Theoretical analysis of the dose dependence of the oxygen enhancement ratio and its relevance for clinical applications
 Tatiana Wenzl^{1}Email author and
 Jan J Wilkens^{1}
DOI: 10.1186/1748717X6171
© Wenzl and Wilkens; licensee BioMed Central Ltd. 2011
Received: 23 September 2011
Accepted: 15 December 2011
Published: 15 December 2011
Abstract
Background
The increased resistance of hypoxic cells to ionizing radiation is usually believed to be the primary reason for treatment failure in tumors with oxygendeficient areas. This oxygen effect can be expressed quantitatively by the oxygen enhancement ratio (OER). Here we investigate theoretically the dependence of the OER on the applied local dose for different types of ionizing irradiation and discuss its importance for clinical applications in radiotherapy for two scenarios: small dose variations during hypoxiabased dose painting and larger dose changes introduced by altered fractionation schemes.
Methods
Using the widespread AlperHowardFlanders and standard linearquadratic (LQ) models, OER calculations are performed for T1 human kidney and V79 Chinese hamster cells for various dose levels and various hypoxic oxygen partial pressures (pO2) between 0.01 and 20 mmHg as present in clinical situations in vivo. Our work comprises the analysis for both low linear energy transfer (LET) treatment with photons or protons and highLET treatment with heavy ions. A detailed analysis of experimental data from the literature with respect to the dose dependence of the oxygen effect is performed, revealing controversial opinions whether the OER increases, decreases or stays constant with dose.
Results
The behavior of the OER with dose per fraction depends primarily on the ratios of the LQ parameters alpha and beta under hypoxic and aerobic conditions, which themselves depend on LET, pO2 and the cell or tissue type. According to our calculations, the OER variations with dose in vivo for lowLET treatments are moderate, with changes in the OER up to 11% for dose painting (1 or 3 Gy per fraction compared to 2 Gy) and up to 22% in hyper/hypofractionation (0.5 or 20 Gy per fraction compared to 2 Gy) for oxygen tensions between 0.2 and 20 mmHg typically measured clinically in hypoxic tumors. For extremely hypoxic cells (0.01 mmHg), the dose dependence of the OER becomes more pronounced (up to 36%). For high LET, OER variations up to 4% for the whole range of oxygen tensions between 0.01 and 20 mmHg were found, which were much smaller than for low LET.
Conclusions
The formalism presented in this paper can be used for various tissue and radiation types to estimate OER variations with dose and help to decide in clinical practice whether some dose changes in dose painting or in fractionation can bring more benefit in terms of the OER in the treatment of a specific hypoxic tumor.
Keywords
Oxygen enhancement ratio hypoxia oxygen effect fractionation high LET radiation therapyBackground
The poor treatment prognosis for tumors with oxygendeficient areas is usually attributed to the decreased radiosensitivity of hypoxic cells. Hypoxia has been identified in many tumor types and the disadvantageous impact of hypoxia on local tumor control has been well recognized [1–4]. Due to the rapid development of noninvasive imaging methods to estimate the spatial distribution of the oxygen partial pressure within the tumor using different hypoxia markers [5–7], patientspecific treatment planning using dose painting or dose escalation in multifraction regimes based on functional hypoxia imaging is now considered a promising approach to overcome hypoxia in radiotherapy [8, 9].
The radioprotective effect of hypoxia can be expressed quantitatively by the oxygen enhancement ratio (OER), which is defined as the ratio of doses D(p_{ 1 }) and D(p_{ 2 }) given under two different oxygenation conditions with oxygen partial pressures p_{ 1 } ≤p_{ 2 } to produce the same biological effect. Depending on the interest of the researcher, the comparison can be done between hypoxicaerobic, anoxichypoxic, aerobichypobaric etc. conditions. Historically the OER has been defined as the radiation dose under anoxic conditions (p_{ 1 } = 0 mmHg = const) divided by the dose under conditions of some partial pressure of oxygen p_{ 2 } = p. In this way, the OER typically increases from unity at anoxic conditions to approximately 3 for normoxic conditions in vitro. Alternatively, OER is often stated as the ratio of the dose to hypoxic cells (at different levels of p_{ 1 } = p) to the dose to aerobic cells (usually in air, p_{ 2 } = 160 mmHg = const). In this case the OER decreases with increasing oxygen partial pressure in the cell environment, and the term hypoxia reduction factor (HRF) is sometimes used instead of OER here. In this paper, we employ the most general definition of OER with variable p_{ 1 } and p_{ 2 } at the same time.
It has been observed that the OER depends on many factors such as the oxygen partial pressures (pO_{2}) for hypoxic and aerobic conditions, the tissue type, the linear energy transfer (LET) of the radiation and the chosen cell survival level (or alternatively the applied local dose). In this work we focus on the dose dependence of the OER and investigate if and how this dependence is important for clinical applications in radiotherapy. To open the discussion we review the experimental data of the dose dependence of the OER published from 1975 up to 2010. In contrary to the relative biological effectiveness (RBE), which was observed to decrease with increasing dose per fraction independent of radiation type and cell line, no clear tendency for the dose dependence of the OER (decreasing, increasing or constant) was reported in the literature.
The purpose of this paper is to estimate the amount of potential OER variations with dose in low and highLET radiotherapy and to assess the potential impact for clinical applications. Based on the previously developed OER model [10] and parameters for several tissue types, we will evaluate two clinically relevant scenarios: the first scenario deals with dose painting strategies, where an inhomogeneous dose within the target volume according to the oxygenation status is prescribed. Typically this involves relatively small local changes of the dose in the order of ± 50% of the mean dose per fraction. The second scenario considers much larger changes in the dose per fraction which occur if the overall fractionation scheme is modified (hyper or hypofractionation).
Methods
The survival of cells after exposure to a radiation dose is often described by the linearquadratic (LQ) model [11]. This model is now in widespread use in both experimental and clinical radiobiology and generally works well in reproducing experimental results both in vitro and in vivo. We employed the standard LQ model with its two radiosensitivity parameters α and β in its most simplified form (without reoxygenation between fractions). Dose rate effects [12, 13] were also not taken into account.
Summary of references to published mammalian cell survival data measured in experiments at low LET
Ref.  Cells  Rad. type (LET)  α_{ a }/α_{ h }  (β_{ a }/β_{ h })^{1/2}  OER(D) 

[30]  V79379A  xrays  1.4  3.8  ↗ 
[31]  V79379A  xrays  2.0  3.6  ↗ 
[32]  V79753B  xrays  3.9  2.8  ↘ 
[25]  V79379A  xrays  →  
[33]  V79B  xrays  ↗  
[28]  V79171  xrays  2.3  3.6  ↗ 
[26]  V79  protons (0.7 keV/μm)  1.7  4.5  ↗ 
V79  protons (1.9 keV/μm)  1.9  3.2  ↗  
V79  xrays  3.6  2.5  ↘  
[34]  V79  xrays  ↘  
[27]  V79  xrays  3.2  3.1  → 
[33]  V79  xrays  →  
CHO6  xrays  →  
[35]  CHOK1  ^{60}Co γrays  ↗  
CHOxrs6  ^{60}Co γrays  ↗  
[36]  CHOK1  xrays  ↗  
[37]  CHLF  ^{60}Co γrays  →  
[38]  R1  xrays  →  
[39]  R1  xrays  →  
[40]  FSaII  ^{60}Co γrays  1.6  2.8  ↗ 
FSaII  protons (1.9 keV/μm)  1.9  2.6  ↗  
[41]  T1  xrays  5.0  2.3  ↘ 
[33]  AA8  xrays  ↘  
[42]  EMT6  ^{60}Co γrays  ↗  
[43]  B14 FAF28  ^{60}Co γrays  ↘  
[44]  U251  ^{60}Co γrays  4.0  2.5  ↘ 
Summary of references to published mammalian cell survival data measured in experiments at high LET
Ref.  Cells  Rad. type  LET (keV/μm)  α_{ a }/α_{ h }  (β_{ a }/β_{ h })^{1/2}  OER(D) 

[26]  V79  argon  94  1.5  3.4  ↗ 
[45]*  V79  carbon  102  2.7  1.1  ↘ 
V79  neon  110  2.0  1.3  ↘  
HSG  carbon  88  2.2  1.3  ↘  
HSG  neon  84  2.9  1.6  ↘  
[38]  R1  αparticles  110  →  
[39]  R1  carbon  95  →  
[46]  R1  carbon  90  2.0  1.6  ↘ 
R1  carbon  95  1.8  1.7  →  
R1  neon  90  1.4  1.9  ↗  
R1  neon  120  1.7  1.5  →  
R1  argon  95  2.1  1.3  ↘  
[41]  T1  carbon  85  2.7  1.4  ↘ 
T1  neon  100  1.8  2.9  ↗  
T1  argon  81  2.3  2.4  →  
T1  argon  91  2.0  2.5  ↗  
T1  argon  117  1.7  1.7  →  
[44]  U251  8 keV xrays  > 50  1.3  > 2.9  ↗ 
Since the first derivative of OER with respect to D never vanishes for D > 0 (unless α_{ a }/α _{ h } = (β_{ a }/β_{ h })^{1/2}), the OER increases with dose per fraction if the ratio α_{ a }/α _{ h } is smaller than (β_{ a }/β_{ h })^{1/2} for a specific cell line, and decreases with dose for α_{ a }/α _{ h } > (β_{ a }/β_{ h })^{1/2}. In the case of α_{ a }/α _{ h } = (β_{ a }/β_{ h })^{1/2}, the OER is independent of dose. In contrast to the RBE, where the ratio of the α values is typically much larger than the square root of the ratio of the β values (leading to a decreasing RBE with dose), these ratios show larger variability for the OER.
with oxygen partial pressures p_{ a } and p_{ h } under hypoxic and aerobic conditions (p_{ h } ≤p_{ a } ). As mentioned in Background these oxygen partial pressures are two independent values of the oxygen tension and the OER depends on the choice of both of them. Equation (10) can be used to describe the OER for different radiation types (lowLET and highLET) and for various oxygen levels relevant for cell experiments in vitro and clinical situations in vivo. This OER model is a simple tool to quantify the oxygen effect in a practical way. The results of our model for the dependence of OER on LET and pO_{2} as discussed in our previous paper are in good agreement with preclinical and clinical studies [10].
In this paper we deal with the dependence of OER on dose per fraction for different irradiation types and degrees of hypoxia. The OER calculations were done for two cell types (V79 Chinese hamster cells and T1 human kidney cells) because only for these cell lines there was sufficient experimental data both in the low and highLET area (Tables 1 and 2). For low LET (0.22 keV/μm), the mean values of α and β found in the literature were taken (V79: α _{ a } = 0.135 Gy^{1}, β _{ a } = 0.032 Gy^{2}, α _{ h } = 0.06 Gy^{1}, β _{ h } = 0.003 Gy^{2}; T1: α _{ a } = 0.10 Gy^{1}, β _{ a } = 0.047 Gy^{2}, α _{ h } = 0.02 Gy^{1}, β _{ h } = 0.009 Gy^{2}). The values for V79 are similar to the data used by Carlson et al. [19] in an OER modeling study for prostate cancer and might therefore be relevant for clinical applications as well. The tissue parameters for high LET were estimated by an LETdependent fitting of the aerobic and hypoxic experimental data in vitro for α and β in the full highLET region between 10 and 260 keV/μm [10] to determine the parameters at 100 keV/μm (V79: α _{ a } = 0.75 Gy^{1}, β _{ a } = 0.061 Gy^{2}, α _{ h } = 0.41 Gy^{1}, β _{ h } = 0.014 Gy^{2}; T1: α _{ a } = 0.62 Gy^{1}, β _{ a } = 0.067 Gy^{2}, α _{ h } = 0.35 Gy^{1}, β _{ h } = 0.019 Gy^{2}).
Results
For V79 cells, the OER increases with dose at low and high LET, whereas it decreases with dose for T1 cells at low LET and stays almost constant for high LET. The dashed lines in Figure 2 show the OER for a typical cell experiment in vitro under extreme hypoxia (p_{ h } = 0.01 mmHg, p_{ a } = 160 mmHg). The ratios of the employed LQ parameters in vitro under hypoxic and aerobic conditions, which determine how and how strong the OER changes with increasing dose per fraction, were α_{ a }/α _{ h } = 2.25, (β_{ a }/β_{ h })^{1/2} = 3.27 (V79 cells) and α_{ a }/α _{ h } = 5.0, (β_{ a }/β_{ h })^{1/2} = 2.29 (T1 cells) at low LET and α_{ a }/α _{ h } = 1.83, (β_{ a }/β_{ h })^{1/2} = 2.09 (V79) and α_{ a }/α _{ h } = 1.77, (β_{ a }/β_{ h })^{1/2} = 1.88 (T1) at high LET. They can be compared with the ratios listed in Tables 1 and 2. If for example the value α_{ a }/α _{ h } for some experiment from Table 1 or 2 is lower than the value given above for V79 and the value (β_{ a }/β_{ h })^{1/2} higher as above for the same cell line and LET range, the curves of the dose dependence of OER will be steeper than in Figure 2 and the dose dependence is more pronounced.
OER variations with dose per fraction
Low LET (1 keV/μm)  High LET (100 keV/μm)  

Dose (Gy)  OER  Dose (Gy)  OER  
V79  T1  V79  T1  
Baseline  2.0  2.16  2.74  1.0  1.70  1.66 
Scenario 1  1.0  2.09 (3%)  3.05 (+11%)  0.5  1.69 (1%)  1.66 (0%) 
(dose painting)  3.0  2.21 (+2%)  2.57 (6%)  1.5  1.70 (0%)  1.66 (0%) 
Scenario 2  0.5  2.05 (5%)  3.32 (+21%)  0.25  1.69 (1%)  1.66 (+0%) 
(hyper/hypofractionation)  20  2.52 (+17%)  2.14 (22%)  10  1.77 (+4%)  1.68 (+1%) 
Discussion
In previous modelling studies regarding the OER [6, 16, 17, 24, 25], both doseindependent and dosedependent implementations of the OER were used, mostly based on the AlperHowardFlanders formula of relative radiosensitivity (Eq. 8) or some modifications of this equation. With respect to experimental data published in the literature, decreasing, increasing and constant OERs with increasing dose for various cell lines and various radiation types were reported (Tables 1 and 2). Corresponding to our calculation in the framework of the LQ model, the OER depends on dose and its behavior is determined by the ratios of the LQ parameters α_{ a }/α _{ h } and (β_{ a }/β_{ h })^{1/2} under aerobic and hypoxic conditions (Eq. 7). Since these ratios can vary considerably with tissue/cell type, LET and pO_{2} (at least experimentally), this could explain the controversial findings from the publications in Tables 1 and 2. Only if these ratios are equal, the dose dependence disappears. This can be implemented on the modelling side if the pO_{2} dependence of α and β is given by α(p)= α_{ a }/f(p) and β(p) = β_{ a }/f(p)^{2} (with the same function f(p) for both α and β) as used by Malinen et al. [6] or Carlson et al. [25]. One can argue whether or not this special case is actually realized in all cell lines or tissue types [25] (at least within experimental uncertainties), and whether a mechanistic interpretation of the LQ parameters and the underlying microscopic processes of radiation damage (see e.g. [12]) supports this situation. Given the caveats of mechanistic interpretations in radiation biology and the relatively large experimental error bars, a dose dependence of the OER can certainly not be excluded, which motivated us to study the potential clinical impact of a dose dependent OER.
The calculations performed in this work can be used to estimate  for a certain cell line or tissue type, irradiation type and oxygenation condition  whether, how and with which magnitude the OER varies with dose. Although the underlying OER model [10] used here was primarily based on experimental data from the literature in vitro, due to the implemented AlperHowardFlanders concept it can also provide reasonable predictions for preclinical and clinical situations in vivo (as discussed in detail in [10]). We therefore conclude that our analysis of the dependence of OER on dose per fraction has also some value for the assessment of realistic clinical situations, at least qualitatively.
Our investigation of the oxygen effect at clinically measured median oxygen tensions p_{ h } between 0.2 and 20 mmHg shows a relatively moderate dependence of the OER on dose per fraction (Table 3). The higher the oxygen partial pressure in a tumor, the less pronounced is the variation of the OER with dose (Figure 2). For hypoxiabased dose painting in the target volume (scenario 1, with local dose variations in the target between 1 and 3 Gy for low LET), the changes of the local OER in vivo were below ± 11% relative to the baseline of 2 Gy per fraction (Table 3). For high LET (dose variations between 0.5 and 1.5 Gy), these variations were much smaller (up to 1%). If the dose per fraction is varied over a larger range (by changing the fractionation scheme, scenario 2), OER variations in the order of ± 21% (0.5 Gy vs. 2 Gy per fraction) and ± 22% (20 Gy vs. 2 Gy per fraction) can be seen for low LET. Again, this was much smaller for high LET (up to 1% difference in OER for 0.25 Gy vs. 1 Gy and up to 4% for 10 Gy vs. 1 Gy per fraction). For a potential portion of extremely hypoxic cells within a tumor the dose dependence of the OER becomes more pronounced in lowLET treatment (see Results), but stays still moderate for high LET. One has to note that the values above depend strongly on the oxygenation conditions and tissue types, and the direction and magnitude of the OER variation could differ from the OERs for V79 and T1 cell lines. Nevertheless, the formalism presented in this paper can be used to estimate OER variations with dose and help to decide in clinical practice whether some changes in fractionation (hyper or hypofractionation) can bring more benefit in the treatment of patients with a specific hypoxic tumor. For example, for human salivary gland (HSG) tumor cells the OER decreases with increasing dose under highLET irradiation (Table 2). This means that a hypofractionated treatment in highLET radiotherapy with heavy ions could be more advantageous with respect to the oxygen effect for patients with this kind of tumor. However, this can only be confirmed by clinical studies using high LET and various fractionation schemes, which (to the best of our knowledge) are currently not available for this tumor type. Alternatively, small animal experiments with implanted human tumors (which often exhibit large hypoxic fractions) could be a means to evaluate the impact of OER for various fractions schemes in a preclinical setting, and to validate the findings of our modeling study.
Another example can be based on the data published by Malinen et al. [6]. The authors investigated a spontaneous sarcoma of a dog with different hypoxic compounds within the tumor. In the most hypoxic area the measured average oxygen partial pressure p_{ h } was 0.2 mmHg. Using their data we calculated for this pO_{2} value α_{ a }/α _{ h } = 1.75 and (β_{ a }/β_{ h })^{1/2} = 3.25 (Eq. 4 and Table 1 in [6]). Compared with our analysis for V79 cell line under lowLET irradiation (α_{ a }/α _{ h } = 2.25 and (β_{ a }/β_{ h }_{ h })^{1/2} = 3.27 and Figure 2A) the dependence of OER on dose per fraction for sarcomas from the study by Malinen et al. is also relatively moderate. The higher the oxygen partial pressure in a tumor, the less significant is the variation in the OER with the choice of dose per fraction. However, if hypoxia based treatment planning shall be performed, a larger number of fractions with smaller doses (hyperfractionation) could  due to the increasing OER with dose  bring more benefit with respect to the oxygen effect for the treatment of the sarcoma presented by Malinen et al. [6].
Although the OER seems to be only moderately dependent on the choice of dose per fraction, tumor hypoxia itself has a large negative effect on cell killing and there is potential for large errors in calculation of alternative dose fractionation schemas using models that do not account for tumor hypoxia at all [19], even if the dose dependence is not considered explicitly. In the case where measurements of oxygen partial pressures in a tumor are possible (e.g. with an Eppendorf histograph or noninvasive methods), the additional dose to hypoxic areas of the tumor required to achieve a constant biological effect in the whole target can be calculated in the framework of our model (Eq. 10) for various radiation types, oxygen tensions and tissue types. In the long run, this may help to overcome the adverse effects of low oxygen concentrations in many tumors.
Conclusions
Since tumors with hypoxic areas exist and the treatment outcome of patients with hypoxic tumors is relatively poor, new predictive models are required to individualize and improve the treatment strategies for radiotherapy. In this work we investigated theoretically the importance of the dose dependence of the OER for clinical applications in radiotherapy. The analysis was performed for two scenarios: small dose variations within the target during dose painting and larger changes of the dose per fraction for different fractionation schemes. The calculations were performed for both lowLET treatment with photons or protons and highLET treatment with heavy ions.
According to our analysis, the OER in clinical practice is moderately sensitive to the choice of dose per fraction. The OER can increase, decrease or remain a constant with increasing dose per fraction, and this behavior is determined by the ratios of the LQ parameters under hypoxic and aerobic conditions. These effects should be taken into account in hypoxia based treatment plan optimization. The formalism presented in this paper can be used to estimate OER variations with dose and to help with clinical decisions about any changes in dose prescription or treatment planning with respect to the oxygen effect. If simplified models without explicit consideration of the dose dependence are used for optimization in dose painting or changes of the fractionation scheme, our methods can be used to estimate the potential error in OER due to dose variations.
List of abbreviations used
 EF:

Enhancement factor
 HRF:

Hypoxia reduction factor
 LET:

Linear energy transfer
 LQ model:

Linearquadratic model
 OER:

Oxygen enhancement ratio
 pO_{2}:

Oxygen partial pressure
 RBE:

Relative biological effectiveness
 RR:

Relative Radiosensitivity.
Declarations
Acknowledgements
Supported by DFG Cluster of Excellence: MunichCentre for Advanced Photonics and by DFG grant WI 3745/11.
Authors’ Affiliations
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