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Dose dependence of the growth rate of multicellular tumour spheroids after irradiation

¹ Servicio de Radiofísica, Hospital Universitario "San Cecilio", Avda. Dr. Olóriz, 16, E-18012 Granada, ² Departamento de Radiología, Universidad de Granada, E-18071 Granada and ³ Departamento de Física Moderna, Universidad de Granada, E-18071 Granada, Spain

Correspondence: M Villalobos, Laboratorio de Investigaciones Médicas y Biología Tumoral, Departamento de Radiología, Universidad de Granada. Avda. Madrid s/n, 18071-Granada, Spain

	Abstract

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The present study investigated differences in the growth rate of multicellular tumour spheroids of the MCF-7 line of human breast cancer before and after their irradiation. Growth of the spheroids was analysed according to a model based on a Gompertz function. In this model, normalization to a common initial volume is achieved in a way that enables meaningful comparisons to be made between the results obtained for each spheroid. For irradiated spheroids the model includes an additional term to take account of sterilized cells. We found that the growth rate observed before irradiation is not fully recovered by irradiated spheroids and that growth recovery reduces with higher irradiation doses. Surviving fractions obtained at doses below 3 Gy are comparable with those found in clonogenic assays on spheroids of the same cellular line. At larger doses, discrepancies between the different studies are considerable.

	Introduction

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Multicellular tumour spheroids (MTS) have attracted considerable attention [1–3] in recent years, largely because they mimic in vitro the micrometastases and growth of real tumours. MTS are composed of mixed cell populations and permit description of the avascular stage of tumour development. Thus, MTS are highly relevant to adjuvant therapy and are considered a good scenario to investigate tumour biology and to evaluate effects of different therapies.

Much of the work in this field has focused on the determination of cell radiosensitivity, expressed in terms of the cell surviving fraction (SF). This appears to be one of the most important parameters for treatment purposes because the radiosensitivity of human tumour cells in culture [4–6] and the radiation response of MTS [7] have been correlated with the clinical radiocurability of the corresponding tumour type.

However, the use of irradiated MTS to obtain the SF remains controversial. The common assumption of many studies is that irradiated MTS recover the growth rate of untreated MTS at some time after irradiation. This time period, known as the growth delay (GD), has been variously considered the time required for treated MTS to regrow to four-fold [8] or eight-fold [9, 10] their initial volume. If the growth recovery assumption is correct, these discrepancies have little relevance because GD will be independent of MTS volume. In addition, some studies [8] showed that GD may not be appropriate to compare the radiosensitivity of tumours of different sizes.

Other methodologies [11] obtain the SF as the ratio between the zero time intercept of the extrapolated linear part (on a logarithmic scale) of the regrowth curve and the initial volume. However, the regrowth curve does not generally follow an exponential shape, so that it is often difficult to unequivocally define the linear part of the curve. In addition, the volumes of different MTS do not undergo any kind of normalization, making comparisons between them inadequate and therefore negating the significance of the characteristic parameters of the growth model.

The present study aimed to test the hypothesis that the growth rate recovers after irradiation, using a novel method to investigate both untreated and irradiated MTS. The MTS are considered individually by a technique that avoids the usual procedure of averaging the volumes of the different MTS at each time point. Thus, the growth curve of each spheroid can be normalized to a given initial volume, allowing valid comparisons to be made. The method also takes account of the coexistence of two different cell populations after irradiation; sterilized and surviving cells. The former do not proliferate and it is the growth rate of the latter that is of interest. Finally, the SF is directly obtained by fitting a mathematical model to the experimental data, so that GD or extrapolation procedures are unnecessary.

	Materials and methods

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Cell culture
Cells of the MCF-7 human breast cancer line established by Soule et al [12] and obtained from Dr G Leclercq, Institut Jules Bordet, Brussels were used. Cells were grown routinely in minimum essential medium supplemented with 10% foetal bovine serum. Monolayer cultures were maintained at 37°C in a humidified atmosphere of 95% air and 5% CO₂. Cells used to develop the spheroids were obtained by trypsinization from exponentially growing monolayers and cultured in the medium described above.

Spheroid initiation
Spheroid cultures were initiated by seeding 1000–1500 cells into each well of six 48-well plates. Each well was previously coated with a thin layer of 1% agar (Bacto agar; Difco, Detroit, MI) and contained standard culture medium. The six plates were agitated for 18–20 h under the conditions described above for the monolayer cultures. In this way, a single spheroid of approximately 100 µm diameter was obtained in each well. Once the MTS were formed, the medium was changed every 3 days.

Ionizing radiation
The diameter of the MCF-7 spheroids was measured every 1–2 days using an inverted phase contrast microscope. The volume of each spheroid was calculated as V=(d³)/6, where d is diameter. At the time of irradiation the spheroids presented diameters of 220–280 µm. One of the six plates was kept as a control and the other five were exposed to single doses of 18 MV X-rays (range 1–5 Gy) using an electron linear accelerator (Mevatron KDS; Siemens, Erlangen, Germany). Electronic equilibrium was ensured by placing the biological material between solid water slides (RW3 of PTW Freiburg). The total depth was 3 cm above and 20 cm below the samples.

Spheroid selection
Only spherical spheroids were considered in this study. In previous studies [13, 14] our group estimated volumes by assuming spheroids to be ellipsoidal and measuring two perpendicular diameters. In the present work all spheroids with evident asymmetry in the initial growth stages were discarded. Spheroids also become asymmetric when they grow to considerable size. In order to maximize the number of spheroids in the sample, time limits for their evaluation were set: 19 days for the control spheroids and those exposed to 1 Gy or 2 Gy, and 21 days for those exposed to 3–5 Gy.

In order to guarantee homogeneity for analysis, only spheroids that showed no growth problems, i.e. they were not the result of the fusion of two or more initial spheroids, included no agar fibres or other foreign matter and were not in a contaminated well, were considered.

A very small number of spheroids failed to regrow after irradiation and were excluded from analysis.

At the time of irradiation, the MTS had a homogeneous composition and showed no necrotic nuclei, fulfilling the condition that the relationship between spheroid volume and number of cells be proportional. 18 control MTS and 15, 28, 22, 20 and 14 MTS irradiated with 1 Gy, 2 Gy, 3 Gy, 4 Gy and 5 Gy, respectively, were considered valid for inclusion in the study.

Mathematical approach
The mathematical approach used permits analysis of growth data for both control and irradiated MTS.

Control spheroids
In order to test the assumption that treated MTS recover the growth rate of untreated MTS at some time after irradiation, it is necessary to correctly determine the parameters of the untreated spheroids. The usual procedure [9, 11, 15] requires two steps. First, the mean volume of the spheroids at each time t_j is calculated as:

where n is the number of spheroids included in the sample, V_i(t_j) is the measured volume for the i-th spheroid at the time t_j and k is the number of time points at which the volumes are measured. These mean values are then fitted to a given sigmoid function.

As previously mentioned, valid comparisons are compromised by differences in volume of different spheroids at the time of first measurement, and it is necessary to normalise data to a common initial volume (CIV). Some authors [14] carried out this normalization by dividing the volume data of each spheroid by V₀, the corresponding volume at the initial time point. The resulting data were fitted to the model function for each spheroid individually, and the corresponding mean parameters were evaluated to obtain a "standard growth curve". However, this procedure is not adequate because it does not permit comparison between different MTS at the same growth stage, as we shall demonstrate below. The present paper proposes a novel approach to CIV normalization that addresses this issue and obtains meaningful characteristic parameters. Thus, each control spheroid in the sample was considered individually and the increase in its volume described using a Gompertz function, following a previously published procedure [13], which relates the volume at a given time t₁ to the volume of the MTS at a previous time t₀:

Here is the initial specific growth rate and a is the proportional rate of decay of .

This formula was developed by Gompertz [16] for actuarial assessment of England's population. A century later this formula was proposed as a possible model for biological growth [17, 18]. In 1964 it was empirically found that Gompertz formula sucessfully described the growth of individual organisms and tumours [19].

The normalization to CIV proposed here is based on the assumption that the growth of each spheroid occurs in two stages. In the first stage, which is not actually observed, the MTS develops from CIV V_CI to the volume observed at the time of the first measurement t=0, V₀ after time T has elapsed.

It is important to note that time T is particular to each spheroid and can be readily calculated with this equation when the parameters have been determined.

In the second stage, which is observed, the MTS grows from volume V₀ to that measured at time t. Hence, the total time elapsed from the CIV to the measured volume at the time t is t+T, giving:

Thus, the normalization of all the spheroid growth curves to the CIV is characterized, in practice, by a new value of parameter of the Gompertz model:

Conversely, parameter a is independent of the normalization procedure. For convenience we used V_CI =10⁶ µm³, which was smaller than all the volumes measured. However, any other choice of value would not have modified the results obtained in our analysis.

Once this normalization is performed, MTS volumes can be compared. The growth data of each spheroid are fitted by means of Equation (4), and the parameters characterizing the growth of the control MTS are obtained as the mean value of parameters and a for each spheroid.

Irradiated spheroids
Growth curves for irradiated spheroids have specific characteristics that must be incorporated into the theoretical model that describes them:

1. Once the spheroid is irradiated, its volume either increases slowly, remains constant or diminishes, and the spheroid can subsequently regrow or the regrowth can fail. The number of failures increases with higher doses [9, 11, 14, 15].
   
2. Once the stagnation or regression phase is overcome, the slope of the growth curve appears to be similar to that shown by the control spheroids, at least for a range of doses (the range depends on the cell line under study) [9, 14, 15]. However, more rigorous study of the data casts doubt on this apparent similarity.
   

Our method avoids this issue, based on a previously reported approach to the problem [22]. It is assumed that two sets of cells coexist after irradiation: surviving cells, responsible for the regrowth of the spheroids; and sterilized cells, including dead cells and those that have lost their proliferation capacity. In our model, the total volume of the irradiated spheroids is expressed as the sum of the volumes of these two subsystems:

where V^surv is the volume of surviving cells and V^ster is the volume of sterilized cells.

We assume that growth of surviving cells follows the same pattern (the Gompertz model) as growth of cells in control spheroids:

We further assume that the sterilized cells will maintain or decrease the volume they show at the time of irradiation. In our model, cell loss is negligible and data are compatible with a virtually constant value for V^ster (t). Indeed we observed no appreciable loss of cellular material during the time the spheroids were evaluated.

Taking the above into account, the following model was used for analysis of the irradiated MTS:

the moment of the first measurement V₀ being very close in time to irradiation. We considered that V(t=0)=V₀. The normalization to CIV is included in the procedure, as for control spheroids. This equation is used to analyse growth data of irradiated spheroids and to determine whether characteristic parameters remain the same as for control MTS.

As mentioned above, our spheroid selection procedure allows us to consider volume of a spheroid and the number of clonogenic cells it contains to be proportional [23]. Hence, the fitting procedure directly yields the SF(s):

which allows us to obtain the radiosensitivity parameters of the cell population.

It is very important to note that normalization to CIV, a basic element of our approach, does not modify the SF values found, because it affects only the parameter and leaves both V₀ and V₀^surv unchanged.

Statistical methods
Uncertainties in measurement of spheroid volumes must be considered. Sources of uncertainty in this experiment include loss of the spherical shape of the cellular aggregates and the effects of periodic changes in the culture medium, among others. However, with our method, only uncertainties linked to instrumental error can be evaluated. The smallest division in the ocular of the microscope was 10 µm, so that uncertainty in the volume u(V) was (in µm³) with d given in micrometres, assuming a uniform distribution for the uncertainty of the measured diameter [24].

The experimental data obtained were fitted to Equation (4) for control spheroids and Equation (8) for irradiated spheroids. The Levenberg–Marquadt method was used as the fitting procedure. This method provides the parameters of the model function by giving the fit with the smallest ².

	Results and discussion

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Control spheroids
Parameters for control spheroids determined by this method are shown in Table 1. Figure 1a depicts the fit obtained for a representative spheroid. For all the MTS in the control sample we obtained a mean ² per degree of freedom of 1.3±0.2, where the uncertainty corresponds to one standard deviation.

View this table: [in this window] [in a new window]   Table 1. Values of the parameters of interest for the control multicellular tumour spheroids obtained in the different approaches. The last column corresponds to the value of the initial specific growth rate of the Gompertz equation renormalized to common initial volume (CIV). Uncertainties correspond to one standard deviation

View larger version (18K): [in this window] [in a new window]   Figure 1. Representative examples of control (a) and irradiated spheroids (b–f). Uncertainties correspond to 2 standard deviations. ---, fits obtained by fixing the initial specific growth rate and the proportional rate of decay to the values obtained for the control spheroids; —, fits obtained by fixing only the proportional rate of decay to the values obtained for the control spheroids.

Dose-dependence of growth rate
The principle issue with treated MTS is whether the growth rate recovers or not. In Figure 1b–f, volumes of a representative spheroid for each of the doses administered are shown and the results of fitting Equation (8) to the measured volumes plotted, considering the parameters and a to be the same as those obtained for the control MTS. In general, the fit was rather poor (mean ² per degree of freedom ranged from 4.3±0.5 for 1 Gy to 27±1 for 5 Gy). This finding appears to challenge the common assumption of growth rate recovery after irradiation.

We investigated this issue further by directly fitting the data to the model function, Equation (8). In principle, if both and a are considered as free parameters, the function includes four parameters rather than the three included for control spheroids in Equation (4), which complicates the fit. We determined that the optimal procedure is to establish parameter a as the mean value obtained in analysis of the control sample (a=0.086 days^-1), bearing in mind that a was not changed by the CIV normalization procedure, as shown above. Thus, we tested the growth rate recovery assumption by studying the variations of parameter . The fit found with this method was considerably better than those shown above; mean ² per degree of freedom ranged from 1.7±0.4 for 1 Gy to 3.8±0.5 for 5 Gy. Some examples are depicted in Figure 1. Figure 2 shows the variation in values obtained in this analysis according to dose; uncertainties correspond to one standard deviation. It can be concluded that growth rate of the control spheroids was not fully recovered by the irradiated spheroids. Our analysis procedure reveals that parameter decreased linearly with increased dose. Figure 2a displays the linear regression that describes the dependence of on dose d. The corresponding characteristic parameters are given in Table 2. Thus, we can state that growth of the exposed spheroids was perturbed by irradiation.

View larger version (7K): [in this window] [in a new window]   Figure 2. Variation of the initial specific growth rate with dose administered. Uncertainties correspond to one standard deviation. (a) Linear fit corresponding to the method used in this study. (b) Linear fit of results obtained in a similar analysis. , results obtained without normalization; , results obtained from the latter analysis and normalized to a common initial volume, as discussed in the text.

View this table: [in this window] [in a new window]   Table 2. Results of the linear regression (d)=md+n of the initial specific growth rate as a function of dose d with and without the normalization to common initial volume (CIV). The bottom row gives the results obtained after a renormalization of the latter data. The parameters characterizing the regression m and n and the correlation coefficient r are given. Uncertainties correspond to one standard deviation

Figure 2b also depicts the values of parameter found when results were renormalized by means of Equation (5), using mean volumes obtained in the fitting procedure. As in the case of control spheroids, these new values were similar to those obtained with our procedure. The main difference is that the uncertainties were clearly larger. The corresponding linear regression, whose parameters are given in Table 2, was also plotted. The results again showed the diminution of the parameter with increased dose. However, the fact that uncertainties are now larger than in our approach makes the decrease of less evident. The advantage of our approach is clear because normalization to CIV performed for each MTS individually produces very accurate results.

Our results do not support the commonly accepted hypothesis of recovery of pre-irradiation growth rate. This assumption is based on the apparently similar slopes shown by the growth curves of both treated and untreated spheroids. We have seen that values of the parameter found with the usual fitting procedures appear to support the growth rate recovery hypothesis. However, we have unequivocally demonstrated that after normalization to the CIV, varies with dose. It is important to state that this variation of cannot be considered an effect of the "volume change" inherent to the CIV normalization because by fixing the same initial volume for all the spheroids, growth data can be compared and conclusions are meaningful.

Surviving fraction
Figure 3 shows the values of the SF obtained in our regrowth assay using Equation (9). Following the standard approach to describe this SF, we fitted the data obtained by means of the linear-quadratic model:

where the parameters and ß are characteristic of the cell line considered. This model has been widely used [26, 27] and despite its simplicity, provides a satisfactory description of the survival curve of a homogeneous cell population.

View larger version (13K): [in this window] [in a new window]   Figure 3. The natural logarithm of the surviving fraction (s) as a function of the imparted dose. , results of the method used in this study; —, corresponding regression to the linear-quadratic model; -.-., results quoted by Villalobos et al [21] (•, 2 Gy dose); ..., linear-quadratic fit obtained by Núñez et al [28] (, 2 Gy dose); and - - -, results corresponding to reanalysis of data of Villalobos et al [21]. Uncertainties correspond to one standard deviation.

Parameters obtained in the fit to the linear-quadratic model in our case are shown in Table 3, where these results are compared with those obtained for the same cell line in a monolayer culture performed by Núñez et al [28] and those obtained previously by our group [21] for spheroid growth, both using clonogenic assays (CA). The values of the SF for a 2 Gy dose are also shown. Figure 3 also depicts this comparison. Results obtained in the two CA (monolayer and MTS) were very similar and differ from the present findings, even at low doses.

View this table: [in this window] [in a new window]   Table 3. Comparison of , ß and the surviving fraction at 2 Gy, s(2 Gy), obtained in this work with those found in clonogenic assay (CA) for monolayer culture and for multicellular tumour spheroid (MTS) growth. The values obtained from the growth delay (GD) for 5- and 8-fold increase in initial volume are also given. Uncertainties correspond to one standard deviation. and ß are characteristic of the cell line considered

It is important to note that the clonogenic and regrowth assays produce comparable results for doses up to 3 Gy, because CA are known to be generally simpler to develop than regrowth with MTS and subsequent analysis is less complicated.

Although GD is not necessary in the method used here, SF results were compared with those obtained when the current GD definitions are applied to the data from this experiment. The corresponding SF from the GD was calculated by modifying the equation given by Rofstad et al [20] for an exponential growth model, in order to obtain one appropriate to the Gompertz model. If a N_V-fold increase in initial volume is considered we have:

where t_delay represents GD.

The values of the parameters of the regression to the linear-quadratic model are given in Table 3. Differences between the values obtained with the two definitions of the GD are again clear. There is also an apparent difference with our results, mainly at high doses. Nevertheless, it is interesting to note how different methods provide similar values of SF for low doses. In any case, this comparison clarifies the indetermination in the SF produced by the GD approach.

	Conclusions

Top Abstract Introduction Materials and methods Results and discussion Conclusions References

 
The hypothesis of regrowth rate recovery by MTS after irradiation was investigated. An analysis of the growth data of multicellular spheroids that includes normalization to a common initial volume was developed, to allow meaningful comparison between the different spheroids. As a direct result, the model gives the surviving fraction of the sample.

This model was applied to spheroids of the MCF-7 line of human breast cancer. The most important finding was that normal growth rate is not fully recovered by the irradiated spheroids, and that the difference in growth rate is greater at higher doses. Both treated and control MTS show apparently similar growth rates when data are not normalized to common initial volume, which accounts for the commonly held assumption of recovery after irradiation. The fact that the growth rate is not fully recovered may be of clinical relevance in planning therapy protocols for patients.

The surviving fractions obtained are comparable with those found in different experiments with MTS of the same cellular line, provided dose is low. For higher doses, discrepancies between different experiments are considerable. At any rate, we can affirm that clonogenic assays are sufficiently accurate to provide the surviving fraction at doses below 3 Gy, and offer clear advantages in experimental procedure and data analysis.

The accuracy of the approach used is comparable with that usually achieved in clonogenic assays with spheroids, and the regrowth assay is a less disturbing method to analyse the surviving fraction. Nevertheless, the regrowth assay is both less laborious and requires lesser disturbance of the cells, making our novel methodology of special interest.

	Acknowledgments

 
The authors thank "San Cecilio" University Hospital of Granada, Spain for permitting the use of the electron linear accelerator to irradiate the spheroids.

	Footnotes

 
This work has been supported in part by the Junta de Andalucía (group FQM0225) and by grant from the Ministerio de Sanidad (Spain) (FIS 01/1090).

Received for publication December 18, 2001. Revision received July 2, 2002. Accepted for publication August 12, 2002.

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Top Abstract Introduction Materials and methods Results and discussion Conclusions References

 

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This Article Abstract Figures Only Full Text (PDF) Services Similar articles in this journal Similar articles in PubMed Alert me to new issues of the journal Download to citation manager Citing Articles Citing Articles via Google Scholar Google Scholar Articles by Guirado, D Articles by Lallena, A M Search for Related Content PubMed PubMed Citation Articles by Guirado, D Articles by Lallena, A M