Radio/ Oncol 1995; 29: 240-7. 
Analytical representations of dinical electron beam central axis depth doses 
Wieslaw Wierzbicki1 and Ervin B. Podgorsak2 
Departement de physique biomedicale, Hopital Natre-Dame; 2 Department of Medica! Physics, Montreal General Hospital Montreal, Quebec, Canada 


Analytical expressions proposed to date to approximate central axis electron beam depth dose distributions are reviewed and their quality of fitting discussed. A recently developed analytical expression based on only four fitting parameters is analyzed. The expression approximates well the measured electron beam data from two commercial Zinem· accelerators in the field size range from 4X4 cm2 to 25X25 cm2 and in the energy range from 4 Me V to 22 Me V in all four regions of the depth dose curve: build-up, dose maximum, dose fall-off, and bremsstrahlung contamination. 
Key words: electrons, particle accelerators, depth doses 
Introduction 
The particular energy loss characteristics of electrons as they penetrate into tissue make electrons suitable for use in treatment of super­ficial malignant diseases. Advantages of elec­trons over superficial x-rays and brachytherapy are a better dose homogeneity in the target volume and a lower dose in tissues surrounding the target. The electron beam depth dose distri­butions consist of four regions: buildup, dose maximum, dose fall-off, and bremsstrahlung contamination. Ever since the first depth dose distributions of clinical electron beams were measured in water, attempts have been made to describe the measured distributions with analytical expressions. In the individual dose 
Correspondence to: Wieslaw Wierzbicki, Ph.D., De­partement de physique biomedicale, H6pital Notre­Dame, 1560, rue Sherbrooke Est, Montreal, Quebec H2L 4Ml. 
UDC: 615.849.5:621.384.64 
regions, it is relatively easy to approximate the dose distributions analytically; however, the distributions are difficult to describe accurately with a single expression covering ali four re­gions simultaneously. 
In 1953 Laughlin et al. 1 proposed the first analytical expression to reproduce electron depth doses in water. Since then, attempts to describe analytically depth doses of clinical elec­tron beams have continued with varying degrees of success.2-12 With each subsequent new pro­posal, the analytical expressions became more accurate but also, to a certain degree, more complicated, as they depended on an ever­increasing number of empirical parameters involved in the curve fitting process. 
In this note, we present a summary of expres­sions that were proposed to date by various · authors to describe electron beam depth dose data analytically. For each expression, we show the fit it provides to a typical measured depth dose distribution. We also provide an analysis of an analytical expression which we developed 

recently for description of electron beams of various energies and field sizes. 12 The expres­sion reproduces electron beam central axis depth doses well in ali four regions of the depth <lose curve, and it achieves this with a smaller number of empirical parameters than does the most accurate approximation proposed pre­vious!y .11 
Analytical expressions for eledron beam depth doses 
The first analytical representation of clinical electron depth doses was proposed by Laughlin et al. 1 as follows: 
D(x) = 11 O I O exp [µ (x 
-Xm)] , (1) 
where D(x) is the percentage depth <lose at depth x in the medium, µ is an attenuation 
measured <lata only for relatively Iow energy electrons in the <lose fall-off region. The equa­tion is valid neither for the <lose build-up region nor for the bremsstrahlung region. 
To provide a better analytical description of electron depth doses, Bagne2 proposed a modi­fication to Eq. (1) through the addition of a cubic term in the exponent: 
D(x)=I 10-10 exp[µm ((x-Xm) p -A. (X-Xm)' p1)] 

(2) 

where D(x), x and Xm were defined above, µm is a mass attenuation coefficient, Q the density of the medium, and A a third adjustable parame­ter. This equation, although an improvement over Eq. (1), is also only valid for the <lose fall-off region. 
Pacyniak and Pagnamenta3 derived the following equation based on physical argu­ments: 
A = µ (RJE), where µ represents an attenu­ation coefficient and E the average electron energy. In contrast to Equations (1) and (2), Eq. (3) provides a fairly good approximation to the measured depth doses in the build-up region, <lose maximum region as well as in the <lose fall-off region. At depths beyond the range R, however, the equation generates high nega­tive percent depth doses and therefore cannot account for the bremsstrahlung contamination of the electron beam. 
Further attempts to parametrize electron depth <lose <lata were based on different types od mathematical functions which can mimic the dependence of measured depth doses on the depth in phantom. One group the these propo­sed approximations4-7 uses polynomial func­tions as follows: 
mum. The equation is simple, depends on only D(z) = 100 ..2 am zrn wilh z= 
x-;m two parameters (µ and xm), but agrees with 
D(x)=l00 +.(X-Xm)2[a, + a, x + a, x2 + a, x'] (4) 
coefficient, and Xm is the depth of <lose maxi­
D(x) = 10qao +a1 f.;-+ •.t"J21f.:-') , and (6) 
D(x) = ao (a, -2 a, s+ a, a, s<•,-1> + a, a. s(a.-1)) 
exp[ -(a, s + a, s' + a, s (a,) + a, s (a,)] + B (7) 

where Xm again is the depth of the <lose maxi­mum, ai are adjustable parameters, and para­meter B in Eq.(7) is a bremsstrahlung dose­related function. Variables z and s in Equations 
( 
5) and (7), respectively, are normalized depths defined as z = (x -Xm)/E where E is the elec­tron beam energy, and s = x/RP where Rr is the measured practical range of electrons. 

It was shown4-6 that Equations (4), (5) and 

( 
6) fit the measured electron depth doses rela­tively well in the build-up region, <lose maxi­mum region, and in the sharp <lose fall-off region for electron energies from 5 Me V to 20 Me V and for field sizes from 5 x 5 cm2 to 


25x25 cm2• However, they fail, similarly to Eq. 
D(x)=loo(.R..:.x)A(l +A.) 
R-Xm R-xm 


(3), to provide an acceptable approximation in where R is the practical electron range in the the bremsstrahlung background region in which medium and Xm again the depth of maximum they generate high negative depth <lose values, <lose. The third parameter A is defined as a property typical of ali proposed polynomial 


,and 


D(x) 
· 
The various approaches to analytical descrip­
D (x) 1 + exp [a, {x -as)] (10) 
where x50 in Eq.(8) is the depth in the dose fall-off region at which the dose reaches 50 % of the rnaxirnurn value and x8 in Eq. (9) is the depth in the fall-off region at which the dose is equal to the surface dose. Parameter a in Equations (8) and (9) and pararneters ai in Eq. 
(10) are adjustable pararneters. Equations (8) and (9) give rnuch better approxirnations to · electron depth doses in the build-up region than Equations (1) and (2), but they are consi­derably less efficient than the polynornial Equa­tions (3 through 7) in sirnultaneously approxi­rnating both the build-up region and the dose fall-off region. Moreover, they still tend to generate negative percent depth doses in the brernsstrahlung contarnination region. 

An excellent quality of curve fitting was achieved with Eq. (10), which has five fitting pararneters and gives a very good approxirna­tion for electron depth doses in the build-up region, dose rnaxirnurn region, and the dose fall-off region. However, at large depths Eq. 
(10) yields either zero or negative values and thus cannot reproduce the dose behaviour in the brernsstrahlung region. 

A more recent and very successful analytical representation of electron depth doses as a function of the depth in phantorn was proposed by Strydorn11 as follows: tions of electron bearn depth dose distributions discussed above and given by Equations (1) though (11) are illustrated in Figure l. A typical elecron bearn depth dose curve rneasured in water (9MeV, field size: 10x10 crn2), and shown as data points, is approxirnated by va­rious expressions (solid curves) proposed to date, starting (a) with the rudirnentary initial proposal of Laughlin et al. 1 with two pararneters and ending with (j) the excellent fitting based on six pararneters proposed by Strydorn. 
The varying quality of the curve fitting results for each to the approaches proposed to date is clearly evident frorn Figure 1, which also gives for each of the approaches the nurnber of pararneters required for the curve fitting proce­dure and the year of the proposal. It is evident that the curve fitting proposals irnproved with tirne but they also becarne considerably more cornplicated as they depended on ever-increas­ing nurnbers of fitted and rneasured pararneters. 
As shown in Figure 1 (j), Strydorn's equation based on six pararneters provides an excellent approxirnation to rneasured electron depth do­ses with four fitted pararneters in addition to two rneasured pararneters: the depth of dose rnaxirnurn and the brernsstrahlung contarnina­tion. Thus, the objective of an accurate appro­xirnation of the whole electron bearn central axis dose distribution has been met successfully with six pararneters in Eq. (11). It is clear that new approaches will not be able to irnprove the quality of fitting; however, they rnight sirnplify the fitting procedure by using equations which representations of electron depth doses. Equa­tion (7) approxirnates ali four electron depth dose regions but is obviously quite cornplicated as it depends on a large nurnber of fitting pararneters. 
Another group of rnathernatical approxirna­tions to electron bearn depth doses is based on a rnodified Fermi-Dirac distribution function:­


D(x) = --1QQ (8) 
{100-B) D(x) 1 -a, {x-xm) 
exp(-(x -xm)'[a2 +a3 (x -Xm) + "4 (x-xmJ2Jl + B (11) 
where Xm again is the depth of the dose rnaxi­rnurn, B represents the brernsstrahlung dose bakcground, and ai are adjustable pararneters. This equation has in effect six varying pararne­ters: (four fitted pararneters: a1, a2, a3 , a4 and two rneasured pararneters: Xm and B), and describes very well the electron depth doses in ali dose regions, including the brernsstrahlung dose background region. 




Depth in water (mm) 
10 20 30 40 50 60 O 10 20 30 40 50 60 
120 120 
100 80  oO" O 00000  -Equation l 2 parameters  00 o 0000 o . o  Equ.uion:?  !00 80  
60  (1953)  (1976)  60  
40  40  
20 o :  (a) :  :  :  o ) "? . . :  :  :  (b)  u.  20  

100  
-Equation 3  -Equation 4  
3 parameters  Sparameters  80  
(1974)  (1981)  60  

40 

20  R  20  
(c)  (d)  
100  100  
<I.) CI) o "O <I.)  80 60  -Equation 5 5 parameters (1983)  -Equa1ion 6 (1988)  80 60  <I.) CI) o "O <I.)  

.::: 
40 
d 
.::: 
d 
v v 
20 20 

(e) (f) 
l 00 .--oO-Q.o 100 00000 -Equation 8 -Equation 9 
80 80 
3 parameters 3 paramelm 
60 (1979) (!984) 60 
20 

(g) (h) 
100 
40 
Xwd
20 
100 

-Equation 10 -Equa1ion 11 
80 
80 

5 parameters 6 parameters 
60 

(!987) (1991) 60 
40 
40 
20 
20 

(i) (j) 
JO 20 40 50 60 O 10 20 30 40 50 60 


Depth in water (mm) 
Figure l. Central axis electron beam depth doses analytical expression, the number of required parame­
calculated from analytical expressions given by Equa­ters and the year the expression was developed are tions (1) through (6) and (8) through (11), compared also given. Equations (1) through (6) correspond to to data measured for a 9 Me V electron beam with a parts (a) through (f), respectively. Equations (8) field size of 10 x 10cm2 . Calculated <lata are shown through (11) correspond to parts (g) through (f), with solid curves, measured <lata as points. For each respectivcly. 
achieve a similar quality fit with a lower number of parameters. 
We have recently proposed12 a new analytical equation which contains only four parameters and is able to fit the measured electron depth doses in all four regions for various nominal energies of the electron beam as well as for various field sizes. The equation is given as follows: 
(12) 
(IOO -B) 

D(x) = 
-5 exp(-1x) + B , 
c l +a(x+c)(x-c)2exp(bx(x+c)(x-c)2] 

where a, b, and c are the fitted parameters, B is a measured parameter representing the bremsstrahlung contamination, and x is the 
depth in medium. 
Materials and methods 

The validity of the approximation given by Eq. 
(12) was verified with electron beam depth dose data which were measured in water at a source-surface distance (SSD) of 100 cm using a 3-D isodose plotter with a p-type semi-con­ductor detector. In the build-up region the percentage depth doses were measured in poly­styrene with a parallel-plate ionization chamber (Markus-type PTW, model 329). Two linear accelerators (Philips SL-25 and Varian Clinac 2300C/D) were used as sources of electron beams with square field sizes in the range from 4 x 4 cm2 to 25 X 25 cm2 and beam energies in the range from 4MeV to 22 MeV. 
Nonlinear curve-fitting was performed with a commercially available graphics software pack­age (KaleidaGraph by Abelbeck Software) on a Maclntosh computer. The general curve fit­ting space is missing program, based on a Marquardt algorithm, 13 is both powerful and efficient, able to fit any arbitrary single variable function containing up to nine fitted parame­ters. Moreover, the program allows fitting of weighted data as well as the use of partial derivatives. In our curve fitting procedure we used wqually-weighted data points. 

Results and discussion 

The fitting of Eq. (12) to measured electron beam percent depth doses resulted in a set of optimized numerical values for parameters a, b and c as a function of field size and nominal electron beam energy. The numerical values of parameters a, b, c and B depend on the field size as well as on the nominal energy of the beam. The second term in the right-hand side of Eq. (12) adjusts the shape of the function to the dose measured at the phantom surface and in its vicinity. 
For curve fitting purposes, the initial values of the four parameters are set as follows: the initial value of c is set equal to Xm, the measured depth of dose maximum; the initial value of a is obtained from Eq. (12) for the phantom surface, i.e., x = O and D(O) = Ds as: 

100-(D,+5) 
(13) 

a 
c' (D, + 5 -B) 

The initial value of b is set a few (typically six) times smaller than the initial value of a; and B is set equal to the measured bremsstrah­lung contamination which is assumed constant for a given electron beam energy. The non-li­near curve fitting starts with the initial values for a, b, and c and reaches the optimal values for a, b, and c through an iterative process. Note that Eq. (13) is used only for estimation of the initial value of parameter a using meas­ured values for the surface dose Ds and brems­strahlung contamination B, and the initial value for c as equal to Xm. The fina! optimal values for parameters a and c are generally not related through Eq. (13). 
An example of Eq. (12) used in fitting expe­rimental electron depth doses is shown in Figure 2 for a field size of 10 X 10 cm2 and electron beams produced by two of our high energy lineai· accelerators. Measured data are shown as data points and the corresponding calculated depth doses by solid curves. Parts (b) and ( d) of Figure 2 show on an expanded scale the build-up regions of parts ( a) and ( c), respective­ly. The agreement between the measured data and the fitted data in all four regions of the 
electron depth dose curves is excellent, proving that Eq. (12) with four parameters offers an excellent analytical approximation to measured data. Optimized parameters a, b, and c as well as the two measured parameters B and D, for beams of Figure 2 are given in Table 1. 
The quality of fitting Eq. (12) to the electron beam data set of Table 1 was evaluated for percentage depth doses above 20 % . The results of a statistical comparison between calculated and measured data representing 655 analyzed points ( 10 x 10cm2 field size, six beam energies for each of the two linear accelerators) are as follows: at the phantom surface, in the build-up region, and in the dose fall-off region, 61 % of calculated points matched the measured data within 1 % , 92 % within 2 % , and 98 % within 3 % . In the fall-off region, the difference be­tween the measured and calculated depths cor­responding to the 50 % depth dose was within 0.4mm for all electron energies produced by the two lineai· accelerators. 
Fitting of Eq. (12) to other sets of measured electron beam data gave results similar to those shown in Figure 2. for the 10 x 10 cm2 field size. Thus a conclusion can be made that the quality of fitting is independent of electron beam machine, field size, or electron beam energy, and all four regions of the electron depth dose curves are approximated well with 
Depth in water (mm) 
o 
20 

40 60 80 100 120 o 10 20 25 30 35 40 
Ficld size 


(a) (b) 
10x!Ocm2 

100 110 Elcctron cncrgy (Me V) 100 
80 o6 90 
„ 9 

60 o12 80 
6 18 
Electron cnergy (Me V) 0 6 m 
22 
„ 9 
o 12 60 

20 6 18 
II 22 

o :=::::::::::...:::='!.:!:::::.=!=:=::::::=!:=:::::=:::::!=:::::::::::=: :::::::::=:::::::::::::=::.:::::=::::::=:::::::::::::=::.:::::=:.::::::::=::: 40 
<1) 
"' 
o 
"O 
<1) 
> 
Ficki sizc 


(c) ·;g 
!Oxl0cm2 110 c:5 
o::: 
4100 
Electron cncrgy (McV) 

80 
0 4 

" 8 60 
o 12 
6 18 
II 22 

40 o 4 
„ 8 
. 12 20 6 18 
" 22 
90 80 70 60 50 

0 1 l 'f(it), 
1 JU!!lt:-: 1 ":1...r 1 1 '--'-.'--'--'.--'-'.._L_._,__!_..L,_.c.L._._,,.-C.J..J 40 

Depth in water (mm) 
Figure 2. Central axis pereentage depth doses for a eorresponding dose distributions calculated with Eq. field size of 10 x 10 cm2 and various clinical electron (12) are represented by solid curves. Parts (b) and ( d)beams produced by two linear accelerators: (a) and show in greater detail the build-up and dose maximum
(b) Varian Clinac 2300C/D; and (c) and (d) Philipsregions of parts (a) and (c), respectively. SL-25. Measured data are shown as data points and 
Table l. Optimized values of fitting parameters a, b, and e and measured parameters B and D, for 10 x cm2 electron bcams with various nominal energies for two commercial linear accelerators: a Varian Clinac 2300C/D and a Philips SL-25. 
Nominal  Parameter  
elcctron encrgy (McV)  a X 107 (mm-3)  b X 108 (mm-4)  C (mm)  B  D,  
VARIAN  6  1283.3  595.82  13.65  0.65  70.6  
CLINAC  9  278.29  146.30  19.89  1.31  77.3  
2300 C/D  12 15  76.647 24.050  42.505 15.052  25.96 27.71  2.14 3.44  83.4 90.3  
18  11.810  6.1579  25.88  4.13  93.1  
22  7.1493  2.2432  20.44  5.14  94.4  

PHILIPS  4  3652.7  1295.1  8.86  0.036  74.7  
SL-25  6  1104.0  406.77  12.5  0.54  77.3  
8  369.59  135.71  16.74  1.08  80.5  

10 178.34 78.031 20.54 81.8 
12 85.550 39.118 23.54 2.49 85.2 15 33.882 16.287 24.52 3.12 90.4 18 15.922 8.5102 27.14 3.31 92.0 20 8.3512 4.1550 23.33 3.68 94.0 22 4.7616 l.9890 15.79 4.06 94.8 
the fitting procedure. Equation (12) with four fitting parameters thus provides a relatively simple yet precise means for expressing clinical electron beams analytically. 
Conclusions 
Ever since electron beams have been used clinically, attempts have been made to describe analytically the measured central axis depth dose distributions. These distributions consist of four regions: dose buildup, dose maximum, dose falloff and bremsstrahlung contamination. Numerous analytical expressions to approxi­mate electron depth doses in ali four regions have been proposed to date. The quality of fitting generally improved with each new propo­sal but the curve fitting equations were beco­ming increasingly more complex as they de­pended on larger and larger numbers of fitting parameters. 
Analytical expressions developed in recent years for descriptions for electron beam depth doses provide an excellent fit to measured data. Improvements in this area can in the future sions which rely on a smaller number of para­meters. We have recently developed a relatively simple expression based on only four parame­ters. We show in this paper that the expression represents, with a high degree of precision, measured electron beam depth doses for various beam energies from two commercial linear ac­celerators. The conclusion can be made that the expression may be applied to describe the electron beam depth doses generally for any linear accelerator, any field size, and any beam energy. 


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