Lima J.J.Pedroso, de (ed.). Nuclear Medicine Physics

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498 Nuclear Medicine Physics

0.01

0.1

510

Absorbed dose (Gy)

FIGURE 8.34

Plot of Equation 8.123.

This is the survival equation of a cell with h sensitive zones

(multitarget/single-hit model) that, in Cartesian coordinates, is a sigmoid.

Expanding Equation 8.123 by applying the binomial theorem

∗

and assum-

ing that D has large values, we obtain

= he

−aD

= he

−D/D

, (8.124)

with a = 1/D

. As can be seen in Figure 8.34 in a semilogarithmic plot, the

survival equation is linear for high doses. The fraction of cells surviving is

plotted on a logarithmic scale against dose on a linear scale.

The extrapolation of this linear portion back to the ordinate (S axis) gives

h, the number of sensitive zones per cell Figure 8.4).

The value D

is the absorbed dose that corresponds to S = 1 in the extra-

polated straight line. D

is called the quasi-threshold dose and is a measure

of the threshold length that precedes the linear portion of the survival curve

for the situation multitarget or single-hit model.

This coefﬁcient is related with the intrinsic capacity of cells to be repaired

from sublethal lesions, and it gives an indication of the damage that has to be

accumulated before cell death occurs.

Applying logarithms to Equation 8.124, we obtain

ln S

=−D/D

+ln h, (8.125)

or in decimal logarithms

log

=−2.304 D/D

+log

h. (8.126)

∗

A binomial of power n can be expanded using the equation (x +a)

= C

; pa

n−p

with n and

p integer and 0 < p < n.

Dosimetry and Biological Effects of Radiation 499

Then, when S

= 1

log

1 = 0 =−2.304 D

+log

h (8.127)

and

= D

log

h/2.304, (8.128)

= ln h. (8.129)

Using Equations 8.126 and 8.129, the linear portion of Equation 8.124 is

log

=−2.304(D −D

)/D

. (8.130)

The slope of tangent to the curve of Equation 8.123 (8.123), for dose D,is

=−ahe

−aD



1 −e

−aD



h−1

. (8.131)

For D = 0, the slope

of function S

is zero. At this point, curve S(D) is

parallel to the Ds axis, meaning the absence of effects for very small doses.

Some experimental sigmoidal curves do not ﬁt the model of equation 8.123,

simply because for t = 0, the slope of tangent is not zero. Most experimental

survival curves have an initial slope, whereas the multitarget or single-hit

model predicts no initial slope.

8.12.1.3 Mixed Model

A single-target single hit or multitarget single hit model, also termed a mix

model, gets closer to the experimental situation discussed earlier. This model

includes one lethal sensitive region and n sublethal sensitive regions.

Cellular death will occur either by one hit in the lethal region or by hits in

all the n sublethal regions. For these conditions, the survival probability of

one cell is

= e

−D/D



1 −



1 −e

−D/D



. (8.132)

is the mean lethal dose of lethal zone, and D

is the mean lethal dose for

sublethal regions.

The slope of this function for every value of D is

=−1/D

−D/D



1 −



1 −e

−D/D





−n/D

−D/D



1 −e

−D/D



n−1

(8.133)

500 Nuclear Medicine Physics

0.01

0.1

510

Absorbed dose (Gy)

Slope 1/D

+ 1/D

FIGURE 8.35

Survival curve for the mixed model, Equation 8.132.

For D = 0,

=−1/D

. (8.134)

The plot of log

versus D is given in Figure 8.5.

For high values of D, the slope of the curve in Figure 8.35 becomes constant

and equal to −(1/D

+1/D

). The initial slope −1/D

indicates the action of

sudden death due to hits on lethal target. Intersection with the ordinate of

the extrapolation of the linear portion of Equation 8.132 gives the value n, the

number of sublethal regions.

Considering two samples with distinct cell populations, satisfying the

model of Equation 8.132, but with different numbers of sublethal zones, which

we can call a

and b

, the corresponding survival curves 1 and 2 would have

intersections on the ordinates at points a

and b, respectively (Figure 8.36).

The mixed model frequently applies to cell populations of mammals irradi-

ated with low LET radiation. Typical values of D

(lethal target) are 4.5–5 Gy

and for D

(sublethal death), they are 1–2 Gy.

For high TLE radiation, such as particles, neutrons, deuterons, etc., the

curves are frequently simple exponential survival curves.

In a sample with two distinct sublethal populations and assuming that

the mixed mammalian model prevails, the survival curve looks similar to

Figure 8.37.

Finally, for situations in which the number of hits necessary to produce cell

death is added to several cell populations with sensitive lethal and sublethal

zones, the survival curve could present the form of Figure 8.38 with no linear

portion for the doses with interest.

The major problem with the single-target, single hit or multitarget, single-

hit model is that there are three parameters, too many to handle in practical

situations.

Dosimetry and Biological Effects of Radiation 501

0.1

510

Absorbed dose (Gy)

FIGURE 8.36

Survival curves for two distinct cellular populations satisfying model Equation 8.132.

A mathematically simpler model with fewer unknown parameters is

needed.

8.12.1.4 The Linear–Quadratic Model

Since there is strong evidence that most of the important biological effects of

ionizing radiation are direct or indirect consequences of DNA double-strand

breaks, it makes sense to analyze the action of radiation on cells to develop a

model based on the direct measurement of chromosome damage. The deve-

lopment of models to simulate chromosome damage and its consequences is

an alternative to studying survival curves using target theory models [39,40].

0.1

Absorbed dose (Gy)

FIGURE 8.37

Survival curve for the mixed model with two distinct sublethal populations.

502 Nuclear Medicine Physics

0.1

510

Absorbed

dose (Gy)

FIGURE 8.38

Survival curve for a complex situation with several cell populations with different sensitive lethal

and sublethal zones the number of hits necessary to produce cell death is variable.

The probability of DNA double-strand breaks strongly depends on the ion-

izing power of the radiation used, as can be seen in Figure 8.39.As an example,

the probability of a double break occurring with electrons is much smaller

than with α particles of the same energy.

The simpliﬁed models for DNA damage from ionizing radiation, shown

above in Figures 8.30 and 8.31, recognizes two types of damage. In type B

damage, a single particle breaks only one strand (Figures 8.30b and 8.31d).

The chromosome only suffers a double break if another particle breaks the

other strand close to the ﬁrst break before repair has taken place (which is not

the case in Figure 8.30c). In type A damage, the ionizing particle, in a single

Photons

(a) (b)

Neutrons

Heavy ions

Tissue

α-particle

LET-120 keV/μm

Electron

LET-0.4 keV/μm

~3 nm

Double

helix

DNA

FIGURE 8.39

(a) Ionization produced by photons, neutrons and heavy ions. (b) Schematic representation of

relative dimensions of ADN and ionizing range of an α particle (120 keV/μm) and an electron

(0.4 keV/μm).

Dosimetry and Biological Effects of Radiation 503

hit, breaks both strands of the DNA, and the chromosome is split into two

fragments Figure 8.30d and 8.31e).

The probability of type A damage is proportional to the dose. The average

number of cells with type A damage, after dose D,ism = αD = D/D

, and

the probability of survival (with no damage) is the Poisson probability P(0) =

−m

= e

−αD

For double-strand break with type B damage, one strand is damaged by

one ionizing particle, and the other is damaged by a second ionizing article.

The probability of fragmenting the DNA molecule is, therefore, proportional

to the square of dose. The average number of molecules with type B damage

is βD

, and the survival curve for type B damage alone is e

−αD

At low doses of x- or γ-rays, double-strand breaks are the consequence of

the action of a single secondary electron. This behavior is represented by the

dashed straight line in Figure 8.40 and corresponds to type A damage. At

higher doses, the two chromosome breaks result from two separate electrons.

The survival curve bends when the quadratic, type B damage component

dominates, as is shown in Figure 8.40.

For x- or γ-rays, the linear-quadratic model for cell survival includes A and

B type damage and is

S = e

−(αD+βD

)

, (8.135)

where S is the fraction of cells surviving dose D, and α and β are constants.

This model assumes that there are two components to cell killing by radiation,

one that is proportional to the dose and a second that is proportional to the

square of the dose.

–3

–2

–1

4 8 12 16

x-rays

α-particle

neutrons

FIGURE 8.40

The important parameters in L–Q curve. The curve is described by the initial slope (D

), the ﬁnal

slope (D

), and a parameter that represents the width of the shoulder D

. Curve 1: Cell survival

under x- or γ-rays (A and B type damage). Curve 2: Type A damage for high LET radiation.

504 Nuclear Medicine Physics

Since for equal components of A and B, the damage is

−αD

= e

−βD

, (8.136)

this becomes

D = α/β, (8.137)

The ratio α/β is the dose at which linear and quadratic components of cell

killing are equal.

∗

For low LET radiation, α/β varies between 1.0 and 2 Gy. For densely ionizing

(high LET) radiation, such as α-particles or low-energy neutrons, the cell-

survival curve is a straight line from the origin; that is, survival approximates

to an exponential function of dose. The greater the LET the higher the constant

α, that is, the greater the slope of the straight line. In this case, the survival

curve can be described by just one parameter, the slope.

The linear-quadratic curve continuously bends but is a good ﬁt to exper-

imental data for the ﬁrst part of survival curve. The curve is described by

the initial slope (D

), the ﬁnal slope (D

), and a parameter that represents the

width of the shoulder (D

Constant α represents the probability of α-type damage, which is irrepara-

ble. Constant β represents the probability that independent, repairable

β-type events have combined to produce lethal events, for example, double-

chromosome breaks.

The linear-quadratic model is currently the model of choice for cell survival

curves in a substantial number of situations (Figure 8.40).

8.12.1.5 Comparison of the L–Q and Target Theory Models

Survival curves of irradiated cell cultures from different tissues and of tumors

show a wide variation in their shapes. Frequently, data are not precise enough

to allow a comparison between the ﬁts of L–Q and target theory models.

Survival curves for high LET radiations, such as α-particles, neutrons, and

very slow electrons, are linear in both cases (irreparable damage dominates).

For low LET radiation, the L–Q curves keep the curvature even at very high

doses, whereas target theory curves become linear at high doses.

For x- and γ-rays and other poorly ionizing radiations, survival curves have

a shoulder at low doses with a characteristic initial slope. At higher doses,

these curves tend to either straighten or continue curved, and survival data

can be ﬁtted by either the target theory or the L–Q models.

The advantage of the L–Q model is that it uses only two parameters, α and

β, comparing with three in the case of target theory models. The initial slope

of the survival curve is represented by either 1/D

or α. Repair is represented

by either D

(target theory) or β (L–Q model). The ratio α/β is the dose at

which α-damage equals β-damage.

∗

α has dimensions of reciprocal of dose; β has dimensions of reciprocal of square of dose.

Dosimetry and Biological Effects of Radiation 505

8.13 Nontargeted Effects Complementary to Direct Action

of Ionizing Radiation for Low Doses

Biological systems generate defence and biological response mechanisms

when subjected to ionizing radiation.

For high doses of ionizing radiation, the associated biological effects are

relatively well known and include cellular death (apoptosis and necrosis),

somatic mutations (especially those that result in cancer), and the mutations

at germinative cells (transmissible genetic defects).

It is found that the biological response for low doses of ionizing radiation

cannot be obtained from extrapolation of high dose data.

Evidence and accumulated experimental results exist, indicating that for

small doses the action of ionizing radiation on tissues occurs through two

simultaneous processes. The ﬁrst corresponds to direct aggression on DNA

with identical laws to those applicable to high doses, and a second consists

in several forms of complementary action with characteristics that, in some

cases, are still far from completely understood. The most important of these

processes, called nontargeted, engulf the DNA repair mechanisms, genomic

instability, bystander effects, and adaptive response.

It is generally accepted that damage to DNA can be the most devastating

event in cells irradiated with ionizing radiation and that the nonrepaired or

badlyrepaired double-strandbreaksinDNAmoleculearethedamagerespon-

sible for the most severe biological processes. The consequences of biological

damage in DNA led to the long-accepted paradigm of cytotoxic action, as

inducer of mutagenesis and malignancy occurring in cells after ionizing radi-

ation irradiation. Target theory postulates that a direct collision (excitation or

ionization) of radiation with a critical cellular constituent, generally chromo-

some DNA, can produce a biological effect. In particular, the incidence of a

microbeam

∗

of ionizing radiation with a nucleus of a cell can rapidly induce

cellular death.

Recent studies, however, question the paradigm of genetic alterations being

exclusively associated with direct DNA damage after exposure to ionizing

radiation.

Experiments have been carried out showing that cells not directly exposed

to ionizing radiation may exhibit responses identical to those subjected to

direct irradiation.

The collision of a microbeam of ionizing radiation with the nucleus of a cell

may induce analogous effects in neighboring nonirradiated cells [41]. This

process is known as the bystander effect and indicates the phenomena in

which irradiated cells transmit damage signals to nonirradiated cells, leading

to genetic effects in neighboring, nondirectly hit cells.

∗

These studies are now possible, thanks to new techniques of microbeaming that target thin

beams (∼2 μm

) at cells with an accuracy of the order of 1–2 μm.

506 Nuclear Medicine Physics

Two possible mechanisms have been suggested to explain transmission

in the bystander effect: cellular communication by junctions through inter-

cellular spaces and processes depending on soluble factors produced in the

medium [42,43].

Bystander effects depend on several factors such as radiation quality, dose

and dose rate, cell culture conditions, and cellular lines; and they can be stud-

ied according to several perspectives (clonogenic activation, apoptosis and

micronucleus, mutations, genetic expression, chromosome aberrations, etc.)

Genomic instability is a process induced by radiation that consists on the

transmission of signals to the descendants of irradiated cells, which, eventu-

ally, many generations after, leads to the occurrence of genetic effects, such as

mutations and chromosome aberrations in descendants distant in time from

the irradiated cells [44].

Adaptive response, another phenomenon, consists on the cellular response

to an external aggression through an intrinsic mechanism that generates

resistance to a new aggression.

The signals transmitted by the hit cells disturb the normal function of neigh-

boring cells (bystander effects) and stimulate them to send new signals that

hit those initially hit and their neighbors. The signals sent by the bystander

cells can help in repairing hit cells or start the process that leads them to

commit suicide.

Aconclusionthatcomes fromthe characteristicsof theeffectsoflow ionizing

radiation doses in tissues under normal conditions is that individual cells

cannot be considered to be individual isolated entities in most of the tissues

in multicellular organisms.

Apart from the effects referred to earlier, the interpretation of the ﬁnal

expression of cellular damage, for low doses, also has to consider the

redistribution, restocking, and mobilization cellular processes.

On the basis of the present knowledge, one cannot yet state whether these

effects increase the health risk or otherwise. A better understanding of the

mechanisms involved is necessary to evaluate possible action with regard to

radiological protection or applications in radiotherapy.

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