Radiotherapy and Oncology, 20 (1991) 166-176 Elsevier
166 RADION 00810
Calculation of normal tissue complication probability and dose-volume histogram reduction schemes for tissues with a critical element architecture Andrzej Niemierko and Michael Goitein Division of Radiation Biophysics, Department of Radiation Medicine, Massachusetts General Hospital and Harvard Medical School Boston, MA, U.S.A.
(Received 22 November 1989; revision received 8 June 1990, accepted 26 November 1990)
Key words: Normal tissue complication probability; Complication probability; Biophysical models; Dose-volume histogram; Tissue architecture; Critical element
Summary We investigate a model of normal tissue complication probability for tissues that may be represented by a critical element architecture. We derive formulas for complication probability that apply to both a partial volume irradiation and to an arbitrary inhomogeneous dose distribution. The dose-volume isoeffect relationship which is a consequence of a critical element architecture is discussed and compared to the empirical power law relationship. A dose-volume histogram reduction scheme for a "pure" critical element model is derived. In addition, a point-based algorithm which does not require precomputation of a dose-volume histogram is derived. The existing published dose-volume histogram reduction algorithms are analyzed. We show that the existing algorithms, developed empirically without an explicit biophysical model, have a close relationship to the critical element model at low levels of complication probability. However, we also show that they have aspects which are not compatible with a critical element model and we propose a modification to one of them to circumvent its restriction to low complication probabilities.
Introduction In current clinical practice the prescriptions for, and evaluations of, radiation treatments are based on statements of the dose distribution within the patient. However, in these contexts dose is really used as a surrogate for the consequences of radiation. In particular, the clinician is implicitly concerned with tumor eradication and treatment morbidity - for which estimates of the tumor control probability (TCP) and normal tissue complication probabilities (NTCP) would be more directly relevant. Recently there has been increasing interest in developing ways of estimating such biophysical end points, both to facilitate the evaluation of complex treatment plans featuring three-dimensional dose
distributions and as tools in developing a scoring function on which plan optimization can be based [1,7-9,11,17,19,23,24,26]. Of course, such TCP and NTCP models can only be as good as the data on which they are based, and the clinical data are sparse and of inconsistent quality [3]. However, used as relative measures of a plan's impact, that is, for comparing rival plans, such models have considerable interest. In this paper we address the estimation of NTCP. A number of models have been proposed for this [7], but it is not always clear upon what biological basis the models were developed and to which tissues they should be expected to apply. Normal tissues and organs differ markedly from one another in their architecture. As Withers at al. [22] pointed out, these differences will
Address for correspondence: Andrzej Niemierko, Ph.D, Department of Radiation Medicine, Massachusetts General Hospital, Boston, MA 02114, U.S.A.
0167-8140/91/$03.50 © 1991 Elsevier Science Publishers B.V. (Biomedical Division)
167 likely result in very different responses to radiation. They suggested a division of tissues into three fundamentally different architectures: (a) Critical element (e.g. spinal cord, nerves, peritoneum): the structure is composed of many functional subunits, and damage to any one will cause a complication expressed by the entire structure. (b) Integral response (e.g. kidney, liver, lung): a complication occurs when a substantial fraction of the functional subunits is damaged. (c) Graded response (e.g. skin, mucosa): radiation reaction occurs on a continuous scale. It is noteworthy that almost all published models of normal tissue complication probability are designed for the critical element architecture [ 1,17,19,24,26 ]. In three dimensional treatment planning there can be a huge amount of data describing dose distributions (typically 5 0 . 5 0 . 3 0 = 75000 points or more). Dosevolume histograms (DVH) provide a useful tool for the quantitative summarization of 3-D dose distributions [ 5,18 ]. DVHs may be used directly for visual evaluation of treatment plans and, by overlaying DVHs, for comparing rival plans. They may also be used as input dosimetric information to models that estimate biological responses to radiation, as was suggested for normal tissues by Lyman [11] and Lyman and Wolbarst [12,13] and was adapted by Kutcher and Burman [9]. Both Lyman and Wolbarst's and Kutcher and Burman's schemes involve reduction of the DVH of the inhomogeneous dose distribution to a one-step DVH with the same probability of complication. Neither DVH reduction algorithm was explicitly developed from a biophysical model of tissue response. Our object has been to calculate tissue complication probability for an inhomogeneously irradiated normal tissue for a specific biological model, namely, the critical element model..We found that the existing DVH reduction schemes [9,11-13] are closely related to the critical element model when the complication probability is small. Methods
The critical element model is based on the assumptions that: (1)an organ consists of a number of identical elements; (2) the response of one element is not correlated with that of any other, and (3) a complication is expressed when one or more elements are incapacitated, that is, every element of the organ is critical. Such elements can be identified with the functional subunit
(FSU) introduced by Withers [22] or the regenerative unit (RU) defined by Archambeau [1] and are related to the tissue rescuing unit (TRU) evolved from studies of Lange and Gilbert [ 10] and refined by Thames and Hendry [20]. Let us assume that the organ consists of Nsuch critical elements (let them be FSUs) and that the probability of injury of a single F S U is equal top. This probability, for a given type of tissue and fractionation scheme, is only a function of the dose D; delivered to the i'th FSU. That is, we assume that FSUs are small enough to neglect inhomogeneity of the dose distribution within an F S U (this assumption will be mentioned again). The dose distribution to the entire organ is represented by the set of the D;, which we write as {Di}. The probability, P*, that the i'th F S U escapes injury is then 1 -p(D~) and, using binomial statistics for uncorrelated events, the complication probability for the entire organ is equal to: N
P(N, {D;}) = 1 - 1-I [1 - p(De)]
(1)
i=l
where the product is taken over all FSUs; that is, over the entire volume of the organ. Equation (1) is a very general one. It takes into account inhomogeneity of the dose distribution and gives the relationship between the probability of complication and the irradiated volume. Schultheiss [17] showed that for an arbitrary inhomogeneous dose distribution defined by the set o f M subvolumes of volume vr within each of which the dose can be considered essentially uniform and equal to D , the complication probability can be expressed as follows: M
P({ Vr}, {Dr}) = 1 - 1-I [1 - P(1, Dr)]vr
(2)
r=l
The computation of normal tissue complication probabilities from DVHs using the critical element model is straightforward. Equation (2) gives the NTCP for an inhomogeneous dose distribution defined by a set of M subvolumes { vr} with corresponding doses {Dr}. This is precisely what a differential DVH constitutes - each bin of the DVH is such a subvolume. Therefore, Eqn. (2) expresses the way in which, in the critical element model, an NTCP is to be calculated for a DVH. It is apparent that Eqn. (2) is independent of the order in which the terms in the product (i.e. the bins of the DVH) are considered. For partial volume irradiation, when only part of an organ (v) is homogeneously irradiated and the rest of the
* We use the symbols NTCP and P equivalently, prefering P in mathematical expressions for compactness.
168 organ receives virtually no dose, Eqn. (2) reduces to the simple form:
P(v,D) = 1 - [1 - P(1, D)]"
(3)
where v is a fraction of the reference volume. The critical element model, per se, says nothing about the dose-response characteristics of individual FSUs (p(D)) nor of the entire organ (P(1, D)). The available clinical and laboratory data suggest that the dose-response relationship is sigmoidal. It can reasonably be described by one of a number of functions. One is the normal probability function [ 11 ]:
l ft:: e-~- dx;
e(1, D) - x / ~ .
We have not done so here mainly for reasons of mathematical simplicity and because the results in terms of the phenomena explored here would not be clinically distinguishable. However, there are important time-dosefractionation issues which need to be addressed in models of the normal tissue response. For such effects, which we do not further address here, one indeed does need a biophysical model of dose response. Discussion
We would first like to emphasize that there is nothing original in our development of the critical element model which has been developed previously by others [10,17,22,24]. Our goal here is to clarify its functional behaviour and its relationship to published DVH reduction schemes.
Functional behaviour of the critical element model t(D)=I(DD O- 1)
(4)
where Dso is the dose which leads to a 50~o complication probability for uniform whole organ irradiation and m is a parameter which governs the slope of the function. The dose-response relationship can equally be represented by the logistic function: 1
P(1, D) =
For the case of partial volume irradiation, Eqn. (3) gives the behaviour of the complication probability as a function of the irradiated volume - as illustrated in Fig. 1 for three levels of dose which correspond to complication probabilities P(1, D) of 0.2, 0.5 and 0.8. For small complication probabilities Eqn. (3) reduces to:
P(v, D) ~- vP(1, D)
(6)
That is for small complication probabilities, the compli-
(5)
l+(Dff) k
where Dso has the same meaning as above, and the parameter k describes the slope of the dose-response curve and, as can easily be shown, is related to the parameter m in Eqn. (4) (for the case when the two functions have the same slope atp = 50 ~o) through the relationship:
cation probability is a linear function of the (uniformly) irradiated volume for constant dose. This is a fundamental property of the critical element model. Any tissue which does not have this linear behaviour, for example, one which exhibits a threshold behaviour with respect 1.0-
0.8'
P(I, D) = 0.2 P(1, D) -- 0.5 P(1, D) = 0.8 I
..... I ................
P(v, D)
k-
x/~m
~
1.6 m
(5a)
These, and several other functional descriptions of dose-response which have been proposed, are clinically indistinguishable. Equations (4) and (5) are phenomenological rather than biophysical representations of the dose-response relationship and, to that extent, our formulation of the critical element model is not a full biophysical model. It is trivial to substitute for Eqn. (5) a biophysical model - such as a linear-quadratic model.
~. "
.....--'~"
...~..-~
0.6'
4
~'~"~
0.40.2' 0.0 0.00
•
,
0.20
•
,
•
0.40
,
0.60
•
,
0.80
•
,
1.00
Partial Volume (v)
Fig. 1. Complication probability as a function o f the irradiated volume at c o n s t a n t dose for partial volume irradiation o f an organ [Eqn. (3)] for three doses leading to complication probabilities o f 20, 50 and 80% when the entire volume is irradiated.
169 to volume at constant dose, can not have a critical element architecture (unless the threshold volume is smaller than the size of a FSU). The N T C P vs. volume relationships at constant dose [Eqns. (2), (3) and (6)] do not depend on the model adopted for dose response. This is not the case, however, for the isoeffect behaviour of complication probability, that is, the graph of dose versus irradiated volume for constant complication probability. To characterize this relationship information about the dose-response characteristics of the tissue or organ is required. For mathematical simplicity we follow Schultheiss [ 17 ] and use the logistic model [Eqn. (5)] in the following development. Using it and Eqn. (3), one can easily show (Appendix A) that the isoeffective dose-volume relationship for partial volume irradiation can be expressed as follows:
D(v) = 05o(Ii+(D(1)'~k~l ; - 1 t ~
(7)
\Dso/d It should be emphasized that D5o is always referred to the entire organ (or some reference volume) and, like k, is a constant parameter that is characteristic of a given organ or tissue and a given endpoint. It is the dose which, if delivered uniformly to the entire organ, would result in a 50~o probability of complication. An important property of the critical element model is that the shape parameter of the dose-volume isoeffect curve is determined by the slope parameter of the doseresponse curve, that is, by the parameter k of the logistic model [Eqn. (5)] or by the parameter m of the normal probability model [Eqn. (4)]. This means that, for these models of dose response, the critical element model contains only two independent parameters, D5o, and one parameter [m in Eqn. (4) or k in Eqn. (5)] which defines both slope of the dose-response characteristic and the slope of the dose-volume isoeffect relationship. Expansion of Eqn. (7) into a Taylor series for small doses (i.e. for D(1) ~ D5o) gives:
O(v) = O(1)
1
v- ~
(8)
This is the familiar power law relationship between isoeffective dose and volume which has been expressed by many authors [e.g. 3,4,9,11,20]. As should be clear from our derivation, it represents the critical element model only for small NTCPs (actually for small values of D(1)/05o, an even more restrictive condition as discussed below). Equations (7) and (8) are compared in Fig. 2 for two levels of complication probability, 0.1 (where they differ little) and 0.5 (where the differences are substantial).
lO00 Isoeffective Dose
(V(v)) (Gy)
~x ••
.........
cndcalelement powe~law
PC1, D)=0.1
..... ~
criticalclement powerlaw
P(1, D) = 0.5
100 .......
. _
10
.01
. . . . . . . .
.'1
;
. . . . . . . .
Partial Volume (v)
Fig. 2. D o s e leading to a constant complication probability as a function o f partial volume irradiation illustrating the deviation o f the p o w e r law dose-volume isoeffect relationship [Eqn. (8)] from that predicted by the critical element model [Eqn. (7)] when the complication probability is larger.
The power law relationship [Eqn. (8)] leads to a straight line on a log-log plot of iso-effectivc dose vs. volume. Deviations from lincarity on such a plot have been reported in animal experiments on brain [2] and spinal cord [25] damage. These tissues are considered probable examples of the critical element architecture [22]. Schulthciss [17] compared both the critical element model and the power law with Berg and Lindgrcn's brain data [2] and showed that their results were well fit by the critical clement model and not by the power law. Van dcr Kogers spinal cord data (private communication and [25]) are shown in Fig. 3 together with our fit to the critical element model [Eqn. (7), solid line] and two fits to the power law model [Eqn. (8)], one de-emphasizing the small volume data (dashed line), the other de-emphasizing the large volume data (dotdashed line). Clearly, neither power law fit is acceptable, and the critical element model is superior. Yacs [26] has suggested that the small volume deviation from the power law fit may be due to the length of the field being comparable to the size of a F S U of the spinal cord
100
• van der Kogcl'sdata - - " powerlawfit(D50 (1)=21.8 k=28.1) ..... power law fit (D 50 (1) = 11.1 k= 1.7) critical ~elementfit (D 50 (1) = 20.1 k = 11)
~. m. k "'"k
DSO
:m_~nm %%
(Gy)
"%%
10
. . . . . . . .
10 u
. . . . . . . .
length of cord (mm)
100 n
Fig. 3. C o m p a r i s o n o f experimental data for the rat spinal cord (van der Kogel, pers. commun, and [25]) with fits to the p o w e r law model [Eqn. (8)] and to the critical element model [Eqn. (7), see text].
170 in the rat. While this is possible, our analysis shows that one does not need to invoke such an explanation; the critical element model predicts just such a deviation from linearity for small volumes. Moreover, the size at which the deviation from linearity occurs (between 5 and 10 mm) seems quite large to represent a single spinal cord FSU in the rat. It is worth observing that, although the critical element model implies the existence of a critical element such as an FSU, the formulation of the model in Eqn. (2) does not contain any information about the FSU. For example, Eqn. (2) has no parameters which depend on the size and dose-response characteristics of a single F S U or on the number of FSUs in a particular organ. This remains true as long as the sub-volumes considered in Eqn. (2) are larger than a single FSU. The experimental data concerning the size of an F S U are modest [ 1,20,21,22,25,26]. However, that size is probably generally smaller than the dimension over which important changes in dose occur (ultimately determined by the size of the radiation beam penumbra).
a histogram could lead to additional inaccuracies in the computation of NTCP. Equation (10a) can be trivially extended to point-based computations where points may represent unequal volumes. If the i'th point represents a relative volume, v;:
P(( z),}) 1 =
-
i= 1
[1+
(lOb)
\Dso} d
Relationship of the critical element model to published D VH reduction algorithms We now discuss three algorithms which have been proposed for calculating NTCPs from DVHs in order to illuminate their relationship to the critical element model and to one another. These algorithms are as follows: Lyman and Wolbarst's original model [11,12] - noted further as LW 1, Lyman and Wolbarst's "preferred" model [13] - LW 2 and Kutcher and Burman's model [9] - which we identify as KB.
Point-based estimate of complication probability
L yman and Wolbarst's algorithms
Our development allows a direct point-based, rather than DVH-based, approach to the calculation of the complication probability for an inhomogeneous dose distribution. Provided the calculational points are distributed uniformly (but not necessarily on a regular grid), each point can be considered to represent a region of fixed volume, and Eqn. (2) can be rewritten as follows:
Lyman and Wolbarst's original DVH reduction algorithm (LW 1) consists of replacing an N step DVH with a simpler N-1 step DVH which would lead to the same complication probability. The procedure is repeated until only one step is left, which corresponds to homogeneous irradiation of the entire volume to an "effective" dose, Deer. For each iteration of the algorithm the two rightmost bins of the cumulative histogram, defined by the pairs of numbers (V1, D1) and (V2, D2) (Fig. 4) are replaced by one-bin (1,'2, Deer) according to the relationship (note the different notation used for partial volume in cumulative histograms (V) and in differential histograms (v):
N
1
P(N, {D,}) = 1 - I-I [1 - P(1, D,)]~
(9)
i=1
where N is now to be understood as the number of calculation points inside the organ and the product is taken over the entire organ. Using the logistic model of dose response, Eqn. (9) can be expressed in the simple form: 1
P(N,{D/})=X-
~ [l+(D;lkl-u kDso} A
i= 1
(lOa)
Expression (10a) can be easily and quickly calculated for any set of points which define a dose distribution. Obviously, a DVH is not necessary to perform these calculations, although it may be convenient. The pointbased approach can provide a mechanism for fast estimation of complication probability using doses calculated at only a few points, not necessarily distributed on a grid in space [ 14], where the process of binning into
P(V2, Deff)= V2- VI p(V2, D2)-I.-V1 P(V2,DI) m
(lla)
The partial volume complication probabilities in Eqn. ( l l a ) are calculated using a model consisting of the normal probability function [Eqn. (4)] and a power law dose-volume relationship for the tolerance dose Dso:
Dso(V) = D5o(1 ) v-"
(11b)
where n is a tissue-specific parameter, obtained by fitting the tolerance dose data as a function of partial volume, v. A subsequent elaboration of this model, LW2,
171 LW 1
LW 2
and
V2=l
The critical element model is defined in terms of differential volumes. For a cumulative DVH Eqn. (13) becomes:
KB
l
P( Vz, Deft) = 1 - [1 - P(1, D2)] v~-v' [1 - P(1, D~)] v' V 2 -- V 1
= 1 - [1 - P(V2, D2)]--W-~
vl
VI
[1 - P(V2, D1)]~ (14)
I
LW D 2 Deft
D
~ Dma x
Fig. 4. A two-bin DVH ( ) and supposedly equivalent "effective" DVHs ( ............. ) obtained using Lyman and Wolbarst's (LW) and Kutcher and Burman's (KB) DVH reduction schemes.
includes: (1) a different interpolation scheme than Eqn. (1 la), i.e. interpolation in the space of the function t(D) [which is defined in Eqn. (4)]; (2) allowance for a critical volume effect, i.e. a minimal volume below which there is no complication regardless of the dose level; and (3)a test to ensure internal consistency of the model, i.e. to force the effective probability, P(V2, Deft) to be intermediate to its two components. For the twobin DVH shown in Fig. 4, the effective dose is the larger of the two doses: De* = D 2 + v 1 (DI De** = v]' D~
Deer=Dso{~-I [i + (D;']l~'i= 1
\Dso/A
i} ~
(15)
For D e < Dso (hence, small probabilities), expansion of the above expression into a Taylor series gives"
Deft =
- D2)
vi (Di)k
k
(16)
i=1
(12)
In this new approach, for the case of partial volume irradiation, one finds that if the dose D*eris greater than ** the complication probability does not depend on Deer, the volume parameter n. If the opposite is true, the complication probability is equal to the complication probability calculated only for the rightmost bin of the DVH, i.e. is totally driven by the dose to the hot spot regardless of the dose to the rest of the organ and size of the hot spot. In both the LW 1 and LW2 models, the parameters n [Eqn. (lib)] and m [Eqn. (4)] are considered to be independent. As noted above, this is inconsistent with the critical element model. For some tissues, LW 2 includes a threshold volume concept (a volume below which the complication probability is equal to zero for any dose delivered to it). Such a threshold, while it may well characterize the response of some tissues, is also incompatible with the critical element model; stochastic effects do not exhibit a threshold behaviour. If we used the critical element model to perform a histogram reduction in the same manner as in LW 1 (expression 1 la), we would obtain from Eqn. (2):
P(V2, Deer) = P ( v 2 + Vl, Deer) = 1 - [1 - P(1, D2)]v: [1 - P(1, D1) ] ~'
It is easy to show, using a linear expansion of Eqn. (14) into a Taylor series, that Eqn. (14) reduces to Lyman and Wolbarst's expression (11) for small probabilities. The equivalent of the dose, Dem which Lyman and Wolbarst obtain through a multi-step procedure can be directly calculated using the critical element formulation (see Appendix A):
(13)
Kutcher and Burman's algorithm Kutcher and Burman's "effective volume" method [9] (KB) transforms a dose volume histogram into a onestep histogram of height veerand dose Dmax somewhat arbitrary chosen to be the maximum dose in the histogram (Fig. 4). Each i'th bin of the differential histogram is transformed to an "equivalent" one at the dose D m a x using the power law relationship:
(veer), = v,
.
(17)
where n is the same parameter as used in both LW~ and L W 2 models [Eqn. (1 lb)]. The effective volume for the entire histogram is the sum of the equivalent volumes:
D i ~! -
Veff= Z(Vefr)il = ~.• Vi \ D m a x , /
(18)
The complication probability is then calculated using Lyman's model for a uniformly irradiated partial volume veer [Eqns. (4) and (llb)]. The critical element formalism [Eqn. (2)] can be
172 used to determine the effective volume, in parallel with Kutcher and Burman's approach, and gives the following formula (Appendix B): In [ 1 - P(1, D;)] veff = ,~. V~ln [1 - e(1, Dmax) ]
(19)
We show in Appendix B that Eqn. (19) reduces to Kutcher and Burman's Eqn. (17) for small complication probabilities. Relationship between L yman and Wolbarst's and Kutcher and Burman's algorithms for small probabilities of complication The effective volume obtained by Kutcher and Burman's formula (18) can be written as follows:
Z v~(D,) k ve~ut cher --
i
k Omax
from which, using Eqn. (16), one can write: Kutcher
1"-FlLyman-[ k ~tJeff
(20)
This, obviously, is the power law relationship between dose and volume for constant complication probability and shows the relationship, at low levels of complication, between the LW~ and KB histogram reduction techniques, which appear, superficially, to be unrelated. Equation (20) holds because both histogram reduction techniques incorporate the power law dose-volume relationship. Relationship between the critical element model and the D VH reduction schemes Equations (18) and (20) are low probability approximations, stemming from the critical element model and the use of the logistic function to describe the doseresponse relationship. The general relationship, based on the critical element model without specification of the dose-response characteristics, which holds for any probability, is given by Eqn. (19) where D m a x C a n be understood as any reference dose. Expression (20) gives the relationship (for small probabilities) between the LW 1 and KB histogram reduction techniques. There is, however, an important
difference between these two empirical schemes. The KB algorithm is formulated in such a way that it would be trivial to incorporate the "correct" (for a critical element architecture) dose-volume relationship expressed in Eqn. (19). This is not true for either the LW l, and LW 2 algorithms which, in addition to the power law relationship, use linear interpolation in the space of complication probabilities [Eqn. (lla)] for LWl, or in the space of dose [Eqn. (12)] for LW 2. A basic property of the critical element architecture is the dependence of the complication probability on the product of probabilities calculated for subvolumes. It is easy to see that, in all three DVH reduction algorithms [9,11-13], the total probability is a weighted sum of probabilities calculated for each bin of the histogram. Although, for small probabilities, the product can be well approximated by a sum, summation does not correctly describe the critical dement architecture and falls for large complication probabilities. This can be seen in Fig. 5, in which we show the effective volume and complication probabilities calculated for the twostep DVH shown in Fig. 4 for the parameters: m = 0.16 and Dso = 66.5 Gy - which roughly correspond to spinal cord dose-response data [3] - and n ( -- 1/k) is chosen to obey Eqn. (5a), in order to respect the critical element relationship between the dose-volume and dose-response relationships, and therefore has a value of m/1.6 = 0.1. Figure 5B corresponds to small complication probabilities and Fig. 5C to large complication probabilities. The calculations presented in Fig. 5 confirm that both the LW 1 and KB reduction schemes converge to the critical element model for small complication probabilities but differ for large probabilities. All models except LW 2 exhibit linear behaviour of the complication probability as a function of the size of the hot spot for small probabilities (Fig. 5B). The LW 1 algorithm is also linear for large probabilities, as opposed to the other algorithms (Fig. 5C). In these examples, the critical element model predicts a larger effective volume (Fig. 5A), but smaller complication probability (Fig. 5C), than the L W 2 and KB algorithms, which in turn give similar results for large probabilities but differ for small complication probabilities. We have made several references to the fact that aspects of LW~ and KB histogram reduction schemes closely parallel the critical element model for small probabilities. Since large complication probabilities are assiduously avoided in clinical practice, one might ask whether the distinction is important. However, the small probability limit must be understood in a rather special sense. It is often invoked through the requirement that D(1) ~ Dso - as in the derivation of Eqns. (8), (16) and (20). This requirement indeed leads to small
173
~
Effective Volume
m
ent }D(1) Os0
u 1
,1
KB
I .001 .001
volume of hot spot (v)
........
critical element
I
.
.
.
.
.
.
.
.
.01
I
!
,1
volume of hot spot (v) 0.07 0.06 -
NTCP
small N
T
C
~
~
~
_B)
s J
,,.,"
0.04 -
..sS s
s
NTCP ~-
0.03 -
-[] •
0.02 -
critical element
KB LWI
•
0.01
s/SS/'SSSSS~'S
0.05' 0.00
0.0
0.4
012
0.6
0.8
1.0
~
volume of hot spot (v) 0.00
0.00
e
n
---
L W ( l o r 2 ) or KB
t
I
I
I
I
I
0.02
0.04
0.06
0.08
0.10
volume of hot spot (v)
0.7 NTCP
B)
,.,"
0.15'
0.05 -
large NTCP's
lil~
o.~ 0.5 0.4
~
•
,,/ 0.3
C)
-
"[]
[~/•
0.2
t-1 / t..a~[P" • 0.1 ~ l
•
•
cridcalelement KB LW1 LW2
0.0 0.0
012
Fig. 6. Effective volume and normal tissue complication probability as a function of the size of a small volume irradiated to a high dose (D1 = 70 Gy) while the rest of the organ receives only a small dose (D2 = 10 Gy), (Dso = 66.5 Gy, m = 0.16, n = 0.1).
0'.4
016
018
110
volume of hot spot (v) Fig. 5(A) Effective volume of a two-step histogram as a function of the volume of the hot spot, calculated using the critical element model [Eqn• (19)] and the power law [Eqn. (18)] for 0.6 and 1.05; (B) normal tissue complication probability as a function of the volume of the hot spot for the four analyzed algorithms, for small probabilities (Dr = 40 Gy, D2 = 30 Gy, Dso = 66.5 Gy, m = 0 . 1 6 , n = 0 . 1 ) ; (C) as (B) but for large probabilities (Dr = 70 Gy, D2 = 50 Gy, Dso = 66.5 Gy, m = 0.16, n = 0.1)2
D(1)/Dso=
probabilities but is in fact more restrictive. The requirement is often not satisfied, for example, when there is a small hot spot. Figure 6 shows the effective volume and the complication probability for a case with a small hot spot (the dose to the hot spot is equal to 70 Gy and of course does not satisfy requirement that D(1)
166 RADION 00810
Calculation of normal tissue complication probability and dose-volume histogram reduction schemes for tissues with a critical element architecture Andrzej Niemierko and Michael Goitein Division of Radiation Biophysics, Department of Radiation Medicine, Massachusetts General Hospital and Harvard Medical School Boston, MA, U.S.A.
(Received 22 November 1989; revision received 8 June 1990, accepted 26 November 1990)
Key words: Normal tissue complication probability; Complication probability; Biophysical models; Dose-volume histogram; Tissue architecture; Critical element
Summary We investigate a model of normal tissue complication probability for tissues that may be represented by a critical element architecture. We derive formulas for complication probability that apply to both a partial volume irradiation and to an arbitrary inhomogeneous dose distribution. The dose-volume isoeffect relationship which is a consequence of a critical element architecture is discussed and compared to the empirical power law relationship. A dose-volume histogram reduction scheme for a "pure" critical element model is derived. In addition, a point-based algorithm which does not require precomputation of a dose-volume histogram is derived. The existing published dose-volume histogram reduction algorithms are analyzed. We show that the existing algorithms, developed empirically without an explicit biophysical model, have a close relationship to the critical element model at low levels of complication probability. However, we also show that they have aspects which are not compatible with a critical element model and we propose a modification to one of them to circumvent its restriction to low complication probabilities.
Introduction In current clinical practice the prescriptions for, and evaluations of, radiation treatments are based on statements of the dose distribution within the patient. However, in these contexts dose is really used as a surrogate for the consequences of radiation. In particular, the clinician is implicitly concerned with tumor eradication and treatment morbidity - for which estimates of the tumor control probability (TCP) and normal tissue complication probabilities (NTCP) would be more directly relevant. Recently there has been increasing interest in developing ways of estimating such biophysical end points, both to facilitate the evaluation of complex treatment plans featuring three-dimensional dose
distributions and as tools in developing a scoring function on which plan optimization can be based [1,7-9,11,17,19,23,24,26]. Of course, such TCP and NTCP models can only be as good as the data on which they are based, and the clinical data are sparse and of inconsistent quality [3]. However, used as relative measures of a plan's impact, that is, for comparing rival plans, such models have considerable interest. In this paper we address the estimation of NTCP. A number of models have been proposed for this [7], but it is not always clear upon what biological basis the models were developed and to which tissues they should be expected to apply. Normal tissues and organs differ markedly from one another in their architecture. As Withers at al. [22] pointed out, these differences will
Address for correspondence: Andrzej Niemierko, Ph.D, Department of Radiation Medicine, Massachusetts General Hospital, Boston, MA 02114, U.S.A.
0167-8140/91/$03.50 © 1991 Elsevier Science Publishers B.V. (Biomedical Division)
167 likely result in very different responses to radiation. They suggested a division of tissues into three fundamentally different architectures: (a) Critical element (e.g. spinal cord, nerves, peritoneum): the structure is composed of many functional subunits, and damage to any one will cause a complication expressed by the entire structure. (b) Integral response (e.g. kidney, liver, lung): a complication occurs when a substantial fraction of the functional subunits is damaged. (c) Graded response (e.g. skin, mucosa): radiation reaction occurs on a continuous scale. It is noteworthy that almost all published models of normal tissue complication probability are designed for the critical element architecture [ 1,17,19,24,26 ]. In three dimensional treatment planning there can be a huge amount of data describing dose distributions (typically 5 0 . 5 0 . 3 0 = 75000 points or more). Dosevolume histograms (DVH) provide a useful tool for the quantitative summarization of 3-D dose distributions [ 5,18 ]. DVHs may be used directly for visual evaluation of treatment plans and, by overlaying DVHs, for comparing rival plans. They may also be used as input dosimetric information to models that estimate biological responses to radiation, as was suggested for normal tissues by Lyman [11] and Lyman and Wolbarst [12,13] and was adapted by Kutcher and Burman [9]. Both Lyman and Wolbarst's and Kutcher and Burman's schemes involve reduction of the DVH of the inhomogeneous dose distribution to a one-step DVH with the same probability of complication. Neither DVH reduction algorithm was explicitly developed from a biophysical model of tissue response. Our object has been to calculate tissue complication probability for an inhomogeneously irradiated normal tissue for a specific biological model, namely, the critical element model..We found that the existing DVH reduction schemes [9,11-13] are closely related to the critical element model when the complication probability is small. Methods
The critical element model is based on the assumptions that: (1)an organ consists of a number of identical elements; (2) the response of one element is not correlated with that of any other, and (3) a complication is expressed when one or more elements are incapacitated, that is, every element of the organ is critical. Such elements can be identified with the functional subunit
(FSU) introduced by Withers [22] or the regenerative unit (RU) defined by Archambeau [1] and are related to the tissue rescuing unit (TRU) evolved from studies of Lange and Gilbert [ 10] and refined by Thames and Hendry [20]. Let us assume that the organ consists of Nsuch critical elements (let them be FSUs) and that the probability of injury of a single F S U is equal top. This probability, for a given type of tissue and fractionation scheme, is only a function of the dose D; delivered to the i'th FSU. That is, we assume that FSUs are small enough to neglect inhomogeneity of the dose distribution within an F S U (this assumption will be mentioned again). The dose distribution to the entire organ is represented by the set of the D;, which we write as {Di}. The probability, P*, that the i'th F S U escapes injury is then 1 -p(D~) and, using binomial statistics for uncorrelated events, the complication probability for the entire organ is equal to: N
P(N, {D;}) = 1 - 1-I [1 - p(De)]
(1)
i=l
where the product is taken over all FSUs; that is, over the entire volume of the organ. Equation (1) is a very general one. It takes into account inhomogeneity of the dose distribution and gives the relationship between the probability of complication and the irradiated volume. Schultheiss [17] showed that for an arbitrary inhomogeneous dose distribution defined by the set o f M subvolumes of volume vr within each of which the dose can be considered essentially uniform and equal to D , the complication probability can be expressed as follows: M
P({ Vr}, {Dr}) = 1 - 1-I [1 - P(1, Dr)]vr
(2)
r=l
The computation of normal tissue complication probabilities from DVHs using the critical element model is straightforward. Equation (2) gives the NTCP for an inhomogeneous dose distribution defined by a set of M subvolumes { vr} with corresponding doses {Dr}. This is precisely what a differential DVH constitutes - each bin of the DVH is such a subvolume. Therefore, Eqn. (2) expresses the way in which, in the critical element model, an NTCP is to be calculated for a DVH. It is apparent that Eqn. (2) is independent of the order in which the terms in the product (i.e. the bins of the DVH) are considered. For partial volume irradiation, when only part of an organ (v) is homogeneously irradiated and the rest of the
* We use the symbols NTCP and P equivalently, prefering P in mathematical expressions for compactness.
168 organ receives virtually no dose, Eqn. (2) reduces to the simple form:
P(v,D) = 1 - [1 - P(1, D)]"
(3)
where v is a fraction of the reference volume. The critical element model, per se, says nothing about the dose-response characteristics of individual FSUs (p(D)) nor of the entire organ (P(1, D)). The available clinical and laboratory data suggest that the dose-response relationship is sigmoidal. It can reasonably be described by one of a number of functions. One is the normal probability function [ 11 ]:
l ft:: e-~- dx;
e(1, D) - x / ~ .
We have not done so here mainly for reasons of mathematical simplicity and because the results in terms of the phenomena explored here would not be clinically distinguishable. However, there are important time-dosefractionation issues which need to be addressed in models of the normal tissue response. For such effects, which we do not further address here, one indeed does need a biophysical model of dose response. Discussion
We would first like to emphasize that there is nothing original in our development of the critical element model which has been developed previously by others [10,17,22,24]. Our goal here is to clarify its functional behaviour and its relationship to published DVH reduction schemes.
Functional behaviour of the critical element model t(D)=I(DD O- 1)
(4)
where Dso is the dose which leads to a 50~o complication probability for uniform whole organ irradiation and m is a parameter which governs the slope of the function. The dose-response relationship can equally be represented by the logistic function: 1
P(1, D) =
For the case of partial volume irradiation, Eqn. (3) gives the behaviour of the complication probability as a function of the irradiated volume - as illustrated in Fig. 1 for three levels of dose which correspond to complication probabilities P(1, D) of 0.2, 0.5 and 0.8. For small complication probabilities Eqn. (3) reduces to:
P(v, D) ~- vP(1, D)
(6)
That is for small complication probabilities, the compli-
(5)
l+(Dff) k
where Dso has the same meaning as above, and the parameter k describes the slope of the dose-response curve and, as can easily be shown, is related to the parameter m in Eqn. (4) (for the case when the two functions have the same slope atp = 50 ~o) through the relationship:
cation probability is a linear function of the (uniformly) irradiated volume for constant dose. This is a fundamental property of the critical element model. Any tissue which does not have this linear behaviour, for example, one which exhibits a threshold behaviour with respect 1.0-
0.8'
P(I, D) = 0.2 P(1, D) -- 0.5 P(1, D) = 0.8 I
..... I ................
P(v, D)
k-
x/~m
~
1.6 m
(5a)
These, and several other functional descriptions of dose-response which have been proposed, are clinically indistinguishable. Equations (4) and (5) are phenomenological rather than biophysical representations of the dose-response relationship and, to that extent, our formulation of the critical element model is not a full biophysical model. It is trivial to substitute for Eqn. (5) a biophysical model - such as a linear-quadratic model.
~. "
.....--'~"
...~..-~
0.6'
4
~'~"~
0.40.2' 0.0 0.00
•
,
0.20
•
,
•
0.40
,
0.60
•
,
0.80
•
,
1.00
Partial Volume (v)
Fig. 1. Complication probability as a function o f the irradiated volume at c o n s t a n t dose for partial volume irradiation o f an organ [Eqn. (3)] for three doses leading to complication probabilities o f 20, 50 and 80% when the entire volume is irradiated.
169 to volume at constant dose, can not have a critical element architecture (unless the threshold volume is smaller than the size of a FSU). The N T C P vs. volume relationships at constant dose [Eqns. (2), (3) and (6)] do not depend on the model adopted for dose response. This is not the case, however, for the isoeffect behaviour of complication probability, that is, the graph of dose versus irradiated volume for constant complication probability. To characterize this relationship information about the dose-response characteristics of the tissue or organ is required. For mathematical simplicity we follow Schultheiss [ 17 ] and use the logistic model [Eqn. (5)] in the following development. Using it and Eqn. (3), one can easily show (Appendix A) that the isoeffective dose-volume relationship for partial volume irradiation can be expressed as follows:
D(v) = 05o(Ii+(D(1)'~k~l ; - 1 t ~
(7)
\Dso/d It should be emphasized that D5o is always referred to the entire organ (or some reference volume) and, like k, is a constant parameter that is characteristic of a given organ or tissue and a given endpoint. It is the dose which, if delivered uniformly to the entire organ, would result in a 50~o probability of complication. An important property of the critical element model is that the shape parameter of the dose-volume isoeffect curve is determined by the slope parameter of the doseresponse curve, that is, by the parameter k of the logistic model [Eqn. (5)] or by the parameter m of the normal probability model [Eqn. (4)]. This means that, for these models of dose response, the critical element model contains only two independent parameters, D5o, and one parameter [m in Eqn. (4) or k in Eqn. (5)] which defines both slope of the dose-response characteristic and the slope of the dose-volume isoeffect relationship. Expansion of Eqn. (7) into a Taylor series for small doses (i.e. for D(1) ~ D5o) gives:
O(v) = O(1)
1
v- ~
(8)
This is the familiar power law relationship between isoeffective dose and volume which has been expressed by many authors [e.g. 3,4,9,11,20]. As should be clear from our derivation, it represents the critical element model only for small NTCPs (actually for small values of D(1)/05o, an even more restrictive condition as discussed below). Equations (7) and (8) are compared in Fig. 2 for two levels of complication probability, 0.1 (where they differ little) and 0.5 (where the differences are substantial).
lO00 Isoeffective Dose
(V(v)) (Gy)
~x ••
.........
cndcalelement powe~law
PC1, D)=0.1
..... ~
criticalclement powerlaw
P(1, D) = 0.5
100 .......
. _
10
.01
. . . . . . . .
.'1
;
. . . . . . . .
Partial Volume (v)
Fig. 2. D o s e leading to a constant complication probability as a function o f partial volume irradiation illustrating the deviation o f the p o w e r law dose-volume isoeffect relationship [Eqn. (8)] from that predicted by the critical element model [Eqn. (7)] when the complication probability is larger.
The power law relationship [Eqn. (8)] leads to a straight line on a log-log plot of iso-effectivc dose vs. volume. Deviations from lincarity on such a plot have been reported in animal experiments on brain [2] and spinal cord [25] damage. These tissues are considered probable examples of the critical element architecture [22]. Schulthciss [17] compared both the critical element model and the power law with Berg and Lindgrcn's brain data [2] and showed that their results were well fit by the critical clement model and not by the power law. Van dcr Kogers spinal cord data (private communication and [25]) are shown in Fig. 3 together with our fit to the critical element model [Eqn. (7), solid line] and two fits to the power law model [Eqn. (8)], one de-emphasizing the small volume data (dashed line), the other de-emphasizing the large volume data (dotdashed line). Clearly, neither power law fit is acceptable, and the critical element model is superior. Yacs [26] has suggested that the small volume deviation from the power law fit may be due to the length of the field being comparable to the size of a F S U of the spinal cord
100
• van der Kogcl'sdata - - " powerlawfit(D50 (1)=21.8 k=28.1) ..... power law fit (D 50 (1) = 11.1 k= 1.7) critical ~elementfit (D 50 (1) = 20.1 k = 11)
~. m. k "'"k
DSO
:m_~nm %%
(Gy)
"%%
10
. . . . . . . .
10 u
. . . . . . . .
length of cord (mm)
100 n
Fig. 3. C o m p a r i s o n o f experimental data for the rat spinal cord (van der Kogel, pers. commun, and [25]) with fits to the p o w e r law model [Eqn. (8)] and to the critical element model [Eqn. (7), see text].
170 in the rat. While this is possible, our analysis shows that one does not need to invoke such an explanation; the critical element model predicts just such a deviation from linearity for small volumes. Moreover, the size at which the deviation from linearity occurs (between 5 and 10 mm) seems quite large to represent a single spinal cord FSU in the rat. It is worth observing that, although the critical element model implies the existence of a critical element such as an FSU, the formulation of the model in Eqn. (2) does not contain any information about the FSU. For example, Eqn. (2) has no parameters which depend on the size and dose-response characteristics of a single F S U or on the number of FSUs in a particular organ. This remains true as long as the sub-volumes considered in Eqn. (2) are larger than a single FSU. The experimental data concerning the size of an F S U are modest [ 1,20,21,22,25,26]. However, that size is probably generally smaller than the dimension over which important changes in dose occur (ultimately determined by the size of the radiation beam penumbra).
a histogram could lead to additional inaccuracies in the computation of NTCP. Equation (10a) can be trivially extended to point-based computations where points may represent unequal volumes. If the i'th point represents a relative volume, v;:
P(( z),}) 1 =
-
i= 1
[1+
(lOb)
\Dso} d
Relationship of the critical element model to published D VH reduction algorithms We now discuss three algorithms which have been proposed for calculating NTCPs from DVHs in order to illuminate their relationship to the critical element model and to one another. These algorithms are as follows: Lyman and Wolbarst's original model [11,12] - noted further as LW 1, Lyman and Wolbarst's "preferred" model [13] - LW 2 and Kutcher and Burman's model [9] - which we identify as KB.
Point-based estimate of complication probability
L yman and Wolbarst's algorithms
Our development allows a direct point-based, rather than DVH-based, approach to the calculation of the complication probability for an inhomogeneous dose distribution. Provided the calculational points are distributed uniformly (but not necessarily on a regular grid), each point can be considered to represent a region of fixed volume, and Eqn. (2) can be rewritten as follows:
Lyman and Wolbarst's original DVH reduction algorithm (LW 1) consists of replacing an N step DVH with a simpler N-1 step DVH which would lead to the same complication probability. The procedure is repeated until only one step is left, which corresponds to homogeneous irradiation of the entire volume to an "effective" dose, Deer. For each iteration of the algorithm the two rightmost bins of the cumulative histogram, defined by the pairs of numbers (V1, D1) and (V2, D2) (Fig. 4) are replaced by one-bin (1,'2, Deer) according to the relationship (note the different notation used for partial volume in cumulative histograms (V) and in differential histograms (v):
N
1
P(N, {D,}) = 1 - I-I [1 - P(1, D,)]~
(9)
i=1
where N is now to be understood as the number of calculation points inside the organ and the product is taken over the entire organ. Using the logistic model of dose response, Eqn. (9) can be expressed in the simple form: 1
P(N,{D/})=X-
~ [l+(D;lkl-u kDso} A
i= 1
(lOa)
Expression (10a) can be easily and quickly calculated for any set of points which define a dose distribution. Obviously, a DVH is not necessary to perform these calculations, although it may be convenient. The pointbased approach can provide a mechanism for fast estimation of complication probability using doses calculated at only a few points, not necessarily distributed on a grid in space [ 14], where the process of binning into
P(V2, Deff)= V2- VI p(V2, D2)-I.-V1 P(V2,DI) m
(lla)
The partial volume complication probabilities in Eqn. ( l l a ) are calculated using a model consisting of the normal probability function [Eqn. (4)] and a power law dose-volume relationship for the tolerance dose Dso:
Dso(V) = D5o(1 ) v-"
(11b)
where n is a tissue-specific parameter, obtained by fitting the tolerance dose data as a function of partial volume, v. A subsequent elaboration of this model, LW2,
171 LW 1
LW 2
and
V2=l
The critical element model is defined in terms of differential volumes. For a cumulative DVH Eqn. (13) becomes:
KB
l
P( Vz, Deft) = 1 - [1 - P(1, D2)] v~-v' [1 - P(1, D~)] v' V 2 -- V 1
= 1 - [1 - P(V2, D2)]--W-~
vl
VI
[1 - P(V2, D1)]~ (14)
I
LW D 2 Deft
D
~ Dma x
Fig. 4. A two-bin DVH ( ) and supposedly equivalent "effective" DVHs ( ............. ) obtained using Lyman and Wolbarst's (LW) and Kutcher and Burman's (KB) DVH reduction schemes.
includes: (1) a different interpolation scheme than Eqn. (1 la), i.e. interpolation in the space of the function t(D) [which is defined in Eqn. (4)]; (2) allowance for a critical volume effect, i.e. a minimal volume below which there is no complication regardless of the dose level; and (3)a test to ensure internal consistency of the model, i.e. to force the effective probability, P(V2, Deft) to be intermediate to its two components. For the twobin DVH shown in Fig. 4, the effective dose is the larger of the two doses: De* = D 2 + v 1 (DI De** = v]' D~
Deer=Dso{~-I [i + (D;']l~'i= 1
\Dso/A
i} ~
(15)
For D e < Dso (hence, small probabilities), expansion of the above expression into a Taylor series gives"
Deft =
- D2)
vi (Di)k
k
(16)
i=1
(12)
In this new approach, for the case of partial volume irradiation, one finds that if the dose D*eris greater than ** the complication probability does not depend on Deer, the volume parameter n. If the opposite is true, the complication probability is equal to the complication probability calculated only for the rightmost bin of the DVH, i.e. is totally driven by the dose to the hot spot regardless of the dose to the rest of the organ and size of the hot spot. In both the LW 1 and LW2 models, the parameters n [Eqn. (lib)] and m [Eqn. (4)] are considered to be independent. As noted above, this is inconsistent with the critical element model. For some tissues, LW 2 includes a threshold volume concept (a volume below which the complication probability is equal to zero for any dose delivered to it). Such a threshold, while it may well characterize the response of some tissues, is also incompatible with the critical element model; stochastic effects do not exhibit a threshold behaviour. If we used the critical element model to perform a histogram reduction in the same manner as in LW 1 (expression 1 la), we would obtain from Eqn. (2):
P(V2, Deer) = P ( v 2 + Vl, Deer) = 1 - [1 - P(1, D2)]v: [1 - P(1, D1) ] ~'
It is easy to show, using a linear expansion of Eqn. (14) into a Taylor series, that Eqn. (14) reduces to Lyman and Wolbarst's expression (11) for small probabilities. The equivalent of the dose, Dem which Lyman and Wolbarst obtain through a multi-step procedure can be directly calculated using the critical element formulation (see Appendix A):
(13)
Kutcher and Burman's algorithm Kutcher and Burman's "effective volume" method [9] (KB) transforms a dose volume histogram into a onestep histogram of height veerand dose Dmax somewhat arbitrary chosen to be the maximum dose in the histogram (Fig. 4). Each i'th bin of the differential histogram is transformed to an "equivalent" one at the dose D m a x using the power law relationship:
(veer), = v,
.
(17)
where n is the same parameter as used in both LW~ and L W 2 models [Eqn. (1 lb)]. The effective volume for the entire histogram is the sum of the equivalent volumes:
D i ~! -
Veff= Z(Vefr)il = ~.• Vi \ D m a x , /
(18)
The complication probability is then calculated using Lyman's model for a uniformly irradiated partial volume veer [Eqns. (4) and (llb)]. The critical element formalism [Eqn. (2)] can be
172 used to determine the effective volume, in parallel with Kutcher and Burman's approach, and gives the following formula (Appendix B): In [ 1 - P(1, D;)] veff = ,~. V~ln [1 - e(1, Dmax) ]
(19)
We show in Appendix B that Eqn. (19) reduces to Kutcher and Burman's Eqn. (17) for small complication probabilities. Relationship between L yman and Wolbarst's and Kutcher and Burman's algorithms for small probabilities of complication The effective volume obtained by Kutcher and Burman's formula (18) can be written as follows:
Z v~(D,) k ve~ut cher --
i
k Omax
from which, using Eqn. (16), one can write: Kutcher
1"-FlLyman-[ k ~tJeff
(20)
This, obviously, is the power law relationship between dose and volume for constant complication probability and shows the relationship, at low levels of complication, between the LW~ and KB histogram reduction techniques, which appear, superficially, to be unrelated. Equation (20) holds because both histogram reduction techniques incorporate the power law dose-volume relationship. Relationship between the critical element model and the D VH reduction schemes Equations (18) and (20) are low probability approximations, stemming from the critical element model and the use of the logistic function to describe the doseresponse relationship. The general relationship, based on the critical element model without specification of the dose-response characteristics, which holds for any probability, is given by Eqn. (19) where D m a x C a n be understood as any reference dose. Expression (20) gives the relationship (for small probabilities) between the LW 1 and KB histogram reduction techniques. There is, however, an important
difference between these two empirical schemes. The KB algorithm is formulated in such a way that it would be trivial to incorporate the "correct" (for a critical element architecture) dose-volume relationship expressed in Eqn. (19). This is not true for either the LW l, and LW 2 algorithms which, in addition to the power law relationship, use linear interpolation in the space of complication probabilities [Eqn. (lla)] for LWl, or in the space of dose [Eqn. (12)] for LW 2. A basic property of the critical element architecture is the dependence of the complication probability on the product of probabilities calculated for subvolumes. It is easy to see that, in all three DVH reduction algorithms [9,11-13], the total probability is a weighted sum of probabilities calculated for each bin of the histogram. Although, for small probabilities, the product can be well approximated by a sum, summation does not correctly describe the critical dement architecture and falls for large complication probabilities. This can be seen in Fig. 5, in which we show the effective volume and complication probabilities calculated for the twostep DVH shown in Fig. 4 for the parameters: m = 0.16 and Dso = 66.5 Gy - which roughly correspond to spinal cord dose-response data [3] - and n ( -- 1/k) is chosen to obey Eqn. (5a), in order to respect the critical element relationship between the dose-volume and dose-response relationships, and therefore has a value of m/1.6 = 0.1. Figure 5B corresponds to small complication probabilities and Fig. 5C to large complication probabilities. The calculations presented in Fig. 5 confirm that both the LW 1 and KB reduction schemes converge to the critical element model for small complication probabilities but differ for large probabilities. All models except LW 2 exhibit linear behaviour of the complication probability as a function of the size of the hot spot for small probabilities (Fig. 5B). The LW 1 algorithm is also linear for large probabilities, as opposed to the other algorithms (Fig. 5C). In these examples, the critical element model predicts a larger effective volume (Fig. 5A), but smaller complication probability (Fig. 5C), than the L W 2 and KB algorithms, which in turn give similar results for large probabilities but differ for small complication probabilities. We have made several references to the fact that aspects of LW~ and KB histogram reduction schemes closely parallel the critical element model for small probabilities. Since large complication probabilities are assiduously avoided in clinical practice, one might ask whether the distinction is important. However, the small probability limit must be understood in a rather special sense. It is often invoked through the requirement that D(1) ~ Dso - as in the derivation of Eqns. (8), (16) and (20). This requirement indeed leads to small
173
~
Effective Volume
m
ent }D(1) Os0
u 1
,1
KB
I .001 .001
volume of hot spot (v)
........
critical element
I
.
.
.
.
.
.
.
.
.01
I
!
,1
volume of hot spot (v) 0.07 0.06 -
NTCP
small N
T
C
~
~
~
_B)
s J
,,.,"
0.04 -
..sS s
s
NTCP ~-
0.03 -
-[] •
0.02 -
critical element
KB LWI
•
0.01
s/SS/'SSSSS~'S
0.05' 0.00
0.0
0.4
012
0.6
0.8
1.0
~
volume of hot spot (v) 0.00
0.00
e
n
---
L W ( l o r 2 ) or KB
t
I
I
I
I
I
0.02
0.04
0.06
0.08
0.10
volume of hot spot (v)
0.7 NTCP
B)
,.,"
0.15'
0.05 -
large NTCP's
lil~
o.~ 0.5 0.4
~
•
,,/ 0.3
C)
-
"[]
[~/•
0.2
t-1 / t..a~[P" • 0.1 ~ l
•
•
cridcalelement KB LW1 LW2
0.0 0.0
012
Fig. 6. Effective volume and normal tissue complication probability as a function of the size of a small volume irradiated to a high dose (D1 = 70 Gy) while the rest of the organ receives only a small dose (D2 = 10 Gy), (Dso = 66.5 Gy, m = 0.16, n = 0.1).
0'.4
016
018
110
volume of hot spot (v) Fig. 5(A) Effective volume of a two-step histogram as a function of the volume of the hot spot, calculated using the critical element model [Eqn• (19)] and the power law [Eqn. (18)] for 0.6 and 1.05; (B) normal tissue complication probability as a function of the volume of the hot spot for the four analyzed algorithms, for small probabilities (Dr = 40 Gy, D2 = 30 Gy, Dso = 66.5 Gy, m = 0 . 1 6 , n = 0 . 1 ) ; (C) as (B) but for large probabilities (Dr = 70 Gy, D2 = 50 Gy, Dso = 66.5 Gy, m = 0.16, n = 0.1)2
D(1)/Dso=
probabilities but is in fact more restrictive. The requirement is often not satisfied, for example, when there is a small hot spot. Figure 6 shows the effective volume and the complication probability for a case with a small hot spot (the dose to the hot spot is equal to 70 Gy and of course does not satisfy requirement that D(1)
