- Email: [email protected]
A method to adjust radiation dose–response relationships for clinical risk factors
Radiotherapy and Oncology 102 (2012) 352–354
Contents lists available at SciVerse ScienceDirect
Radiotherapy and Oncology journal homepage: www.thegreenjournal.com
Dose-response relationship
A method to adjust radiation dose–response relationships for clinical risk factors Ane L. Appelt a,b,⇑, Ivan R. Vogelius c a
Department of Oncology, Vejle Sygehus; b University of Southern Denmark; c Department of Radiation Oncology, Rigshospitalet, University of Copenhagen, Denmark
a r t i c l e
i n f o
Article history: Received 28 April 2011 Received in revised form 15 August 2011 Accepted 27 August 2011 Available online 6 October 2011
a b s t r a c t Several clinical risk factors for radiation induced toxicity have been identified in the literature. Here, we present a method to quantify the effect of clinical risk factors on radiation dose–response curves and apply the method to adjust the dose–response for radiation pneumonitis for patients with/without pre-existing pulmonary co-morbidities. Ó 2011 Elsevier Ireland Ltd. All rights reserved. Radiotherapy and Oncology 102 (2012) 352–354
Keywords: Normal tissue complications Radiotherapy Modelling Dose–response relationship Radiation pneumonitis
Understanding dose–response relationships for both malignant and normal tissue remain a central issue when optimizing radiation treatment. However, response to irradiation can be affected by a range of factors not related to the physical dose. These factors can influence the dose–response curve, but are often not taken into account in dose–response relationships reported in the literature. In the recent QUANTEC initiative, the need to quantify the influence of physiological factors and co-morbidities on the risk of radiation induced side effects was identified as an important area for future research [1]. Ideally, the effects of clinical risk factors on a dose–response relationship should be quantified when applying results from the literature in clinical settings. Here we demonstrate that it is possible to quantify and correct for the effect of a clinical risk factor on the dose–response curve if the odds ratio (OR) for developing toxicity for patients with/without the risk factor is known, and the proportion of patients exposed to the risk factor in the original dose–response study can be estimated. As an example, we apply the method to the risk of developing radiation pneumonitis (RP), where a dose–response relationship for a general patient population is available from the QUANTEC review [2], and an OR for developing RP when having pulmonary co-morbidity is known from a recent meta-analysis [3].
⇑ Corresponding author at: Department of Oncology, Vejle Sygehus, Kabbeltoft 25, 7100 Vejle, Denmark. E-mail address: [email protected] (A.L. Appelt). 0167-8140/$ - see front matter Ó 2011 Elsevier Ireland Ltd. All rights reserved. doi:10.1016/j.radonc.2011.08.031
Mathematical method Let the dose be characterized by a simple dosimetric factor, D, condensing the dose–volume histogram into a single point value. Examples of such factors include the equivalent uniform dose, EUD [4], and the mean dose, Dmean. The dose-dependence of the response, p, is then assumed to be sigmoid shaped, and can conveniently be described by the two parameters D50 and c50 [5]. D50 is the dose resulting in a 50% response, p(D50) 0.50, and c50 is dp the normalized dose–response gradient at D50, c50 dD Djp¼0:5 . Here we assume that p follows a logistic dose–response function,
p¼
1 ; 1 þ expðxÞ
D : x ¼ 4c50 1 D50
ð1Þ
where x may be any arbitrary function with a linear dependence on dose, but it is usually parameterized using D50 and c50 as shown. Assuming that the relationship between D and p is known for a group of patients N, we consider the case where N is made up of two subgroups N0 (the low risk group) and N1 (the high risk group), and where the odds ratio OR between the complication probabilities in the two groups is independent of D. The prevalence of the factor determining high risk, s = N1/(N0 + N1), is also assumed to be independent of dose. Hence,
p ¼ sp1 þ ð1 sÞp0 ORp0 þ ð1 sÞp0 ; ¼s p0 ðOR 1Þ þ 1
ð2Þ ð3Þ
where p0 and p1 are the probabilities of complications in the two subgroups N0 and N1, and p is the overall probability of
353
A.L. Appelt, I.R. Vogelius / Radiotherapy and Oncology 102 (2012) 352–354
P pjp0 ¼0:5 ¼
1 OR 1 1þs : 2 OR þ 1
ð4Þ
P is a modifying factor monotonically dependent on OR and s. Second, D050 is extracted from Eq. (1) as the dose resulting in the complication probability P
P 1 ln 1P D50 D050 ¼ 1 þ 4 c50
P þ 4 ln c 50 1P s ð2P 1Þ2 sPð1 PÞ
ð6Þ
The parameters describing the dose–response curve for the group N1 can be found from Eq. (1) and the definition of the odds ratio:
1 D150 ¼ D050 1 0 ln OR 4c50 1 1 0 c50 ¼ c50 ln OR: 4
ð8Þ
t ie s
idi t ie s
tc om
ou
N
T A ti it h Pa QU s w t n t ie Pa
or b
co mo rb f it
wi
EC
th
0.3
ΔMLD 5
10
0.02
0.4
5
7.5
10
0.3 0.2 0.1
0
10
20 Dose
30
40
Fig. 2. Demonstration of the validity of the logistic approximation. The bold line shows the QUANTEC fit used in Fig. 1. Dose–response curves for patients with/ without risk factors were then generated from the QUANTEC fit using the logistic approximation with parameters s = 0.1, 0.5, 0.9 and OR = 0.2, 0.5, 2.5, 5.0. Finally, the total population dose response was reconstructed from the approximated curves using Eq. (3) (thin lines). The quality of the logistic approximation with the parameters tested here is illustrated by the close match of the reconstructed curves and the original QUANTEC fit.
15 20 MLD [Gy]
As an example, consider the relationship between mean lung dose (MLD) and radiation-induced pneumonitis (RP), as reported in the recent QUANTEC paper [2]. A number of clinical factors are known or suspected to change the risk of RP [3], for example preexisting co-morbidities, smoking status and the chemotherapy regimen. Fig. 1 shows the effect of pulmonary co-morbidities as a clinical risk factor: the QUANTEC fit is assumed to be valid for s = 0.5 (50% of patients having co-morbidities), while the surrounding curves show the estimated dose–response for patients with comorbidities and for patients without co-morbidities. Here, we used OR = 2.27 for developing radiation pneumonitis for patients with/ without co-morbidities [3]. Note that MLD50 can vary between 27.7 Gy and 33.9 Gy (DMLD in the figure).
Conclusion
i di
0.4
en ts
Probability of Pneumonitis
0.5
0 0
0.05
0.5
Application
0.6
0.1
0.08
0.6
ð7Þ
Hence the dose–response relationship can be corrected for the effect of the clinical factor, allowing the relationship to be applied to differently composed patient populations – or compared to other reported dose–response relationships for the same tissue.
0.2
0.7
0
ð5Þ
The definition of c050 is the normalized gradient of the dose response 0 at 50% complication probability: c050 ¼ dp D050 jp0 ¼0:5 . Some algebra dD using this definition and Eqs. (1) and (3) leads to the expression
c050 ¼
0.8
Probability of complication
complication. Note that p, p0 and p1 will all be dose-dependent. The dose–response for the entire group is described by the parameters D50 and c50. Similarly, the dose–response of the low risk group, p0, can be described by parameters D050 and c050 , and the dose–response of the high risk group is described by D150 and c150 . We note that p0 and p1 will not follow logistic functions, but that they can be approximated by ones except for very large/small values of OR. First, the average complication probability for the total group of patients at a dose causing 50% complication probability in the low risk group is found from Eq. (3):
25
30
35
Fig. 1. Relationship between mean lung dose (MLD) and incidence of radiation pneumonitis. Solid line: dose dependence of response as reported in [2] (D50 = 30.8 Gy, c50 = 0.97). Dashed line: the corresponding relationship for a patient population without co-morbidities, assuming a prevalence of 50% (s = 0.5) in the cohort used to generate the QUANTEC fit and OR = 2.27 for developing radiation pneumonitis for patients with/without co-morbidities. D050 ¼ 33:9Gy; c050 ¼ 1:11. Dotted line: dose–response for patients with co-morbidities. D150 ¼ 27:7Gy; c150 ¼ 0:91.
The method presented above allows assessment of the effect of a dose-independent risk factor (OR > 1) or a protecting factor (OR < 1) when evaluating dose–response relationships. Since the corrected relationship is not perfectly logistic, the description in terms of D050 and c050 is not exact, but it is a very good approximation to the full expression in the ranges of D, s and OR usually relevant (see Fig. 2). The correction is potentially useful when interpreting published dose–response relationships and when comparing the results of independent investigations with different patient populations. Also, the results of a dose–response analysis that does not consider clinical risk factors can be combined with the results of an analysis of clinical risk factors that does not consider radiation dose, as in the application presented here.
Conflict of interests statement The authors have no conflicts of interest to disclose.
354
Dose response and clinical risk factors
Acknowledgements A.L.A. and I.R.V. are supported by CIRRO – The Lundbeck Foundation Center for Interventional Research in Radiation Oncology and The Danish Council for Strategic Research. A.L.A. acknowledges support from the Region of Southern Denmark. References [1] Bentzen SM, Constine LS, Deasy JO, et al. Quantitative analyses of normal tissue effects in the clinic (QUANTEC): an introduction to the scientific issues. Int J Radiat Oncol Biol Phys 2010;76:3–9.
[2] Marks LB, Bentzen SM, Deasy JO, et al. Radiation dose–volume effects in the lung. Int J Radiat Oncol Biol Phys 2010;76:70–6. [3] Vogelius IS, Bentzen SM. Clinical factors associated with risk of radiation pneumonitis: a literature based meta-analysis (abstract). Radiother Oncol 2010;96:125–6. [4] Niemierko A. Reporting and analyzing dose distributions: a concept of equivalent uniform dose. Med Phys 1997;24:103–10. [5] Bentzen SM, Tucker SL. Quantifying the position and steepness of radiation dose–response curves. Int J Radiat Biol 1997;71:531–42.
Contents lists available at SciVerse ScienceDirect
Radiotherapy and Oncology journal homepage: www.thegreenjournal.com
Dose-response relationship
A method to adjust radiation dose–response relationships for clinical risk factors Ane L. Appelt a,b,⇑, Ivan R. Vogelius c a
Department of Oncology, Vejle Sygehus; b University of Southern Denmark; c Department of Radiation Oncology, Rigshospitalet, University of Copenhagen, Denmark
a r t i c l e
i n f o
Article history: Received 28 April 2011 Received in revised form 15 August 2011 Accepted 27 August 2011 Available online 6 October 2011
a b s t r a c t Several clinical risk factors for radiation induced toxicity have been identified in the literature. Here, we present a method to quantify the effect of clinical risk factors on radiation dose–response curves and apply the method to adjust the dose–response for radiation pneumonitis for patients with/without pre-existing pulmonary co-morbidities. Ó 2011 Elsevier Ireland Ltd. All rights reserved. Radiotherapy and Oncology 102 (2012) 352–354
Keywords: Normal tissue complications Radiotherapy Modelling Dose–response relationship Radiation pneumonitis
Understanding dose–response relationships for both malignant and normal tissue remain a central issue when optimizing radiation treatment. However, response to irradiation can be affected by a range of factors not related to the physical dose. These factors can influence the dose–response curve, but are often not taken into account in dose–response relationships reported in the literature. In the recent QUANTEC initiative, the need to quantify the influence of physiological factors and co-morbidities on the risk of radiation induced side effects was identified as an important area for future research [1]. Ideally, the effects of clinical risk factors on a dose–response relationship should be quantified when applying results from the literature in clinical settings. Here we demonstrate that it is possible to quantify and correct for the effect of a clinical risk factor on the dose–response curve if the odds ratio (OR) for developing toxicity for patients with/without the risk factor is known, and the proportion of patients exposed to the risk factor in the original dose–response study can be estimated. As an example, we apply the method to the risk of developing radiation pneumonitis (RP), where a dose–response relationship for a general patient population is available from the QUANTEC review [2], and an OR for developing RP when having pulmonary co-morbidity is known from a recent meta-analysis [3].
⇑ Corresponding author at: Department of Oncology, Vejle Sygehus, Kabbeltoft 25, 7100 Vejle, Denmark. E-mail address: [email protected] (A.L. Appelt). 0167-8140/$ - see front matter Ó 2011 Elsevier Ireland Ltd. All rights reserved. doi:10.1016/j.radonc.2011.08.031
Mathematical method Let the dose be characterized by a simple dosimetric factor, D, condensing the dose–volume histogram into a single point value. Examples of such factors include the equivalent uniform dose, EUD [4], and the mean dose, Dmean. The dose-dependence of the response, p, is then assumed to be sigmoid shaped, and can conveniently be described by the two parameters D50 and c50 [5]. D50 is the dose resulting in a 50% response, p(D50) 0.50, and c50 is dp the normalized dose–response gradient at D50, c50 dD Djp¼0:5 . Here we assume that p follows a logistic dose–response function,
p¼
1 ; 1 þ expðxÞ
D : x ¼ 4c50 1 D50
ð1Þ
where x may be any arbitrary function with a linear dependence on dose, but it is usually parameterized using D50 and c50 as shown. Assuming that the relationship between D and p is known for a group of patients N, we consider the case where N is made up of two subgroups N0 (the low risk group) and N1 (the high risk group), and where the odds ratio OR between the complication probabilities in the two groups is independent of D. The prevalence of the factor determining high risk, s = N1/(N0 + N1), is also assumed to be independent of dose. Hence,
p ¼ sp1 þ ð1 sÞp0 ORp0 þ ð1 sÞp0 ; ¼s p0 ðOR 1Þ þ 1
ð2Þ ð3Þ
where p0 and p1 are the probabilities of complications in the two subgroups N0 and N1, and p is the overall probability of
353
A.L. Appelt, I.R. Vogelius / Radiotherapy and Oncology 102 (2012) 352–354
P pjp0 ¼0:5 ¼
1 OR 1 1þs : 2 OR þ 1
ð4Þ
P is a modifying factor monotonically dependent on OR and s. Second, D050 is extracted from Eq. (1) as the dose resulting in the complication probability P
P 1 ln 1P D50 D050 ¼ 1 þ 4 c50
P þ 4 ln c 50 1P s ð2P 1Þ2 sPð1 PÞ
ð6Þ
The parameters describing the dose–response curve for the group N1 can be found from Eq. (1) and the definition of the odds ratio:
1 D150 ¼ D050 1 0 ln OR 4c50 1 1 0 c50 ¼ c50 ln OR: 4
ð8Þ
t ie s
idi t ie s
tc om
ou
N
T A ti it h Pa QU s w t n t ie Pa
or b
co mo rb f it
wi
EC
th
0.3
ΔMLD 5
10
0.02
0.4
5
7.5
10
0.3 0.2 0.1
0
10
20 Dose
30
40
Fig. 2. Demonstration of the validity of the logistic approximation. The bold line shows the QUANTEC fit used in Fig. 1. Dose–response curves for patients with/ without risk factors were then generated from the QUANTEC fit using the logistic approximation with parameters s = 0.1, 0.5, 0.9 and OR = 0.2, 0.5, 2.5, 5.0. Finally, the total population dose response was reconstructed from the approximated curves using Eq. (3) (thin lines). The quality of the logistic approximation with the parameters tested here is illustrated by the close match of the reconstructed curves and the original QUANTEC fit.
15 20 MLD [Gy]
As an example, consider the relationship between mean lung dose (MLD) and radiation-induced pneumonitis (RP), as reported in the recent QUANTEC paper [2]. A number of clinical factors are known or suspected to change the risk of RP [3], for example preexisting co-morbidities, smoking status and the chemotherapy regimen. Fig. 1 shows the effect of pulmonary co-morbidities as a clinical risk factor: the QUANTEC fit is assumed to be valid for s = 0.5 (50% of patients having co-morbidities), while the surrounding curves show the estimated dose–response for patients with comorbidities and for patients without co-morbidities. Here, we used OR = 2.27 for developing radiation pneumonitis for patients with/ without co-morbidities [3]. Note that MLD50 can vary between 27.7 Gy and 33.9 Gy (DMLD in the figure).
Conclusion
i di
0.4
en ts
Probability of Pneumonitis
0.5
0 0
0.05
0.5
Application
0.6
0.1
0.08
0.6
ð7Þ
Hence the dose–response relationship can be corrected for the effect of the clinical factor, allowing the relationship to be applied to differently composed patient populations – or compared to other reported dose–response relationships for the same tissue.
0.2
0.7
0
ð5Þ
The definition of c050 is the normalized gradient of the dose response 0 at 50% complication probability: c050 ¼ dp D050 jp0 ¼0:5 . Some algebra dD using this definition and Eqs. (1) and (3) leads to the expression
c050 ¼
0.8
Probability of complication
complication. Note that p, p0 and p1 will all be dose-dependent. The dose–response for the entire group is described by the parameters D50 and c50. Similarly, the dose–response of the low risk group, p0, can be described by parameters D050 and c050 , and the dose–response of the high risk group is described by D150 and c150 . We note that p0 and p1 will not follow logistic functions, but that they can be approximated by ones except for very large/small values of OR. First, the average complication probability for the total group of patients at a dose causing 50% complication probability in the low risk group is found from Eq. (3):
25
30
35
Fig. 1. Relationship between mean lung dose (MLD) and incidence of radiation pneumonitis. Solid line: dose dependence of response as reported in [2] (D50 = 30.8 Gy, c50 = 0.97). Dashed line: the corresponding relationship for a patient population without co-morbidities, assuming a prevalence of 50% (s = 0.5) in the cohort used to generate the QUANTEC fit and OR = 2.27 for developing radiation pneumonitis for patients with/without co-morbidities. D050 ¼ 33:9Gy; c050 ¼ 1:11. Dotted line: dose–response for patients with co-morbidities. D150 ¼ 27:7Gy; c150 ¼ 0:91.
The method presented above allows assessment of the effect of a dose-independent risk factor (OR > 1) or a protecting factor (OR < 1) when evaluating dose–response relationships. Since the corrected relationship is not perfectly logistic, the description in terms of D050 and c050 is not exact, but it is a very good approximation to the full expression in the ranges of D, s and OR usually relevant (see Fig. 2). The correction is potentially useful when interpreting published dose–response relationships and when comparing the results of independent investigations with different patient populations. Also, the results of a dose–response analysis that does not consider clinical risk factors can be combined with the results of an analysis of clinical risk factors that does not consider radiation dose, as in the application presented here.
Conflict of interests statement The authors have no conflicts of interest to disclose.
354
Dose response and clinical risk factors
Acknowledgements A.L.A. and I.R.V. are supported by CIRRO – The Lundbeck Foundation Center for Interventional Research in Radiation Oncology and The Danish Council for Strategic Research. A.L.A. acknowledges support from the Region of Southern Denmark. References [1] Bentzen SM, Constine LS, Deasy JO, et al. Quantitative analyses of normal tissue effects in the clinic (QUANTEC): an introduction to the scientific issues. Int J Radiat Oncol Biol Phys 2010;76:3–9.
[2] Marks LB, Bentzen SM, Deasy JO, et al. Radiation dose–volume effects in the lung. Int J Radiat Oncol Biol Phys 2010;76:70–6. [3] Vogelius IS, Bentzen SM. Clinical factors associated with risk of radiation pneumonitis: a literature based meta-analysis (abstract). Radiother Oncol 2010;96:125–6. [4] Niemierko A. Reporting and analyzing dose distributions: a concept of equivalent uniform dose. Med Phys 1997;24:103–10. [5] Bentzen SM, Tucker SL. Quantifying the position and steepness of radiation dose–response curves. Int J Radiat Biol 1997;71:531–42.