

REVIEW ARTICLE 

Year : 2013  Volume
: 9
 Issue : 3  Page : 348350 

Fractionated beam radiotherapy is a special case of continuous beam radiotherapy when irradiation time is small
Jayanta Biswas^{1}, Tapan K Rajguru^{2}, Krishnangshu B Choudhury^{3}, Sumita Dutta^{4}, Shyam Sharma^{1}, Aniruddha Sarkar^{5}
^{1} Department of Radiotherapy, Institute of Post Graduate Education and Research (IPGMER), Kolkata, West Bengal, India ^{2} Department of Radiotherapy, Bankura Sammilani Medical College, Bankura, West Bengal, India ^{3} Department of Radiotherapy, R. G. Kar Medical College and Hospital, Kolkata, West Bengal, India ^{4} Department of Anatomy, Calcutta National Medical College, Kolkata, West Bengal, India ^{5} Department of Anatomy, Medical College, Kolkata, West Bengal, India
Date of Web Publication  8Oct2013 
Correspondence Address: Jayanta Biswas ARITRI Apartment, 155, R. N. Guha Road, Kolkata, West Bengal India
Source of Support: None, Conflict of Interest: None  Check 
DOI: 10.4103/09731482.119300
Fractionated beam radiotherapy, in other terms, external beam radiotherapy (EBRT) and continuous beam radiotherapy or Brachytherapy are two modes of radiotherapy techniques. Although in many ways, they appear to be different, radiobiologically, with the help of mathematics, it can be proved that the biological effective dose (BED) of EBRT is similar to BED of Brachytherapy, when irradiation time is small. Here an attempt is made to correlate these two predominant modes of radiotherapy techniques. Keywords: Continuous radiotherapy, fractionated radiotherapy, irradiation time
How to cite this article: Biswas J, Rajguru TK, Choudhury KB, Dutta S, Sharma S, Sarkar A. Fractionated beam radiotherapy is a special case of continuous beam radiotherapy when irradiation time is small. J Can Res Ther 2013;9:34850 
How to cite this URL: Biswas J, Rajguru TK, Choudhury KB, Dutta S, Sharma S, Sarkar A. Fractionated beam radiotherapy is a special case of continuous beam radiotherapy when irradiation time is small. J Can Res Ther [serial online] 2013 [cited 2019 Nov 19];9:34850. Available from: http://www.cancerjournal.net/text.asp?2013/9/3/348/119300 
> Introduction   
The main effects of radiation on cell survival are lethal, sublethal, and potentially lethal damages on chromosomes, modulated by 5 "R"  repair, repopulation, reoxygenation, reassortment, and radiosensitization. ^{[1],[2]} However, of the 5 R's of radiotherapy that exist, it has emerged from clinical studies that repopulation is one of the most significant factors that can provide insight into the lack of efficacy of radiation treatment. Kirkpatrick and Marks ^{[3]} stated that simple radiobiologic models that fail to incorporate the heterogeneity of radiosensitivity and or tumor cell repopulation will not adequately describe clinical outcomes. In addition, Kim and Tannock ^{[4]} proposed that during chemotherapy or radiation treatment repopulation of cancer cells often has a dominant effect on treatment outcome. The kinetics of repopulation offer insight into the underlying mechanisms of tumor cell death and regrowth, and as such, these models may be clinically useful in predicting response to therapy. ^{[5]}
The hypothetical cell survival curves for describing the mechanisms behind lethality in response to radiation have been explained by number of biomathematical models, like "Multi Target  Single Hit", Linear Quadratic (LQ), Linear QuadraticLinear (LQL), and quadratic models.
> Discussion   
The ultimate effect of radiation culminates in cell death. The fundamentals of radiobiology are described in terms of timedose relationships. The repair and cell death following radiation exposure are time and doserate dependent. Brenner et al. explained that the quantitative predictions of dose/fractionation dependencies in radiotherapy is the mechanistically based LQ model. ^{[6]} LQ model was originally proposed by Keller and Rossi as a result of microdosimetry of radiationinduced cellular damages. ^{[7]} LQ formalism describes the fractionation and doseprotraction effects through a particular functional form, the generalized time factor, G. ^{[8],[9]} The LQ formalism is now almost universally used for calculating radiotherapeutic isoeffect doses for different fractionation/protraction schemes. ^{[10],[11]} This model considers the effect of both irreparable damage and repairable damage susceptible to misrepair, which ultimately leads to mitotic cell death.
The biological effect (E) per fraction (n) of fractional dose (D) can be expressed as:
E_{n} = (αD + βD ^{2} )
The biologically effective dose is an approximate quantity by which different radiotherapy fractionation regimens may be intercompared. For instance, for an external beam radiotherapy (EBRT) regimen employing n equal fractions of conventional size the BED will be:
where n = number of fractions, D = dose/fraction, and nD = total dose.
This formulation assumes that full repair occurs between fractions so that the biological effect of each fraction is the same. Many extensions to the LQ and BED formulations have been developed to account for temporal phenomena such as sub lethal repair and repopulation, which can occur during lengthy fractionated regimens.
> The Precondition   
The precondition here is that the duration of fractionated beam radiotherapy should be less than one and half hour and for continuous beam radiotherapy it should be more than one and half hour. This is because, as repair half life is 0.52 hours, whenever duration of treatment is more than one and half hour or 90 minutes, repair of sub lethal damage goes on simultaneously with actual cell kill.
> Mathematical Derivation   
At first, the BED also known as extrapolated response dose (ERD) of continuous beam therapy is calculated. According to LQ model, the repair of sub lethal damage is supposed to be exponential. e ^{μt} is the amount of damage existing after time t and therefore, the amount of repair after time t is equal to (1 e ^{μt} ) (where, μ = repair constant).
From the LQ model, relative effectiveness,
where A type damage signifies linear component damage and B type damage is quadratic component damage.
Calculation for A type of damage is a bit straightforward, that is, αD = αRT, where D = Dose, R = dose rate, α = coefficient of linear damage, and T = total time. When explaining the B type damage, the probability of hitting one target of a cell by a single hit is PD.
But, as there are two targets in a cell and as both are to be hit and by separate radiation events, the probability of hitting one target among two targets will be double, 2PD = 2PRdt, where dt is a small time and R = dose rate.
Therefore, probability of damage existing after time
For getting the total probability of damage existing after entire time t, integrating the equation 2
Probability of the other target being damaged after additional time dt would be
For getting total probability of both the targets being damaged in time T, integrating the left hand side of eq (3),
^{[12]}
This is the equation for calculating BED (or, ERD) of continuous beam radiotherapy where duration of treatment is more than 90 minutes or one and half hour.
When T is small, μT is also small (less than 90 minutes duration of radiotherapy), that is, μT ≤ 1;
This is the equation of RE of fractionated beam radiotherapy. Hence, mathematically, it can be proved that fractionated beam radiotherapy is a special case of continuous beam radiotherapy.
> Conclusion   
Radiobiologically, when irradiation time is short, fractionated beam radiotherapy behaves like continuous beam radiotherapy.
> References   
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3.  Kirkpatrick JP, Marks LB. Modelling killing and repopulation kinetics of subclinical cancer: Direct calculations from clinical data. Int J Radiat Oncol Biol Phys 2004;58:64154. [PUBMED] 
4.  Kim JJ, Tannock IF. Repopulation of cancer cells during therapy: An important cause of treatment failure. Nat Rev Cancer 2005;5:51625. [PUBMED] 
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6.  Brenner DJ, Hlatky LR, Hahnfeldt PJ, Huang Y, Sachs RK. The linearquadratic model and most other common radiobiological models result in similar predictions of timedose relationships. Radiat Res 1998;150:8391. [PUBMED] 
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8.  Lea DE. Actions of Radiations on Living Cells. London: Cambridge University Press; 1946. 
9.  Lea DE, Catcheside DG. The mechanism of the induction by radiation of chromosome aberrations in Tradescantia. J Genet 1942;44:21645. 
10.  Brenner DJ. The linearquadratic model is an appropriate methodology for determining isoeffective doses at large doses per fraction. Semin Radiat Oncol 2008;18:2349. [PUBMED] 
11.  Barendsen GW. Dose fractionation, dose rate and isoeffect relationships for normal tissue responses. Int J Radiat Oncol Biol Phys 1982;8:198197. [PUBMED] 
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