Introduction. One of the important problems of

modern radiology is to develop mathematical models

(MM) for calculation of the probability of post�radiation

complications (PPC) in organs and tissues subjected to

radiation therapy with a given scheme of dose fractionat�

ing (DF). Such MMs can be designated as synthesized

MMs (SMMs), because they can be obtained by combin�

ing MMs for calculation of PPC for fixed DF schemes

and MMs for determination of tolerant doses for arbitrary

DF modes and fixed values of PPC.

This work consists of several parts, each of them

being published in Biomedical Engineering as a separate

article. In the first part, synthesis of the Ellis MM and

Klepper MM, as well as the Ellis model and modified

Lyman model, was described. The second part considered

synthesis of the LQ�model and the Klepper model, and of

the LQ�model and modified Lyman model. Synthesis of

two population–phenomenological (PP) MMs and the

Klepper MM were considered in the third part. The syn�

thesis of the two PP MMs and the modified Lyman MM

is to be considered in the fourth part (this article). Several

examples of calculation of SMM parameters on the basis

of clinical information will be given in the fifth part of this

work.

Equations, statements, and assumptions are num�

bered continuously throughout the five parts of this work.

See part 1 for introduction, description of the Klepper

and Lyman MMs and the dependence of the tolerant dose

on irradiated tissue volume for arbitrary PPC value, Eqs.

(1)�(71), statements (1)�(4), and assumptions (1)�(6);

part 2 for Eqs. (72)�(133), statements (5)�(9), and

assumptions (7)�(12); part 3 for Eqs. (134)�(186), state�

ments (9)�(13), and assumptions (13)�(16).

In this work, the LQ�model is extended by introduc�

tion of two time parameters: radiation therapy duration T

and time interval t between sessions. According to

assumption 8 (part 2), only the parameter E of the LQ�

model depends on Q. Thus, it is assumed that PPC

depends on the relative number of surviving cells. Indeed,

the PPC is generally considered as a function of the

amount of cells surviving X�ray exposure, i.e., cellularity.

In [4] various post�radiation complications on skin and

their probability values are measured in cellularity units.

In [5, 6] tolerant doses are considered in their relation to

cellularity (although the relative tolerant number of sur�

viving cells is determined as a function of the irradiated

tissue volume). According to assumption 9 (part 2), in the

LQ�model, the dependence of parameters α and β on the

irradiated tissue volume V can be described by the follow�

ing equations:

α(V) = α(1)Vb, (187)

β(V) = β(1)V 2b, (188)

where the parameter b depends only on the type of tissue

(organ).

Assumptions 8 and 9 do not contradict assumption 1.

Indeed, it follows from assumptions 8 and 9 that the LQ�

model can be rewritten as follows:

D(Q, V) = E(Q)/((α(V) + β(V))d(V)) =

= E(Q)V –b/((α(1) + β(1))d(1)). (189)

This corroborates assumption 1.

The parameter E of the LQ�model depends on the

tissue cellularity and is used to describe the radiation

damage to tissue at the cellular level. The value of E varies

for different cell types, which makes it difficult to deter�

mine its value. In practice, the effective value of E is used.

Biomedical Engineering, Vol. 40, No. 5, 2006, pp. 251�255. Translated from Meditsinskaya Tekhnika, Vol. 40, No. 5, 2006, pp. 36�40.

Original article submitted July 15, 2003.

2510006�3398/06/4005�0251 2006 Springer Science+Business Media, Inc.

Central Institute of Mathematical Economics, Russian Academy of

Sciences, Moscow, Russia.1 The third part of this work was published in Biomedical Engineering,

No. 3, 2006.

Synthesis of Radiological Models and Radiological Constants.Part 4: Synthesis of Population–Phenomenological Modelsand the Lyman Model1

L. Ya. Klepper

252 Klepper

It can be suggested to introduce in the model, in addi�

tion to (or instead of) E, another radiological parameter,

the single�exposure tolerant dose DR [1]. This dose is a

generalized characteristic of the radiation effect on organ

or tissue that can be experimentally determined. The dose

DR depends of P (or Q) and V [2]. The relationship between

E and DR can be described by the following equation:

E(P) = α(V)DR(P, V) + β(V)DR2(P, V) =

= α(1)DR(P, 1) + β(1)DR2(P, 1) = U(DR(P)). (190)

PP Models. Usually, mathematical descriptions of

the fraction of surviving cells exposed to ionizing radia�

tion are constructed on the basis of the single�target mul�

tistroke model, the multitarget multistroke model [17,

18], the LQ function, etc. Strictly speaking, there are an

infinitely large number of functions (mathematical mod�

els) providing more or less adequate description of experi�

mental and clinical data. It was shown in [1, 3] that an

adequate choice of means of mathematical description

allows model transition from lower (cellular or popula�

tion) to higher (tissue) level. As a result, we obtain a phe�

nomenological model, which allows major parameters of

fractionated irradiation (number of fractions N and treat�

ment duration T) to be put in correlation with tolerant

dose D. This description is correlated with population

models. The mathematical models constructed on the

basis of phenomenological models are suggested to be

called population–phenomenological (PP) models.

In [2], I suggested various methods for construction

of PP models on the basis of the LQ function, a simple

exponential model of cell population growth, and a mod�

ification of this model providing more flexible description

of the repopulation rate. Two of these PP models, PP3

and PP4 [1], are considered in this work.

For simplicity, let us consider only uniform irradia�

tion schemes, i.e., schemes with a constant dose value

and a constant time interval between irradiation sessions.

Equation (73) can be rewritten as follows:

C(DR, 0) = C(d)NH(t)N – 1, (191)

where t = T(N – 1) is the time interval between irradia�

tion sessions, DR is the single�exposure tolerant dose, and

H(t) is the function describing cell repopulation.

Let us assume that:

C(d) = exp(–αd), H(t) = exp(–λk), (192)

where λ and k are parameters of the model H(t). Thus we

obtain the PP3 model:

D(N, T) = DR + (λ/α)(N – 1)1 – k[(N – 1)⋅t]k =

= DR + (λ/α)(N – 1)1 – kTk, (193)

where λ characterizes the repopulation rate. According to

assumption 13 (part 3), Eq. (193) is valid for any value of

PAPC = Q (or PPC = P = Q – 1) and any volume of irra�

diated tissue, DR depends only on Q and V, λ is independ�

ent of Q and V, whereas α depends on V and is independ�

ent of Q. Taking into account also assumption 9 (part 2),

we can obtain MM PP3 from Eq. (193):

D(Q, V, N, T) = DR(Q, V) + (λ/α(V))(N – 1)1 – kT k. (194)

Assuming that:

C(d) = exp(–αd – βd2), H(t) = exp(–λtk), (195)

and taking into account assumption 14 (part 3), MM PP4

can be constructed in a similar manner:

D(Q, d(V), V, N, T) = [α(1)DR(Q, 1) + β(1)DR2(Q, 1) +

+ λ(N – 1)1 – kT k]⋅V –b/[α(1) + β(1)d(1)]. (196)

The goal is to construct a SMM on the basis of PP3,

PP4, and Lyman models. The Lyman model is described

by the following formula:

D(P, V, N, T) = D(0.5, V, N, T) + F(P)σ(V, N, T) =

= [D(0.5, 1, N, T) + F(P)σ(1, N, T)]V –b. (197)

It can be seen from Eq. (197) that D(0.5, V) and σ(V)

decrease with increasing V. Thus, the PPC curve shifts to

the left and becomes steeper with increasing V.

Synthesized PP3–Lyman Model. The dose value

corresponding to PPC = P for fixed N and T is described

by Eq. (197), while the dose value corresponding to

PPC = P for an arbitrary DF mode is described by MM

PP3:

D(P, V, N, T) =

= [DR(P, 1) + (λ/α(1))(N – 1)1 – kT k]V –b. (198)

Equations (197) and (198) determine equal dose val�

ues. Therefore,

D(0.5, 1, N, T) + F(P)σ(1, N, T) =

= DR(P, 1) + (λ/α(1))(N – 1)1 – kT k. (199)

Synthesis of Radiological Models and Radiological Constants 253

It is assumed that MM (198) and, therefore, MM

(199) are valid for all values of N and T. Therefore, they

are valid for N = T = 1. Thus, the dependence of the sin�

gle�exposure dose on PPC is described by the following

linear equation:

DR(P, 1) = D(0.5, 1, 1, 1) + F(P)σ(1, 1, 1). (200)

Assumption 17. Synthesis of MMs (197) and (198) is

possible if the following equations are valid:

K 71 = D(0.5, 1, 1, 1) = const, (201)

K 72 = σ(1, 1, 1) = const, (202)

DR(P, 1) = E(P) = K 71 + K 72F(P), (203)

where K 71 and K 72 are the synthesizing constants (invari�

ants).

Equation (203) determines the single�exposure dose

(reduced to unit volume) corresponding to PPC = P. It

can also be used to determine the dependence of relative

cellularity C on P:

C(P) = exp[–(K 71 + K 72F(P))]. (204)

The SMM is described by two equivalent equations:

D(P, V, N, T) =

= [K 71 + K 72F(P) + (λ/α(1))(N – 1)1 – kT k]V –b, (205)

F(P) = [D(V)V b – K 71 – (λ/α(1))(N – 1)1 – kT k]/K 72. (206)

It should be noted that the synthesis of MMs is based

on the assumption that the single�exposure dose (reduced

to unit volume) corresponding to PPC = P can be

described by Eq. (203). The following statements describe

characteristics of SMMs (205) and (206).

Statement 14. In SMM (205), the curve P(D) for

fixed irradiated volume shifts to the right with increasing

N and/or T. The curve slope remains unchanged.

Proof. It follows from Eq. (205) that D increases with

increasing N and/or T for fixed P and V. Therefore, the

curve P(D) shifts to the right. The parameter K 72 = const

is used in Eq. (206) as coefficient σ. Therefore, the P(D)

curve slope remains unchanged.

Statement 15. In SMM (205), the curve P(D) for

fixed N and T shifts to the left with increasing V, while the

curve becomes steeper.

Proof. It follows from Eq. (205) that D increases with

increasing V for fixed P, N, and T. Therefore, the curve

P(D) shifts to the left. To prove that the curve becomes

steeper, let us show that the dose interval decreases with

increasing V for two arbitrary values of PPC, P1 and P2.

We obtain for V1 > V2:

D(P1, V1, N, T) =

= [K 71 + K 72F(P1) + (λ/α(1))(N – 1)1 – kT k]V1–b, (207)

D(P2, V1, N, T) =

= [K 71 + K 72F(P2) + (λ/α(1))(N – 1)1 – kT k]V1–b, (208)

D(P1, V2, N, T) =

= [K 71 + K 72F(P1) + (λ/α(1))(N – 1)1 – kT k]V2–b, (209)

D(P2, V2, N, T) =

= [K 71 + K 72F(P2) + (λ/α(1))(N – 1)1 – kT k]V2–b. (210)

Therefore,

[D(P1, V1, N, T) – D(P2, V1, N, T)]/[D(P1, V2, N, T) –

– D(P2, V2, N, T)] = (V2/V1)–b < 1. (211)

Thus, the dose interval corresponding to P1 and P2 de�

creases with increasing V, which makes the curve steeper.

The parameters of SMM can be determined by solv�

ing the following nonlinear extremal problems:

(212)

where M is the number of clinical observations. The prob�

lem of determination of the SMM parameters can be

considered in terms of dose (Eq. (205)):

(213)

Problems (212) and (213) can be modified if certain

parameters of the MMs (205) and (206) are known.

Synthesized PP4–Lyman Model. The dose value cor�

responding to PPC = P for fixed V, d(V), N, and T is

described by the Lyman model:

254 Klepper

D(P, V, d(V), N, T) =

= [D(0.5, 1, d(1), N, T) + F(P)σ(1, d(1), N, T)]V–b, (214)

while the dose value corresponding to PPC = P for an

arbitrary DF mode is described by MM (196):

D(P, V, d(1), N, T) = [α(1)DR(P, 1) + β(1)DR2(P, 1) +

+ λ(N – 1)1 – kT k]V –b/[α(1) + β(1)d(1)] =

= [U(DR(P)) + λ(N – 1)1 – kT k]V –b/[α(1) + β(1)d(1)] =

= [E(P) + λ(N – 1)1 – kT k]V –b/[α(1) + β(1)d(1)]. (215)

Assumption 18. Synthesis of MMs (214) and (215) is

possible if:

1) parameters D(0.5, 1, d(1), N, T) and σ(1, d(1), N,

T) of MM (214) do not depend on P;

2) the following parameters of MM (214) depend on

the DF scheme: D(0.5, 1, d(1), N, T) and σ(1, d(1), N, T);

3) the PP4 MM has only two parameters depending

on P (or Q = 1 – P): E(P) and U(DR(P)). For N = T = 1:

D(P, V, d(1), 1, 1) =

= [D(0.5, 1, d(1), 1, 1) + F(P)σ(1, d(1), 1, 1)]V –b; (216)

4) synthesis of MMs (214) and (215) is possible if the

following conditions are met:

D(0.5, 1, d(1), 1, 1) = E(0.5)/[α(1) + β(1)d(1)] =

= U(DR(0.5))/[α(1) + β(1)d(1)], (217)

σ(1, d(1), 1, 1) = K 8/[α(1) + β(1)d(1)], (218)

U(DR(P)) = U(DR(0.5)) + F(P)K 8, (219)

E(P) = E(0.5) + F(P)K 8. (220)

Equating the right parts of Eqs. (215) and (216) and

taking into account Eq. (16), we obtain for N = T = 1:

D(0.5, 1, d(1), 1, 1) + F(P)σ(1, d(1), 1, 1) =

= [α(1)DR(P, 1) + β(1)DR2(P, 1)]/[α(1) + β(1)d(1)] =

= U(DR(P))/[α(1) + β(1)d(1)] = E(P)/[α(1) + β(1)d(1)]. (221)

The first term in the right part of Eq. (216) corre�

sponds to PPC = 0.5, while the second term completes it

to PPC = P. Representing the right parts of Eq. (221) as a

sum of the two terms we obtain, taking into account that

E(P) and U(DR(P, 1)) are additive values, that:

U(DR(P)) = U(DR(0.5)) + U(DR(P)) – U(DR(0.5)), (222)

E(P) = E(0.5) + E(P) – E(0.5). (223)

Substitution of Eqs. (222) and (223) into Eq. (221)

gives:

D(0.5, d(1), 1, 1) + F(P)σ(d(1), 1, 1) = U(DR(0.5))/(α(1) +

+ β(1)d(1)) + (U(DR(P)) – U(DR(0.5)))/(α(1) +

+ β(1)d(1)) = E(0.5)/(α(1) + β(1)d(1)) + (E(P) –

– E(0.5))/(α(1) + β(1)d(1)). (224)

Thus, if follows from Eq. (217) that:

D(0.5, d(1), 1, 1) = U(DR(0.5))/(α(1) + β(1)d(1)) =

= E(0.5)/(α(1) + β(1)d(1)). (225)

Taking into account Eq. (225), we obtain from Eq.

(224) that:

F(P)σ(d(1), 1, 1) = (U(DR(P, 1)) – U(DR(0.5, 1)))/(α(1) +

+ β(1)d(1)) = (E(P) – E(0.5))/(α(1) + β(1)d(1)). (226)

Grouping of terms gives:

σ(1, d(1), 1, 1)[α(1) + β(1)d(1)] = (U(DR(P, 1)) –

– U(DR(0.5, 1)))/F(P) = (E(P) – E(0.5))/F(P) = K8. (227)

The left and right parts of this equation and, conse�

quently, K 8, are dimensionless (σ, Gy; α(1) + β(1)d(1),

Gy–1). Thus, it follows from Eq. (227):

σ(1, d(1), 1, 1) = K 8/[α(1) + β(1)d(1)], (228)

U(DR(P)) = U(DR(0.5)) + F(P)K 8, (229)

E(P) = E(0.5) + F(P)K 8. (230)

It can be seen from Eq. (228) that the random vari�

ance σ for fixed N decreases with increasing single�expo�

sure dose, while the PPC curve becomes steeper. It also

follows from Eq. (225) that D(0.5, d(1), 1, 1) decreases

Synthesis of Radiological Models and Radiological Constants 255

with increasing single�exposure dose, while the PPC

curve shifts to the left.

The synthesized MM can be presented in the follow�

ing form:

D(P, V, d(1), N, T) = {U(DR(0.5)) + K8F(P) +

+ λ(N – 1)1 – kT k}V–b/[α(1) + β(1)d(1)] = {E(0.5) +

+ K8F(P) + λ(N – 1)1 – kT k}V –b/[α(1) + β(1)d(1)], (231)

F(P) = {D(V)V b[α(1) + β(1)d(1)] – U(DR(0.5)) –

– λ(N – 1)1 – kT k}/K8 = {D(V)V b[α(1) + β(1)d(1)] –

– E(0.5) – λ(N – 1)1 – kT k}/K8. (232)

The statements below describe the characteristics of

the obtained MM.

Statement 16. In SMM (76), the curve P(D) shifts to

the right and becomes flatter with increasing N and/or T

for fixed P, V, d.

Proof. It follows from Eq. (231) that D increases with

increasing N and/or T for fixed P, V, d. Therefore, the

curve P(D) shifts to the right. To prove that the curve

becomes flatter, let us consider two arbitrary values of

PPC, P1 and P2. Let us use the following designation:

H(N1, T1) = λ(N1 – 1)1 – kT k1 > λ(N2 – 1)1 – kT 2

k =

= H(N2, T2). (233)

Let us consider the ratio between the dose intervals

for the two fixed values of PPC and different values of

H(N1, T1):

[D(P1, V, d(1), N1, T1) – D(P2, V, d(1), N1, T1)]/

/[D(P1, V, d(1), N2, T2) – D(P2, V, d(1), N2, T2)] =

= H(N1, T1)/H(N2, T2) > 1. (234)

Thus, the dose interval is greater for H(N1, T1), so

that the curve P(D) becomes flatter.

Statement 4. The curve P(D) shifts to the left and

becomes steeper with increasing V for fixed P, d, N, and

T.

Proof. It follows from Eq. (76) that the total dose

decreases with increasing V. Therefore, the curve shifts to

the left. To prove that the curve becomes steeper, let us

consider, in the same manner as above, the ratio between

the dose intervals:

[D(P1, V, d(1), N1, T1) – D(P2, V, d(1), N1, T1)]/

/[D(P1, V, d(1), N2, T2) – D(P2, V, d(1), N2, T2)] =

= (V1/V2)–b < 1. (235)

Thus, it follows from Eq. (80) that the curve P(D)

becomes steeper with increasing V.

Statement 17. The curve P(D) shifts to the left and

becomes steeper with increasing d for fixed P, V, N, and T.

Proof. It follows from Eq. (231) that the total dose

decreases with increasing d. Therefore, the curve shifts to

the left. For arbitrary d1(1) > d2(1) and fixed P, V, N, and

T, Eq. (231) takes the following form:

[D(P1, V, d1(1), N1, T1) – D(P2, V, d1(1), N1, T1)]/

/[D(P1, V, d2(1), N2, T2) – D(P2, V, d2(1), N2, T2)] =

= [α(1) + β(1)d2(1)]/[α(1) + β(1)d1(1)] < 1. (236)

It can be seen from Eq. (236) that the dose interval

decreases with increasing d(1). Therefore, the curve P(D)

becomes steeper.

The statements considered above can be used at the

stage of preliminary processing of clinical information to

assess the efficiency of use of the developed SMMs in

radiological practice.

This study was supported by the Russian Foundation

for Basic Research, project No. 05�01�00326.

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