Increasing Number of Fractions in the Case of Non-Uniform Dose and Fixed Nominal BED Leads to Increased Cell Killing in the Treatment Target

Probability of Survival and BED in the Case of Non-Uniform Dose

Consider first a course of radiotherapy with f N treatment fractions, (uniform) dose per fraction d and total dose f D N d = . In the LQ model, the probability of survival for irradiated cells is (e.g., [5, 6])

2 exp( ) exp( ) D S D dD D N β α β α = − − = − −

(7) f Suppose that PTV V denotes the planning target volume (PTV). In the case of heterogeneous target dose, let ( ) V D denote volume of the target that receives a dose equal to or greater than a given dose D. The ratio ( ) / PTV V D V is a function of dose and is referred to as (cumulative) dose-volume histogram or DVH [17]. From the definition of DVH it follows that the difference d ( ) ( ) d DVH DVH D DVH D D D D − + ∆ ≈− ∆       equals relative volume of the PTV with dose ranging between D and D D + ∆ . Consequently, the probability of survival averaged over the dose distribution in the target is

2 d exp( ) d d D

D DVH S D D N D β α = − − − ∫ , (8)

,max tar f D ,min tar

where ,min tar D and ,max tar D denote minimum and maximum doses in the PTV. Note that the minus sign before the integral in Eq. (8) is due to the fact that DVH is a monotonically decreasing function of dose.

Computation of S requires knowledge of the entire DVH for the target volume. For the purpose of our discussion, it is convenient to rewrite Eq. (8) as D

2 exp( ) ( )d D S D f D D N β α = − − ∫ (9)

,max tar f D ,min tar

where d ( ) d DVH f D D ≡− represents density distribution of the target dose. The corresponding BED and TCP can be expressed as follows (e.g., [13, 19])

0 1 ln andTCP exp( ) BED S N S α = − = −

(10)

Considered Dose Distribution in the PTV

In the derivations below, we show that under conditions in Eqs. (5) and (6) S decreases with increasing f N assuming fixed nom BED . Note that this conclusion is valid for all SBRT dose distributions in the target volume which we refer to as realistic. Specifically, we consider distributions of the target dose with

1 nom BED α > . (11)

Since the initial number of malignant cells in the target is typically very large1, the condition in Eq. (11) is necessary to attain TCP reasonably close unity (see Eq. (2)). Another feature of a realistic dose distribution in SBRT is that minimum dose ( min D ) in the PTV is of the same order of magnitude as the mean dose. For example, we examined 60 SBRT plans with the treatment schedule 12 Gy 5 60 Gy × = and found min 30 Gy D > for each plan. In the following discussion we employ a condition for min D :

min ref D rD > (12)

where ref D is the mean dose in the PTV for a reference

1 N0 is given by the product of no and VT, where no is the initial concentration of malignant cells in the tumor and VT is the tumor volume. A reasonable estimate for no is 107 cm-3 [1].

regimen with , f ref N fractions and r is the real solution of the following equation:

ln 1 nom r BED r α − = − (13)

Because ln 1 r r ≤ − for 1 r ≥, the real solution of Eq. (13) for nom BED α > 1 satisfies 1 r < . Eq. (13) can be numerically solved by iterations:

( ) exp( ), exp (1 ) ,

r BED r BED r α α = − = − × −

0 1 0 nom nom = − × − = − × − (14)

( ) ( ) exp (1 ) ,..., exp (1 ) ,...

r BED r r BED r

α α −

2 1 1 nom n nom n

Several examples are as follows: 5 nom BED α = ,

3 6.977 10 r − ≈ × ; 10 nom BED α = ,

5 4.542 10 r − ≈ × ; 100 nom BED α = ,

44 3.720 10 r − × ≈ . Note also that due to condition in Eq. (12),

( ) ref f D is zero in the interval 0, ref rD    .

Proof that S Decreases with Increasing f N if

nom BED const = The probability of survival averaged over the dose distribution ( ) f D can be expressed as follows (see Eqs. (4) - (6)):

( ) 2 2 ∞ ∞   − = − − = − −       ∫ ∫ , (15)

x BED D D Dx S D f D D f x x N D D

α β α α

2 0 0 exp( ) ( )d exp ( )d nom ref f ref ref

ref D D x cD D = = In turn, the derivatives S D ∂ ∂ and

2

2 S D ∂ ∂ are where ( ) 2 2

∞     − ∂ = − − −         ∂     ∫ and (16)

x BED D S x x Dx f x x D D D D D

α α α

2 2 0 exp ( )d nom ref ref ref ref ref

( ) 2 2 2 2

α α ∞     − ∂ = − − −         ∂     ∫ . (17)

x BED D S x x Dx f x x D D D D D

2 2 2 0 exp ( )d nom ref ref ref ref ref

2 S D ∂ ∂ is an

2 S D ∂ ∂ >0. Consequently,

It is apparent that

S ∂ increasing function of D . If we can show that ∂ is negative D

for nom D BED = , then S D ∂ ∂ is negative for all nom D BED < . From the condition nom BED const = (see Eq. (4)), it follows that D increases as f N increases. Consequently, if 0 S ∂ < ∂ we can then D S N ∂ ∂ is negative for all f N .

conclude that f Let 1 nom BED λ α = > . Since the mean dose for the distribution ( ) ref f D is ref D , we have for nom D BED = (see Eq.(16))

      ∂ = − −           ∂               = − − − − −                   ∞

2 BED x S x x f x x D D D D

α α ∫ exp ( )d

nom ref ref ref ref

2 0 ∞

2 x x x x f x x D D D D

λ λ ∫ exp ( 1)exp( ) ( )d

ref ref ref ref ref rD

2 ref (18) By using / 1 ref s x D = −, we have ∞

S D s s s f D s s D

∂   = − + − − +   ∂

( ) ( ) ∫ exp( ) 1 exp 1 ( ( 1))d

λ λ α ref ref ref r

(19)

1 − ∞ ∫ exp( ) ( ) ( ( 1))d

D sQ s f D s s

λ ≡ − + ref ref ref r

1 − where ( ) ( ) ( ) 1 exp 1 Q s s s λ = + − −. To show that S D ∂ ∂ is less than zero, it is sufficient to show that ( ) 0 sQ s < for all 1 s r > − . To accomplish this, we will show that ( ) Q s is negative for all (0, ) s∈ ∞ and positive for all ( 1,0) s r ∈ − .

Clearly, (0) 0 Q = . Note that derivative ( ) d ( ) 1 ( 1) exp( ) d Q s s s s λ λ = − + − (20) is negative for 0 s ≥ . Consequently, ( ) 0 Q s < and ( ) 0 sQ s < for 0 s > . Note also that equation ˆ d ( ) 0 d Q s s = has a single root ˆ 1/ 1 s λ = −. From the definition of r in Eq. (13) it follows that ( ) ( 1) exp ( 1) 1 0 Q r r r λ − = − − −= . (21) Does ( ) Q s have zeroes in the closed interval [ 1,0] r − besides its endpoints? If ( ) Q s had another zero, then by the Rolle’s Theorem d ( ) d Q s s would have at least two zeroes in the open interval ( ) 1,0 r − which is impossible. Consequently, ( ) Q s is either positive or negative on ( 1,0) r − . Note that since d ( ) 0 at 0 d Q s s s < = , ( ) Q s is positive on( ) 1,0 r − . As a result, we conclude that the product ( ) sQ s is negative for ( ) 1,0 s r ∈ − .

This, in turn, implies that 0 S

∂ < ∂ . Since 0

D ∂ > ∂ , S decreases D

N f with increasing f N . The proof is complete.

It should also be realized that since S decreases with increasing f N , 1 ln BED S α = − increases with increasing number of fractions.

Elucidation of the Dependence of S on f N

The previous Section contains a rigorous proof that S decreases with increasing f N if nom BED const = for a realistic dose distribution and uniform radiosensitivity in the target. Unfortunately, this proof doesn’t easily yield a qualitatively clear explanation of the claimed dependence of S on f N . Here, we consider a simpler and more intuitive approach previously outlined in [21], which leads to the same conclusion.

2 exp( ) D D N

β α − + in a power series around D , we obtain By expanding f

2 2 2

2 ( ) ( ) ( ) exp( ) ( ) ( ) ... 2 f

D S D D D S D D S D D D N D D β ∂ − ∂ − + = + − + + ∂ ∂ (22) In the case when the variance of the target dose σ is small (i.e., / 1 D σ << ), we can restrict series expansion in Eq. (22) to the second order term. Substituting expression for

2 exp( / ) f D D N α β − + from Eq. (22) into Eq. (9) and considering the fact that the average value of D D − is zero, we obtain the following equation for S :

2 2 ( , ) ( , ) 2

S D S S D D

σ α α ∂ = + = ∂

2 + − × −          +              

2 2 2

2 2 1 exp( ) 2 ( / ) 1 nom

D BED N N σ β α α α β . (23)

f f One can verify that under the condition 1 D α >> , we

2   << +       . Consequently, expression for the

2 2 2 1 ( / ) f f N D N

β α α β have probability of survival in Eq. (23) is reduced to σ α α α β = + × −       +          

2 2 2 2 1 exp( ) 2 ( / ) 1 nom

D S BED N . (24)

f In the case nom BED const = , from Eq. (4) it follows that dose per fraction D N d = decreases with increasing f N .

f Consequently, Eq. (24) implies that decreases with increasing f N . Unfortunately, truncation of the infinite series for S (which leads to Eq. (24)) is difficult to justify rigorously. Note that the proof in the previous section doesn’t use series expansion for S. As a result, it is free of limitations inherently present in the intuitive but non-rigorous derivation of Eq. (24) presented above.

2 2 2 2 1 2 ( / ) 1

  +       << , we have D N σ α α β Finally, in the case f

2 2 2 2 nom f

  = − +      

D BED BED N σ β α α (25)

Eq. (25), in turn, indicates that for fixed / D σ , reduction in BED due to dose heterogeneity rapidly increases in magnitude with increasing dose per fraction:

2 D BED BED BED N α α β

2 ~ ( ) . nom nom

− + ( ) (26) f

Effect of non-uniform Radiosensitivity on Cell Survival

The discussion in the preceding Sections is focused on the effect of target dose heterogeneity on BED while radiosensitivity of malignant cells is assumed to be uniform. It is potentially clinically important to incorporate the effect of heterogeneous radiosensitivity in the analysis of BED dependence on number of fractions. In this Section we employ the following assumptions: (a)α and β are independent random variables with probability density functions ( ) fα α and ( ) fβ β ; (b) the joint probability for alpha and beta is described by the Gamma probability density function; i.e., , ( , ) ( ) ( ) f f f α β α β α β α β ≡ is given by α β α β θ θ α β α β θ θ − −  − −   > > =  Γ Γ  

1 1 exp( ) , 0 and 0 ( , ) ( ) ( )

t t β α (27) α β t t f t t

( ) ( ) α β α β α β

0, otherwise In Eq. (27) Γ denotes the so-called Gamma function [18]

∞ − − Γ = ∫ , (28)

1

0 ( ) t x t x e dx where 0 t > . It should be mentioned that parameters , , and t t α β α β θ θ are defined by the mean values of alpha (α ) and beta ( β ), and their variances 2 2 ( ) and ( ) α β σ σ [18]; i.e.,

2 2 2 2 σ σ α β θ θ σ σ α β = = = = . (29)

2 2 , , and t t β α α β α β α β Note that ( , ) f α β in Eq. (27) is normalized so that ∞∞ = ∫∫ .

0 0 ( , ) 1 f d d α β α β In the case of non-uniform dose and non-uniform radiosensitivity in the target, the probability of survival average over the distributions of the target dose and radiosensitivity is (see Eq. (8) for comparison)

2 d exp( ) ( , )d d d d f

D DVH S D f D N D β α α β α β   = − − −     ∫∫∫ . (30)

By using substitutions (see Eq. (27))

2 1 1

1 1 f N d β β θ θ = + ′ (31)

f N d α α θ θ = + ′ and and integrating over alpha and beta, we can reduce the triple integral in Eq. (30) to a single integral t t D − −      = − + +               ∫ . (32) β α

2 d 1 1 d d D D DVH S D t t N D

α β max f D α β min Note that biologically effective dose for non-uniform radiosensitivity is defined as [19]

1 ln BED S α = − . (33)

Radiobiological Assumptions

The LQ model for cell killing forms the foundation of our work. The applicability of this model for hypofractionation has been disputed in several studies which described a number of alternative models (e.g., [22, 23, 24, 25]). The main feature of the proposed non-LQ models is that on the log- linear plot the survival curve becomes linear at high doses. By analyzing tumor control data for SBRT and SRS, recent studies, however, concluded that the LQ model fits the clinical TCP data best [20, 26, 27]. It is important to realize that our model doesn’t consider the effect of accelerated (i.e., faster than before commencement of radiotherapy) repopulation of malignant cells which begins after a delay $T_k$ following the first fraction of radiation (e.g., [28, 29]). Since reported $T_k$ for non-small cell cancer of the lung ranges between 14 and 35 days [29], the effect of accelerated repopulation is likely to be small in the case $N_f \leq 20$ considered in our study.

Dependence of BED on the Variance of the Target Dose and $N_f$

As shown in Figure 1, small variance of the target dose (i.e., $\sigma \leq 0.1\bar{D}$) can cause a significant reduction in BED. In this work it is analytically shown that for a given ratio $\sigma/\bar{D}$ and uniform radiosensitivity BED always increases with increasing $N_f$ if $BED_{nom}$ is fixed. It is important to establish whether this increase can be significant for clinical cases.

The results in Figure 1a demonstrate that transitioning from a frequently used SBRT schedule $12\mathrm{Gy} \times 5 = 60\mathrm{Gy}$ with 5 fractions [16] and $BED_{nom} = 132\mathrm{Gy}$ to a schedule $4.54\mathrm{Gy} \times 20 = 90.8\mathrm{Gy}$ with 20 fractions and the same $BED_{nom}$ can lead to 7-8% increase in the corresponding BED. Unlike our previous study [13], these results were obtained by considering clinical cases of SBRT without making additional assumptions regarding the dose distribution in the target.

Effect of Non-Uniform Radiosensitivity on Dependence of BED on $N_f$

The analytically derived conclusion in Section 2.3 that BED increases with increasing number of fractions doesn't consider variations in radiosensitivity. To study the effect of small variations in $\alpha$ and $\beta$ in the tumor on the dependence of BED on $N_f$, we assumed (in contrast to our previous study [21]) that these variations were uncorrelated. This assumption can be justified as follows. In the LQ model, alpha term describes lethal damage produced on the nanometer level. Conversely, beta term relates to damage caused by interactions of double-strand breaks on a significantly larger scale [30]. Another assumption employed in this study is that joint probability distribution of alpha and beta can be approximated by the Gamma density probability function (see Eq. (27)). The rationale for Gamma distribution is threefold:

• Gamma distribution is a smooth, bell-shaped distribution which, in contrast to Gaussian distribution, doesn't allow negative values of alpha and beta;
• Gamma distribution approaches Gaussian distribution when $\sigma_a << \bar{\alpha}$ and $\sigma_\beta << \bar{\beta}$ [18];
• Recently, Gamma distribution was successfully employed to model tumor control in almost 3000 patients [20].

The results from Figures 1b and 1c indicate that BED increases with increasing $N_f$ for non-zero $\sigma_a$ and $\sigma_\beta$. This conclusion was confirmed by computing BED for a Gaussian (joint) distribution of alpha and beta. The resulting values of BED (not shown here for brevity) were within 1% of those in Figures 1b and 1c. The observed good agreement between BEDs computed for Gamma and Gaussian distributions is not surprising because, as mentioned above, Gamma distribution approaches Gaussian distribution for relatively small values of $\sigma_a$ and $\sigma_\beta$.

**Clinical Implications**

Reduction in BED Due to Dose non-uniformity: According to the obtained results, for SBRT regimens with $N_f = 5$, the corresponding BED can decrease by about 30% while $\sigma$ varies between zero and 8% of the mean dose (see Figure 1a). Such significant changes in BED can present a problem for radiobiological comparison of different hypofractionation regimens because clinical reports do not normally contain variance of the target dose for each case. For example, consider two regimens $12\mathrm{Gy} \times 5 = 60\mathrm{Gy}$ and $4.54\mathrm{Gy} \times 20 = 90.8\mathrm{Gy}$ characterized by the same $BED_{nom} = 132\mathrm{Gy}$. In the case $\sigma/\bar{D} = 0.08$, the former regimen with 5 fractions can yield BED of 90Gy while the regimen with 20 fractions achieves BED of 97Gy (see Figure 1a). As shown in this work, in the case of uniform radiosensitivity in the tumor, reduction in BED due to inhomogeneity of the target dose is always smaller for a schedule with a number of fractions $N_{f,1}$ as compared to that for a treatment schedule with $N_{f,2} < N_{f,1}$ for the same ratio $\sigma/\bar{D}$ and $BED_{nom}$.

One possibility to decrease discrepancy between BEDs for different clinical cases is to limit acceptable dose non-uniformity to less than 5% (i.e., $\sigma \leq 0.05\bar{D}$) as proposed in [13, 21]. However, our clinical experience indicates that in some cases this condition is difficult to accomplish. The results of this work suggest that increasing number of fractions can be radiobiologically beneficial for small, well-perfused lung lesions which are frequently targeted in lung SBRT. Specifically, in the case $\sigma > 0.05\bar{D}$ and small variances of alpha and beta, moderate hypofractionation with $N_f = 20$ can yield higher BED (and associated TCP) as compared to treatment schedules with five or fewer fractions used for SBRT of non-small, early-stage lung cancer.