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Computational Tool for Calculation of
Tissue Air Ratio and Tissue Maximum
Ratio in Radiation Dosimetry
Atia Atiq*, Maria Atiq and Saeed Ahmad Buzdar
Department of Physics, The Islamia University of Bahawalpur, Pakistan
Submission: November 27, 2018;Published: January 03, 2019
*Corresponding author: Atia Atiq, Department of Physics, The Islamia University of Bahawalpur, Pakistan
How to cite this article: Added Atia A, Maria A, Saeed A B. Computational Tool for Calculation of Tissue Air Ratio and Tissue Maximum Ratio in Radiation Dosimetry. Curr Trends Biomedical Eng & Biosci. 2019; 17(4): 555966. DOI:10.19080/CTBEB.2019.17.555966.
The radiation therapy field is advancing continuously to achieve higher degrees of accuracy and efficiency. The purpose of this work is to enhance the efficiency of radiotherapy machine commissioning by applying computational tools to interpolate dosimetric quantities Tissue Air Ratio (TAR) and Tissue Maximum Ratio (TMR). Newton Divided Difference interpolation method was used to interpolate data between tabulated values. In order to shorten the commissioning time, technique of interpolation was used; in which calculation of dosimetric quantities were made for certain depths and field sizes with reasonable step size and remaining values at different depths and field sizes were obtained by interpolation. It was found that interpolated results considerably agree with the measured data and so this method could be used efficiently for interpolation of above mentioned dosimetric quantities. The data obtained by this technique is highly reliable and can fill the discrete set of tabulated data to make it continuous. The results obtained in this technique were in agreement with measured data and hence could be used as input data in radiotherapy treatment planning process. The interpolated results were well within accuracy limits and this method can significantly improve the efficiency of machine commissioning and resultantly improve treatment planning process.
Keywords: Radiotherapy; Interpolation; Tissue maximum ratio; Tissue air ratio; Commissioning
Abbrevations: PDD: Percentage Depth Dose; TMR: Tissue Maximum Ratio; TAR: Tissue Air Ratio; SSD: Source to Surface Distance; BJR: British Journal of Radiology
In radiation therapy, it is essential to calculate the dosimetric quantities such as Percentage Depth Dose (PDD), Tissue Air Ratio (TAR), and Tissue Maximum Ratio (TMR). The basic depth dose data was determined by dosimetric measurements taken in dummy patients (phantoms), which have density nearly equal to human body tissues, with ionization chamber placed in them . A system for absorbed dose calculations has been developed to foresee the depth dose distribution in patients going to be treated. The radiation dose deposited by ionizing radiations within the patient or medium varies with the varying depth. This variation is due to different parameters like depth, beam energy, field size, Source to Surface Distance (SSD). While calculating absorbed doses, greater considerations must be given to these parameters as they cause changes to depth dose distributions . Plenty of radiotherapy units, such as linac and cobalt-60 units, accomplish the treatment of cancerous parts. To treat cancerous tissues, cobalt 60 gamma ray beam is used for more than fifty years in radiotherapy .
Nowadays these units are used in conjunction with computer-controlled devices . Determination of dosimetric characteristics of all radiation beams is important so that most appropriate set of treatment planning parameters is chosen. Dosimetry is vital element of treatment using radiation as all the treatment planning is based on data obtained during dosimetry . To ensure that malignant part get the prescribed dose, it is essential to perform dose calculation by managing radiation beams which are characterized by various parameters in the treatment machine. This process of delivering radiations is called treatment planning. In advanced radiation techniques, treatment planning is generally performed with computing software’s. These software’s are used for identifying and locating anatomy of patients and the machine parameters to simulate the actual treatment . We intended to increase the efficiency of the radiation therapy by applying mathematical and computational tools not only to verify the measured dosimetric data but also to aid in rapid machine commissioning.
In the process of treatment planning many methods have been adopted to calculate variation of dose in the patients. Percentage Depth Dose (PDD) and Tissue Phantom Ratio (TPR) are among the two methods used for photon beams. First method devised for the calculation of dose was PDD but it has restriction that it depends on Source to Surface Distance (SSD) and this limitation was overcome by introducing a simpler quantity TPR . Although dependence of PDD on Source to Surface Distance can be overcome by the use of quantity Tissue Air Ratio (TAR) but its usefulness is only for beams having low X-ray energies i.e. in gamma beam
radiotherapy, so this quantity was in use when high-energy beams
were not common.
The quantity TAR can be defined as the quotient of absorbed
dose with a certain depth in phantom to the absorbed dose at the
same depth in free space .
Where Db is the absorbed dose at certain point d in phantom
and Dfs is the absorbed dose in free space at the same point. The
limitation of TAR for megavoltage units is because of the fact that
at high beam energies it becomes cumbersome to make measurements
in air . With high energy beams it become difficult to
make measurements in air so we can say that Tissue Phantom
Ratio TPR and Tissue Maximum Ratio TMR are the alteration of
Tissue Air Ratio, which makes them acceptable to be used for high
energies. Tissue Phantom Ratio is defined as the quotient of absorbed
dose at a given depth d to the absorbed dose at fixed reference
depth in water phantoms [10,11].
Where t0 is taken to be the reference depth. Although TPR can
be normalized to any reference depth but for most clinical purposes,
5cm is taken to be the reference depth. TMR is the special
case of TPR in which reference depth is taken to be the depth of
maximum dose Dd max .
Because of insensitivity of TMR on SSD, it has been found useful
for isocentric treatments whereas for any variation in SSD inverse
square correction must be applied for dose calculation by
PDD . TAR and TMR should be measured with great care as
they vary with varying field sizes and increasing depths. This entire
commissioning process must be performed methodically and
steadily which may require several months. Dosimetric data is
obtained for certain depths and field sizes in phantom. However,
it is time consuming to generate dosimetric values for all depths
and field sizes. As it is desirable to save time and improve the efficiency
of radiotherapy process, so values of TAR and TMR are
calculated for some field sizes and depths with discrete step size
and the rest of data is obtained by interpolation. This work aimed
at enhancing the efficiency of treatment plan and for this purpose
Newton Divided Difference Interpolation method had been used
to calculate absorbed doses at various depths first by keeping
field size constant and then at various field sizes by keeping depth
constant. Newton divided difference method is the polynomial interpolation
technique used for obtaining intermediate values and
is based on difference tables [14,15]. This method was preferable
because it finds no difficulty in calculating interpolated values
for those data points that neither are in definite sequence nor are
equally spaced. Important characteristics of this method are as
i. In interpolation, the values of x may or may not be equally spaced.
ii. The x values used in interpolation can be randomly selected
i.e. increasing suffix of x does not mean that values of x are
For pictorial understanding of this method, table for five data
points is shown in Table 1. This work aimed at calculating, interpolating,
and analyzing Tissue Air Ratio for Cobalt-60 Gamma
ray beams and Tissue Maximum Ratio for 4 MV and 6 MV photon
beams at different depths and field sizes.
One of the most important factors in the process of treatment
using radiation beams was to determine dose characteristics such
as TAR and TMR. Interest was to determine these characteristics
mainly because aim of this work was to use this data in treatment
planning process and also to study physics behind radiation
beams. The main objective of this work was to evaluate, measure
and calculate dosimetric characteristics using Newton Divided
Difference interpolation technique.
The data presented in this work was of Tissue Air Ratio for
Cobalt-60 Gamma ray beams. Tables 2 & 3 represented values of
TAR and TMR at constant field size and varying depth respectively.
Tables were drawn with depths varying from 1 cm to 26 cm.
Step size was taken from 1 cm to 20 cm depths and after that only
even numbered depths are taken i.e. 22 cm, 24 cm and 26 cm. The
depth of 1 cm had been assigned depth of maximum dose i.e. dm
. With increasing depth, values of TAR decreased as was clearly
depicted in Table 2. For the purpose of interpolation, separation
was chosen to be 5 cm. In Table 2, depths were varied keeping
field sizes constant. Field sizes were selected from 4 x 4-cm2 to
75 x 75-cm2, and field size represented in Table 2 was 7 x 7-cm2.
In Table 2, Tissue Air Ratio was interpolated at certain depth.
The calculated TAR data showed the same trend as the measured
data. Similar results were observed for TMR, as represented in Tables 3 & 4, proving that it is possible to interpolate any value
between given measured values. Minor variation was observed in
measured and interpolated values of TAR and TMR. Percentage
difference between interpolated and measured values of two basic
dosimetric quantities were calculated. Interpolation was done
using Newton Divided Difference Method and measured values
were taken from British Journal of Radiology (BJR) supplement
25, and calculated results was found to be within limits of accuracy
i.e. percentage difference was less than 2%. So interpolated
TAR values were consistent when compared with standard data. It
was to be noted that there was no general rule in the variation of
percentage difference of calculated Tissue Air Ratio. They varied
randomly at different depths but remain within the boundaries
of accuracy. The results between percentage difference and depth
for TAR values were also presented graphically in Figure 1. The
figure was plotted between depth on x-axis and percentage difference
For carrying out interpolation of TMR values, 4 MV & 6 MV
photon beams were used. In Table 4, TMR values were calculated
for field size of 12 x 12-cm2 while varying the depth from 1
cm to 25 cm. Calculations were carried out at depths with 5 cm
separation and divided difference method was used to calculate
TMR in between these depths. The field sizes were set from 4 x
4-cm2 to 40 x 40-cm2. It was further observed that for 4 MV X-ray
beams, the calculated results were in close agreement with the measured data. Considering Table 3, by keeping the separation to
be 5 cm, the interpolation of TMR data was performed for 6 MV
X-ray beams. The calculated data was then compared with measured
data and percentage difference between interpolated and
measured data was found. Percentage differences of interpolated
TMR from measured TMR were almost 2% or even less, which is
the limit of accuracy requirement. The results between percentage
difference and depth for TMR values were also presented
graphically in Figure 2. These results can further be extended by
applying interpolation method for varying field sizes and constant
depths to prove the robustness of this technique.
For TAR, depths were kept constant at 5 cm and 10 cm while
field sizes varied from 4 x 4-cm2 to 35 x 35-cm2. Interpolation
points were kept at field sizes 4 x 4-cm2, 8 x 8-cm2, 15 x 15-cm2
and 35 x 35-cm2. The maximum percentage difference in Table 5
was 1.47% at the field size of 25 x 25-cm2. There was no percentage
difference at the field sizes 6 x 6-cm2, 7 x 7-cm2 and 10 x 10-
cm2. Results were also graphically plotted in Figure 3 taking field
size at x-axis and percentage difference at y-axis.
For TMR, the variation for field sizes from 4 x 4-cm2 to 35 x 35-
cm2. The points of interpolation were set at field sizes 4 x 4-cm2, 8
x 8-cm2, 15 x 15-cm2, and 35 x 35-cm2. Table 6 represented TMR
values for 6 MV X-ray beams at constant depth of 5cm and the interpolated
results were observed to be quite satisfactory. Percentage
difference was zero for field sizes 5 x 5-cm2, 7 x 7-cm2 and 12 x
12-cm2. The differences were close to zero for field sizes 6 x 6-cm2,
9 x 9-cm2 and 12 x 12-cm2. The maximum percentage difference
was noted at the field sizes of 30 x 30-cm2 and it was found to
be 4.74%. Results were also graphically plotted in Figure 4 taking
field size at x-axis and percentage difference at y-axis.
By analyzing the results thoroughly it was established that
Newton Divided Difference method could be used as an efficient
method for interpolating accurate values both for varying depths and varying field sizes and also data calculated using this method
was accurate enough to be used as an input data in the process
of radiotherapy, thus increasing the effectiveness of radiotherapy
process while reducing time period.
This work mainly focused on finding the ways to improve
the treatment planning process. This work suggested an efficient
numerical method to calculate dosimetric parameters Tissue Air
Ratio TAR and Tissue Maximum Ratio TMR that could be useful
for medical physicists in increasing the accuracy of radiotherapy
treatment practice. The Divided Difference method was applied in
this work to calculate TAR and TMR at different depths and field
sizes with the reasonable step size and remaining values were interpolated
to obtain continuous data set. The results obtained in
this technique were in agreement with measured data and hence
could be used as input data in radiotherapy treatment planning
process. To facilitate the radiotherapy treatment planning process,
further it was recommended to interpolate different dosimetric
quantities by using different methods of numerical analysis
and to compare those results. Additionally, different interpolation
techniques could be explored to find the best one, which will be
helpful in reducing the error.
All procedures performed in studies involving human participants
were in accordance with the ethical standards of the institutional
research committee Shaukat Khanum Memorial Cancer
Hospital and Research Centre.