Question

4.a In the molecular theory of radiation action (aka. LQ model), derive the functional forms of α and β by expanding the total DNA strand breaks, Q, into Taylor series. That is, obtain the mathematical forms of α = f(k0, n0, f0, δ ) and β = f(k2, n1, n2, f0, f1, f2, 1- δ ).

.b Using the same expansion as in 4.a, derive a 3rd order model of radiation action.

.c Provide a biological explanation of what could give rise to a 3rd-order model.

Answer 1

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  The LQ formalism The LQ model, in its most usual current version, describes cell killing in terms of the following mechanisms: 1. Radiation produces DNA double strand breaks (DSBs) proportionate to the dose. 2. These DSBs can be repaired, with first-order rate constant 2 (=ln2/T42, where Ty2 is the repair half time). In practice, there may be more than one class of DSB that may be repaired with different rate constants; the LQ formalism can be simply extended to take this into account.
  
  3. In competition with DSB repair, binary misrepair of pairs of DSBs produced from different radiation tracks (i.e., different photons) can produce lethal lesions (often identified as predominantly dicentric chromosomal aberrations), the yield being proportional to the square of the dose (see the quadratic term in Eq. 1 and Eq. 3). The two independent radiation tracks can occur at different times during the overall regimen, allowing repair of the first DSB to take place before it can undergo pairwise misrepair with the second; it is this phenomenon which is the heart of the fractionation/protraction
  
  4. In addition, single radiation tracks can produce lethal lesions, possibly by a variety of mechanisms, the yield being linearly proportional to the dose. Overall, in the LQ formalism, the yield (Y) of lethal lesions, and the corresponding survival (S) equation are Y a AD + GBD2. (1) (note that the biologically effective dose, BED, is defined as Y/a(1)). Then, assuming the lethal lesions are Poisson distributed from cell to cell, the surviving fraction will be S = exp(-Y),
  
  S = exp[- (aD + GBD2)]. (3) In Eq. 1 and Eq. 3, G is the generalized Lea-Catcheside time factor, which accounts quantitatively for fractionation/protraction; it is important to note that G acts only on the quadratic component, as described in Point 3 above. The generalized time factor has the form (16) G = (2 1D2 (4) buna fe *-blerar Je-1(t–t) D(t')dt'
  
  Here D(t) describes the variation in dose rate over the entire course of the radiotherapy, and 2 is a characteristic damage repair rate. Generically, the term after the second integral sign refers to the first of a pair of DSBs required to produce a lethal lesion - the exponential term describing the reduction in numbers of such DSB through repair; similarly, the term after the first integral sign refers to the second DSB, which can interact with DSBs, produced earlier, that still remain after repair. The time factor, G, can be calculated for any fractionation/protraction scheme, and systematically accounts for the effects of protracting the dose delivery in any way. G can take values from zero to one, with G=1
  
  Multi-target Model Initial slope measure, D., due to single-event killing Do: the dose that yields a surviving fraction of 37%. Dq: the region of the survival curve where the shoulder starts (indicates where the cells start to die exponentially = quasi-threshold dose). n: extrapolation number (measure of width of shoulder) TTTT Final slope measure, Do, due to multiple-event killing D TUTTTTTTTTT Densely Ionizing (neutrons or a-rays) Sparsely Ionizing X-rays Logen = Dq/D. 12 16 B (Gy)