Equivalence of Quantitative Models for Tumour ...

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Vol. 18

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ONCOLOGY RADIATION PHYSICS BIOLOGY

1979

EQUIVALENCE OF QUANTITATIVE MODELS FOR TUMOUR RESPONSE TO IONIZING RADIATION IN TREATMENT FIELD OPTIMISATION PROCEDURES S. GRAFFMAN, T. GROTH,B. JUNG and G. SKOLLERMO

Modern computer-aided treatment planning systems provide for fast and accurate simulation of external beam irradiation (GLICKSMAN et coll. 1972, STERNICK 1976). For each individual case a multitude of different plans can be considered, which brings up the problem of appropriate techniques for the selection of the plan to be preferred. Presently, this choice is, at most clinics, based on common clinical criteria. These' tend to differ from clinic to clinic and are often also not stringent enough to result in a unique best choice. Attempts at formalising the selection procedure with computer support have been reported (e.g. HOPE et coll. 1967, VAN DER LAARSE & STRACKEE 1976). However, no scheme appears to have reached a wider acceptance. There are good reasons why the computer should be considered also for the selection work. One is that considerable computational power is needed in the calculation of indices of potential value in the selection process. Furthermore, when an adequate algorithm has been established, the computer offers the possibility to search automatically for an optimum dose plan. The use of a simple cell-kinetic model in computer-aided optimisation of treatment parameters has been discussed previously (GRAFFMAN et coll. 1975 a, b). It was shown that the automatic procedure is reasonable with respect to computing time. From the Departments of Oncology &d Radiation Physics, Akademiska Sjukhuset. Uppsala University Data Center, and the Department of Computer Science, University of Uppsala, S-750 10 Uppsala, Sweden. Submitted for publication 20 March 1978. Acta Radiologica Oncology 18 (1979) F a x . I 1 - 195845

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S. GRAFFMAN, T. GROTH, B. JUNG AND G. SKOLLERMO

The results were also found to agree with established clinical facts such as a critical relation between local tumour cure and accumulated target dose. Considerable uncertainty remained, however, concerning the validity of the simple response model for decision-making in radiation therapy. Biologic and clinical research has given a general knowledge about tumour growth and response to radiation and host factors. A complex model with a large number of descriptive parameters would indeed be necessary if all known facets should be taken into account. It can be hoped that, under practical circumstances, a simple model with a small number of parameters would describe the complex situation reasonably well with respect to factors of prime importance for the restricted situation of treatment field optimisation. This idea would be supported if (1) it could be demonstrated that a simple model emulates more complex models based on biologic findings or models which have been evaluated in wider clinical practice, such as NSD (ELLIS1966) or CRE (KIRKet coll. 1971), and (2) if experience shows that the behaviour of certain types of tumour can be simulated with a limited number of lumped parameters. It also appears reasonable that thinking in model terms and searching for factors of prime importance in clinical practice may aid in the development of future quantitative descriptions of tumour response. The first of these two points is discussed in the present report.

Methods Fractionation. Although there are strong indications that the time of dose delivery & BANKS1976, is decisive for an optimum irradiation (FISCHER 1971 by ALMQUIST WHELDON & KIRK 1976), the available quantitative models appear to be insufficiently well established to motivate a radical change from a conventional fractionation scheme. The in numero experiments were therefore based on 30 daily fractions of 2 Gy each to a total target dose of 60 Gy. Tumour dose distribution. The basic problem in an optimisation procedure of the kind described here is to find a valid numerical estimate of the chance of local cure with different dose distributions, these being the result of variations in treatment parameters. Since, in clinical practice, a variation in dose of k 5 and even k 10 per cent can often not be avoided, the different tumour response models were tested for several accumulated doses in the range 54 to 66 Gy. The whole tumour-bearing volume was considered to be homogeneously irradiated at each dose level. If the numerical estimates from the various models do not differ in this dose range, equivalence is established also for the case with a non-homogeneous dose distribution in this or a smaller range. Tumour cell density. Treatment field optimisation should be of special interest when minute changes in dose delivery produce a radical change in local curability. This implies, within the framework of the cell-kinetic approach that the expectation

TUMOUR RESPONSE TO IONIZING RADIATION

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value of the number of cells remaining after the radiation treatment should be close to one. The initial number of clonogenic cells in the different models was hence chosen so as to yield an expectation number of about one for the number of clonogenic cells remaining after the full treatment of a tumour (30 X 2 Gy). The initial number of cells was calculated with this condition from published parameter values of the contrasted models and the assumed treatment schedule and was found to be of the order of lolo. The different cell-kinetic models were thus tested under closely similar conditions. Curability. The local curability was taken to be


C = exp (-NS)

(1)

where NS is the expectation value of the number of clonogenic cells surviving after the treatment series. Models. Four models were considered, referred to as the Fischer I model, the Fischer I1 model, the Cohen model and the Kirk-Gray-Watson (KGW) model after the authors (FISCHER 1969, 1971 a, COHEN1973, KIRKet coll. 1971). The Fischer I model. Cells are assumed to be killed by radiation according to the single-hit-multi-target law S(D, Do,n) = 1 -(1 - exp ( - D/Do))”

(2)

where S is the surviving fraction following irradiation with dose D, n is the hit or extrapolation number and Do is the characteristic or mean lethal dose. Two tumour cell populations are taken into account; one well oxygenated and one hypoxic. The fraction of oxygenated cells, p, was assumed constant during the treatment period, i.e. complete reoxygenation to the initial value was assumed to take place between fractions. Beginning with a tumour with NI cells, the number of surviving tumour cells, NS, after a treatment series of f fractions, each with dose D, is written NS = NI x (p x S(D, DO, NO) +(1 -p) x S(D, DA, NA))‘

(3)

where, following the notation by FISCHER, DO and NO denote characteristic dose and extrapolation number for well oxygenated cells, and DA and NA denote corresponding parameters for hypoxic cells. This simple model contains six parameters, viz. NI, p, DO, NO, DA and NA in addition to the treatment schedule parameters f and D. The Fischer II model. In this model reoxygenation, tumour growth and radiationinduced mitotic delay are also taken into account, viz. (1) the fraction of oxygenated cells is variable and given by exp ( - B X N), where N is the total number of cells and

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S. GRAFFMAN, T. GROTH, B. JUNG AND G. SKGLLERMO

/? is a constant to be specified; (2) assuming an exponential growth for oxygenated cells with a rate constant a,the total number of cells of an untreated tumour is given by the differential equation dN/dt = a x N x exp (-/? x N)

(4)

(3) the radiation-induced mitotic delay, At, is assumed to be proportional to the dose, i.e. At = y x D. The number of cells surviving f fractions, each of dose D, was calculated with the following computational scheme (cf. HETHCOTE & WALTMAN1973): for v =0, 1,2, ..., f - 1 calculate Acta Oncol Downloaded from informahealthcare.com by 112.253.6.182 on 05/20/14 For personal use only.

pcv) = exp ( -

/? x N(Y))

N(v+l)= N(v)x (p(v)x S(D, DO, NO) x exp (ax max (0, t -y x D) +(1 -~ ( " 1 ) x S(D, DA, NA))

(5)

(6)

N(O) is the total number of cells initially (=NI), t =time after latest treatment, and max means the maximum of the two arguments 0 and (t - y x D). This more complex model has eight parameters, viz. NI, a,p, y, DO, NO, DA and NA. Implicit in this model is the concept that hypoxic cells are killed by a single-hit mechanism and well-oxygenated cells are killed by a multi-hit mechanism.

The Cohen model. The cell-population kinetic model by COHEN is built on singleand multi-target, exponential lethality functions and a logistic regeneration mechanism. After f fractions of D Gy, delivered at intervals of t days, the cellular surviving fraction, S,, is given by the logistic function

s,=i=o n

S, x Hi x exp (- J x D)(1- (1 - exp ( - K x D))")

S,+(H,-S,)Xexp(-LxH,xt)

(7)

where S,=l and H,= 1. J and K are single- and multi-target sensitivity constants (Gy-l), n is the extrapolation number and L is the mean regeneration rate (day-l). H, is the final population level approached asymptotically with time: i

H, = exp (G) x

n S,

j=1

where G (cell cycles) is the total reparative capacity of surviving cells. Implicit in eq. (7) is the concept that the tumour cells are killed both by single- and multi-hit mechanisms (cf. the Fischer I1 model).

The KGW model. KIRKet coll. (1971) elaborated on the Nominal Standard Dose (NSD)-concept introduced by ELLIS(1966), and derived the following formula for the cumulative radiation effect: CRE = (T/f)-OJ1 x D x f O 4 5

(9)



I

0.6

I

I

I

I

I

1.0 1.2 Mean lethal dose, DO

0.8

I

I

1

5

Fig. 1. Result of comparison between the Fischer I and Fischer I1 models. Three values of p, the fraction of well-oxygenated cells, in the Fischer I model were tested, i.e. p=O.lO (---), 0.50 (-) and 0.90(---). The curves delineate (DO, NO>values, which in the Fischer I model yield curability values that closely simulate those of Fischer 11.

1X Gy

where T is the total treatment time in days, f is the number of treatments and D is the dose per treatment. Dose fractionation schemes giving similar values for CRE should according to these authors have similar effects on normal connective tissue. Computational experiments. For the Fischer I1 model and the Cohen model the curability values for 9 different doses between 1.8 and 2.2 Gy, given daily for 30 days, were calculated. The initial number of tumour cells was in each case chosen so that the daily dose 2 Gy would give one surviving cell. All other parameters were chosen from values given in the two cited references: For the Fischer I1 model the following values were given to the parameters: B=l/NI, a=0.004 h-l, y = O S h Gy-l, NO=4, D O = 1 Gy, N A = l and DA=2.5 Gy. The initial number of tumour cells, NI, was set equal to 3.2 x log, giving one surviving cell after 30 fractions of 2 Gy. For the Cohen model the following values were used (representing squamous cell carcinoma): G = 1.5 cell cycles, J =0.48 Gy-l, K =0.56 Gy-l, n = 10 and L =0.7 day-l. Expressed as characteristic dose the values given correspond to DA =2.083 Gy, DO=1.786 Gy and a mean characteristic dose (DC=l/(J+K)) equal to 0.962 Gy. In order to get one surviving cell after 30 fractions of 2 Gy the initial number of cells (NI) had to be 1.2 x loll cells. Corresponding calculations were then done with the Fischer I model with the same range of doses as above and with the same number of initial cells as calculated for the Fischer I1 and the Cohen models. NA and DA were set to unity and 2.6 X DO, respectively, and the values of DO, NO and p varied. Finally the sum of squares of the difference in curability between Fischer I and the more complex models were computed for the 9 dose values. The variance was chosen as a measure of the dif-

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s. GRAFFMAN, T. GROTH, B. JUNG AND G. SK~LLERMO

I

(---), 0.50 (-) a i d 0.90 (---). The curves delineate (DO,


NO)-values which in the Fischer I model yield curability values that closely simulate those of Cohen.

w n v

0.6

1.o 1.2 Mean lethal dose, DO

0.8

1.4 Gy

ferznce between the models. Pairs of DO and NO values giving variance of less than 0.01 were stored in the computer. For the KGW model the comparisons were made with a fully oxygenated population of initially 1Olo cells, i.e. the very simple submodel of Fischer I with p = 1. This model has only three parameters, viz. NI, Do and n. The values of CRE and the corresponding local curability levels were chosen as shown in the Table. One irradiation per day was assumed, i.e. T/f = 1 in the CRE formula. For a given value of CRE possible combinations of D and f exist. The concern was whether there exist values of the characteristic dose Do and the extrapolation number n such that the curability estimate C is constant when D and f are varied in such a way that CRE is constant. The calculations were performed in the following way: (1) for f = 20, 30 and 40 compute D to give the desired value of CRE, (2) for n = l to 3 in steps of 0.2 compute Do to give the desired curability estimate using the values of D and f obtained in (1).

Thus for each CRE-value a family of curves in the (Do,n)-plane was obtained, each curve representing one choice of f, the number of fractions.

Results The result of the comparison between the Fischer I and Fischer I1 models appears in Fig. 1. The figure gives the range of DO-NO values in the Fischer I model that gives a calculated variance in curability less than 0.01. This value corresponds to a 10 per cent variation in the local curability, a variation that is whithin the uncertainty of e.g. the dose simulation calculation (GRAFFMAN et coll. 1975 b). The comparison


c

= .-0

-

2-

Y

-

CJ 0

-

Q

e -

Y

x


w

-

1

I

1.0

I

I

1.2

1

I

1.4

II

II

I

7

Fig. 3. Result of comparison between the simplified Fischer I model(p= 1)andtheKGWmodel. Three fractionation schemes were considered, i.e. f, the number of fractions, = 20 (-+, 30 (-) and 40 (---). The dose per fraction was calculated from the KGW formula for CRE =20 (upper right cluster), 16 (lower left cluster) and 18 (central cluster).

1.6 Gy

Mean lethal dose, Do

was made for three values of the fraction of well-oxygenated cells, p=O.l, 0.5 and 0.9. The results imply that there is quite a range of DO-NO values for which both models are equivalent with regard to the final curability value in the dose range 54 to 66 Gy and that the Fischer I model may be reduced further in this respect to the simple model with just one population of tumour cells (p = 1.O) provided Do= 1.04 to 1.05 Gy and n =4.0 to 4.3 in this special case. The comparison of Fischer I and the Cohen model appears in Fig. 2. Also in this case it is apparent that a proper combination of DO-NO pairs yields almost complete equivalence between the two models concerning curability within the selected dose interval and that the simplest model with one cell-population may be used with Do=0.89 Gy to 0.92 Gy and n =4.8 to 5.1 in the situation considered here. The equivalence of the simple Fischer I model (p = 1) to the KGW model bears critically on the existence of Do-n values that make the Fischer I model predict a constant curability under conditions giving a constant CRE. As illustrated in Fig. 3 there is, in fact, for each CRE-value, a small range of Do and n values where all curves lie very close together and intersect each other. (The ranges are tabulated in the Table.) This means that for Do and n in such a region the curability estimate from the simple model is constant when CRE is constant, for all tested values of f. The Do values increase with decreasing curability, as would be expected, but the range of n-values is constant, 2.2 to 2.5.

Conclusions and Discussion

Although no exhaustive investigation of all published tumour response models or of all suggested or reasonable parameter values in the four models was made, it was concluded, that the results of the numerical experiments strongly indicate that the basic hypothesis is sound, i.e., a simple cell-kinetic model is adequate, or at least not

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S. GRAFFMAN, T. GROTH, B. JUNG AND G. SKOLLERMO

Table


CRE-values and the corresponding curability Ievels assumed. The ranges of Do and n are those which make the simple model equivalent to the CRE-model

CRE

Curability (in per cent)

Do-range

n-range

16 18 20

80 50 20

1.1-1.2 1.3-1.4. 1.5-1.6

2.2-2.5 2.2-2.5 2.2-2.5

inferior to other, more complex models for treatment field optimisation when a conventional fractionation regimen is followed. The parameters of any model should be regarded as lumped quantities, which, if properly chosen, reflect some basic characteristics of the real system. The more complex the models become and the more realistic they may seem, the larger is the number of parameters and the more difficult to estimate are the values of the parameters. Cell-kinetic parameters are, at present, not well-known from laboratory experiments or clinical experience. Furthermore, the theoretically defined parameters may not necessarily have a one-to-one correspondence to observable quantities. This circumstance gives further support to the idea of using simple cell-kinetic response models for treatment field optimisation. The present investigation ended with the simple 3-parameter model described by eqs (1, 2, 3) with p = l . The parameters Do and n should in a real situation be varied in quite a large space, e.g. Do= 0.8 to 1.4 and n = 2 to 6 until further evidence is collected. If it could be shown that for an appropriate selection of clinical cases the optimum choice of treatment field parameters is little depending on the particular choice of (Do, n)-values in this wide (Do,n)-subspace, the first objective of the present analysis would be obtained. However, it may be reasonable to speculate a little further. From the diagnostic findings and microscopic measurements, including cytometry for the types of tumour considered, it should be possible to have a rough estimate of the total number of tumour cells in the target volume of the special clinical case. From the simple cellkinetic formula areas in the (Do, n) space might then be determined for which the model predictions of local curability agree with therapeutic results on similar tumours. However, optimisation of treatment field parameters should, in principle, also include cases that have a tumour cell density that is not constant in space. Neither diagnostic tools nor model descriptions of this frequent and important situation are available.


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SUMMARY Three cell-kinetic models of tumour response to ionising radiation were compared with regard to their prediction of variation in curability when the dose to the tumour varied between 54 and 66 Gy given in 30 fractions. It was found that a simple model emulated the results of the more complex ones when the parameters of the simple model were properly adjusted. A comparison of the simple model and the CRE formula gave a similar result.


ZUSAMMENFASSUNG Drei Zell-kinetische Modelle fur die Tumor-Reaktion gegenuber ionisierender Strahlung wurden in Hinblick auf die Vorhersage der Veriinderung in der Kurabilitat verglichen, wenn die Dosis zwischen 54 und 66 Gy variierte, gegeben in 30 Fraktionen. Es wurde gefunden, dass das einfache Modell mit den Ergebnissen der mehr komplizierten konkurrieren kann, wenn die Parameter des einfachen Modells richtig eingestellt waren. Ein Vergleich des einfachen Modells mit der CRE Formel ergab ein ahnliches Resultat.

RBSUMB Trois moddes de cinktique cellulaire de rkponse de tumeur aux radiations ionisantes ont btb comparks en ce qui concerne la prkvision des variations de curabilitk quand la dose 21 la tumeur varie entre 54 et 66 Gy donnks en 30 fractions. Les auteurs ont constatk qu’un modele simple donne des rksultats kquivalents 21 ceux des modkles plus complexes quand les parametres du modkle simple sont convenablement ajustks. La comparaison du modkle simple et de la formule CRE a donnk un rksultat similaire.

REFERENCES ALMQUIST K. J. and BANKSH. T.: A theoretical and computational method for determining optimal treatment schedules in fractionated radiation therapy. Math. Biosci. 29 (1976), 159. COHENL.: An interactive program for standardization of prescriptions in radiotherapy. Comp. Prog. Biomed. 3 (1973), 27. ELLISF.: The relationship of biological effect to dose-time fractionation factors in radiotherapy. In: Modem trends in radiotherapy. Vol. 1. Edited by T. J. Deely and A. P. Wood. Butterworths, London 1966. FISCHER J. J.: Theoretical considerations in the optimisation of dose distribution in radiation therapy. Brit. J. Radiol. 42 (1969), 925. - (a): Mathematical simulation of radiation therapy of solid tumours. I. Calculations. Acta radiol. Ther. Phys. Biol. 10 (1971), 73. - (b): Mathematical simulation of radiation therapy of solid tumours. 11. Fractionation. Acta radiol. Ther. Phys. Biol. 10 (1971), 267. GLICKSMAN A., CEDERLUND J., COHENM., CUNNINGHAM J., JUNGB., OLSENB. and ORR J. S., Editors: Computers in radiation therapy. Proceedings of the Fourth International Conference on the Use of Computers in Radiation Therapy, Uppsala, Sweden 1972.

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GRAFFMAN S., GROTH T., JUNG B., SKOLLERMO G. and SNELL J.-E. (a): Cell kinetic approach to optimizing dose distribution in radiation therapy. Acta radiol. Ther. Phys. Biol. 14 (1975), 54.


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(b): Errors and uncertainty in external radiation therapy. Acta radiol. Ther. Phys. Biol. 14 (1975), 239. HETHCOTE H. W. and WALTMAN P.: Theoretical determination of optimal treatment schedules for radiation therapy. Radiat. Res. 56 (1973), 150. HOPEC. S., LAURIEJ., ORRJ. S. and HALNANK. E.: Optimization of X-ray treatment planning by computer judgement. Phys. in Med. Biol. 12 (1967), 531. KIRKJ., GRAYW. N. and WATSONE. R.: Cumulative radiation effect. Part 1. Fractionated treatment regimes. Clin. Radiol. 22 (1971), 145. STERNICK E. S.: Computer applications in radiation oncology. Proceedings of the Fifth International Conference on the Use of Computers in Radiation Therapy, Hanover, New Hampshire 1974. The University Press of New England, 1976. VAN DER LAARSER. and STRACKEE J.: Pseudo optimization of radiotherapy treatment planning. Brit. J. Radiol. 49 (1976), 450. WHELDON T. E. and KIRKJ.: Mathematical derivation of optimal treatment schedules for the radiotherapy of human tumours. Fractionated irradiation of exponentially growing tumours. Brit. J. Radiol. 49 (1976), 441.