Biomedical Engineering, Vol. 36, No. 2, 2002, pp. 100-107. Translated from Meditsinskaya Tekhnika, Vol. 36, No. 2, 2002, pp. 31-37.Original article submitted February 28, 2001.

0006-3398/02/3602-0100$27.00 2002 Plenum Publishing Corporation

100

Approximate Methods for Calculating Probability of Post-radiationComplications. Poisson�Klepper�LQ Method

Central Institute of Mathematical Economics, Russian Acad-emy of Sciences, Moscow, Russia.

L. Ya. Klepper

Introduction

The main goal of planning of radiation therapy(RT) of malignant tumors is to select the irradiationplan that allows the main problem of RT (creation ofthe effective therapeutic dose field of necessary con-figuration and intensity in the target lesion) to be solvedwith minimal radiation load on normal organs and tissues(less than the level of tolerance). To solve this problem,it is necessary to develop methods for assessment of theprobability of resorption (PR) of the lesion focus andthe probability of post-radiation complications (PPC)in normal organs and tissues. Development of math-ematical models (MM) providing adequate descriptionof PR of the lesion focus and PPC in normal organsand tissues as functions of the irradiation plan is oneof the important tasks of modern radiology.

In current medical practice MM based on normaldistribution of probabilities, logistic function, and amodified Weibull function are used for description ofPPC in normal organs and tissues as a function ofirradiated tissue volume and absorbed dose [1, 2, 13].

The problem of description of PR of the lesion focusas a function of its volume and total absorbed dose ismore complicated. Malignant tissues can have differenthistological structure. Typical (i.e., radiosensitive) andradioresistant cells constituting a malignant tumor differin their radiobiological characteristics. Response of amalignant tumor to irradiation depends on the locationof the lesion focus. These and many other factors makethe problem of description of PR of the lesion focusrather difficult.

The goal of this work was to develop mathematicalmodels providing approximate calculation of PPC innormal organs and tissues as a function of irradiated

volume and absorbed dose and PR of the lesion focusas a function of its volume and total absorbed dose.

This work consists of three parts, each of them beingpublished in Biomedical Engineering as a separatearticle. In the first part, methods for approximatecalculation of PPC (ACPPC) in normal organs andtissues based on the Klepper MM were considered. Amethod for calculation of equivalent conditions of ir-radiation (CECI) of the lesion focus is suggested.

In the second part, an approximate method forcalculating PR of the lesion focus is considered. Thismethod is based on three mathematical models: Poissonmodel, Klepper model, and LQ-model (PKLQ method).It is assumed that the number of typical malignant cellsdetermined using the Poisson model is approximatelyequal to the actual number of malignant cells in thelesion focus, i.e., that radioresistant cells either areabsent or their radiobiological characteristics are simi-lar to those of typical cells.

The third part will be published in one of the sub-sequent issues of Biomedical Engineering. In it, the PKLQmethod is generalized. For this purpose, the effectivenumber of typical malignant cells in the lesion focus(effective volume of the lesion focus) is considered.

The problem of determination of PR of malignanttumors as a function of the total absorbed dose and thevolume of the lesion focus is one of the most urgentproblems of modern radiology. In current medicalpractice, the required cancericide dose is usually de-termined based on the personal experience of the ra-diologist in planning RT and generalized experienceof other physicians described in the medical literature.In this work, several approaches to this difficult prob-lem are discussed.

The probability of resorption of the lesion focuscan be described on cellular and tissue levels. Foradequate description of PR of the lesion focus on thecellular level, the dependence of the survival rate ofmalignant cells on the conditions of irradiation shouldbe known. Presently, the LQ-model is widely used for

101Probability of Post-radiation Complications

roteH

describing the fraction of surviving malignant cells asa function of the total dose received by the lesion focus.

LQ-Model. It was shown in [7, 10, 11, 13] that thefraction of malignant cells surviving exposure to ion-izing radiation can be described using the LQ-functionsuggested in [6, 9]:

C(d) = exp(−αd − βd2), (1)

where α and β are the parameters of the model and dis the single-exposure dose. A mathematical modelsuggested in [13] allows PR of the lesion focus to becalculated using the Poisson distribution. This modelis based on the following assumptions.

1. Malignant tissue consists of typical and radiore-sistant malignant cells.

2. The dependence of the survival rate of typicaland radioresistant cell subpopulations on the radiationdose is described by LQ-functions. Parameters of LQ-functions do not depend on the volume of the malignanttissue.

3. The processes of inactivation of typical and ra-dioresistant malignant cells are considered as indepen-dent.

4. Time intervals between the irradiation sessionsare sufficient for complete recovery of malignant cellsexposed to sub-lethal doses, whereas the total RT durationis short enough to ignore the malignant cell repopulation.In particular, the standard irradiation plan (5 sessionsper week; single-exposure dose of 2 Gy) meets theserequirements.

5. The number of radioresistant malignant cells isproportional to the volume of malignant tissue (totalnumber of malignant cells in the tumor) [8].

6. Parameters of the LQ-function determined in vitro[7, 10, 13] can be used for approximate description ofthe survival rate of typical malignant cells.

7. The density of malignant cells is 109 cells percm3, unless otherwise specified.

8. The single-exposure dose is 2 Gy independent ofthe total dose received by the lesion focus.

Assumptions 7 and 8 allow calculations to be sim-plified. A mathematical model based on assumptions 1-6 is described in [2]. Consider n sessions of irradiationof typical malignant cells with a single-exposure dosed. If assumptions 1-6 are valid, the surviving fractionof malignant cells C(D) is described by the followingequation:

where D = nd and γ = α + βd. Parameters α and βof MM (1) for six histologically different types ofmalignant cells are given in Table 1 [10]. These datawere obtained in vitro.

Poisson Model. The probability of resorption ofmalignant tissue is usually defined as the probabilityof inactivation of all cells constituting this tissue. Ifthe probability of destruction of each malignant cellis supposed to be independent on the total number ofcells, i.e., the tumor is regarded as a cell aggregaterather than a tissue system, PS of malignant cells canbe described using the Poisson distribution [3, 13]. Theprobability of destruction of all malignant cells isdescribed by the following equation:

P(D, N0, d) = exp[−N0⋅C(d)] = = exp[−N0⋅exp(−γD)] = exp[−N0⋅exp(−γdn)], (3)

where N0 is the number of cells constituting the ma-lignant tissue.

Function (3) describes an asymmetric logistic curvethat increases sharply with increasing radiation dose.This property of the function (3) is considered in detailin [2].

Taking into account assumptions 1-6, the math-ematical model for calculation of PS of malignant tissuetakes the following form [2, 7]:

P(D, N0, d) = P(D, N01, d)⋅P(D, N02, d) == exp[−N01⋅exp(−α1D − β1d⋅D)]⋅exp[−N02⋅

⋅exp(−α2D − β2d⋅D)] = exp[−N01⋅exp(−γ1D)]⋅exp[−N02⋅⋅exp(−γ2D)] = exp[−N01⋅exp(−γ1D) − N02⋅exp(−γ2D)], (4)

where P(D, N01, d) is PS of a population of N01 typicalmalignant cells exposed to the total dose D; P(D, N02,d) is PS of a population of N02 radioresistant malignantcells exposed to the total dose D; d is the single-exposuredose; N = N01 + N02 is the number of cells constitutingmalignant tissue; N02 = kN0 and N01 = (1 � k)N0, wherek is the relative number of radioresistant cells in avolume V0 = N0/ρ (k ≥ 0); ρ is the density of malignant)2(,exp(exp()exp(

)(exp)exp()(

21

2

1

2

D Dd D /nD D

d d d d DCn

i

n

i

))baba

baba

γ−= −−= −−=

=

+−=−−= ∑∏

==

TABLE 1. LQ-Function Parameters for Various Types of Malignant Cells

(data taken from [10])

Tumor α, Gy−1 β, Gy−2 α/β, Gyγ, Gy−1

d = 2 Gy

Glioblastoma 0.241 0.029 8.31 0.299

Melanoma 0.255 0.053 4.81 0.361

Squamous cell carcinoma 0.273 0.045 6.07 0.363

Adenocarcinoma 0.311 0.055 5.65 0.421

Lymphoma 0.451 0.051 8.48 0.553

Small-cell carcinoma 0.650 0.081 8.02 0.812

102 Klepper

cells; α1 and β1 are the parameters of the LQ-model fortypical malignant cells; α2 and β2 are the parametersof the LQ-model for radioresistant malignant cells; γ1 =α1 + β1d; γ2 = α2 + β2d. We obtain that:

V0 = ρN01 + ρN02 = V01 + V02, (5)

where V01 is the volume occupied by typical malignantcells and V02 is the volume occupied by radioresistantmalignant cells.

Klepper Model. Accumulated experience in RT showsthat the dependence of PR of the lesion focus of a fixedvolume on the total dose received by the lesion focusis logistic and, therefore, can be described using thefollowing model (Klepper model) [1, 2]:

(6)

where A1 and A2 are the parameters of the model.Examples of the use of MM (6) for description of PRof the lesion focus are given in [1, 2]. It was found thatthe increase in PR of the lesion focus with increasingtotal absorbed dose predicted by MM (3) is sharperthan the increase observed in clinical practice [1, 2].

CECI Method

This method was described in the first part of thiswork, so that we restrict ourselves to a schematicdescription of this method. It was shown that MM (3)can be used for determining the relationship betweenthe dose and volume of the lesion focus for equivalentconditions of irradiation.

Definition. Conditions (D1, V1, d) and (D2, V2, d)of irradiation of identical malignant tissues are equiva-lent if they provide the same PR.

Consider the function P(D, N0, d) = P(D) for a fixedvolume of the lesion focus V0 = N0/ρ and a fixed single-exposure dose d (d ≥ 0, D ≥ 0, k ≥ 0). The CECI methodis based on the following two statements proved in thefirst part of this work.

Statement 1. If assumptions 1-8 are valid and thePR of the lesion focus is described by Eq. (5), the PSof malignant tissue of a volume V1 = rV0 (N1 = rN0)is equal to P(D, V0, d)r.

Statement 2. Consider the function P(D, V0, d) fora given single-exposure dose d, a given volume V0 =N0/ρ of the lesion focus, and an arbitrary total dose D.The equivalent cancericide dose D1 (i.e., providing the

same PR) for an arbitrary volume V1 of the lesion focuscan be determined on the basis of statement 3 using thefollowing equation:

P(D1, V0, d) = P(D, V0, d)1/r. (7)

The equivalent cancericide dose can be determinedfor an arbitrary volume of malignant tissue using Eq.(7) if the function P(D, V0, d) for given d and V0 isknown (analytically or graphically). The CECI methodallows the equivalent cancericide dose to be calculatedwithout determining parameters α1, β1, α2, and β2 ofLQ-functions for typical and radioresistant malignantcells. It is also unnecessary to determine the relativenumber k of radioresistant malignant cells in the lesionfocus. It was shown in the first part of this work thatfor practical application of the CECI method it is moreappropriate to consider the function P(D, V0, d) as aone-variable function. Even in this case, however, D1

is determined only approximately, so that in the gen-eral case the obtained D1 is not a multiple of d1. Thus,the number of sessions is determined with an accuracyof ± 1 (to meet the condition D1 = nd1, D1 should berounded). If the dependence of PR on the total cancericidedose is specified graphically, the CECI method can beused only for the visible part of the curve. Thus, in thegeneral case the dependence of PR on the total cancericidedose cannot be determined completely for another vol-ume of the lesion focus (see the first part of this work).

PKLQ Method for Approximate Calculation of PR ofthe Lesion Focus as a Function of Its Volume and theTotal Absorbed Dose

Parameters of the LQ-function for 6 different typesof malignant cells are given in Table 1. According toassumption 6 these cells are taken to be typical (ra-diosensitive). This information can be used for estimat-ing the relationship between the numbers of the typicaland radioresistant malignant cells in the tumor.

Statement 3. If the parameters of the LQ-functionare known, the Poisson model allows the qualitativerelationship between the numbers of the typical andradioresistant malignant cells in the lesion focus to bedetermined.

Proof. Let us prove this statement for squamous-cellcarcinoma. The systematized clinical information isgiven in Table 2.

Let us assume that the density of malignant cellsρ (cells/cm3) is known, so that the number N of malig-nant cells can be determined from the volume V of the

,exp1,(2

1

−−=

A

A

D V DP )

Probability of Post-radiation Complications 103

lesion focus (tumor tissue) by the formula N = ρV. Letus also assume that the density of malignant cells isconstant over the lesion focus, and that the lesion focuscontains only typical (radiosensitive) malignant cells.In this case, the structure of the tumor tissue and thequalitative relationship between the numbers of the typicaland radioresistant malignant cells can be assessed usingthe Poisson model.

Indeed, let us determine the parameters of MM (3).The single-exposure dose is 2 Gy. In this case, tuningof the parameters of MM (3) involves only determina-tion of the number N0 of typical malignant cells in thelesion focus. Parameters of the LQ-function for typicalsquamous cells are given in Table 1 (α = 0.273, β =0.045). Using the least-squares method, we obtain thatN0 = 1.252⋅109 cells. Thus, if the density of malignantcells is 109 cells/cm3, the tumor volume is 1.252 cm3.

If the values of the total absorbed dose in Table2 are increased by 5 Gy, the number of malignant cellsN+5 is found to be 7.686⋅109 (V+5 = 7.686 cm3). Decreas-ing the values of the total absorbed dose in Table 2by 5 Gy, we obtain that the number of malignant cellsN�5 is 2.038⋅108 (V�5 = 0.204 cm3). Thus, knowing theparameters of the LQ-function for typical malignantcells, density of the malignant cells, and the tumorvolume, the number of radioresistant cells in the tumorcan be estimated. If the tumor volume calculated usingMM (3) exceeds the actual volume of the tumor, thetumor contains radioresistant cells. The greater is thediscrepancy between theoretical and clinical data, themore radioresistant cells are contained in the tumor.Usually, only a range of possible values of the densityof malignant cells is known. In this case, it is impos-sible to reveal the presence of radioresistant cells unlessthe calculated number of malignant cells (or the tumorvolume) falls beyond the range of possible values. Thus,the statement is proved.

Method of Reduction of the Irradiation Conditions

The systematized clinical information about thefrequency of resorption of the lesion focus as a function

of its volume and the total absorbed dose is requiredfor determining the parameters of the model describedby Eqs. (3), (4), and (6). Usually, the number of clinicalobservations is too small, or they do not provide therequired diversity of information. Only seldom thesystematization of clinical information required forobtaining the necessary response characteristics can beperformed without additional procedures.

It is well known that the reduction of clinicalinformation to a given volume of malignant tissue ora given value of the total absorbed dose is one of themain methods of primary processing of clinical infor-mation. This method allows the amount of clinicalinformation used for determining the parameters of themathematical model to be considerably increased. Forexample, in [12] the Ellis model [5] was used forsystematizing the clinical information.

Consider a method of reduction of clinical infor-mation to a given volume of malignant tissue basedon the Poisson model. Consider a lesion focus contain-ing both typical and radioresistant malignant cells.Using the Poisson model, the condition for equiva-lence of two different irradiation conditions is writtenas follows:

P(D, N0, d) = P(D', N'0, d'), (8)

where (D, N0, d) are the given conditions of irradia-tion, and (D', N'0, d') are the reduced conditions ofirradiation. Using Eq. (8), each of the conditions ofirradiation can be determined as a function of theother two. Our task is to determine the reduced valueD' of the total absorbed dose from the given reducednumber of malignant cells (reduced tumor volume) andthe given reduced single-exposure dose. It should betaken into account that V0 = N0/ρ and V'0 = N'0/ρ (ρis the density of malignant cells, α1 and β1 are theparameters of the LQ-function for typical malignantcells, and α2 and β2 are the parameters of the LQ-function for radioresistant malignant cells). Using Eq.(8), we obtain that:

It should be noted that the reduced value of thetotal absorbed dose is calculated from the ratio be-tween the volumes of the lesion focuses (or the ratiobetween the numbers of malignant cells). If the single-exposure doses are equal (d' = d), we obtain that:

TABLE 2. Dependence of PR on the Total Absorbed Dose (stages T2

and T3; unreliable clinical information is marked with asterisks) [11]

Dose, Gy PR

57.4

60.0

63.3

66.1

*69.3*

0.41

0.68

0.79

0.86

*0.66*

.

)(ln

2211

22110

0

d d

Dd d V

V

D'baba

baba

+++

++++

′

= (9)

104 Klepper

(10)

If the single-exposure doses are equal, and the tumorcontains only typical cells, Eq. (10) takes the form:

(11)

Assuming that d = 2 Gy, α = 0.273, β = 0.045(Table 1), and the clinical information is reduced toa volume of 1 cm3 for a single-exposure dose of 2 Gy,we obtain that:

(12)

where D' is the reduced value of the total absorbeddose meeting the condition for equivalence of the fol-lowing irradiation schemes: total dose D', d = 2 Gy,V = 1 cm3, and total dose D, d = 2 Gy, V = V0.Equation (12) is used for reducing the total absorbeddose to a volume of 1 cm3, provided that the tumorcontains only typical malignant cells. If the tumorcontains both typical and radioresistant malignant cells,only approximate reduction can be performed (themethod of approximate reduction will be considered inthe third part of this work).

Example 1. Let us estimate the likely error of theclinical information reduction to a volume of 1 cm3.The single-exposure dose is taken to be 2 Gy. Let usassume that:

γ = α + βd = 0.273 + 0.045⋅2 = 0.363,γ1 = α1 + 2β1. (13)

Thus, Eq. (10) can be rewritten as follows:

(14)

It follows from Eq. (14) that the accuracy of reduc-tion of the total absorbed dose to a lesion focus volumeof 1 cm3 depends on the parameter γ1. This parametervaries within the range 0 ≤ γ1 ≤ γ = 0.363. The corres-ponding range of variation of H is from 2.755 to 1.377.

The application range of the method of reductionof the irradiation conditions should be assessed. Ob-

viously, this method can be used if the actual numberof malignant cells in the lesion focus is close to thenumber of malignant cells calculated using the Poissonmodel (statement 3). The greater is the number ofradioresistant cells in the lesion focus (or the greateris the discrepancy between the calculated and actualnumbers of malignant cells), the greater is the error ofcalculations by this method.

PKLQ Method

The PKLQ method is based on three mathematicalmodels: Poisson model (3), Klepper model (6), and LQ-model (2). This allows the advantages of the three modelsto be combined. The phenomenological Klepper modelprovides better description of the dependence of PR onthe total absorbed dose than the Poisson model, but itdoes not take into account the volume of tumoral tissue.On the other hand, the Poisson model takes into ac-count the number of malignant cells and, therefore, thetumor volume. The LQ-model provides calculation ofthe number of surviving malignant cells as a functionof the total absorbed dose.

The following assumptions are taken to be valid.Assumption 9. Within the range of therapeutic doses

and PR, the number of typical malignant cells deter-mined using the Poisson model is approximately equalto the actual number of typical malignant cells in thelesion focus.

If assumption 9 is valid, the method of reductionof the total absorbed dose can be used for calculatingPR.

Assumption 10. The Klepper model provides betterdescription of PR of the lesion focus as a function ofthe total absorbed dose than the Poisson model (curveconstructed using the Klepper model is flatter than thecurve constructed using the Poisson model) [4, 13].

Algorithm for Calculation of PR by the PKLQMethod. The algorithm for calculation of PR as afunction of the lesion focus volume and the total ab-sorbed dose consists of two stages.

Preliminary stage. 1. Clinical information is pro-cessed using the method of reduction of the total absorbeddose to a given volume of the lesion focus (for example,1 cm3). This can be done if assumption 9 is valid.

2. Reduced clinical information is systematized, andthe response characteristics (PR for various values ofthe absorbed dose) are determined.

3. Parameters A1 and A2 of the Klepper model (6)and the parameter N0 of the Poisson model (3) aredetermined from the response characteristics. If the

.ln1

0

0

2211

V

V

d d

D D'

′+++

+=baba

.ln1

0

0 V

V

d

D D'

′+

+=ba

( ) ( ),ln755.2ln1

00 V D Vd

D D' ⋅−=

+−=

ba

( ) ( ).g.

VH D V

D D' 00

1

lnln3630

1 ⋅−=+

−=

density of malignant cells is known, the tumor volumecan be determined from N0. Approximate calculationof N0 can be performed using an interactive procedure.For this purpose, a reference point should be selectedon the curve constructed using the Klepper model (6).This procedure is valid only if MM (3) and (6) are ingood agreement within the range of doses and PR underconsideration.

Main stage. 1. The equivalent dose D' reduced toa tumor volume of 1 cm3 and a single-exposure doseof 2 Gy is determined for given tumor volume V, totalabsorbed dose D, and single-exposure dose d.

2. The probability of resorption of the lesion focusis determined using MM (6).

Example 2. By way of example, consider how thedeveloped method can be used for calculating PR ofsquamous-cell throat carcinoma (T2 and T3 stages).The systematized clinical information is given in Table2. At the preliminary stage, the parameters of the Klepperand Poisson model are determined by solving an ex-tremum problem. Omitting unreliable information (point5), we obtain that the Klepper model takes the follow-ing form:

(15)

The dependence of PR on the total absorbed doseobtained using Eq. (15) is shown in Fig. 1 (curve I).A point lying on the curve I and corresponding to adose of 62 Gy and PR of 70% is selected as a referencepoint. The parameter N0 of the Poisson model is foundto be 2.12⋅109 cells. Thus, if the density of malignantcells is, for instance, 109 cells/cm3, the effective volumeof the tumor is 2.12 cm3. The Klepper model corre-sponds to this volume of malignant tissue. The Poissonmodel takes the following form:

P(D) = exp[−2.12⋅109⋅exp(−0.363D)]. (16)

The curve PR vs. dose constructed using Eq. (16)is shown in Fig. 1 (curve II). It can be seen that withinthe dose range D > 60 Gy (PR > 50%), the curves Iand II are close to each other.

Let us assess the accuracy of determination of thetumor volume by the PKLQ method. An approximateclassification of the volumes of the squamous-cell throatcarcinoma was suggested at the Department of Clini-cal Radiology, Russian Medical Academy of Post-graduate Education. According to this classification,the volume of the lesion focus corresponds to the stage

of the disease. The lesion focus is assumed to beellipsoidal in shape.

1. Stage T1: ellipsoid 1 × 0.5 × 0.5 cm of volume0.131 cm3; maximum volume, 2 cm3.

2. Stage T2: ellipsoid 1.5 × 1.0 × 1.0 cm of volume0.8 cm3; maximum volume, 3 cm3.

3. Stage T3: ellipsoid 2 × 3 × 2 cm of volume6.283 cm3; maximum volume, 9 cm3.

4. Stage T4: the volume of the lesion focus exceeds9 cm3.

Thus, clinical observations [11] given in Table 2correspond to a tumor volume within the range from0.8 to 9 cm3. The effective volume of the tumor tissuewas found to be 2.12 cm3 (under the assumption thatthe density of malignant cells is 109 cells/cm3). Thisvalue falls within the specified range.

Example 3. Let us calculate PR of squamous-cellthroat carcinoma by the PKLQ methods. The follow-ing irradiation scheme is used: D = 70 Gy, V = 1 cm3,d = 2 Gy. Using Eq. (12) the total absorbed dose canbe reduced to a volume of 2.12 cm3 (Eq. (15) wasobtained for this volume):

(17)

Substituting D' into Eq. (15), we obtain that:

(18)

Thus, PR of the lesion focus of a volume 1 cm3

exposed to a total dose of 70 Gy (single-exposure doseis 2 Gy) is 99.6%.

Probability of Post-radiation Complications 105

.653.60

exp1(582.9

−−= D

DP )

Fig. 1. Dependence of PR (%) on the total absorbed dosefor squamous-cell throat carcinoma. Curve I is constructedusing the Klepper model; curve II, Poisson model. Clinicalinformation is taken from Table 2 (points 1, 2, 3, 4; point 5is omitted). Analysis shows that curves I and II are close toeach other within the dose range D > 58 Gy.

...

Gy07721

12.2ln

3630

170

D' =

+=

.996.0652.60

07.72exp107.72(

582.9

P =

−−=)

Example 4. Use of the total body of informationcontained in Table 2 deteriorates the agreement be-tween the clinical and theoretical values of PR. TheKlepper model takes the following form:

(19)

The dependence of PR on the total absorbed doseobtained using Eq. (19) is shown in Fig. 2 (curve I). Itis flatter than the corresponding dependence in Fig. 1(curve I) and provides worse agreement with clinicaldata. Selecting as in example 2 D = 62 Gy as a referencepoint, we obtain that the corresponding value of PR is59%, which is 11% less than in example 2. The effectivevolume Veff of the tumor is found to be 3.13 cm3. The

dependence of PR on the total absorbed dose obtainedusing the Poisson model is also shown in Fig. 2 (curveII). Selecting a point D = 64 Gy, PR = 70% as areference point, we obtain that Veff = 4.38 cm3.

Thus, the result of calculation of the effective volumeof the tumor varies from 2.12 to 4.38 cm3 dependingon the selection of the reference point and omission ofunreliable clinical information. This is a permissiblespread, so that the results of calculation by the PKLQmethod are sufficiently stable.

Example 5. Let us compare the efficiencies of thePKLQ and CECI methods for calculation of PR of thelesion focus as a function of its volume and the totalabsorbed dose. For this purpose, consider the curvesPS vs. D for squamous-cell throat carcinoma of volumes1 and 10 cm3 (Fig. 3). Curves 1 and 3 were obtainedusing the CECI method; curves 2 and 4, the PKLQmethod. Within the range of therapeutical doses the curvesare close to each other. For V = 1 cm3 D > 60 Gy (curves1 and 2); for V = 10 cm3 D > 66 Gy (curves 3 and4). For small values of the total absorbed dose, thediscrepancy in the values of PR obtained by differentmethods is considerable. This can be explained by thefact that clinical information corresponds only to thera-peutic values of PR.

A final conclusion about the efficiency of the PKLQand CECI methods can be made only on the basis ofa considerable body of clinical information.

Conclusions

1. The PKLQ method for calculation of PR of thelesion focus as a function of the total absorbed doseand the volume of the lesion focus is described. Themethod is based on the assumption that the lesion consistsmainly of typical (radiosensitive) malignant cells,whereas radioresistant cells either are absent or haveradiobiological characteristics close to those of typicalcells. The PKLQ method is based on three mathemati-cal models: Poisson model, Klepper model, and LQmodel.

2. A method of reduction of clinical information toa given volume of the lesion focus is an important partof the PKLQ method. This method can be used bothfor systematization of clinical information and calcu-lation of PR.

3. The algorithm of implementation of the PKLQmethod is described. The algorithm consists of twostages. The preliminary stage includes preliminaryprocessing of clinical information and tuning of math-ematical models. At the main stage, PR is calculated

106 Klepper

.82.62

exp1(961.8

−−= D

DP )

Fig. 2. Dependence of PR (%) on the total absorbed dosefor squamous-cell throat carcinoma. Curve I is constructedusing the Klepper model; curves II and III, Poisson model.Clinical information is taken from Table 2 (points 1, 2, 3, 4,5). Curve II corresponds to a reference point D = 62 Gy,PR = 59%; curve III, D = 64 Gy, PR = 70%.

Fig. 3. Dependence of PR (%) on the total absorbed dose.Curves 1 and 3 are obtained using the CECI model; curves2 and 4, PKLQ model. Curves 1 and 2 correspond to V =1 cm3; curves 3 and 4, to V = 10 cm3. Within the range oftherapeutic doses the curves are close to each other.

Probability of Post-radiation Complications 107

for arbitrary values of the lesion volume and the totalabsorbed dose.

4. Several examples of application of the PKLQmethod to calculation of PR of squamous-cell throatcarcinoma are given.

REFERENCES

1. L. Ya. Klepper, Formation of Dose Fields by RemoteRadiation Sources [in Russian], Moscow (1986).

2. L. Ya. Klepper, Formation of Dose Fields by Radio-active Sources of Radiation [in Russian], Moscow (1993).

3. V. Feller, Introduction into the Theory of Probability andIts Applications [Russian translation], Moscow (1967).

4. S. P. Yarmonenko, Radiobiology of Man and Animals[in Russian], Moscow (1984), p. 318.

5. K. H. Chadwick and H. P. Leenhouts, Phys. Med. Biol.,18, No. 1, 78-87 (1973).

6. F. Ellis, Clin. Radiol., No. 1, 1-7 (1969). 7. B. Fetil and E. Malaise, Int. J. Radiat. Oncol. Biol.

Phys., 7, No. 5, 621-629 (1981). 8. M. Goitein and J. Busse, Radiology, 117, 407 (1975). 9. A. M. Kellerer and H. H. Rossi, Radiat. Res., 47, No.

1, 14-34 (1971).10. E. Malaise, B. Fetil, N. Chavaudra, and M. Gichard, Int.

J. Radiat. Oncol. Biol. Phys., 12, No. 4, 617-624 (1986).11. L. Peters, H. K. Withers, H. Thames, and G. Fletcher,

Int. J. Radiat. Oncol. Biol. Phys., 8, No. 1, 101-108 (1982).12. L. J. Shukovsky, Am. J. Roentgenol., 108, 27-29 (1970).13. R. J. Yaes, Int. J. Radiat. Oncol. Biol. Phys., 14, No.

1, 147-157 (1988).