British Journal of Radiology (2005) 78, 939-944
© 2005 British Institute of Radiology
doi: 10.1259/bjr/40226390
The potential for mathematical modelling in the assessment of the radiation dose equivalent of cytotoxic chemotherapy given concomitantly with radiotherapy
B Jones, MSc, MD, FRCR1 and
R G Dale, MSc, PhD, FIPEM2
1 Department of Clinical Oncology, Queen Elizabeth University Hospital, Birmingham B15 2TH and 2 Department of Radiation Physics and Radiobiology, Charing Cross Hospital, London W6 8RF, UK
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Abstract
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The linear quadratic (LQ) concept of biological effective dose (BED) is used with Poisson statistics to estimate the radiation equivalent BED of cytotoxic chemotherapy (CBED) that would provide improvements in tumour control probability (TCP) typically achieved in randomized clinical trials of chemoradiation. The concepts of pure radio-sensitization and independent chemotherapy cell kill are represented by mathematical equations. Small values of sensitizer enhancement ratios (s) can provide modest increases in TCP when large numbers of radiotherapy fractions are sensitized; larger s values are required if only a small number of radiotherapy fractions are sensitized. Independent chemotherapy induced cell kill is sufficient to explain the benefits achieved with concomitant chemoradiotherapy in situations where a sufficiently high chemotherapy dose intensity is used (i.e. the dose–time intensity of cytotoxic chemotherapy without radiation is considered to be sufficient to cause significant tumour regression although not cure). Care is required in the use of the Poisson cure probability model because of the associated steep dose–response curves that may underestimate both s and the CBED. By use of random sampling methods and estimation over a theoretical population of different tumours, more robust results are obtained with dose–response curves that correspond better to those in clinical data sets. These predict a 2–4 Gy10 equivalent for each pulse of chemotherapy such as single agent Cis-Platinum when used weekly during radiotherapy for a maximum of 4 cycles. This preliminary paper does not consider normal tissue complication probabilities, of which there are relatively few mature results for modern chemoradiotherapy. The BED concept can be used to estimate the equivalent dose of radiotherapy that will achieve the same cell kill as concomitant cytotoxic chemotherapy. Relatively simple radiobiological modelling can be used to guide decision-making regarding the assessment of the most appropriate combined modality schedules, and has important implications in the design of clinical trials.
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Introduction
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There is, at the present time, widespread clinical use of concomitant cytotoxic chemotherapy with radical radiotherapy [1–4]. Many randomized control trials, analysed separately or collectively, show statistically significant improvements in tumour control resulting from the use of concomitant "chemoradiotherapy", for example in squamous cell cancers of the cervix [5, 6]. There remain concerns as to whether significant increments in acute and late toxicity will occur, in which case the chemotherapy will only have modified the overall effect of the radiation [2, 6].
The prescription of chemotherapy is usually based on body surface area, weight or glomerular filtration rate. Although the cell kill due to chemotherapy will be expected to be proportional to the total drug dose, it is difficult to quantify the cell killing effect of chemotherapy that is used alone or in combination with radiotherapy. In principle, quantification of chemotherapy cell kill can be made in the same way as radiation effects on the surviving fraction of cells, since cell survival curves obtained following exposure to many chemotherapeutic agents are broadly similar to those obtained following radiation exposure [7].
Consequently, it would be practically useful if the effects of cytotoxic drugs could be expressed in terms of equivalent radiation dose or biological effective dose (BED). BED is the total radiation dose required to achieve a specified level of effect if given in infinitely small dose fractions, and is used to assess and compare different radiotherapy schedules [8–11]. BED values are additive so that different components of treatments can be assessed and incorporated in a total value for a treatment schedule where fractionation and overall time may vary. If similar BED assesments could be applied to cytotoxic chemotherapy, oncologists could then assess the relative contributions of radiation and chemotherapy in combined treatments.
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Methods
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We consider that the cytotoxic drug may either enhance the effect of the radiation dose by an average multiplicative effect, or may kill cells by independent means as an additive effect.
Dose sensitization
Dose sensitization implies that the effect associated with each fractional dose of radiation is enhanced such that a dose d including the drug effectively kills the same number of cells as would a dose ds of radiation alone, where s is the chemotherapy-modulated radiation dose enhancement factor and is assumed to represent the average value throughout a series of repeated chemotherapy treatments, in the same way that 
and 
represent the average radiosensitivities during fractionated radiotherapy. The dose of chemotherapy used in this situation is not necessarily sufficient to cause significant cell kill.
The equations for these assumptions are given in Appendix A.
Independent cell kill
We next assume additional cell kill due to the independent cytotoxicity of the chemotherapy. This would apply in situations where the chemotherapy is of sufficient dose intensity (i.e. dose per unit time) to cause substantial cell kill, e.g. in situations where the chemotherapy dose schedule, when used alone, can achieve significant tumour regression.
The equations for these assumptions are given in Appendix B.
Extension of model to include variable overall treatment time
The relevant equations are given in Appendix C.
For the parameters and treatment schedules used in the virtual trial, we assume the dose and overall time details as shown in Tables 1 and 2
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Table 1. Assumed mean parameters (with standard deviates (SD) in parentheses) for virtual clinical trial of different schedules used in Figure 4
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Table 2. Details of four different combinations of external beam radiotherapy and brachytherapy used in the treatment of cancer of the cervix uterus. These schedules are the most commonly used in UK, including the use of high dose rate (HDR) and low dose rate (LDR) brachytherapy. The USA schedules are those used in the randomized trials that used concomitant chemotherapy, which are quoted elsewhere in the text
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Results
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The calculated sensitization factors given in Table 3 were found by solution of Equation (11) for thirty 2 Gy fractions and /=10 Gy and =0.3 Gy–1 for a TCP increment from 45% to 67%. The steep increase in s values, seen as the number of exposed fractions is reduced, reflects the sensitivity of the Poisson TCP model to small increments in dose when uniform values of parameters are used.
For pure dose sensitization, Figure 1 shows the s values required against the number of sensitized fractions with variation of the parameter.
For the assumption of chemotherapy independent cell kill, in Figure 2 the gain in TCP is shown for an individual tumour for assumed additional chemotherapy BED values. The parameter assumptions are the means given in Table 1 in the methods section; the assumed overall time is 57 days. The dose–response curves are steep and increasing the chemotherapy equivalent BED displaces the curves to the right.
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Figure 2. Plot of individual tumour control probability against total radiation dose for variable chemotherapy biological effective shown as ChemoRx BED.
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For simulation of a population of tumours, random sampling methods are used to construct Figure 3. The assumptions made regarding the means and standard deviations of the radiobiological parameters are given in Table 1. The dose–response curves are shallower than in the previous figure, but again the displacements are proportional to the chemotherapy BED equivalent.
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Figure 3. Relationship in a population of 500 tumours between tumour control probability and total radiation dose with assumed additional chemotherapy biological effective dose equivalents, shown as ChemoRx BED.
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A plot of resultant TCP against the chemotherapy equivalent BED is shown in Figure 3 for the parameters given in example 3 of Table 4. In Table 4, steps 1–6 given in Methods have been run separately with changes in some of the radiobiological parameter assumptions to test the sensitivity of the model, as shown in the three examples given in Table 4: it can be seen from the lowest row that subtle increases in the standard deviation of the re-population factor and in the mean value can produce a reduction in the slope of the TCP increment per BED-gray equivalent of chemotherapy.
If the gain in TCP is achieved by several cycles of chemotherapy, then the data presented can be used to estimate the effective BED of each cycle. For example, if a trial shows a gain in tumour control from 69% for radiotherapy alone to say 88% for chemoradiotherapy (as in Figure 4) the additional chemotherapy BED is found to be approximately 12.5 Gy10. If this has been achieved by the use of 4 cycles of chemotherapy, then the chemotherapy BED per cycle is 12.5/4=3.1 Gy10. We here assume that each cycle of chemotherapy has an equal effect – as in radiobiological modelling using the BED concept – where each dose per fraction has the same effect, and the BEDs from all exposures are additive.
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Figure 4. Example of relationship between tumour cure probability and the additional biological effective dose (BED) associated with chemotherapy: the parameters are those in example 3 in Table 2.
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The results of clinical trial virtual simulation are shown in Figure 5, where concomitant Cis-platinum is designated as "P". It can be seen that shorter schedules given to a lower total dose are predicted to give broadly the same results as longer schedules given to a higher total dose. In all cases the addition of Cis-platinum is of potential benefit.
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Discussion
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In principle, it is possible to estimate the separate contribution of cytotoxic chemotherapy to tumour cell kill either as a sensitization factor or in terms of the equivalent BED. These results are dependent on the validity of the linear quadratic model of radiation effect and the parameter assumptions made.
There are three possible approaches to determine the contribution of chemotherapy to radical radiotherapy:
- By intuition and "rules of thumb". There are clinical data sets for certain tumour types, particularly the squamous cell carcinomas, where the tumour control appears to increase by 1% per additional Gy of BED [12, 13]. Other trials may link the same histological class of cancer with chemotherapy effect, e.g. a 10–15% increase in local control by the addition of chemotherapy given in 4 cycles of the drug (or drug combination). By extrapolation from one study to another, an increase in control of (say) 12% is related to 12%/4=3% increase per cycle, which then equates with a 3 Gy BED per cycle. The chemotherapy BED will of course be in the relevant units for tumour control (usually Gy10). The total effect of the chemotherapy would be a BED of 3 Gy per cycle x 4=12 Gy. This approach is simplistic and can be criticised because of the extrapolations of logic used.
- By the Poisson modelling as described above.
- By random sampling methods combined with (2), as described above.
Another potential method to enhance radiotherapy effects is to use biological dose modifiers or drugs that may purely inhibit re-population (i.e. have cytostatic effects) in some cancers, e.g. signal transduction blocking agents/hormones in some cancers. These have been considered in mathematical terms in another publication [14], where integration of the re-population factor was necessary to define the average re-population during the cytostatic agent exposure, which was then discounted from the unopposed re-population. This approach is entirely different from the present study, where additional cell kill due to chemotherapy has been included. However, the benefits in both cases depend on the characteristic features of the dose–response curves; for example the gains in tumour control due to the chemotherapy, or other biological agent, are greater for a tumour with a moderate cure probability than in the case of another tumour with a high expected cure probability.
Assessments of chemotherapy effects in palliative situations are also possible: tumour re-growth times can be predicted from the radiobiological characteristics of the tumour [15].
It has been assumed in the earlier equations in this paper that the chemoradiotherapy schedules have been given over the same overall treatment time as the radiotherapy alone; this may not be true in cases where overall treatment times may be extended because of the side effects caused by increased treatment intensity. The random sampling method did include a re-population term (see step 3), with allowance for increasing treatment times. The occurrence of delayed or accelerated re-population was accommodated by using a time delay factor such that the time over which re-population occurs is modified to be T–Tdel, where Tdel is the time of onset of re-population after the commencement of treatment. If the durations of treatments are known then an appropriate re-population correction term can be included in all of the equations.
The model has not been extended for use with respect to normal tissue toxicity at this stage. The calculated equivalent BED values for chemotherapy are appropriate only for tumour control. The extant trials of chemoradiotherapy have concentrated on tumour control and cure statistics, with relatively scant attention to side effects in many cases. Where good data can be obtained regarding side effects, the methods described above could also be extended to study late tissue side effects, with a lower / ratio (usually between 1.5 Gy and 4 Gy), but might require inclusion of volume effects, although the volumes irradiated should be very comparable in the two arms of large randomized trials that compare radiotherapy with chemoradiotherapy.
Statistical fitting of clinical data sets has not been attempted, but proof of principle has been demonstrated in that the typical clinical results can be simulated. The equations presented could be used in detailed statistical studies by those who have access to entire data sets.
The virtual clinical trial method, which involves random sampling of important radiobiological parameters, reveals interesting results. The quantitative assessment methods described might be useful in the rational design of future clinical trials.
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Appendix A
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In terms of the LQ model [4–6], radiation effect (E), commonly referred to as the "log cell-kill", may be expressed as
where n is the number of fractions, d the dose per fraction and and are the radiosensitivity parameters.
When chemotherapy is also used, pure dose sensitization is assumed for modelling purposes, i.e. that a dose d of radiation becomes a dose ds when chemotherapy is also given, we then obtain:
In terms of BED, the unsensitized BED of
becomes therefore
Re-population is at this stage ignored (we assume at this stage that the same overall time may apply to the overall treatment whether chemotherapy is added or not).
In cases where only some of the fractions are sensitized, we have:
The enhancement of the BED will depend on the number of sensitized fractions and the value of s.
The value of s can be estimated algebraically by transformation of clinical trial results to two simultaneous equations, as follows. From the Poisson statistical model [16], we obtain:
where is the radiosensitivity coefficient of cell kill and c the clonogen number. Let TCP1 and TCP2 be the respective tumour cure probabilities without and with sensitization, respectively. It follows from Equations 3–5 that:
where D is the total radiation dose (=nd).
These equations are relevant for individual tumours; for populations of heterogenous tumours, variation of the parameters such as radiosensitivity and re-population rates are further required (this is discussed further below).
In the context of a randomized control trial, where the same dose of radiation is given in each of two arms, the following conditions exist:
Radiotherapy alone
survival outcome 1 Radiotherapy+chemotherapysurvival outcome 2
These conditions correspond to two simultaneous equations, so that the contribution of chemotherapy to outcome change can be calculated.
Then, by dividing Equation (6) by Equation (7), which eliminates clonogen number, we obtain
the positive root for s is
For example, if TCP1=0.45, TCP2=0.67, =0.3 Gy–1, d=2 Gy given for 30 fractions such that the total dose, D, is 60 Gy and /=10 Gy, the solution for s is=1.03, a small value since all 30 fractions of radiotherapy are sensitized.
Exposure to cytotoxic drugs may only occur during some radiation exposures. Then, if m1 fractions are sensitized and if m2 fractions are un-sensitized, then:
Equations (6) and (9) are combined to obtain
Now since m1+m2=n (the total number of radiotherapy fractions), Equation (10) can be simplified to be:
The roots are
It can be seen that the solution for s does not depend on n, the total number of fractions, but only on m1, the number of sensitized fractions.
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Appendix B
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For chemoradiation, the total "cell kill effect" ET is represented by
where EC is the overall cytotoxic drug related cell kill (i.e. from all cycles of chemotherapy).
Equation (12) also has the implication that:
Total BED=BED of radiotherapy+equivalent BED of chemotherapy
By use of the Poisson model for tumour cure probability, we obtain deterministic equations for TCP1 and TCP2, the tumour control probabilities of each if the trial arms for radiotherapy alone and for radiochemotherapy, respectively:
where ER represents the total radiation cell kill, and
where the total cell kill is a result of the independent action of radiotherapy and that of chemotherapy (EC). By dividing Equation (13) by Equation (14) we obtain
which leads to
If TCP1=0.45 and TCP2=0.67, then EC=0.69. That is, the "log cell-kill" due to chemotherapy is 0.69. Now, as the log cell kill is the same as x BED, then for an value of 0.3 Gy–1, the equivalent BED will be 0.69/0.3=2.3 Gy10 (units of Gy10 refer to the dose in conditions where /=10). If this is associated with chemotherapy given in (say) 4 cycles, the implication is that the chemotherapy BED equivalent is 2.3/4=0.58 Gy10 per cycle.
This is not a particularly large cell kill or BED compared with that achieved by the radiotherapy to a modest radical dose (e.g. 72 Gy10 for 60 Gy in 30 fractions).
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Appendix C
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To study the variation of tumour control probability (TCP) with assumed (mean) values of tumour clonogen re-population parameters during therapy for a Poisson based TCP in an overall treatment time T, with a delay time (Tdel) before the effective onset of re-population at a daily re-population dose equivalent, K, the standard BED are multiplied by for consistency of dimensions in the surviving fraction (SF) portion of the equation, since SF=e-.BED [10, 11]:
where BED(RadioRx) and BED(ChemoRx) are the respective BEDs due to radiotherapy and chemotherapy. The re-population dose equivalent parameter (K) is related to the average clonogen doubling time (Teff) as follows [11]:
If we assume =0.26 Gy–1, T=57 days, Tdel=25 days, Teff=4 days and the re-population daily dose equivalent per day is K=0.6 Gy10/day. A large increment of TCP is achieved by a relatively small chemotherapy BED equivalent. The average increase in TCP per Gy of chemotherapy BED is approximately 3% (see later graphs, Figures 2–4). Because of the very steep dose–response curves generated by the Poisson model with uniform radiosensitivities, this result is appropriate only for a relatively uniform population of cells [17–21]. One alternative approach is to set up a virtual trial, i.e. to computer simulate the trial with allowance for the spread of parameter values that may occur within a population of patients, e.g. using Mathematica software (Wolfram, USA).
By taking the standard Poisson equations for tumour control, and using assumed means and standard deviations for the radiosensitivity parameter , the / ratios, re-population rates (K), T (the overall treatment time), clonogen numbers c, and assuming an increasing value of BED per chemotherapy cycle, a plot of TCP increment with the additional BED due to chemotherapy can be generated. This more sophisticated approach produces broadly similar results as in method 1. The computational steps are as follows:
- Since E/=BED, then E=.BED
- E/=radiotherapy BED+chemotherapy BED–re-population BED
- E=.D(1+d/(/))+.BED(chemo)-.K(T–Tdel)
- TCP=Exp[–c.Exp[–E]]
- Random variation within normal or log normal distributions is introduced to parameters , c and Tdel and K by use of software (see Table 1)
- TCP is then plotted against BED(chemo).
Received for publication November 8, 2004.
Revision received April 22, 2005.
Accepted for publication April 25, 2005.
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Abstract
Introduction
