Analysis of clinical complication data for radiation hepatitis using a parallel architecture model

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Int. .I. Radiation Oncology Bwl. Phyb., Vol. 31, No. 4. pp. 883-X9 I, 199.5 Copyright 0 1995 Elscvier Sc~encc Ltd Printed in the USA. All rights reserved

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l Physics Original Contribution

ANALYSIS OF CLINICAL COMPLICATION DATA FOR RADIATION HEPATITIS USING A PARALLEL ARCHITECTURE MODEL

A. JACKSON, PH.D.,* R. K. TEN HAKEN, PH.D.,-’ J. M. ROBERTSON, M.D.,+ M. L. KESSLER, PH.D.,+ G. J. KUTCHER, PH.D.* AND T. S. LAWRENCE, M.D., PH.D.+

* Memorial Sloan-Kettering Cancer Center, New York, NY 10021; and ‘University of Michigan Medical Center, Anne Arbor, MI 48109

Purpose: The detailed knowledge of dose volume distributions available from the three-dimensional (3D) conformal radiation treatment of tumors in the liver (reported elsewhere) offers new opportunities to quantify the effect of volume on the probability of producing radiation hepatitis. We aim to test a new parallel architecture model of normal tissue complication probability (NTCP) with these data. Methods and Materials: Complication data and dose volume histograms from a total of 93 patients with normal liver function, treated on a prospective protocol with 3D conformal radiation therapy and intraarterial hepatic fluorodeoxyuridine, were analyzed with a new parallel architecture model. Patient treatment fell into six categories differing in doses delivered and volumes irradiated. By modeling the radiosensitivity of liver subunits, we are able to use dose volume histograms to calculate the fraction of the liver damaged in each patient. A complication results if this fraction exceeds the patient’s functional reserve. To determine the patient distribution of functional reserves and the subunit radiosensitivity, the maximum likelihood method was used to fit the observed complication data. Results: The parallel model fit the complication data well, although uncertainties on the functional reserve distribution and subunit radiosensitivy are highly correlated. Conclusion: The observed radiation hepatitis complications show a threshold effect that can be described well with a parallel architecture model. However, additional independent studies are required to better determine the parameters defining the functional reserve distribution and subunit radiosensitivity.

Liver, Radiation hepatitis, NTCP, Parallel architecture, Dose volume histograms, Volume effect.

INTRODUCTION effect in normal organs is to test the predictions of models

The volume effect is defined as the way tolerance doses and parameterizations of NTCP against the available data.

change with the partial volume of normal organs irradi- Recently, methods of calculating NTCP under conditions

ated during radiation therapy (8, 14). It is one of the key of inhomogeneous irradiation have been described (13,

elements in successful treatment planning, and the ability 26, 34) for tissues with parallel architecture (33, 34). The

to exploit it for dose limiting normal tissues is a major parallel architecture model is thought to be applicable to

premise of conformal radiation therapy and many auto- organs (such as the lung, liver, and kidney) whose func-

mated optimization schemes. The successful determina- tion is carried out in parallel by quasi-independent sub-

tion of the volume effect in normal tissues could enable units (33). The parallel model assumes that a given com-

us to improve tumor control probabilities (TCPs) without plication rate cannot occur unless the partial volume dam-

significantly increasing normal tissue complication proba- aged exceeds a threshold value (33,34,35). The existence bilities (NTCPs). of such thresholds is important, becauses it implies that

The only way to refine our understanding of the volume parallel tissues with dose limiting whole volume toler-

Presented at the 35th Annual ASTRO Meeting, New Orleans, LA, October 14, 1994.

Reprint requests to: A. Jackson, Ph.D., Department of Medi- cal Physics, Memorial Sloan Kettering Cancer Center, 1275 York Avenue, New York, NY 10021. - Acknowledgements-This work was supported in part by grant nos. CA547-49, CA59827, and CA42761 from the National Cancer Institute, Department of Health and Human Services,

Bethesda, MD. A. J. would like to thank Ed Melian, Ellen Yorke, Cliff Ling, Y.-C. Lo, John Humm, Michael Lovelock, Lon Marsh, Mary Martel, Rahde Mohan, and Medical Physics Computer Services for their support and useful comments. We are grateful to T. E. Schultheiss’ for pointing out the standard maximum likelihood method used here.

Accepted for publication I9 August 1994.

883

8X4 I. J. Radiation Oncology l Biology l Physics

antes may present no obstacle to radiation therapy when lower partial volumes are irradiated.

Until recently, models of the volume effect in normal tissues have suffered from an acute lack of data, and what data existed were of poor quality. While this situation is manageable as long as treatment stays within known boundaries, the exploitation of novel treatment strategies is badly hampered by our ignorance of volume effects and tolerance doses. Prior to the analysis of the data used

in this paper (20-22), most of our knowledge about the tolerance doses for radiation hepatitis in adults came from the study of whole organ irradiation that originally detined the disease (12), together with other studies with very limited numbers of complications (I, 6, 9, 1 I, 17, 23). Only one of these studies used dose volume histograms (DVHs) ( I), and the others contain comparatively vague information about the doses, the volumes treated, and the fractionation schemes used. Knowledge of the volume effect in the adult liver was virtually nonexistent: the one-third and two-thirds volume tolerance doses given by Emami ef al. (8) are stated to be speculative, and are drawn from accounts of treatments that include doses to unquantified subvolumes together with whole volume background doses (1 I, 27).

The purpose of this paper is to test a parallel architecture model using liver DVHs and clinical complication data for serious radiation hepatitis. These data came from 93 patients treated for tumors of the liver with three- dimensional (3D) conformal radiation therapy (20. 21). A previous analysis of the complications seen in 79 of these patients (22) using the parameterization of Lyman and Wolbarst (5, 18, 19, 23-25), suggested a very large volume effect for the liver, as would be expected if the liver was a parallel organ. We aim to answer two ques- tions. First, can the parallel model fit the data in an acceptable way? Second, can we determine the model parameters from the complication data? To answer these ques- tions we tit the parameters of the model to the data and evaluated this tit using standard statistical techniques.

METHODS AND MATERIALS

Patient treatment und complicution duto We analyzed differential dose volume histograms

(DDVHs) and complication information from 93 patients treated for intrahepatic cancer with concurrent intraarterial hepatic fluorodeoxyuridine (FdUrd) and conformal radiation therapy. All patients had normal liver function, as determined by prothrombin, and partial thromboplastin times. The details of these treatments have been described elsewhere (20-22).

The plans used to treat these patients were generated using limits on the volume of normal liver that could be treated to a given dose, with the aim of delivering higher target doses when smaller volumes of normal liver were irradiated, while satisfying constraints on the dose hetero-

geneity in the target and the dose to the kidney, spinal cord, and other normal structures. These principles, and the response to the pattern of failure of the treatments, resulted in six treatment groups. The doses delivered and partial volumes treated within these groups were roughly homogeneous, and are summarized in Table I and described in more detail in reference 20.

In patients with diffuse liver disease, the volume of normal liver (defined as the total liver volume minus the tumor volume(s)) could not be determined. These patients received whole liver treatment to 33 or 36 Gy. For patients with localized disease, the volume of normal liver irradiated to 50% of the isocenter dose was used to determine the dose delivered to the target. For these patients, treatment consisted of either whole volume treatment to 30 Gy plus a boost treatment to the target, or treatment of the target volume alone.

Doses were delivered twice daily in 1.5 Gy or 1.65 Gy fractions at least 4 h apart (2 1). Intraarterial hepatic FdUrd was delivered concurrently at a dose of 0.2 mg/kg/d. Be- cause an intraarterial hepatic catheter could not be main- tained for more than approximately 2 weeks, treatment other than whole volume was divided into two courses separated by a 2-week interval.

To diagnose radiation hepatitis, follow-up data in the form of physical examinations, serum chemistries, and computed tomography (CT) scans were obtained for 93 patients at 1.5 months and 91 patients at 4 months after treatment. Radiation hepatitis was defined by detection of elevated levels of alkaline phosphatase by at least a factor of two, together with nonmalignant ascites (detected either clinically or by CT scan at I .5 months and 4 months if clinically indicated). Two of the 93 patients were unavailable for follow-up at 4 months. Because both showed no evidence of radiation hepatitis at 1.5 months, they were considered without hepatitis for this analysis. Ten patients developed radiation hepatitis (Table I). Nine of the 10 patients had symptoms of abdominal swelling and clinically evident ascites requiring diuretic and/or ste- roid therapy (clinically significant radiation hepatitis). One patient who was asymptomatic and had transient ascites detected only by CT scan did not receive treatment and we have excluded this complication from our analysis.

A previous statistical analysis of 79 of these patients (22) showed no significant association of the incidence of radiation hepatitis with any of the patient-related variables tested (age, sex, primary tumor type, and total liver volume), where significance was defined by a two-tailed p-

value of less than 0.05.

Thr par&4 model and the pcrrumrtrrs uwd in its implrmentcition

The scarcity of complication data makes it essential to minimize the number of model parameters while max- imizing the flexibility of our description of the volume

Analysis of clinical complication data 0 A. JACKSON ef d. 885

Table 1. Doses, partial volumes treated, and complications occurring in the six treatment groups

Dose (Gy) to whole Dose (Gy) Partial No. of No. of

Group volume to partial volume volume patients complications

1 30 48 0.25-0.5 I1 4 II 30 67 5 0.25 9 2

III 33 17 0 IV 36 15 3 V 53 0.34-0.67 19 0

VI 73 5 0.33 22 (1)

All doses were delivered in 1.5 or I.65 Gy fractions. The partial volumes quoted are the volumes receiving greater than 50% of the isocenter dose. The unbracketed complications shown required treatment, the bracketed complication in group VI did not, as is described in the text.

effect. We used a parallel model without radiosensitivity averaging to describe the complication data. This model is the simplest of the possibilities outlined in reference 13 and contains only four parameters. In the model, a treatment is assumed to cause a complication if the fraction of the organ damaged exceeds the functional reserve of the patient. The NTCP for the treatment is then the proportion of patients with functional reserves less than the fraction of the organ damaged:

NTCP = H(f), (Eq. 1)

wheref is the fraction of the organ damaged by the treatment, and H(f) is the cumulative distribution of functional reserves in the patient population. By implication, if a partial volume u is irradiated, the complication rate cannot exceed the proportion of patients whose functional reserve is less than u.

We assumed that the cumulative functional reserve distribution can be described as a displaced error function and specified by the mean value of the functional reserve Q, and the width of the functional reserve distribution

c,,,

du exp[-(u - uso)*/2cr~]. (Eq. 2)

This is illustrated in Fig. 1. This function was chosen to ensure that unphysical values of the parameters could be easily detected (i.e., if (us0 - 30,) < 0, or if (us0 + 3a,, > 1, then (H(1) - H(0)) may differ significantly from unity).

The fraction of the organ damaged by a treatment was calculated using some additional assumptions (13, 26) that enabled us to use the differential dose volume histograms (DDVHs) rather than the full 3D dose distributions. We assumed that the organ is composed of a large number of subunits that respond to radiation independently, and that these subunits are small enough that they receive

homogeneous dose. We assumed that a sigmoid dose response function, p( d), describes the probability of damaging a subunit at a given biologically equivalent dose. Apart from the assumption that biologically equivalent doses can be calculated from a linear quadratic formula, we did not attempt to connect this probability with any underlying cellular or vascular mechanism of radiation injury, or to identify the subunits involved. Instead we chose to describe the subunit response phenomenologi- tally, using a logistic function of dose parameterized in terms of the dose d,,* at which 50% of the subunits damaged, and the slope parameter k that determines rate at which the probability of damaging subunits

creases with dose (Fig. 2):

are the in-

1

p(d) = [ 1 + (d,,#] . (Eq. 3)

Given the subunit dose response function, we calcu-

lated the fraction of the organ damaged from the DDVH,

cumulative functional reserve 1 .o [7

I L 1,, ,1,, I,, .A:, ,” , 1, __2 01 0.2 0.3 0.4 0.5 06 07 08 oa 1.0

1, fractional volume

Fig. 1. The form of the cumulative functional reserve distribution used to fit the data. The distribution shown has us0 = 0.497, and 0, = 0.047, and was determined by the maximum likelihood analysis described in the text.

886 I. J. Radiation Oncology l Biology 0 Physics Volume 3 I, Number 4, I995

liver sub-unit dose response 10 7--‘,T’-I,1r_7r 1,,111,1 /I I _--a 884

09 I

d, dose I” Gy

Fig. 2. The form of the subunit dose response p(d) used to fit the data. The dose response shown has d1,2 = 41.6 Gy for doses delivered in 1.5 Gy fractions, and k = I .95, and was determined by the maximum likelihood analysis described in the text.

expressed in terms of biologically equivalent doses. We calculated biologically equivalent doses by reexpressing the dose in terms of doses delivered in 1.5 Gy fractions using the linear quadratic model (2). When the number of subunits is large, it has been shown that the total damage can be well approximated by the mean damage (13). Summing over the fractions of the organ damaged at each dose level, we find the simple result:

.f = c w(4), (Eq. 4)

where u, is the fractional volume receiving dose d,. It is natural to use the DDVH rather than the DVH for this calculation. Combining Eqs. 1 and 4 we have (13):

NTCP = H(x u;p(d;)). (Eq. 5)

There are four parameters in Eq. 5, two each to describe the position and slope of the functions, p(d) and H(f).

Because there is no direct evidence to constrain either the subunit dose response or the cumulative functional reserve independently, we regard the variables describing these functions as free parameters to be determined (if possible) from the available complication data. There is a fifth parameter if the fractionation parameter, a/,0, is allowed to vary. However, the results of our analysis are not significantly dependent on the value of alp. We have chosen to fix CY/@ = 2 Gy, a value appropriate for highly differentiated tissues with low repair capacities (2, 32).

Fitting the parallel model parameters from the data We used the method of maximum likelihood to fit the

model to the complication data and find the best values

and confidence intervals for the model parameters (7, 28). This technique is advantageous whenever the data to be tit are sparse. The method asserts that the best fit of the model to the complication data can be obtained by max- imizing the model probability that the observed pattern of complications occurs. The probability that Eq. 5 correctly predicts the observed liver complications can be calculated with a likelihood function, defined as the product of predicted NTCPs for patients with complications, times the products of (I - NTCP) for all patients without complications (Appendix Eq. A 1). More details of this method are given in the Appendix.

It is more convenient to use the logarithm of this likelihood, M. We maximized M using a grid search technique (3); we then deduced the variance-covariance matrix at the maximum likelihood using the quadratic form given

in Appendix Eq. A3. We determined the goodness of fit by using Appendix

Eqs. A4 and A5 to calculate the mean and variance of M, given the best fit parameters, and comparing these with the actual value of M, given both the best fit parameters and the observed complications. Under the assumption that the likelihood is normally distributed about its mean value, we used this comparison to determine the chances of obtaining a smaller value of the likelihood.

As an independent check on the goodness of fit, we determined how well the fitted parameters describe the group complication rates. We have calculated the rates predicted by the fit and used these results to calculate a chi-squared statistic, defined as the weighted sum of the squares of the differences between the observed and calculated complication rates for each group.

Finally, we present a more intuitive picture of the results of the fit that illustrates its meaning. If the subunit dose response were known, then the fraction damaged could be calculated for each patient. We could then de- duce the cumulative functional reserve distribution directly by plotting the observed complication rate as a function of the fraction of the organ damaged (by Eq. 1, this is a measurement of the function H(f)). Therefore, we have binned the fraction damaged as calculated from the fitted subunit dose response function, plotted the observed complications for each bin, and compared this with the fitted cumulative functional reserve. The parallel model predicts that the fraction of the organ damaged should correlate strongly with complication probability. The degree of correlation between complication rate and fractional damage was tested against the null hypothesis (that the complication rate is completely uncorrelated with calculated fraction damaged) by ranking the patients by fraction destroyed and calculating Kendal’s nonparametric 7 statistic (I 6).

RESULTS

We performed an automated search through the four dimensional parameter space (Q,,, o,;, d,,,, k) to find the

Analysis of clinical complication data 0 A. JACKSON et ul. 887

Table 2. Best fit parameter values and individual 68% confidence regions

Parameter 68% confidence

Fitted value interval

Mean functton reserve us0 Width of functional reserve

0.497 kO.043

distribution 0” Dose (in 1.5 Gy fractions)

0.047 20.027

at which 50% of liver subunits are damaged drlz 41.62 Gy ~13.50 Gy

Slope parameter for subunit dose response k 1.95 kO.77

The individual 68% confidence regions assume errors are normally distributed.

best fit parameters using complication data for clinically

significant radiation hepatitis. The best tit parameters, to-

gether with the resultant 68% confidence intervals, are

given in Table 2. The value of us,, was found to be about one-half and that of d1,2 to be about 42 Gy (delivered in 1.5 Gy fractions). The functional reserve distribution and subunit dose response corresponding to the best fit values of the parameters are shown in Figs. 1 and 2.

Given the 93 complication probabilities for the individual patients resulting from the fitted parameters, the expected value of M (observed from a four parameter fit) is -23.34, with a variance of 13.9. The observed value of A4 for the fit is -20.5, and the probability of finding a worse fit (smaller value of M) is 70% under the assumption that M is normally distributed. This indicates that the fit is acceptable. The variance-covariance matrix for the fit is given in Table 3. The 68% confidence ellipsoid for joint variation of us0 and d,,2 is shown in Fig. 3.

As can be seen from both the figure and the covariance matrix, the errors on the parameters are highly correlated. For comparison, the dashed line in Fig. 3 shows the line of constant whole volume 50% tolerance dose, DS,,., = 41.4 Gy delivered in 1.5 Gy fractions. It is apparent that this curve is aligned very well with one of the principle axes of the ellipse. Predictions of the model with best lit parameters for the partial volume response at various doses and the dose response at various partial volumes are shown in Figs. 4 and 5, respectively. The model predictions show a clear threshold effect: at partial volumes below 0.4, the complication probability is near zero for all doses.

Because most of the complications occurred in patients

68% confidence region for v-50 and d-1/2 I ” I ” ” I 1 ‘1 I 1 “I

F t g 0.55 -

z .P ? 0.50 -

2

5 2 0.45 -

5: >’ - 1

0.40 *

whole volume 50% tolerance dose = 41 36 Gy in 1.5 Gy iract1ons

best 111 values of v-50 and d_W with indwidual 66% canildence repns

t#>I sIII l,,~~i~>~~i 35 40 45 50

d-1/2, dose at which prob of sub unit damage = l/2

Fig. 3. The 68% confidence ellipsoid for joint variation of us,, and d,,2 as determined from the covariance matrix in Table 3. Also shown are the individual 68% confidence limits for both variables. The dashed line shows the line of constant whole volume 50% tolerance dose, D5(,,, = 41.4 Gy delivered in 1.5 Gy fractions.

receiving whole volume irradiation to 30 Gy or more (in 1.5 or 1.65 Gy fractions), we expect that the whole volume tolerance dose deduced from these data should be very well determined. The individual 68% confidence regions on the 5% and 50% whole volume tolerance doses, calculated using the model with the best fit parameters in Table 2, are given by DS., = 35.2 (t 1.23) Gy and DsO., = 4 1.4 (-c I .8) Gy respectively, for doses delivered in 1.5 Gy fractions. These whole volume doses agree well with published values (8) considering the uncertainties involved in those estimates.

The predicted complication rates for the six major treatment groups are given in Table 4 along with their standard deviations. These can be compared with the observed complication rates and their errors deduced from binomial statistics. The fitted model predicts the observed complication rates within the standard deviations for all six treatment groups. Chi-squared, calculated as the sum of squares of differences between the observed and calculated complication rates (weighted with the inverse of the squared uncertainty of this difference) is 0.614. Because

Table 3. Variance-covariance matrix for the best tit parameter valuesof Table 2

g‘u J-so d,iz (GY) k

C” 7.1 x 10-4 4.9 x IO_” -9.7 x IO_’ 1.5 x IO_? us0 4.9 x IO 4 1.8 x IO 1 -1.3 x IO_’ 7.7 x lomX dll? (GY) -9.7 x IO_? -1.3 x IO_’ 1.2 x IO’ -4.6 x IO-’ k 1.5 x IO_’ 7.7 x IO 3 -4.6 x IO-’ 6.0 x IO-’

888 1. .I. Radiation Oncology 0 Biology 0 Physics Volume 3 I. Number 4. 1995

partial volume response at various doses Table 4. Observed and calculated complication rates for radiation hepatitis requiring treatment, for the six treatment

groups of Table 1 and best fit parameters in Table 2 k= 1.95, alpha&& = 2 Gy v-50 = 0 496. qma_v = 0 047

dose lo palal volume Y

---- dose=35Gv

No. of No. of NTCP NTCP Group patients complication observed predicted

1 11 4 0.36 ? 0.15 0.26 -t 0.14 II 9 2 0.22 i- 0.14 0.21 -+ 0.16 III 17 0 0.00 ? 0.06 0.05 -t 0.07 IV 1.5 3 0.20 i- 0.10 0.20 + 0.08 V 19 0 0.00 -c 0.05 0.02 ? 0.05 VI 22 0 0.00 2 0.05 0.00 ? 0.00 I

0.4 ; : i :

0.3 ; ;

I), I 1’ ;; 0.2 ; :

' : !' '

0.1 i t ,' ,'

When no complications were observed, we calculated errors on observed complication rates assuming one complication occurred in the group. We calculated the errors on the model group complication rates from the square root of the variance of the calculated complication rates within the group.

I’ ,’ F <ISI 1 #!I/ I I1II IhJL,,,, d,II ,,I, i r ,‘,,’ , _I’ _ ’

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

v, partial volume irradiated

,

1.0 parameters in Table 2. Patient data are collected in bins of similar fraction damaged. The observed complication rate is compared with the fitted cumulative functional reserve. The strong correlation of complication probability with fraction damaged expected from the model is clearly present in the data (Fig. 7), and is well described by the fitted cumulative functional reserve. Using Ken- dal’s nonparametric r statistic, this correlation was found

Fig. 4. The partial volume response at various doses predicted from the model with best fit parameters from Table 2. At high doses the partial volume response tends to the limiting form given by the cumulative functional reserve distribution.

there are 2 degrees of freedom, the probability that chi- squared should have a value larger than this purely by chance is about 74%, indicating that the description of the group complication rates produced by the fit is acceptable.

In Fig. 6 the observed complication rate is plotted as a function of the fraction of the liver damaged, as calculated with the subunit response function using the best fit

complication rate versus fraction damaged 0.3

0.2

0.1

lr d-112 = 41.6 Gy, I” 1.5 Gy fractions k = 1.95, alpha/beta = 2 Gy

* observed compllcatlon rate _ fltted cumulative funcllanal reserve

v-50 ii 0.496, SIQrlL_” = 0 047 ,

7125

dose response at various partial volumes

~~~~

3

-LIZ 0.2 0.3

l.;i,, , , ,I 0.4 0.5 0.6

016

/ I /

0.1

1, calculated fraction damaged

Fig. 6. The observed complication rate as a function of the calculated fraction of the liver damaged using the best fit subunit dose response given in Table 2. Patient data are collected in fraction damaged bins of width 0.1 except for the last two bins, which have width 0.05. The observed complication rate is shown in the center of the bin. Errors shown are binomial. The errors on complication rates of zero were calculated assuming one complication in the bin. The solid line is the cumulative functional reserve distribution with best fit parameters given in Table 2.

10 20 30 40 50 60 70 80 90 100

d, dose in Gy

Fig. 5. The dose response at various partial volumes predicted from the model with best fit parameters of Table 2.

Analysis of clinical complication data 0 A. JACKSON et al. 889

rank, fraction damaged and complications O.~L,~~~,~~~~,~~~,,~~~~,~~~~,~~~~,~~~~,~~~~,~,~~~~~~~,

d-112 = 41 6 Gy I” 1 5 Gy lractms k = 1.95, alphaibeta = 2 Gy

t,,,,I,,,,I,,,,l,,,,l,,,~i,,,,l,,,,I,,,,I,,,,l.,,,I 10 20 30 40 50 60 70 60 90 100

rank

Fig. 7. Patient treatments ranked according to calculated fraction destroyed using the best fit parameters of Table 2. Patients with complications are shown with black triangles, patients without complications are shown with open circles.

to be statistically significant at the p (two-tailed) < 0.0001 level. The observed correlation is robust with respect to variation of the model parameters within their confidence limits, and (irrespective of the correct values of the parameters) calculation of fraction damaged can be used to produce a ranking of rival plans that will strongly correlate with increasing NTCP.

DISCUSSION AND CONCLUSIONS

We have shown that a simple parallel model can be used to produce a satisfactory fit of the radiation hepatitis complication data. Our best fit model of the liver has a mean functional reserve of about one-half, and a local response function that is a relatively shallow function of dose, with a dose that produces 50% local damage of about 42 Gy (in 1.5 Gy fractions).

The best fit value of the mean functional reserve is smaller than two-thirds, the amount of liver that it is safe to resect surgically (15). The shallow gradient of the local response function could arise from several possibilities. The liver moves during treatment, and some subunits nominally in low dose regions move into high dose regions and vice verca. Or, there may be intrapatient variation of subunit tolerance doses. Both these effects could flatten the subunit response deduced in this analysis.

The 5 and 50% whole volume tolerance doses resulting from our analysis are consistent with those given in the literature (given the uncertainties in the treatments used in those studies), and with the 10% & 7% complication rate for hepatitis found in a recent RTOG dose escalation

study of 51 patients given nominal whole volume doses of 33 Gy in 1.5 Gy fractions (30). This suggests that the use of intraarterial hepatic FdUrd does not change the liver tolerance doses substantially.

We have observed a correlation between the calculated fraction damaged and the observed complication probability that can be used to rank plans in order of risk of radiation hepatitis. This correlation depends on our assumptions about the existence of a local response function, but is reasonably robust with respect to the exact form of this function. (In this respect, it is interesting to note that a previous analysis of these data found a correlation between mean dose and complication probability (22). This might be expected from the robust correlation observed here, because if the local response function were roughly proportional to dose over the relevant dose interval, then fraction destroyed is proportional to mean dose.) The correlation we have observed does not depend on any assumptions about functional reserves. At the very least, the cumulative functional reserve fit in this analysis can be viewed as summarizing this correlation.

It is important to stress that our results are model dependent. We have assumed the existence of a functional reserve and a subunit response function. While we have found that the data does not rule out our model, this cannot not be taken to prove that the model is correct. For example, the current data set does not rule out the possibility that radiation hepatitis of clinical significance occurs because of nonlocal damage due to whole volume irradiation. This hypothesis has some support from animal data, where radiation hepatitis has been observed in rats only when given whole volume irradiation (10). This possibility cannot be described using Eq. 5.

Only the combination of cumulative functional reserve and subunit dose response given in Eq. 5 is directly related

to the complication data (notice that the line of constant

whole volume 50% tolerance dose in Fig. 3 indicates that the same whole volume tolerance dose is consistent with the entire confidence interval on us,, and d,,*). This has resulted in highly correlated errors on the best fit model parameters. Consequently, these parameters should be in- terpreted with caution. For example, although Figs. 4 and 5 predict that irradiation of one-half the liver is safe even when doses as high as 75-80 Gy are used, the 68% confidence ellipse in Fig. 3 indicates that the threshold volume for a 50% complication rate could easily be as low as 0.43.

To resolve the remaining uncertainties in the model parameters requires either independent information on the subunit response and cumulative functional reserve, or knowledge of the complication rates in patient groups receiving partial volume irradiation without whole volume treatment. SPECT studies of the liver carried out before and after irradiation (analogous to those performed for the lung (4)), might make it possible to determine the subunit dose response independently. In this case, a plot

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of the complication rate as a function of calculated frac- treatments with partial volume irradiation (groups V and tion damaged (such as Fig. 6) would constitute a direct VI)). Because tumor response in these patients has been measurement of the cumulative functional reserve distri- found to correlate with dose (21, 29), dose escalation bution. Dose escalation studies using 3D conformal radia- for patients receiving partial volume irradiation is both tion therapy carried out using dose volume protocols (3 1) feasible and desirable. Understanding the tolerance doses have the potential to determine, for example, the 5% toler- for partial volume irradiation of the liver may enable us ante doses for pure partial volume irradiation. to improve tumor response while limiting complication

No complications requiring treatment resulted from the probability.

REFERENCES

1. Austin-Seymour, M. M.; Chen, G. T. Y.; Castro, J. R.; Saunders, W. M.; Pitluck, S.; Woodruff, K. H.; Kessler, M. Dose volume histogram analysis of liver radiation tolerance. Int. J. Radiat. Oncol. Biol. Phys. 12:31-35; 1986.

2. Barendsen, G. W. Dose fractionation, dose rate and iso- effect relationships for normal tissue responses. Int. J. Ra- diat. Oncol. Biol. Phys. 8:1981-1997; 1982.

3. Bevington, P. R.; Robinson, D. K. Data reduction and error analysis for the physical sciences. 2nd Ed. New York: McGraw-Hill Inc.; 1992: 180- 192.

4. Boersma, L. J.; Damen, E. M. F.; de Boer, R. W.; Muller, S. H.; Roos, C. M.; Valdes Olmos, R. A.; Van Zandwijk, N.; Lebesque, J. V. Dose-effect relations for local functional and structural changes of the lung 3 to 4 months after irradiation for malignant lymphoma. Radiother. Oncol. (in press).

5. Burman, C.; Kutcher, G. J.; Emami, B.; Goitein, M. Fitting normal tissue tolerance data to an analytic function. lnt. J. Radiat. Oncol. Biol. Phys. 21:123-135; 1991.

6. Calkins, A. R.; Rosenshein, N. B.; Fox, M. G.; Order, S. E. Delayed split whole abdominal irradiation in the combined modality treatment of ovarian cancer. Int. J. Radiat. Oncol. Biol. Phys. 20:661-665; 1991.

7. Cox D. R.; Snell, E. J. Analysis of binary data. 2nd Ed. London: Chapman and Hall; 1989: 179- 183.

8. Emami, B.; Lyman, J.; Brown, A.; Coia, L.; Goitein, M.; Munzenrider, J. E.; Shank, B.; Solin, L. J.; Wesson, M. Tolerance of normal tissue to therapeutic radiation. Int. J. Radiat. Oncol. Biol. Phys. 21:109-122; 1991; and refs. therein.

9. Feigen, M.; Lavrin, L.; Mameghan, H.; Peters, L. Radiation hepatitis following moving strip radiotherapy. Int. J. Radiat. Oncol. Biol. Phys. 9.397-400; 1983.

IO. Geraci, J. P.; Mariano, M. S.; Jackson, K. L. Hepatic radiation injury in the rat. Radiat. Res. 125:65-72; 1991.

11. Haddad, E.; Le Bourgeois, J. P.; Kuentz, M.; Lobo, P. Liver complications in lymphomas treated with a combination of chemotherapy and radiotherapy: Preliminary results. Int. J. Radiat. Oncol. Biol. Phys. 9:1313-1319; 1983.

12. Ingold, J. A.; Reed, G. B.; Kaplan, H. S.; Bagshaw, M. A. Radiation hepatitis. Am. J. Roentgenol. 93:200-208; 1965.

13. Jackson, A.; Kutcher, G. J.; Yorke, E. Probability of radiation induced complications for normal tissues with parallel architecture subject to nonuniform irradiation. Med. Phys. 20:613-625; 1993.

14. Jolles, B.; Mitchell, R. G. Optimal skin tolerance dose levels. Br. J. Radio]. 20:405-409; 1947.

15. Kemeny, N.; Sugarbaker, P. H. Treatment of metastatic cancer to the liver. In: DaVita, V. T.; Helman, S.; Rosen- berg, S. A., eds. Cancer, Principles and Practice of Oncol- ogy. 3rd Ed. Philadelphia, PA: J. B. Lippencott Co; 1989:2275-2298.

16. Kendal, M. G.; Stuart, A. The advanced theory of statistics.

Vol. 2. Inference and relationships. 2nd Ed. New York: Hafner; 1967.

17. Kim, T. J.; Panahon, A. M.; Friedman, M.; Webster, J. H. Acute transient radiation hepatitis following whole abdominal irradiation. Clin. Radio]. 27:449-454; 1976.

IX. Kutcher, G. J.; Burman, C. Calculation of complication probability factors for nonuniform normal tissue irradiation: The effective volume method. Int. J. Radiat. Oncol. Biol. Phys. 16:1623-1630; 1989.

19. Kutcher, G. J.; Burman, C.; Brewster, L.; Goitein, M.; Mo- han, R. Histogram reduction method for calculating complication probabilities for three dimensional treatment planning evaluations. Int. J. Radiat. Oncol. Biol. Phys. 21:137- 146; 1991.

20. Lawrence, T. S.; Tesser, R. J.; Ten Haken, R. K. An appli- cation of dose volume histograms to the treatment of intrahepatic malignancies with radiation therapy. Int. J. Radiat. Oncol. Biol. Phys. 19:1041-1047; 1990.

21. Lawrence, T. S.; Dworzanin, L. M.; Walker-Andrews, S. C.; Andrews, J. C.; Ten-Haken, R. K.; Wollner, 1. S.; Lichter, A. S.; Ensminger, W. D. Treatment of cancers involving the liver and porta hepatis with external beam irradiation and intraarterial hepatic fluorodeoxyuridine. Int. J. Radiat. Oncol. Biol. Phys. 20:555-561; 1991.

22. Lawrence, T. S.; Ten Haken, R. K.; Kessler, M. L.; Robert- son, J. M.; Lyman, J. T.; Lavigne, M. L.; Brown, M. B.; DuRoss, D. J.; Andrews, J. C.; Ensminger, W. D.; Lichter, A. S. The use of 3-D dose volume analysis to predict radiation hepatitis. Int. J. Radiat. Oncol. Biol. Phys. 23:78 l- 788; 1992.

23. Lyman, J. T. Complication probabilities as assessed from dose-volume histograms. Radiat. Res. 104:s 13-S 19; 1985.

24. Lyman, J. T.; Wolbarst, A. B. Optimization of radiation therapy. III. A method for assessing complication probabilities from dose-volume histograms. Int. J. Radiat. Oncol. Biol. Phys. 13: 103-109; 1987.

25. Lyman, J. T.; Wolbarst, A. B. Optimization of radiation therapy. IV. A dose-volume histogram reduction algorithm. Int. J. Radiat. Oncol. Biol. Phys. 17:433-436; 1989.

26. Niemierko, A.; Goitein, M. Modeling of normal tissue response to radiation: The critical volume model. lnt. J. Ra- diat. Oncol. Biol. Phys. 25:135-145; 1993.

27. Poussin-Rosillo, H.; Nisce, L. Z.; D’Anglo, G. J. Hepatic radiation tolerance in Hodgkin’s disease patients. Radiol- ogy 121:461-464; 1976.

28. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; Vetterling, W. T. Numerical recipes, FORTRAN edition. Cambridge, England: Cambridge University Press; 1986:454-546.

29. Robertson, J. M.; Lawrence, T. S.; Dworzanin, L. M.; An- drews, J. C.; Walker, S.; Kessler, M. L.; DuRoss, D. J.; Ensminger, W. D. Treatment of primary hepatobiliary cancers with conformal radiation therapy and regional chemotherapy. J. Clin. Oncol. 11:1286-1293; 1993.

Analysis of clinical complication data 0 A. JACKSON et al. 891

30.

31.

32.

33.

34.

35.

Russell, A. H.; Clyde, C.; Wasserman, T. H.; Turner, S. S.; Rotman, M. Accelerated hyperfractionated hepatic irradiation in the management of patients with liver metastases: Results of the RTOG dose escalating protocol. Int. J. Ra- diat. Oncol. Biol. Phys. 27: 1 l7- 123; 1993. Ten Haken, R. K.; Martel, M. K.; Kessler, M. L.; Hazuka, M. B.; Lawrence, T. S.; Robertson, J. M.; Turrisi, A. T.; Lichter, A. S. Use of Veff and iso-NTCP in the implementa- tion of dose escalation protocols. Int. J. Radiat. Oncol. Biol. Phys. 27:689-695; 1993. Thames, H. D.; Hendry, J. H. Fractionation in radiotherapy. Taylor and Francis; 1987:72-74. Withers, H. R.; Taylor, J. M. G.; Maciejewski, B. Treatment volume and tissue tolerance. Int. J. Radiat. Oncol. Biol. Phys. 14:751-759; 1988. Wolbarst, A. B.; Chin, L. M.; Svensson, G. K. Optimization of radiation therapy: Integral-response of a mode1 biologi- cal system. Int. J. Radiat. Oncol. Biol. Phys. 8: l76l- 1769; 1982. Yorke, E.; Kutcher, G. J.; Jackson, A.; Ling, C. C. Probabil- ity of radiation induced complications for normal tissues with parallel architecture under conditions of uniform whole or partial organ irradiation. Radiother. Oncol. 26226-237; 1993.

APPENDIX: THE MAXIMUM LIKELIHOOD METHOD, VARIANCE-COVARIANCE MATRICES, AND THE LIKELIHOOD

DISTRIBUTION

The probability that the parallel model (Eq. 5) correctly described the pattern of complications that occurred in the 94 patients is (7, 28):

L(y,, y2r . .) = n NTCP,,(?/,> 727 . . .)

m complxxtwn

x n (1 - NTCP,,(?/,> 72, . .)I, (Eq. Al) II

no complxarwn

where the products over m and IZ run over patients for

which complications occurred and did not occur, respectively, and the model prediction of the complication probability for each patient is written NTCP,(y,, -y2, . . .). The dependence on the model parameters is denoted as y,, y2, . . for notational convenience. It is more useful to take the logarithm of L:

M(yr, y2, . . .I = c Ln[NTCP,(y,, y2, . . .>I ,?I

CO,“plUltK>”

+ c Ml - NTCP,,(y,, y2, . . .)I. (Eq. A21 n

no compllcatK,n

Expanding around the optimal values of the parameters y,, y2, . . , M can be written:

MY,, Y2r . . .) = MT,, 72, . . .I

4

+ C$cr, - 7,)(Y, - 7,) d2M r.,= I ariar, ;; 1;;

+ . . . . , (Eq. A3)

The variance-covariance matrix C,,, can be obtained from the second derivative matrix in Appendix Eq. A3. Under the assumption that the errors involved are normally distributed, this matrix can be used to determine confidence intervals on any single parameter, or confidence ellipsoids on any combination of parameters taken together. Individual 68% confidence regions are given by the square root of the diagonal elements. Ellipsoidal confidence regions in any number of parameters can be deduced by inverting the appropriate submatrices as outlined in reference 28.

To determine the goodness of fit, we need to assess the probability that the complication probabilities deduced for the 93 individuals using best fit model parameters would produce a value of the likelihood smaller that the observed one. Typically speaking, if this turns out to be too large, the model overfits the data; if it turns out to be too small, the model does not lit the data well. Averaging over all treatment outcomes given the 93 best fit patient complication probabilities, $, = NTCP,,(p,, . . . p4), it is straight- forward to show that (M), the mean value of M is

(M) = c ((1 - li,)lnll - b,,l + A, W,,l), @q. A4) m

and S,, the variance of M, is given by

sM = z(,..(l - B,J(ln[(, f”‘p,,I)‘) , 0%. A51

where the sums in Appendix Eqs. A4 and A5 run over all patients. Under the assumption that the value of M is normally distributed, Appendix Eqs. A4 and A5 can be used to determine the goodness of fit.

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Analysis of clinical complication data for radiation hepatitis using a parallel architecture model