Biomedical Engineering, Vol. 35, No. 2, 2001, pp. 65-70. Translated from Meditsinskaya Tekhnika, Vol. 35, No. 2, 2001, pp. 13-17.Original article submitted June 5, 2000.

0006-3398/01/3502-0065$25.00 2001 Plenum Publishing Corporation

65

Methods of Transition from Nonuniform Dose Distributionin Tissue to Adequate Dose of Uniform Irradiation with the SameProbability of Post-radiation Complications

L. Ya. Klepper

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

Analysis of the dosimetric plan of radiation therapy(RT) and selection of the optimum irradiation planinvolves comparative assessment of the effects of irra-diation on normal and tumoral tissues. One of the mainproblems of RT is to form a uniform dose field ofnecessary configuration and intensity in the lesion focus.An estimate of the value of the uniform dose field inthe lesion focus can be regarded as an estimate of theirradiation effect. The problem of assessment of theirradiation effect on normal organs and tissues is moreintricate because the dose distribution over normal organsand tissues can be significantly nonuniform. This isbecause RT is mainly targeted toward the lesion focusand the radiation should be concentrated within thelesion, whereas the dose field at the interface of lesionand normal tissues should decrease sharply.

Currently used methods for assessment of the non-uniform irradiation effect on normal organs and tissuesare inexact. The dose distribution in tissues is describedin terms of the mean dose, predominant dose, modaldose, values of maximum, mean, and minimum dose intissue, etc. The method of describing the nonuniformdose distribution using dose�volume histograms [8] findsincreasing application. However, these methods do notprovide an unambiguous description of the dose distri-bution. For example, the same mean dose correspondsto infinitely many nonuniform dose distributions. A dose�volume histogram can be regarded as a partially or-dered vector characteristic of nonuniform dose distri-bution. This characteristic also fails to provide anunambiguous description of the dose distribution pat-tern.

The lack of an unambiguous characteristic of non-uniform dose distribution hinders the use of mathemati-cal simulation methods for assessment and prognosis ofthe response of normal tissues and organs to radiation.

The goal of this work was to describe two math-ematical models (MM) for assessing nonuniform dosefield in tissue in terms of the adequate doses (AD) ofuniform irradiation of tissue (scalar characteristics ofthe dose field). The probability of post-radiation com-plications (PPC) in tissue is considered as a criterion ofadequacy of nonuniform and uniform dose distributions.

The mathematical models are based on the follow-ing assumptions.

Assumption 1. Let the following equation be valid:

P[D(V), V] = P[D(1), 1], (1)

where D(V) is the dose of uniform irradiation of thevolume V of given tissue, D(1) is the adequate dose ofuniform irradiation of the same tissue per unit volume,and P[D(V), V] is the probability of post-radiationcomplication as a function of the radiation dose andirradiated volume. Let us assume that the dose D(1) perunit volume can be calculated by the following formula[2, 3, 7]:

D(1) = D(V)V b, (2)

where b is a parameter dependent on the type of tissue.The well-known formula for recalculation of the

tolerant dose values of level P for different volumes ofirradiated tissue follows from this assumption:

D1(V1)/D2(V2) = (V1/V2)−b, (3)

where D1(V1) and D2(V2) are the tolerant doses forvolumes V1 and V2 of irradiated tissue, respectively.

66 Klepper

roteH

The tolerant dose of tissue irradiation is usually takenat the level P = 5%.

As shown in previously [2-5], the probability of post-radiation complication P(D, V) of a certain type causedby uniform irradiation of an organ (tissue) is deter-mined by the modified Weibull function:

where D is the dose of uniform irradiation of tissue, Vis the irradiated volume of tissue, D(1) is the dose perunit volume calculated by Eq. (2), and A1 and A2 arethe model parameters. The probability Q(D, V) of theabsence of post-radiation complication (PAPC) is:

(5)

The mathematical model (5) allows the irradiationdose to be written as a function of radiological param-eters:

D(V) = A1⋅|ln(Q)|1/A2⋅V −b, (6)

D(1) = D(V)⋅V b = A1⋅|ln(Q)|1/A2. (7)

Transition from nonuniform dose distribution intissue to the adequate dose of uniform irradiation isbased on calculation of PPC in nonuniformly irradiatedtissue. As shown in one of the preceding works (1986),under certain assumptions an expression for AD can beobtained using the mathematical model describing PPCin uniformly irradiated tissue.

Mathematical Model 1. Let the irradiated volumeV < V0 be divided into m equal elementary volumes gso that the dose distribution within each elementaryvolume can be regarded as quasi-uniform (V0 is thevolume of tissue under consideration; V = mg is theirradiation volume; V0 = Mg, where M is the numberof elementary volumes).

Equations (4) and (5) can be applied in the case ofnonuniform dose distribution under the following twoassumptions [2, 3].

Assumption 2. The probability of the absence ofpost-radiation complications in the ith elementary volume

(i = 1, ..., m) depends on the dose Di in the elementaryvolume g and the total volume of irradiated tissue Vbut does not depend on the dose distribution in otherelementary volumes.

Assumption 3. The probability of the absence of post-radiation complications in the ith elementary volume gexposed to quasi-uniform irradiation dose Di is describedby the following mathematical model based on the modifiedWeibull function [2, 3] (provided that V > g):

Let the dose distribution in the irradiated volumeV or, to be more precise, in m control points distributedwithin the irradiated volume be described with a vectorDi, ..., Dm. The relative volume of irradiated tissue canbe approximately determined as V = m/M, where M isthe total number of control points within the tissue volumeV0 and m is the number of control points in which thedose exceeds a certain threshold value Dt characteristicof the irradiated tissue. Determination of the values ofDt for various organs and tissues is a serious problemthat should be solved using both experimental and theo-retical methods. The probability of the absence of post-radiation complications in an elementary volume is:

(9)

It follows from assumptions 2 and 3 that the PAPCin the elementary volume g is equal to PAPC in thetissue volume V exposed to irradiation dose Di

g1, whereg1 = g/V ≤ 1 is the relative elementary volume of ir-radiated tissue.

Taking into account assumptions 2 and 3, the PAPCin tissue exposed to nonuniform irradiation can bedetermined by:

(10)

,1

)1(exp

1exp1,(

22

A

D

A

VD V DP

AAb

−=

⋅−−=) (4)

.1

)1(exp

1exp

,(1,(

22

A

D

A

VD

V DP V DQ

AAb

−=

⋅−=

=−= ))

.)(1

exp,,(2

−=

A

ii VA

D

V

g g V DQ ) (8)

.),(),(1

exp

1exp,,(

1/

/2

2

gi

Vgi

VgAb

i

Abi

i

V DQ V DQ A

VD

A

VD

V

g g V DQ

==

−=

=

−=)

.,(

,(,,(),(

1

1

1

1

1

gm

ii

m

i

gi

m

ii

V DQ

V DQ g V DQ V DQ

=

===

∏

∏∏

=

==

)

))

67Transition from Nonuniform Dose Distribution

Thus, the PAPC in tissue exposed to nonuniformirradiation dose Di, ..., Dm is the geometric mean of thevalues of the PAPC in the elementary volumes exposedto uniform irradiation doses Di, i = 1, ..., m. Equation(10) can be rewritten as follows:

(11)

or

(12)

where DAD is the adequate dose. Equation (12) describesthe PAPC in the case of tissue exposure to a uniformadequate dose DAD:

The adequate dose is an equidosimetric scalar value[1]. It is determined as a dose of uniform irradiationof an organ or given volume of tissue providing thesame PPC as a given nonuniform irradiation of thesame organ (tissue). Characteristics of AD (see Eq. (10))were discussed in [6]. The following specific featuresof AD are the most important.

1. For a given total dose of irradiation of a givenvolume of tissue (i.e., for a given mean dose in theirradiated volume), uniform irradiation provides mini-mum PPC.

2. If the parameters used in the model (4) aresubstituted into model (13), the value of PPC deter-mined using model (9) is overestimated. This is causedby the fact that model (1) is based on the mean dosevalues and, therefore, the dose value in this model isunderestimated.

3. As follows from the specific features of MM 1described above, the radiation load on normal organsand tissues can be reduced by increasing the uniformityof distribution of the total dose within these organs andtissues. The total dose received by normal organs and

tissues should be minimized. Model (11) allows pos-sible values of the radiation load on normal organs andtissues to be determined from the dose distribution andthe total dose value.

4. Radiation therapy of malignant tumors is moreefficacious with nonuniform dose distribution withinthe lesion, because this results in higher values of theadequate dose and, therefore, higher probability ofsterilization of the malignant tissue. It should be noted,however, that this statement is only valid if the malig-nant tumor is regarded as a tissue system rather thana cell aggregate.

5. If the parameters A2 and b are related by theequation

Ψ = A2⋅b = 1,

(i.e., b = 1/A2 or A2 = 1/b), the radiobiological effecton elementary volumes of tissue does not depend on thetotal volume of irradiated tissue [6].

6. Post-radiation recovery of normal organs andtissues usually involves the whole tissue system (sys-temic recovery). Therefore, for normal organs and tissuesA2⋅b is greater than 1 [6]. The greater is A2⋅b, thegreater is the ability of the tissue to recover, i.e., themore flattened is the curve P(D) until reaching its pointof inflexion (P is the probability of post-radiationcomplications caused by dose D).

7. If Ψ is taken to be 1 (A2 = 1/b), the value ofPPC is overestimated.

8. The adequate dose decreases with increasingA2. Therefore, if the value of AD is overestimated,the parameter A2 used in calculations should be in-creased.

Mathematical Model 2. Nonuniform dose distribu-tion in tissue can be described approximately in termsof n isodose surfaces. Consider the set Rn of dose valuescorresponding to isodose surfaces:

Rn = {D1(V1), D2(V2), �, Dn(Vn)}, (14)

where

Di(Vi) < Di + 1(Vi + 1), i = 1, �, n − 1,Vi > Vi + 1, i = 1, �, n − 1,

where Vi is the volume bounded by the ith isodose surface(isodose surface corresponding to the dose Di), i = 1,�, n.

Assumption 4. The radiation dose between the iso-dose surfaces corresponding to the doses Di(Vi) andDi + 1(Vi + 1) is uniform and equal to Di(Vi).

DA

V

V

g

g V DQ V DQ

M

ii

Ab

m

ii

⋅

⋅

−=

==

∑

∏

=

=

1

2A

2

1

1exp

,,(),( )

,1

1exp

1exp),(

2

AD

2

AD

−=

−=

AAb

A

D

A

VVD V DQ

)()(

.2/1

1

2AD

Am

i

AiD

V

g VD

−= ∑

=)( (13)

68 Klepper

Therefore, the maximum relative error in the dosevalue determined using assumption 4 is:

δ = max[Di + 1(Vi + 1) − Di(Vi)]/Di + 1(Vi + 1). (15)

Thus, the set Rn can be arranged so that:

Di + 1(Vi + 1) = Di(Vi) + ∆D, i = 1, �, n, (16)

where ∆D is the dose increment.Assumption 5. The irradiated tissue volume is equal

to V1 and bounded by the isodose surface correspondingto the dose D1(V1). Outside this volume, the radiationdose is zero.

The greater the number n, the more justified is theassumption of uniformity of the dose field between theisodose surfaces. The number n and the set Rn can alwaysbe selected to provide the required relative uniformityδ of the dose field.

Assumption 6. If the dose field has only one extre-mum, the isodose surface corresponding to a smallerdose is enclosed within the isodose surface correspond-ing to a greater dose. Therefore,

Di(Vi) < Di + 1(Vi + 1), Vi > Vi + 1, i = 1, �, n − 1. (17)

The case of many extrema is considered below.The nonuniform dose distribution corresponding to

the set Rn can be considered as a superposition of uniformdose fields. The uniform dose field D1(V1) is producedwithin the volume V1. Then, the additional uniform dosefield

D2(V2) = D2(V2) − D1(V1) (18)

is produced within the volume V2 < V1, etc., until theadditional uniform dose field

Dn(Vn) = Dn(Vn) − Dn − 1(Vn − 1) (19)

is produced within the volume Vn.It is reasonable to determine AD per unit volume

(elementary volume) of irradiated tissue. In this case,the values of AD for the total volume of irradiatedtissue can be determined using Eqs. (1) and (2). Thedose distribution within the selected elementary vol-ume should be uniform (quasi-uniform) with a givenaccuracy.

Thus, the nonuniform dose distribution Rn can bereduced to an adequate total dose of uniform irradia-tion of the elementary volume of tissue:

D(1) = D1(V1)V 1

b + [D2(V2) − D1(V1)]V 2

b ++ [D3(V3) − D2(V2)]V

3b + � =

The total irradiated tissue volume was taken to beV1. Therefore, the adequate dose of uniform irradiationof the volume V1 is:

DAD(V1) = D(1)V 1

−b. (21)

Let us prove the following statement.Statement 1. The adequate dose of uniform irradia-

tion of tissue calculated by Eq. (21) and the nonuni-form dose distribution determined by the set Rn providethe same PPC.

Proof. It is sufficient to prove that Eq. (20) can beobtained by applying Eq. (11) to the nonuniform dosedistribution determined by the set Rn. Let the set Rn

contain only two elements:

R2 = {D1(V1), D2(V2)}. (22)

Let the functions P(D, V) and Q(D, V) be describedby Eqs. (4) and (5). Assume that tissue volume V1 isexposed the dose D1(V1), whereas tissue volume V2 isexposed to dose:

D2'(V2) = D2(V2) − D1(V1). (23)

If the effects of these exposures are taken to beindependent, the following formula for PAPS is ob-tained using Eq. (5):

i

.)()(2

111

∑∑=

−−=

−=n

i

bi i i

n

i

biii VVD VVD (20)

,1

111exp

1

}{exp

1exp

2

1221

2

21122

2

111

−+

−=

=

−−⋅

−=

A

AbAb

A

D D D

A

VVD VD

A

VVD

)()()(

)()()(

where D12(1) is the dose D1 in volume V2 per unit volume(D1(V2)). Taking the logarithms of the left and rightsides of Eq. (24), we obtain (taking into account Eq.(7)) the following equation:

DAD(1) = D1(1) + D2(1) − D12(1) = D1(V1)V 1

b +

+ [D2(V2) − D1(V1)]V 2

b = A1⋅|ln[Q(DAD, 1)]|1/A2. (25)

Q(DAD, 1) = Q(D1(V1), V1)⋅Q(D2'(V2), V2) =

(24)

69Transition from Nonuniform Dose Distribution

Extending this equation to the case of an arbitrary setRn equation (20) is obtained. Thus, the statement is proven.

It can be shown that calculation of DAD using Eq.(20) does not require additional assumptions concerningvariations in the exposure time or dose power. Indeed,the total dose received by the tissue volume Vn is:

Dn = D1(V1) + D2'(V2) + � + Dn(Vn') == [D2(V2) − D1(V1)] + [D3(V3) − D2(V2)] + � +

+ [Dn(Vn) − Dn − 1(Vn − 1)] = Dn. (26)

The accuracy of calculation of DAD depends on thedose value corresponding to the isodose surface bound-ing the irradiated volume of tissue. Let the minimumdetectable value of dose received by tissue be Dmin andthe corresponding irradiated volume of tissue Vmin. Itis necessary to determine the tissue volume consideredas irradiated if the accuracy ε% of calculation of DAD

is specified. Using Eq. (20), we obtain:

(27)

where n = (Dmax � Dmin)/d, Dmax is the maximum dosereceived by tissue, and d is the isodose distribution step(it can be made indefinitely small). The specified ac-curacy of calculation of DAD can be attained as follows.Let us calculate DAD (1) for a certain index k > 1:

(28)

The index k can be determined from the followinginequality:

(29)

The maximum value of k, for which inequality (29)is valid, is taken as determining the boundary dosevalue Dk(Vk) and the minimum permissible volume Vk

of irradiated tissue.

Calculation of the Adequate Dose of Uniform Irradiationof Tissue in the Case of a Multi-extremum Dose Field

A multi-extremum nonuniform dose field can bedescribed in terms of an isodose distribution Rn (a set

of dose values and corresponding volumes of irradiatedtissue; see Eq. (14)). For simplicity, let us consider adose field with two extrema. The method of calculationof DN described below can be easily extended to thecase of an arbitrary number of extrema.

R2, n = {D1(V1), D2(V2), ..., DL(VL), D1, 1(V1, 1), D1, 2(V1, 2), ...,D1, N1(V1, N1), D2, 1(V2, 1), D2, 2(V2, 2), �, D2, N2(V2, N2)}, (30)

where DL(VL) is the dose corresponding to the isodosesurface separating the single-extremum regions of thedose field. This isodose surface bounds the volume VL.The first maximum of the dose field is equal to D1, N1

(V1, N1) and corresponds to uniform irradiation of a tissuevolume V1, N1 < VL. The second maximum of the dosefield is equal to D2, N2 (V2, N2) and corresponds to uniformirradiation of a tissue volume V2, N2 < VL. The firstsingle-extremum region of the dose field contains N1isodose surfaces, and the second single-extremum re-gion contains N2 isodose surfaces. The adequate doseof uniform irradiation of tissue per unit volume iscalculated by the following formula:

(301)

(302)

(303)

where D1, 0 (V1, 0) = DL (VL), D2, 0 (V2, 0) = DL(VL). Thefollowing components of the dose field are taken intoaccount in calculating the adequate dose DL, N1, N2 (1)per unit volume: the dose field outside the single-ex-tremum regions (301); the first single-extremum region(302); and the second single-extremum region (303).

Comparative Analysis of Mathematical Models 1 and 2

Consider an isodose distribution specified by a setR3 and containing three isodose surfaces:

R3 = {D1(V1), D2(V2), D3(V3)}. (31)

Assume that assumptions 4-6 are valid for thisdistribution. It should be taken into account that thetissue volume corresponding to the dose D1 is V1 � V2,the tissue volume corresponding to the dose D2 is

,)()()1(2

111

AD ∑∑=

−−=

−=n

i

bi i i

n

i

biii VVD VVD D

.)()()1(1

11AD ∑∑+=

−−=

−=n

k i

bi i i

n

k i

biii VVD VVD D

%.%100)1(

)1()1(

AD

ADADe

D

D D ≤′−

+−= ∑∑=

−−=

L

i

bi i i

L

i

biiiN N L VVD VVD D

211

121, )()()1(,

∑∑=

−−=

+−+1

211111

1

1111 )()(

N

i

bi i i

N

i

bi i i VVD VVD ,,,,,,

∑∑=

−−=

−+2

221212

2

1222 ,)()(

N

i

bi i i

N

i

bi i i VVD VVD ,,,,,,

V2 � V3, and the tissue volume corresponding to thedose D3 is V3. Using Eq. (11) (MM 1), we find that:

(32)

Under the same assumptions, using Eq. (18) (MM2), we find that:

(33)

Comparison of Eqs. (32) and (33) shows that the twoMM have the same structure. The only differencebetween the two MM is the method of calculation ofPAPC in volumes V2 and V3.

The main advantage of MM 2 over MM 1 is thatMM 1 is based on assumptions 2 and 3 of quasi-independence of PAPC in tissue volumes exposed touniform irradiation, whereas MM 2 does not requirethese assumptions because it is based on superposition(summation) of dose fields. However, only testing in

Klepper70

clinical practice can show which of the two mathemati-cal models is more effective for processing clinical data.

Conclusions

1. A mathematical model (MM 1) for calculatingan adequate dose of uniform irradiation of tissue (doseof uniform irradiation providing the same PPC as givennonuniform irradiation dose) was developed. The modelis based on the assumption of quasi-independence ofPPS in separate tissue volumes exposed to radiation.The dependence of the parameters of MM 1 on thetissue ability for recovery was studied.

2. A mathematical model (MM 2) for calculatingthe adequate dose per unit volume was developed. Thismodel is based on superposition of dose fields.

3. The comparative analysis of the two mathemati-cal models is performed. It is shown that the two MMhave the same structure. The only difference betweenthe two MM is the method of calculation of PAPC indifferent volumes of irradiated tissue. The mathemati-cal model 2 does not require assumptions 2 and 3,because it is based on superposition of dose fields.

This study was supported by the Russian Founda-tion for Basic Research, project No. 980100057.

REFERENCES

1. I. B. Keirim-Markus, Equidosimetry [in Russian], Mos-

cow (1980).2. L. Ya. Klepper, Formation of Dose Fields by Remote

Radiation Sources [in Russian], Moscow (1986).3. L. Ya. Klepper, Formation of Dose Fields by Radio-

active Sources of Radiation [in Russian], Moscow (1993).4. L. Ya. Klepper, Med. Tekh., No. 2, 24-27 (1997).

5. L. Ya. Klepper, Med. Radiol., No. 1, 47-50 (1997).6. L. Ya. Klepper, Med. Tekh., No. 5, 6-10 (1999).

7. J. T. Lyman and A. B. Wolbarst, Int. J. Radiat. Oncol.Biol. Phys., 13, No. 1, 103-109 (1987).

8. T. E. Schultheiss, C. G. Orton, and R. A. Peck, Med.Phys., 10, No. 3, 410-415 (1983).

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]),[

]),[

]),[

1exp

1exp

1exp

,,())1((

1/3111/3

112

1/2112

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3332221112AD2

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