Page 63 - ARNM-2-4
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Advances in Radiotherapy
& Nuclear Medicine Different approaches for the computation of BED
(VMAT) and dynamic conformal arcs (DCAs) were β D 2
n exp −α
analyzed. The methodology employed in this study and the N = ∫ 0 D − N dV (5)
results are presented in Sections 2 and 3. The discussion of f
the results and conclusions are included in Section 4.
where integration is performed over the target volume.
2. Methods In Eq. (5), N and n represent the total number of surviving
0
malignant cells and the initial (i.e., before commencement
2.1. BED for a non-uniformly irradiated target of radiotherapy) concentration of malignant cells,
For fractionated radiotherapy delivered in N fractions respectively. When n is uniform, Eq. (5) yields
0
f
with uniform dose per fraction d and total dose D, N β D 2
the probability of cell survival in the LQ model can be N = 0 ∫ exp −α D − dV = NS mean (6)
0
expressed as follows 27-29 V tar N f
β D 2 Where N = n V denotes the total initial number of
0
tar
0
S = exp( −α D − β dD = exp) − ( α D − ) (1) clonogenic cells in the target.
N f
Let us now assume that the target is irradiated with a
where parameters α and β define cellular radiosensitivity. negligibly small, uniform dose per fraction. Using Eq.
Although α and β can vary in the tissue (e.g., due to (6) and assuming that the total equivalent dose results
different phases of the cell cycle and spatially varying in the same cell kill as a given, that is, non-uniform dose
tissue oxygenation), information about their values in vivo distribution, we obtain
is currently not available. For simplicity, in the following
30
discussion, the parameters α and β for the anatomical Nexp − ( α BED ) = N S = N 0 ∫ exp −α D − β D 2 dV
structure of interest will be assumed to remain spatially 0 nud 0 mean V tar N f
and temporally uniform throughout radiotherapy. In (7)
addition, Eq. (1) does not describe the repair of sublethal
damage, 31-34 which may be important in the case of long Eq. (7) is equivalent to Eq. (3) with the average
fraction time (e.g., 0.3 – 0.5 h) characteristic of treatments probability of survival defined in Eq. (4).
delivered using multiple non-coplanar beams. 35 The degree of treatment success can be predicted if
The BED is defined as the equivalent total dose at the corresponding TCP is known. In the framework of
15,36-38
an infinitely low-dose rate or with infinitely small well- the LQ model and Poisson statistics for cell kill, the
relationship between BED and TCP is as follows:
spaced-out fractions yielding the same log cell kill as nud
the schedule being studied. 11-13 Equation (1) yields the TCP = exp(−N S ) = exp[−N exp(−αBED )] (8)
nud
0 mean
0
following expressions for BED 12,13 :
2.2. Relationship between BED and EQD 2
d 1
BED = D (1 + ) or BED =− ln S (2) Instead of BED, one can use the equivalent dose in
/
αβ α 2-Gy fractions, EQD . For a given non-uniform dose
14
2
distribution, EQD produces the same cell kill as the given
where S is described in Eq. (1). dose distribution. Straightforwardly, the relationship
2
In the case of non-uniform dose (“nud”) in the treatment between BED and EQD is as follows :
19
2
target with volume V , one can employ a more general BED
tar
definition of BED, i.e., EQD = 2 (9)
2
1+
1 α
BED nud =− α ln( S mean ) (3) ( β )
where S mean is the probability of survival averaged over For brevity, the subsequent discussion focuses on the
the dose distribution in the target, 19,30 i.e., BED. Because BED is proportional toEQD , the conclusions
2
of this study are also valid for the latter quantity.
1 β D 2
S mean = ∫ exp −α D − dV (4) 2.3. Employing the dose-volume histogram (DVH) to
V tar N f compute the BED
To derive Eq. (3), one can consider the equation for cell Let V(D) denote the volume of the target receiving a dose
survival in the target equal to or greater than a given dose D. The ratio V(D)/V PTV
Volume 2 Issue 4 (2024) 3 doi: 10.36922/arnm.4826
