3.6.1 Serial and parallel architecture of organs
Organs are classified into serial and parallel organs depending upon either their structural or functional basis as explained by Witherset al. [138]. The fundamen- tal functional reserve of an organ is defined by functional subunits (FSUs). The damage caused by irradiating an organ depends upon its architecture in addition
(a)
(b)
Figure 3.4: Effect of varying (a) mean radiosensitivity, ¯α(b) variation in radiosensitivity over a patient population,σα on the TCP curve by keeping the other parameters constant.
Figure 3.5: Organ architecture models (a) serial (b) parallel (c) serial-parallel (reproduced from K¨allman, 1992).
to other factors. A serial organ has its FSUs organised like a chain and loss of one of these FSUs can result in dysfunction of the organ. On the other hand, the FSUs of a parallel organ are independent of each other and a damage caused to one of its FSUs will not result in total organ failure.
The functional reserve of an organ following radiotherapy is preserved or dam- aged depending upon the dose delivered and the behavioural response of an organ based on its architecture. Spinal cord is an example of a serial organ, the tol- erance of which is determined by the maximum dose whereas lung is a parallel organ and the toxicity caused is usually a function of mean dose. Some organs have partly serial and partly parallel architecture like the nephrons of the kidney. Figure 3.5 shows the serial, parallel, serial-parallel organisational architecture of organs [26].
3.6.2 Lyman-Kutcher-Burman (LKB) Model
The LKB model proposed by Lyman et al.[139] employs an error function given by equation 3.5 to represent the sigmoidal behaviour of dose-reponse following irradiation of a partial volume of an organ.
NTCP = √1
2π
Z t
−∞
e−t2/2dt (3.5)
The probability of causing a complication to the organ of interest is a function of both dose and volume. The LKB model is a four parameter model and is given by the formulas expressed in equations 3.5 to 3.7.
t = D−T D50(V /Vref)
mT D50(V /Vref)
(3.6)
TD50(1) = T D50(V /Vref)(V /Vref)n (3.7) HereTD50(1) is the dose to the whole organ which would result in a complication
in 50% of the patient population, Vref, is the reference volume which is usually the whole organ volume, m, a parameter representing the steepness or the slope of the dose-response curve, n represents the volume effect parameter; a power law relationship is assumed between the tolerance doses for uniform whole and partial organ irradiation, D is the reference dose. A low n value indicates the serial behaviour of an organ whereas a highern value corresponds to organs that have parallel architecture. For example, lung which is a parallel organ has a n
value close to 1.
The above equations are valid for uniform irradiation of an organ. In order to account for inhomogeneous dose distribution in treatment plans it is essential to reduce the DVHs into single dose and define the reference volume of the organ receiving this dose. In our study, we use the effective volume method proposed by Kutcher and Burman. The DVH of the organ of interest is converted into an effective volume Vef f which is the volume of the organ as a whole or less than the total volume that receives the maximum dose Dmax. The effective dose Def f and volume Vef f are derived by equations 3.8 and 3.9 respectively.
Veff = k X i=1 ∆Vi(Di/Dmax)1/n (3.8) Deff = X i " Di1/n Vi VT #n (3.9)
Here, Di is the dose in each bin, Vi is volume in each bin, Dmax is the maximum dose received by the organ of interest, VT is the whole organ volume, n is the volume effect parameter.
3.6.3 Logistic Model
The NTCP of ribs with rib fracture as the end point was calculated for the SABR patient plan as the tumour was adjacent to one of the ribs. NTCP of the ribs were calculated using the logistic model and the parameters suggested by Petterson
et al. [140]. In this study, the authors have evaluated the risk of rib fracture in terms of two models, one based on cut-off volume and the other based on cut-off dose. Results show that the cut-off volume model resulted in a better fit than the cut-off dose model. The NTCP of rib is calculated by logistic-model which is given in equation 3.10. NTCP(DV) = 1 1 +e 4γ50,V (D50,V 3 +αβ) (2D50,V n +αβ) " 1− DV D50,V (DV n +αβ) (D50,V n +αβ) # (3.10)
Here, DV is the cut-off volume descriptor which is the maximum dose received by the high dose volume,D50 is the dose at which there is 50% risk of complication,
γ50 is the normalised dose-response gradient and n is the number of fractions.
3.6.4 NTCP parameters for LKB model and logistic model End point: radiation pneumonitis
The parameters to calculate NTCPs for lung were taken from De Jaegeret al.[40] which are shown in the table 3.2.
Table 3.2: LKB model NTCP parameters for lung with radiation pneumonitis as the end point.
NTCP parameters Values
TD50 29.2 Gy
m 0.45
n 1
End point: rib fracture
calculate the probability of rib fractures for the dose received by 2 cm3 cut-off
volume.
Table 3.3: Logistic model NTCP parameters for ribs with rib fracture as the end point.
NTCP parameters Values
D50 49.8 Gy
γ50 2.05
End point: xerostomia
Xerostomia is a common toxicity following radiotherapy of head and neck tumours which is caused by irradiation of salivary glands such as parotids, submandibular and sublingual glands. Deasy et al. [53] have given a detailed report on the ra- diotherapy dose-volume effects on the salivary glands. NTCPs with xerostomia as an end point has been evaluated and reported for NPC IMRT in the study ex- plained in chapter 8. The parameter sets given in table 3.4 derived by Semenenko
et al. [141] for LKB model is used to calculate the NTCPs of parotids.
Table 3.4: LKB model NTCP parameters for parotids with xerostomia as the end point.
NTCP parameters Values
TD50 31.4 Gy
m 0.54
n 1