29(4), pp. 262–280. [CrossRef] [PubMed]
He, X. , and Bischof, J. C. , 2005, “ The Kinetics of Thermal Injury in Human Renal Carcinoma Cells,” Ann. Biomed. Eng., 33(4), pp. 502–510. [CrossRef] [PubMed]
Bhowmick, S. , Swanlund, D. J. , and Bischof, J. C. , 2000, “ Supraphysiological Thermal Injury in Dunning AT-1 Prostate Tumor Cells,” ASME J. Biomech. Eng., 122(1), pp. 51–59. [CrossRef]
Feng, Y. , Oden, J. T. , and Rylander, M. N. , 2008, “ A Two-State Cell Damage Model Under Hyperthermic Conditions: Theory and In Vitro Experiments,” ASME J. Biomech. Eng., 130(4), p. 041016. [CrossRef]
Wright, N. T. , 2013, “ Comparison of Models of Post-Hyperthermia Cell Survival,” ASME J. Biomech. Eng., 135(5), p. 51001. [CrossRef]
Weinberg, W. A. , 2007, The Biology of Cancer, Garland Science, Taylor & Francis, New York.
Vandenabeele, P. , Galluzzi, L. , Berghe, T. V. , and Kroemer, G. , 2010, “ Molecular Mechanisms of Necroptosis: An Ordered Cellular Explosion,” Nat. Rev. Mol. Cell Biol., 11(10), pp. 700–714. [CrossRef] [PubMed]
Vanlangenakker, N. , Bertrand, M. J. M. , Bogaert, P. , Vandenabeele, P. , and Berghe, T. V. , 2011, “ TNF-Induced Necroptosis in L929 Cells is Tightly Regulated by Multiple TNFR1 Complex I and II Members,” Cell Death Dis., 2(11), p. e230. [CrossRef] [PubMed]
Gozuacik, D. , and Kimchi, A. , 2004, “ Autophagy as a Cell Death and Tumor Suppressor Mechanism,” Oncogene, 23(16), pp. 2891–2906. [CrossRef] [PubMed]
Zhang, Y. , and Calderwood, S. K. , 2011, “ Autophagy, Protein Aggregation and Hyperthermia: A Mini-Review,” Int. J. Hyperthermia, 27(5), pp. 409–414. [CrossRef] [PubMed]
Bergsbaken, T. , Fink, S. L. , den Hartigh, A. B. , Loomis, W. P. , and Cookson, B. T. , 2011, “ Coordinated Host Responses During Pyroptosis: Caspase-1-Dependent Lysosome Exocytosis and Inflammatory Cytokine Maturation,” J. Immunol., 187(5), pp. 2748–2754. [CrossRef] [PubMed]
Eissing, T. , Conzelmann, H. , Gilles, E. D. , Allgower, F. , Bullinger, E. , and Scheurich, P. , 2004, “ Bistability Analyses of a Caspase Activation Model for Receptor-Induced Apoptosis,” J. Biol. Chem., 279(35), pp. 36892–36897. [CrossRef] [PubMed]
Mackey, M. , and Roti, J. L. R. , 1992, “ A Model of Heat-Induced Clonogenic Cell Death,” J. Theor. Biol., 156(2), pp. 133–146. [CrossRef] [PubMed]
O'Neill, D. P. , Peng, T. , Stiegler, P. , Mayrhauser, U. , Koestenbauer, S. , Tscheliessnigg, K. , and Payne, S. J. , 2011, “ A Three-State Mathematical Model of Hyperthermic Cell Death,” Ann. Biomed. Eng., 39(1), pp. 570–579. [CrossRef] [PubMed]
Alberts, B. , Johnson, A. , Lewis, J. , Raff, M. , Roberts, K. , and Peter, W. , 2008, Molecular Biology of the Cell, Garland Science, New York. [PubMed] [PubMed]
Cravalho, E. G. , 1992, “ Response of Cells to Supraphysiological Temperatures: Experimental Measurements and Kinetic Models,” Electrical Trauma. The Pathology, Manifestations and Clinical Management, R. C. Lee , E. G. Cravalho , and J. F. Burke , eds., Cambridge Press, Cambridge, UK.
Peschke, P. , Lohr, F. , Wolber, G. , Hoever, K.-H. , Wenz, F. , and Lorenz, W. J. , 1992, “ Response of the Rat Dunning R3327-AT1 Prostate Tumor to Treatment With Fractionated Fast Neutrons,” Radiat. Res., 129(1), pp. 112–114. [CrossRef] [PubMed]
Sapareto, S. A. , Hopwood, L. E. , Dewey, W. C. , Raju, M. R. , and Gray, J. W. , 1978, “ Effects of Hyperthermia on Survival and Progression of Chinese Hamster Ovary Cells,” Cancer Res., 38(2), pp. 393–400. [PubMed]
Qin, Z. P. , Balasubramanian, S. K. , Wolkers, W. F. , Pearce, J. A. , and Bischof, J. C. , 2014, “ Correlated Parameter Fit of Arrhenius Model for Thermal Denaturation of Proteins and Cells,” Ann. Biomed. Eng., 42(12), pp. 2392–2404. [CrossRef] [PubMed]
Rosenberg, B. , Kemeny, G. , Switzer, R. C. , and Hamilton, T. C. , 1971, “ Quantitative Evidence for Protein Denaturation as the Cause of Thermal Death,” Nature, 232(5311), pp. 471–473. [CrossRef] [PubMed]
Barrie, P. J. , 2012, “ The Mathematical Origins of the Kinetic Compensation Effect: 2. The Effect of Systematic Errors,” Phys. Chem. Chem. Phys., 14(1), pp. 327–336. [CrossRef] [PubMed]
Barrie, P. J. , 2012, “ The Mathematical Origins of the Kinetic Compensation Effect: 1. The Effect of Random Experimental Errors,” Phys. Chem. Chem. Phys., 14(1), pp. 318–326. [CrossRef] [PubMed]
Yelon, A. , Sacher, E. , and Linert, W. , 2012, “ Comment on ‘The Mathematical Origins of the Kinetic Compensation Effect’ Parts 1 and 2 by P. J. Barrie, Phys. Chem. Chem. Phys., 2012, 14, 318 and 327,” Phys. Chem. Chem. Phys., 14(22), pp. 8232–8234. [CrossRef] [PubMed]
Barrie, P. J. , 2012, “ Reply to ‘Comment on ‘The Mathematical Origins of the Kinetic Compensation Effect’ Parts 1 and 2' by A. Yelon, E. Sacher and W. Linert, Phys. Chem. Chem. Phys., 2012, 14, DOI: 10.1039/c2cp40618g,” Phys. Chem. Chem. Phys., 14(22), pp. 8235–8236. [CrossRef]
He, X. , and Bischof, J. C. , 2003, “ Quantification of Temperature and Injury Response in Thermal Therapy and Cryosurgery,” Crit. Rev. Biomed. Eng., 31(5–6), pp. 355–421. [CrossRef] [PubMed]
Wright, N. T. , 2003, “ On a Relationship Between the Arrhenius Parameters From Thermal Damage Studies,” ASME J. Biomech. Eng., 125(2), pp. 300–304. [CrossRef]
Eyring, H. , and Polyani, M. , 1931, “ Über Einfache Gasreaktionen (On Simple Gas Reactions),” Z. Phys. Chem. B, 12, pp. 279–311.


Grahic Jump Location
Fig. 1

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

Grahic Jump Location
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.

Grahic Jump Location
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.

Grahic Jump Location
Fig. 4

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

Grahic Jump Location
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.

Grahic Jump Location
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.

Grahic Jump Location
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.

Grahic Jump Location
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.

Grahic Jump Location
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.

Grahic Jump Location
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.

Grahic Jump Location
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.



Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In