A Three-State Mathematical Model of Hyperthermic Cell Death

DAVID P. O’NEILL,1 TINGYING PENG,1 PHILIPP STIEGLER,2 URSULA MAYRHAUSER,2 SONJA KOESTENBAUER,2

KARLHEINZ TSCHELIESSNIGG,2 and STEPHEN J. PAYNE1

1Institute of Biomedical Engineering, Department of Engineering Science, University of Oxford, Oxford, UK; and 2Division ofTransplantation Surgery, Department of Surgery, Medical University of Graz, Auenbruggerplatz 29, 8036 Graz, Austria

(Received 13 July 2010; accepted 24 September 2010; published online 6 October 2010)

Associate Editor Gerald Saidel oversaw the review of this article.

Abstract—Thermal treatments for tissue ablation rely uponthe heating of cells past a threshold beyond which the cellsare considered destroyed, denatured, or killed. In this article,a novel three-state model for cell death is proposed wherethere exists a vulnerable state positioned between the aliveand dead states used in a number of existing cell deathmodels. Proposed rate coefficients include temperaturedependence and the model is fitted to experimental data ofheated co-cultures of hepatocytes and lung fibroblasts withvery small RMS error. The experimental data utilized includefurther reductions in cell viabilities over 24 and 48 h post-heating and these data are used to extend the three-statemodel to account for slow cell death. For the two cell linesemployed in the experimental data, the three parameters forfast cell death appear to be linearly increasing with %content of lung fibroblast, while the sparse nature of the datadid not indicate any co-culture make-up dependence for theparameters for slow cell death. A critical post-heating cellviability threshold is proposed beyond which cells progress todeath; and these results are of practical importance withpotential for more accurate prediction of cell death.

Keywords—Cell death, Hyperthermia.

INTRODUCTION

For the treatment of cancer, an increasingly com-mon alternative to surgical resection is the use ofthermotherapy. In all forms—radiofrequency, ultra-sound, laser, or microwave—a minimally or non-invasive technique is used to deposit heat energy withinthe cancerous tumor. However, in order for thermaltreatments to provide a reliable alternative to resec-tion, they must outperform surgical procedures, whichis not always the case.

The underlying principle of thermotherapy is thatcells have a thermal tolerance, such that if sufficient

heat is supplied, or if a critical temperature is exceeded,cells die. At the tissue level this process manifests itselfby the production of an ablated zone of dead cellssurrounded by tissue that is unaffected by the ablatingprocedure.

Treatment protocols for power levels, durations,and probe particulars for thermal treatments areestablished by equipment manufacturers.1 The decisionto deviate from these or to repeat ablations is at thediscretion of the radiotherapist. The learning curve forradiotherapists has been shown to be steep andstrongly to influence patient survival rates.12 Examin-ing the 1-year survival rate of patients, those treated byradiotherapists with less than 2 years of experience hada survival rate of 69%, rising to 93% for the sameradiotherapists after 3–4 years of experience. The2-year survival rate rose from 46 to 89%.

In order to aid radiotherapists in their treatmentplanning much work has been done to model heattransfer in biological tissue with external heat sources2;however, such models are highly dependent upon anaccurate model for cell death to be of clinical use. Themost simple models use a single temperature thresholdabove which cells are instantaneously declared dead,and below which cells remain fully functioning.4,29

These isotherm models take no account of the durationfor which cells are at elevated temperatures despite itbeing known that this factor has an influence on thecellular response.

Across the field of thermal damage, the vastmajority of models assume cells can be assigned to twomodel compartments, containing (1) all alive and (2)all dead cells. Transitions between such compartmentsare typically modeled with first-order rate processes tocharacterize biological state changes. In 1947, Henri-ques and Moritz proposed applying the temperature-dependent Arrhenius law for chemical reaction ratesto cutaneous skin burns,11,16,17 an idea that has

Address correspondence to David P. O’Neill, Institute of Bio-

medical Engineering, Department of Engineering Science, University

of Oxford, Oxford, UK. Electronic mail: david.oneill@eng.ox.ac.uk

Annals of Biomedical Engineering, Vol. 39, No. 1, January 2011 (� 2010) pp. 570–579

DOI: 10.1007/s10439-010-0177-1

0090-6964/11/0100-0570/0 � 2010 Biomedical Engineering Society

570

dominated the field since. The rate of cell damage isproportional to exp(2Ea/RT), where Ea is an activa-tion energy, R is the universal gas constant, and T isthe temperature, and although limitations exist withArrhenius-based models, it has become the yardstickto which other models are compared and contrasted.

The two main limitations of such Arrhenius-basedmodels are first that the two governing parameters arevery sensitive to experimental values. The constant ofproportionality is especially sensitive with reportedvalues ranging over tens of orders of magnitude19;requiring large safety margins. The second limitation isthat cell survival rates for different temperatures showa marked discontinuity at temperatures around 43 �C,the exact temperature break depending upon species oforigin and cell line.4,22

The concept of a thermal dose starts with the pre-mise that if tissue held at a constant temperature T diesafter time s(T) and under the correct transformation,heating times at other temperatures can be convertedto equivalent minutes at 43 �C. This transformationwas proposed in 1984 by Sapareto and Dewey23 basedon earlier experimental work.24,25 The expression forequivalent heating time at 43 �C is shown in Eq. (1),where ti and Ti are the heating time and heatingtemperature and a is a constant that is set to 0.5 forheating at temperatures above 43 �C and 0.25 forheating at temperatures below 43 �C.

te43 ¼ a 43�Tið Þti: ð1Þ

Sapareto and Dewey only deal with discrete, singletemperature heating epochs, however, a continuousform of Eq. (1) that can be used for dynamicallyvarying temperatures was derived by Whiting30 basedon different experimental data.5,7 They note thatanalysis of the cells killed by thermal treatments showsa ‘‘marked departure from the rule used in conven-tional hyperthermia to determine the region treated’’.In particular, temperatures above 55 �C cause thegreatest deviation from the standard model. Modernthermal treatments, however, frequently involve sig-nificant regions of tissue exceeding 55 �C, with needletip temperatures reaching 105 �C or higher.10

Other methods use empirical models with differentparameters in different temperature regimes.20 Fenget al.,8 however, emphasize the need for a unified modeland propose an expression for cell viability as a func-tion of heating temperature (T) and time (s), which wasderived from fundamental thermodynamic principles:

AðT; sÞ ¼ e� c=Tþasþbð Þ

1þ e� c=Tþasþbð Þ; ð2Þ

where A is the cell viability, and c, a, and b are con-stants determined from experimental data. This form

models well the ‘‘shoulder region’’ exhibited by theexperimental data for a temperature up to 56 �C and aheating time of up to 30 min. However, the model isnot dynamic as (2) contains a dependence upon thetotal heating time s. The consequence of this is that themodel cannot be implemented during a dynamic tem-perature simulation.

It has been speculated that such drastically differentrates of cell death at different temperatures point to thecomplex sequential steps involved in biological pro-cesses20 and two-state models can never sufficientlycapture the behavior. There exist a number of modelsthat incorporate mechanistic events, including thework of Jung,13–15 who postulates that the path to celldeath requires two sequential events, first that a non-lethal lesion is created in an otherwise healthy cell andsecond that one such non-lethal lesion transforms,creating a lethal event and killing the cell. There is nolimit on the number of lesions in a cell and thereby themodel contains a potentially infinite number of com-partments.

From a mechanistic point of view, models contain-ing three or four compartments hold significantpotential advantage over standard two-state modelswhist avoiding the undesirable nature of models com-prising of an infinite number of compartments. Breenet al. 3 propose such a three-state model where cells arefirst moved to an intermediary state, a process that isreversible, before then suffering fatal, irreversibledamage. Similarly Szasz and Vincze27 incorporate an‘‘excited’’ state between intact and lethal states. Uchidaet al.28 consider a two step process of damage similarto that found in the Jung model, however, to better fitSapareto’s data25 they propose that sub-lethal damageis repaired within a certain time.

Aside from the models mentioned above, we notethat there exist a number of statistical models thatoriginate in the radiation field. Fowler9 clarifies theapplicability of the single-hit, multi-target and themulti-hit, single-target models and Roti Roti andHenle21 make a comparison to the linear-quadratic.Although statistical models can be shown to fitexperimental data, their statistical nature limits insightinto mechanistic events.

It must be mentioned that the majority of the lit-erature deals with first-order reaction kinetics. Higher-order models can be used but as Dienes6 notes, ‘‘thereis no conceptual difficulty in introducing bimolecular,or higher-order, reactions.’’

In this article, we propose a new model that incor-porates all the features outlined above: a single andsimple continuous function that models both fast andslow cell death over a temperature range extending to100 �C and exhibits the ‘‘shoulder region’’ found inexperimental data.

Three-State Model of Hyperthermic Cell Death 571

EXPERIMENTAL DATA

Constant temperature heating experiments werecarried out on cultures from two cell lines: human liverhepatocellular carcinoma Hep G2 and human lungfibroblast MRC-5. Cells were cultured overnight at37 �C in a culture medium of Minimum EssentialMedium with 10% fetal calf serum and 1% penicillin/streptomycin. The volume concentration of cells was3 9 104 cells/100 lL, and the cells were seeded in96-well plates and cultivated overnight in a humifiedatmosphere (5% CO2).

Experiments were performed on two pure culturesand three co-cultures (25/75, 50/50, and 75/25) of HepG2 and MRC-5. For each condition, a plate had fourseparate wells of each culture, the culture medium wasreplaced with a preheated medium and the platesincubated in a preheated heating cabinet. The pre-heated temperatures were 55, 65, 75, 85, and 100 �Cand the heating times were 300, 600, or 900 s. Beforeuse, the temperature of the heated medium was mea-sured with a HI 935005 K-Type waterproof ther-mometer (resolution 0.1 �C, accuracy ±0.2%); themedium was also spot checked during heating. Afterheating, the heating medium was replaced with amedium at 37 �C and spiked with 20 lL MTS reagent(Promega). After 2 h of post-heating incubation at37 �C the fluorescence at wavelengths of 490 and650 nm was measured in SpectraMax Plus 384 (accu-racy: < ±0.006 OD ±1.0%, precision: < ±0.003 OD±1.0%). Fluorescence measurements were also taken26 and 50 h after heating; the recorded fluorescence

signifies the metabolic activity of living cells. For eachtime point, the fluorescence data were normalized to acontrol value at 37 �C.

For statistical analysis, each experiment had five orsix repeats; each datum was taken to be the mean ofthe readings from the four wells across all experiments.The data described in this section thus provides cellviability information as a function of heating time,heating temperature, incubation time as well as varia-tions in co-culture cell line make-up.

For the initial data fitting described below, the fivedata points from the different cultures for each heatingtemperature (5 + 1 control), heating time (3), andincubation time (3) were averaged. The resulting 54data points from combinations of three heating times,six heating temperatures, and three post-heating incu-bation times are shown in Table 1.

MATHEMATICAL MODEL

Fast Cell Death

The component of the model for fast cell deathduring thermal treatment is presented in this section.The model includes a third ‘‘vulnerable’’ compartment,which is located between the fully alive (A) and dead(D) compartments in the same way as proposed byBreen et al.3 and Szasz and Vincze.27 In biologicalterms, it is intended to represent a state, where thecell’s normal functions may be impaired but where thecell is not yet fully dead. As shown in Eq. (3), the three-compartment model has three possible compartmenttransitions: alive to vulnerable, vulnerable to alive, andvulnerable to dead:

A�!kf

�kb

V�!kf

D: ð3Þ

The transition from alive to vulnerable representsinitial damage caused to an injured state, possibly astate where the cell is no longer functioning normally.Once past a critical point the cell progresses to a deadstate from which the cell cannot return. The transitionfrom vulnerable to alive represents the self-healingprocess whereby an injured cell can recover to itsfully functional alive state similar to Jung’s lesionmodels.13–15

As proposed, all complex stages of the path to celldeath are modeled by a single damage mechanism, thusas the processes from alive to vulnerable and fromvulnerable to dead are taken to be two stages of thesame result of state change, a single forward rateconstant, kf, is assumed for simplicity. The biologicalprocess for healing (vulnerable to alive) is separate

TABLE 1. Experimental data (% cell viability).

Heating

time (s)

Heating

temperature

(�C)

Post-heating incubation time (h)

2 26 50

300 37 100.0 100.0 100.0

55 100.0 98.02 98.02

65 100.0 98.02 98.02

75 89.66 89.00 86.82

85 52.89 25.48 9.56

100 4.23 1.00 0.00

600 37 100.0 100.0 100.0

55 94.71 83.75 83.75

65 56.94 22.33 13.38

75 7.89 1.47 1.47

85 2.45 0.00 0.00

100 1.19 0.00 0.00

900 37 100.0 100.0 100.0

55 66.99 53.80 10.61

65 10.75 1.33 1.65

75 2.65 0.00 0.00

85 2.51 0.00 0.00

100 0.00 0.00 0.00

O’NEILL et al.572

from the mechanism for thermal damage and is thusassigned a different, backward, rate constant, kb,assumed to be invariant with temperature.

Using standard reaction kinetics, the system can bedescribed mathematically by three simultaneous dif-ferential equations, however, as all cells are accountedfor in one of the three compartments, i.e., A +V + D = 1, the system can be reduced to two simul-taneous differential equations, the solution to whichdescribes the system with two degrees of freedom:

dA

dt¼ �kfAþ kb 1� A�Dð Þ; ð4Þ

dD

dt¼ kf 1� A�Dð Þ; ð5Þ

where A and D are, respectively, the fractional popu-lation of cells in the alive and dead states. To representthe fact that at normal body temperature there is nosignificant thermal damage, at 37 �C the forward rateconstant must be small, thereafter increasing withtemperature. A simple exponential curve fits thesecriteria of temperature dependence, mimicking the factthat tissue that is already highly damaged is mostsusceptible to further damage. To acknowledge the factthat the damaged state of surrounding cells may have agreater than linear influence on the reaction dynamicsthe volume fraction of cells in the vulnerable and deadcompartments (1 2 A) is incorporated into the for-ward rate constant:

kf ¼ kfeT=Tkð1� AÞ; ð6Þ

where kf is a scaling constant and Tk is a parameterthat sets the rate of the exponential increase withtemperature; both Tk and T have units of �C. Thus, foreach experiment with a fixed heating time and tem-perature, there are three independent variables dictat-ing the dynamics of the system: kf; kb; and Tk:

One result of the inclusion of (1 2 A) in the forwardrate constant is that an initial condition of all cellsbeing in the alive compartment would result in a staticsolution. The initial conditions were thus set such thata small portion of cells start in the vulnerable com-partment. This was set to be 1%, which is sufficientlysmall to have negligible effect on the intendeddynamics of the system. The portion of cells initially inthe dead compartment was set at 0%.

The inclusion of (1 2 A) in Eq. (6) makes the paireddifferential equations nonlinear. To fit the three-parameter model to the experimental data, the paireddifferential equations were first solved numericallyover time for a fixed heating temperature using afourth-order Runge–Kutta solver in MATLAB. Atypical time course for fixed temperature heating isshown in Fig. 1; a state of dynamic equilibrium is

sought between the alive and vulnerable compartmentswhile there is a continual leakage of cells from thevulnerable to the dead compartment. This increase lagsbehind the increase in the fraction of cells in the vul-nerable state, thus providing the ‘‘shoulder region’’found in experimental data as the observed experi-mental data is the total of the cells in the alive andvulnerable compartments, or (1 2 D).

To determine values for the three parameters theMATLAB function ‘‘fminsearch’’—which finds a min-imum of an unconstrained multivariable function usinga derivative-free method—was used to find the set ofparameters that minimized an error function consistingof the root-mean-square error between each experi-mental data point and the value of ‘‘alive + vulnera-ble’’ output from the numerical solution inMATLABatthe relevant time point. Experimental results were ini-tially averaged across the five co-cultures for eachheating temperature and heating duration.

Table 2 shows the optimized parameter values,while in Fig. 2 the optimized results are shown in termsof cell viability against heating time, one line for eachheating temperature. The experimental data and cor-responding error bars are also included to demonstratethe closeness of the fit. The RMS error between themodel with the optimal parameter set and the data is1.40%; this value is well within the tolerance of the

0 100 200 300 400 500 600 700 800 9000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time (s)

Cel

l fra

ctio

n

AliveVulnerableDead

FIGURE 1. Typical time course for fixed temperature heatingdisplaying fractions of cells in alive, vulnerable, and deadstates.

TABLE 2. List of fast cell death parameters and valuesfor minimum error.

Parameter Optimized value

kf 3.33 9 1023 s21

kb 7.77 9 1023 s21

Tk 40.5 �C

Three-State Model of Hyperthermic Cell Death 573

experimental data at 2 h incubation time which has anmean standard error of 3.39%.

Slow Cell Death

The part of the model that incorporates the mod-eling of cell death evident during post-treatmentincubation is now presented. Although rapid and largechanges in cell viability occur during thermal treat-ments, the experimental data show that cell viabilitycontinues to decrease in the 26 and 50 h after heattreatment. The mathematical model presented in theprevious section is temperature dependent and there-fore cannot fully account for this additional cell deathwhile the cells are in normothermia and so an exten-sion is now proposed. Closer examination of the datareveals that cultures with significant reductions in cellviability (approximately <70% cell viability) causedby thermal heating subsequently progress over time toa fully dead state with 0% cell viability. However, forlesser levels of thermal damage, the final value of cellviability is somewhat greater than zero.

The proposed extension to the model to account forcell death via a secondary, slow mechanism requirestwo further transitions in the model. These are bothsingle direction transitions to a dead state—one fromthe alive compartment and one from the vulnerablecompartment—as shown in Fig. 3. Cells that are notdead can gradually progress from their current state tothe dead state; there is no facility for dead cells to behealed. The conditions post-heating (incubation at37 �C) are such that the rate constant for fast celldeath, kf, is zero. Thus, when considering slow cell

death, cells need only be considered in two compart-ments: ‘‘dead’’ and ‘‘not dead’’ as is shown in rateequation form in Eq. (7):

1�Dð Þ�!ks D: ð7Þ

The resultant reaction dynamics of (7) are thus:

dD

dt¼ ksð1�DÞ: ð8Þ

To fit the model to the data, the rate constant ks wasassumed to be solely a function of the fraction of cellsin the dead compartment. The following restrictionswere imposed on the rate constant, based on theexperimental data:

(1) ks > 0 for all D so that no dead cells return toan alive state.

(2) ks = 0 at D = 0 such that no cells die whenthe tissue is 100% alive.

(3) ks = 0 at some threshold value, Ds, such thatthere is a value of maximum cell death forcultures which have suffered only minimalthermal damage.

Various forms of curve that satisfied these condi-tions were tried and qualities of fit examined; the formof curve found to give the best fit to the data was aquartic expression in D, where conditions 1 and 4combine to give a repeated root:

ks ¼ ksD 1�Dð Þ D�Dsð Þ2; ð9Þ

where ks is a baseline scaling value for the rate constantand Ds is the threshold value described in the thirdcriterion above. The final form for ks is thus charac-terized by just two parameters, ks and Ds; an exampleplot for this rate constant is shown in Fig. 4.

To determine the two parameters ks and Ds, anerror minimization was again performed in MATLABas in the previous section. The data were extracted as 18sets of cell viabilities with three incubation time values;one set for each combination of heating temperatureand heating time—as shown in Table 1. Since the ratefor slow death is far smaller than the rate for fast celldeath, it was assumed that negligible slow death wouldoccur during the heating time and so the models wereapplied successively. The data point for 2 h incubation

0 100 200 300 400 500 600 700 800 9000

10

20

30

40

50

60

70

80

90

100

Heating time (s)

% v

iabi

lity

37 °C model55 °C model65 °C model75 °C model85 °C model100 °C model

37 °C data55 °C data65 °C data75 °C data85 °C data100 °C data

FIGURE 2. Predicted cell viability (continuous lines) andexperimental data (symbols) with error bars.

FIGURE 3. Full model schematic incorporating transitionsfor both fast and slow cell death model parts.

O’NEILL et al.574

was used each time as the initial condition required tosolve (9) numerically. The MATLAB function fmin-search was applied to a root-mean-square error func-tion for the errors for all 54 data points. The resultingparameter values are shown in Table 3.

Figure 5 shows the 18 resulting theoretical plotsfrom the model using the error minimizing parametervalues as well as the corresponding experimental datapoints and associated error bars. The RMS errorbetween the fitted model and the experimental data for

the entire data set of slow death plots is 2.80%; this isvery similar to the mean standard error for the entireexperimental data set (2.71%).

Figure 5 shows the different cell viabilities duringincubation at 37 �C. Cultures with cell viabilitiesgreater than 80% at 2 h of incubation remain above80% cell viability, whereas cultures starting below 80%cell viability all progress toward a final state with 0%viability. This 80% threshold is shown in the figure bythe solid line and can be interpreted as a critical cellviability, one that can be utilized to determine the finalcell viability from the cell viability at 2 h of incubationand no other information.

Cell Line Dependence

Re-examining the experimental data detailed inthe second section, before averaging across differentco-cultures, some variation is apparent in the data:co-cultures with higher contents of the lung fibroblastMRC-5 are recorded to have higher cell viabilities thanco-cultures with higher contents of hepatocellularcarcinoma HepG2 cells for the same temperature andtime. This feature suggests that fibrous tissue may bemore resilient to heat treatments. The model is thusnow fitted to the data at 2 h of post-heating incubationfor individual co-cultures to examine the dependenceof each parameter upon the MRC-5 content of theco-culture.

Using the same fitting procedures and the samemathematical model proposed previously, the data foreach separate co-culture was used to determine thethree parameters for fast cell death (kf; kb; and Tk)optimized for each co-culture. Table 4 shows theoptimized parameter values, while Fig. 6 displays plots

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Fraction of cells in dead state

Rat

e co

nsta

nt (

x10−

3 s−1 )

FIGURE 4. Example form for rate constant for slow celldeath, ks ; as defined by Eq. (10).

TABLE 3. List of slow cell death parameters and values forminimum error.

Parameter Optimized value

ks 0.316 9 1023 s21

Ds 0.208

2 26 500

10

20

30

40

50

60

70

80

90

100Heating time 300s

Incubation time (hr)

% v

iabi

lity

2 26 500

10

20

30

40

50

60

70

80

90

100Heating time 600s

Incubation time (hr)

% v

iabi

lity

2 26 500

10

20

30

40

50

60

70

80

90

100Heating time 900s

Incubation time (hr)

% v

iabi

lity

37°C Model

55°C Model

65°C Model

75°C Model

85°C Model

100°C Model

37°C Data

55°C Data

65°C Data

75°C Data

85°C Data

100°C Data

FIGURE 5. Predicted cell viability reduction during incubation at 37 �C (continuous lines) with experimental data (symbols) witherror bars and critical cell viability (solid line).

Three-State Model of Hyperthermic Cell Death 575

of these values normalized with respect to the valuesreported in ‘‘Fast cell death’’ section against the per-centage content of MRC-5.

As Table 4 shows, the RMS errors between fittedvalues and experimental data are of the order of themean standard error of the data (3.39%); consideringthe sparse nature of the data, this is a good quality fit.Separately, using the optimization of the parametersfor fast cell death, the slow cell death model and the 26and 50 h heating data were considered, however, theresults were inconclusive. No firm conclusions aboutthe variation of slow cell death parameters or athreshold value with cell type could be drawn on thebasis of the experimental data used here.

DISCUSSION

The mathematical model proposed in the previoussection contains a number of assumptions. The mostsignificant limitation is that there are only two possibleprocesses leading to cell death—heat induced fast celldeath and subsequent cell viability dependent slow celldeath; however, due to the good nature of the fit, thislimitation does not seem overly restrictive. Othermodels consider the dynamic progression toward celldeath over the time span of thermal treatments (103 s),

but they cannot account for the continued progressionto death exhibited by the experimental data post-heat-ing. To model cell death over this extended timescale(105 s) during which there is no additional thermalinput, either the rate constants controlling instant ther-mal cell death must be reduced to zero magnitude, or aseparate, slow death model must be introduced. A dis-continuous pair of successive models is not ideal andthus the proposed model directly halts compartmenttransitions due to fast cell death by including a temper-ature dependence in the rate constant for fast cell death.

Common to the transitions for both fast and slowcell death in the proposed model, as well as in themajority of the literature is the assumption that thetransition to death is irreversible. This has beenaccounted for here through use of a vulnerable com-partment from which cells can heal and return to thealive compartment. This corresponds to the ‘‘Margin-ally Damaged’’ state proposed by Breen et al.,3 or the‘‘excited’’ form proposed by Szasz and Vincze.27 In allthree models, only partially damaged cells can progressto death. In the slow cell death part of the model,however, it is assumed that past the criterion thatdetermines that an individual cell will die there is noability to reverse or halt the process.

The value of using a three-state model can beassessed by comparing the results of ‘‘Fast cell death’’section and Fig. 2 with results from fitting the com-monly used Arrhenius model as shown in Fig. 7. Usingthe standard form for the rate constant of a two state,irreversible process:

k ¼ Ae �Ea=RTð Þ; ð10Þ

the two parameters A and Ea were determined by fit-ting to the same data with the same fitting process asdescribed previously. The optimal values for minimumerror were found to be 769 s21 and 37,886 J kg21; theRMS error was 20.59% indicating a poor fit. Quali-tatively this poor fit is shown by Fig. 7.

It is thus shown that the simple two-state Arrheniusmodel is incapable of modeling the no-treatment con-dition of a continuous incubation temperature of37 �C as the nature of the model predicts significantcell death even in normothermia. As a second com-parison,18 we fitted the model by Jung,13–15 which has

TABLE 4. List of optimized fast cell death model parameters for each co-culture.

Parameter

Co-culture cell line make-up (% content of MRC-5)

0% 25% 50% 75% 100%

kf 0.80 9 1023 s21 2.62 9 1023 s21 3.52 9 1023 s21 4.54 9 1023 s21 9.07 9 1023 s21

kb 0.25 9 1023 s21 5.74 9 1023 s21 8.46 9 1023 s21 10.8 9 1023 s21 19.2 9 1023 s21

Tk 24.6 �C 36.7 �C 41.6 �C 46.3 �C 63.5 �CRMS error 4.27% 2.40% 1.79% 1.59% 3.59%

0 10 20 30 40 50 60 70 80 90 1000

0.5

1

1.5

2

2.5

3

% MRC−5

Nor

mal

ised

par

amet

er v

alue

FIGURE 6. Variations of fast cell death model parameterswith co-culture content.

O’NEILL et al.576

infinitely many states, but again no backward, recov-ery, process. The RMS error achieved for this modelwas 2.3% again indicating a poorer fit than our pro-posed model. It is therefore the inclusion of at leastthree states and a form of cellular recovery that pro-vides an acceptable fit with the data, also a rate con-stant dependence on temperature that does not predictsignificant cell death at 37 �C.

Despite the low RMS error calculated here, indi-cating a high quality of model fit with only a fewparameters, there are some limitations. The fast celldeath part of the proposed model requires an initialfraction of cells to be in the vulnerable compartment.The choice of A0 = 0.99, V0 = 0.01, D0 = 0, is a rea-sonable choice since 1% of cells in the vulnerablecompartment is a sufficiently small fraction to inducenegligible ‘‘acceleration’’ of the system. A smaller valuewould, however, generate greater dynamic ‘‘inertia’’and result in significantly higher values of the fittedparameter ks: Re-optimizing the parameter values withV0 = 0.1%, the RMS error for the fast cell death datafitting becomes 2.71%, while with V0 = 5%, the RMSerror becomes 3.02%, significantly worse fits than thatproposed (RMS error of 1.4%). Two issues arise fromthis inclusion of a non-zero V0. First, there is a prob-lem in the physical interpretation: in well controlledsmall culture experiments of discrete cells it is theo-retically possible to ensure that V0 = 0; in this sce-nario, the proposed model would remain at its initialconditions of A0 = 1, V0 = 0, D0 = 0, and no celldeath would be possible. The second possible issue isthat initial damage might be higher than V0 = 1%. Inthese cases the dynamic behavior of the system wouldbe significantly different to that simulated above. Sucha scenario might arise in clinical thermal treatmentswhere certain fractions of cells have already entered avulnerable or dead compartment due to non-thermal

causes such as physical trauma (needle-induced dam-age) or chemical injury (chemotherapy drug-induceddamage). However, this would be very difficult toquantify.

The fast cell death model has three possible transi-tions. To minimize the number of free parameters inthe model, it was assumed that the forward rate con-stants governing the transitions A fi V andV fi D are the same. The physical interpretation ofthis is that there is a single damage process whichincorporates all physiological damage mechanisms.This mathematical simplification is justified as thevulnerable state is an arbitrarily set position along thefast path to cell death representing the ‘‘point of noreturn’’ rather than a change in the mechanism ofthermal damage, or a defined, physically different stateof the cell. As previously shown18 in a comparisonbetween Jung’s infinite-state lesion based model13–15

and the proposed model, this vulnerable state usedhere could be viewed as incorporating all cells that areneither lesion free, nor yet dead; i.e., all cells in anystate with a non-zero probability of transitioning to adead state.

The final assumption made in the mathematicalmodel for slow cell death is that there is some thresholdvalue of surrounding cell viabilities below which cellsprogress to death, and above which cells settle at anon-zero level of viability. The concept for thisthreshold came from inspection of clinical radiofre-quency ablation zones. Regions of ablated tissue werefound to increase in size during a period of days fol-lowing thermal therapies, however, the zones did notincrease in size indefinitely. Inspection of the distri-bution of alive and dead cells around the ablatedzones’ border regions revealed that isolated alive cellswith an ablated zone were not observed in tissue leftfor several days. Further, isolated dead cells in mostlyalive tissue outside an ablated zone were still presentmany days later. A link has been shown between thepresence of heat-shock proteins (created due to ther-mal stress) and apoptotic dead cells.26 From this it wasassumed that the variable governing slow cell deathwas the local cell viability, which may be modeled by athreshold value.

This cell viability threshold is of immense practicalimportance computationally. Cell death models aretypically derived to complement heat transfer modelsin predictive computational simulations for plannedablative treatments. By defining a critical cell viability,a threshold is set such that following the simulation ofthe thermal therapy, the simulation can be halted andthe final ablative zone can be calculated. This is pos-sible using our proposed model since slow cell death issolely a function of the conditions at the end of thethermal treatment.

0 100 200 300 400 500 600 700 800 9000

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% v

iabi

lity

37°C model55°C model65°C model75°C model85°C model100°C model37°C data55°C data65°C data75°C data85°C data100°C data

FIGURE 7. Arrhenius model predicted cell viability (contin-uous lines) and experimental data (symbols) with error bars.

Three-State Model of Hyperthermic Cell Death 577

The data used for fitting the model described in‘‘Experimental data’’ section were obtained from cellcultures assumed to be experiencing step change tem-perature heating. This raises an important consider-ation: tissue is neither isotropic nor homogenous; sofar, the proposed model has only been fitted to datafrom cultured media, there is no information availableon the possible effects of specific geometries and dis-tributions of differing cell lines. Further work is plan-ned to investigate how the optimizing parameters varywhen fitted to data from cultures of different cell lines.

There remains further work to be done with regardto the investigation of parameter dependency upondifferent cell lines and co-culture mixes, particularlyupon cell lines not included in the experimental datadetailed in ‘‘Experimental data’’ section. This more indepth work will lead to significant practical imple-mentation benefits regarding different tissue types inthe application of simulating predictions for cell deathof real tumor tissue.

CONCLUSIONS

A novel mathematical model for cell death underhyperthermic condition has been developed and fittedto experimental data with optimized parameters forerror minimization presented. The model is capable ofpredicting fast cell death as a function of temperatureduring thermal treatments and is capable of accountingfor further, slow death of cells in the days followingtreatments. Simulation results from the fast cell deathmodel fit experimental data well as a function of bothtreatment temperature and treatment duration, andexhibit the ‘‘shoulder region’’ of low dosage thermaltolerance reported in the literature. Slow cell deathmodel simulation results also fit the experimental datawell and confirm the hypothesis that there is a criticalcell viability that determines whether cell cultures atnon-zero, non-unity cell viabilities progress to a com-pletely dead state or not. Results of data fitting showthis critical cell viability to be ~80% alive; this result isof practical importance with potential for more accu-rate prediction of cell death, which could lead to reli-able hyperthermia and ablation treatment planning.

ACKNOWLEDGMENTS

Katja Schick for her contributions to work in thecell culture. The research leading to these results hasreceived funding from the European Community’sSeventh framework Programme under grant agree-ment no. 223877, project IMPPACT.

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