The probability of resorption (PR) of the lesion focus

and the probability of post�radiation complications

(PPCs) in normal organs and tissues can be effectively

used by radiologists as criteria for assessing the efficiency

of radiation therapy (RT). One of the important problems

of modern radiology is to develop mathematical models

(MMs) describing PPC in organs and tissues (models for

criterial assessment of RT).

The parameters of MMs for describing PPC in nor�

mal organs and tissues as a function of the irradiation

dose and volume should be determined on the basis of

clinical information (tuning of MMs). The goal of this

work was to develop methods for determining parameters

of MMs used for calculating PPC in organs and tissues.

The parameters of MMs are determined on the basis

of the following clinical information:

(Di, Vi, Pi), i = 1, …, M1, (1)

where Di is the dose in irradiated tissue volume Vi (Gy), Pi

is the PPC, and M1 is the number of systematized clinical

observations. It is assumed that the dose distribution in

the irradiated volume is uniform. To determine Pi, it is

necessary to perform preliminary processing (systemati�

zation) of clinical information.

The initial clinical information usually includes the

dose value, the irradiated tissue volume, and the irradia�

tion effect (presence or absence of clinically identifiable

response to irradiation). The clinical information is pre�

sented as follows:

(Di(V), Vi, ki), i = 1, …, M, (2)

where M (M1 < M) is the number of non�systematized

clinical observations. The parameter k is a Boolean vari�

able:

1 − response to irradiation,

0 − no response to irradiation. (3)

This clinical information is used to determine fre�

quency characteristics [3, 4], which is a rather complicat�

ed task for the following reasons.

1. Extensive clinical information is needed to deter�

mine frequency characteristics as functions of the irradi�

ation dose and tissue volume.

2. Radiological information is usually insufficiently

full and/or diversified because it is obtained using stan�

dard irradiation methods.

The problem of determination of frequency charac�

teristics and approximate PPC values is considered below

(Task 4).

Mathematical Models for Calculation of PPC inNormal Organs and Tissues. Let us consider two MMs for

calculation of PPC in normal organs and tissues:

– modified Weibull function [1�3, 6];

– power function for calculation of approximate

PPC [1�3].

The tolerant level of irradiation of organ or tissue is

usually assumed to correspond to PPC = 3�5% (P = 0.03�

0.05) of a clinically identifiable type. Let us adopt the fol�

lowing definition.

Definition 1. Tolerant dose of level P is the dose of

irradiation of fixed volume (area) of tissue or organ lead�

ing to PPC = P.

Assumption 1. For two MMs under consideration,

the dependence of the tolerant dose D(P, V) of level P on

the volume V meets the following relationship:

D(P, V1)/D(P, V2) = (V1/V2)–b, (4)

where b does not depend on tissue volume and PPC in tis�

sue.

It follows from assumption 1 that:

D(Pi, Vi)Vib = D(Pi, 1) = C(Pi) = const(Pi), (5)

Biomedical Engineering, Vol. 42, No. 1, 2008, pp. 32�36. Translated from Meditsinskaya Tekhnika, Vol. 42, No. 1, 2008, pp. 33�37.

Original article submitted January 20, 2005.

320006�3398/08/4201�0032 2008 Springer Science+Business Media, Inc.

Central Institute of Mathematical Economics, Russian Academy of

Sciences, Moscow, Russia.

Aggregation Method for Determination of Parameters of Mathematical Models

L. Ya. Klepper

k =

Method for Determination of Parameters of Mathematical Models 33

where D(P, 1) is the tolerant dose of level P reduced to

unit volume of irradiated tissue, C(P) is the constant for

tissue tolerance level P.

Calculation of PPC in Organs and Tissues UsingModified Weibull Function. The developed MM for calcu�

lation of PPC in organs and tissues using the modified

Weibull function is based on the following equation [1�3,

6]:

P(D, V) = 1 – Q(D, V) = 1 – exp[–(DV b/A1)A2], (6)

where 0 ≤ P(D, V) ≤ 1 is the PPC in tissue; 0 ≤ Q(D, V) ≤1 is the probability of absence of post�radiation complica�

tions (PAPC) in tissue; A1, A2, b are the model parameters

(6) independent of the irradiated tissue volume.

Q(D, V) = exp[–(DV b/A1)A2]. (7)

It follows from Eq. (5) that:

D(Q, V)V b = D(Q, 1) = C(Q) = const(Q), (8)

where D(Q, 1) is the dose reduced to unit volume. This

value is constant for given tissue and fixed value of Q. At Q

= 0.95 (P = 0.05) Eq. (5) takes the following well�known

form relating tolerant doses to the irradiated tissue volume:

D1/D2 = (V1/V2)–b. (9)

The MM (4) is a generalization of Eq. (9) for 0 <

Q ≤ 1.

The MM (7) includes three interdependent radio�

logical parameters Q, D, V:

D = A1|ln(Q)|1/A2V –b, (10)

V = [A1|ln(Q)|1/A2/D]1/b. (11)

For a fixed Q and two equivalent irradiation modes

(D1, V1, Q) and (D2, V2, Q) Eq. (5) is obtained. For a fixed

V and two equivalent irradiation modes the following

equation for doses and PAPC is obtained using Eq. (10):

D1/D2 = [ln(Q1)/ln(Q2)]1/A2. (12)

For a fixed dose the equation for irradiated volume

and PAPC is obtained using Eq. (10):

V1/V2 = [ln(Q1)/ln(Q2)]1/A2b. (13)

Equations (4), (12), and (13) can be used for deter�

mining constants included in MM (6) and assessing thus

the adequacy of the MM to the radiological information

used.

Task 1. Determination of Parameters of MM (7). The

parameters of MM (7) can be determined by solving the

following nonlinear extremal problem:

(14)

under the following restrictions:

A1 > 0, A2 > 0, (15)

0 ≤ b < 1. (16)

It is rather difficult to solve the problem (14)�(16)

using direct numerical methods.

Task 2. Method of Problem Linearization. Logarith�

mation of MM (10) for calculation of PPC gives:

ln(D) = ln(A1) + (1/A2)ln|ln(Q)| – b⋅ln(V). (17)

Let us assume that:

X1 = ln(Ai), X2 = 1/A2, X3 = b. (18)

For M1 clinical observations:

X1 + ln|ln(Qi)|X2 – ln(Vi)X3 = ln(Di), i = 1, …, M1. (19)

The values of MM parameters can be found by solv�

ing the following extremal problem:

(20)

X2 > 0, (21)

0 ≤ X3 < 1. (22)

There are no restrictions on the variable X1, because

A1 can be either less or greater than 1.

The problem (20)�(22) can be solved as a condition�

al nonlinear problem. It can also be solved by the least�

squares method without taking into account restrictions

(21) and (22). The solution is acceptable if the variables

X2 and X3 automatically meet restrictions (21) and (22).

Task 3. The problem of determination of the param�

eters of MM (10) can be considered as a linear program�

ming (LP) problem:

34 Klepper

(23)

X1 + ln|ln(Qi)|X2 – ln(Vi)X3 + Xi + 3 = ln(Di), i = 1, …, M1, (24)

Xi ≥ 0, i = 2, …, M + 3, (25)

where Xi + 3, i = 1, …, M are artificial variables. The lower

limit of the sum of all artificial variables is equal to zero.

If the sum is actually zero, the M restrictions (24) become

strict equations. The set of restrictions (24) can be

expanded if the restrictions on variables X1, X2 and X3 are

known.

Task 4. Method for Aggregation of Non�systematizedClinical Information. Let us consider a method for deter�

mining MM parameters that can be effectively used in

cases when the amount of clinical information is insuffi�

cient for constructing reliable frequency characteristics.

Consider the following clinical information (2):

(Di(V), Vi, ki), i = 1, …, M, (26)

where M is the number of clinical observations and ki is

the Boolean variable equal to 1 in case of radioreaction;

otherwise it is equal to 0.

It is necessary to systematize clinical information

and construct frequency characteristics to determine

approximate value of PPC in tissue as a function of dose

and tissue volume. For this purpose, the dose and volume

values are divided into intervals with selected equal or

unequal steps. Then, the number (frequency) of clinical

observations within a given interval is determined. For

example, nsf clinical observations can fall within the fol�

lowing dose–volume interval:

Ds ≤ D < Ds + ∆Ds, Vf ≤ V < Vf + ∆Vf. (27)

To determine PPC in tissue volume Vf exposed to

dose Ds, it is necessary to calculate the number of cases

with radiation complications. If this number is ksf, the

approximate value of PPC in tissue is P(Ds, Vf) = ksf/nsf.

The efficiency of the described procedure for sys�

tematization of clinical information increases with

increasing amount and diversity of clinical information. If

the amount and diversity of clinical information are

insufficient for determining acceptable PPC values, the

interval should be increased, which reduces the accuracy

of PPC determination. However, even increasing the

interval does not always allow the PPC to be determined

in the case of insufficient clinical information.

Below, a method for increasing the useful informa�

tion volume and determining the MM parameters is sug�

gested. The method is based on aggregation (reduction)

of clinical information.

Let us assume that the parameter b of MM (6) falls

within the interval b1 ≤ b ≤ b2, and that b assumes G values

within this interval with a given step ∆b:

bg = b1 + (g – 1)∆b, g = 1, …, G. (28)

Let us assume that:

D(1, b) = DV b, (29)

validating thus assumption 1 and reducing the clinical

information to unit volume. This can be done only if the

parameter b is known. The initial clinical information

takes the following form:

(Di(1, bg), ki), i = 1, …, M >> M1. (30)

To construct the frequency characteristics, it is nec�

essary to determine the reduced dose intervals. This leads

to an increase in the number of clinical observations and,

therefore, the accuracy of determination of PPC. The

algorithm for determining the parameters of MMs (6) or

(7) can be described as follows.

Step 1. The initial values of parameters b1, b2 and ∆b

required for solving the problem are set. It is assumed that

g = 0 and F (criterion of quality of obtained parameters)

is large.

Step 2. The value of parameter b is set under assump�

tion that g = g + 1: b = b1 + (g – 1)∆b.

Step 3. Clinical information is reduced to unit vol�

ume according to Eq. (29).

Step 4. Frequency characteristics are constructed

and approximate values of PPC as a function of the

reduced dose are determined. It should be noted that this

procedure is especially effective if the discretization steps

for the reduced dose are selected in interactive mode.

Systematization of clinical observations gives:

(Di(1, bg), Pi), i = 1, …, M2, M1 < M2 ≤ M. (31)

Step 5. The parameters A1 and A2 of MM (7) are

determined by solving, for example, the truncated

extremal problem (20)�(22):

(32)

X2 > 0, (33)

or by solving the truncated LP problem (23)�(25):

Method for Determination of Parameters of Mathematical Models 35

(34)

X1 + ln|ln(Qi)|X2 + Xi + 2 = ln(DiVib), i = 1, …, M, (35)

Xi ≥ 0, i = 2, …, M + 2. (36)

Step 6. The sum of squared disparities is calculated:

(37)

If Fg < F, it is assumed that F = Fg and current

parameters are stored: bopt = bg, A1opt = A1, A2opt = A2.

Step 7. If bg < b2, go to step 2; otherwise, go to

step 8.

Step 8. The problem is solved.

Approximate Method for Calculation of PPC. The

simplest method for approximate description of PPC in

normal organs and tissues is based on tolerant doses [1�3].

This method was used to relate the linear form coefficient

to tolerant doses in organs and tissues. The linear form

coefficient is used for determining the optimal irradiation

plan by LP methods. The approximate value of PPC can

be described by the following power function:

P(D, V) = [D/DT(V)]H, (38)

where D is the uniform radiation dose received by tissue

volume V; DT(V) is the tolerant dose as a function of the

irradiated volume.

Taking into account assumption 1, we obtain that:

DT(V) = DT(1)⋅V–b, (39)

where H is a parameter of MM (38)�(39) depending on

the tissue type, but independent of the irradiated volume.

Numerical analysis shows that MM (38)�(39) pro�

vides good description of PPC for doses below tolerant

level, i.e., to which normal organs and tissues are usually

exposed during RT.

PPC vs. dose curves for normal tissues based on clin�

ical observations are of logistic (S�shaped) type with an

inflection point separating concave (left) and convex

(right) parts. The MM (38)�(39) provides good approxi�

mation of the first part of the PPC curve. Within this

range the MM provides virtually the same results as MM

(6) [2, 3].

The MM (38)�(39) is sufficiently simple, which

makes it useful for solving extremal problems of RT plan�

ning. Besides, if the dose exceeds the tolerant level, func�

tion (38) sharply increases, which makes it possible to use

it to keep the dose in tissue below tolerant level (interior

point method [5]).

It follows from (38) and (39) that:

P(D, V) = [DVb/DT(1)]H. (40)

Tuning of MM (40) involves determination of two

parameters: DT(1) and b. Useful equations for preliminary

assessment of the MM can be obtained from MM (40). It

should be noted, however, that MM (40) is valid for P(D,

V) ≤ 0.5 (50% PPC).

The tolerant dose of level P in tissue volume V can be

obtained from Eq. (40):

D(P, V) = DT(1)|P|1/HV–b. (41)

Using Eqs. (40) and (41) it is possible to obtain use�

ful relationships between P, D, and V. In case of fixed P

the following well�known relationship between tolerant

doses of level P and irradiated tissue volume is obtained:

D1/D2 = (V1/V2)–b. (42)

It follows from Eq. (41) that:

V = [DT(1)|P|1/H/D]1/b, (43)

and, thus, for fixed D and two equivalent in terms of D

modes:

V1/V2 = (P1/P2)1/Hb, (44)

while for fixed V and two equivalent in terms of V modes:

D1/D2 = (P1/P2)1/H. (45)

The relationships between P, D, and V can be used to

tune the parameters of MM (40).

Task 5. Determination of MM Parameters forApproximate Calculation of PPC. The parameters of MM

(40)�(41) can be determined by the same methods as the

parameters of MM (6)�(7), i.e., by solving the following

conditional extremal problem or its modifications:

(46)

under the following restrictions:

DT(1) > 0, (47)

j

36 Klepper

H > 1, (48)

0 ≤ b < 1. (49)

Task 6. Method for Problem Linearization. Logarith�

mation of Eq. (41) gives:

ln(D) = ln(DT(1)) + 1/Hln|(P)| – b⋅ln(V). (50)

Let us assume that:

X1 = ln(DT(1)), X2 = 1/H, X3 = b. (51)

For M1 clinical observations:

X1 + ln(Pi)X2 – ln(Vi)X3 = ln(Di), i = 1, …, M1. (52)

The values of MM parameters can be found by solv�

ing the following conditional extremal problem:

(53)

X2 < 1, (54)

0 ≤ X3 < 1. (55)

Problem (53)�(55) can be solved as a conditional

nonlinear problem or using the least�squares method

without taking into account additional restrictions (54)

and (55). The solution is acceptable if the variables X2 and

X3 automatically meet the restrictions (54) and (55).

Task 7. The problem of determination of parameters

of MM (41) can be considered as a linear programming

problem:

(56)

X1 + ln|ln(Qi)|X2 – ln(Vi)X3 + Xi + 3 = ln(Di),

i = 1, …, M1, (57)

Xi ≥ 0, i = 2, …, M + 3, (58)

where Xi + 3, i = 1, …, M1 are artificial variables. The lower

limit of the sum of all artificial variables is equal to zero. If

the sum is actually zero, the M restrictions (57) become

strict equations. The set of restrictions (57) can be expand�

ed if the restrictions on variables X1, X2 and X3 are known.

Task 8. Method for Aggregation of Initial ClinicalInformation. This method is similar to that described

above for MM (6). It can be applied to MM (40)�(41)

taking into account the specific features of the approxi�

mate MM.

Conclusions

1. Two MMs for calculating PPC in normal organs

and tissues have been considered. The first MM provides

calculation of PPC within the entire range of variation of

dose and tissue volume. The second MM is approximate

and can be used to calculate PPC < 50%. The MMs are

analyzed.

2. Methods for determining the MM parameters by

solving extremal problems are described.

3. A method for determining the MM parameters by

solving a complex optimization problem is described. The

method is based on aggregation (reduction) of clinical

information and includes straightforward enumeration.

Aggregation is performed in terms of parameter b. To

determine the other two parameters, an extremal problem

is solved.

REFERENCES

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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 Radioactive Radiation

Sources [in Russian], Moscow (1993).

4. V. Feller, Introduction into the Theory of Probability and Its

Applications [Russian translation], Moscow (1967).

5. D. Himmelblau, Applied Nonlinear Programming [Russian trans�

lation], Moscow (1975).

6. W. Weibull, J. Appl. Mechan., 18, 293 (1951).