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Why Do Both Mean Dose and V≥x Often Predict for Normal Tissue Outcomes?

Open AccessPublished:July 27, 2022DOI:https://doi.org/10.1016/j.adro.2022.101039
      In the QUANTEC, HyTEC and PENTEC reviews, mean dose was identified as a reasonable predictor for toxicity risk in some normal organs (e.g. 1-5), particularly in organs classically considered to have a “parallel-like architecture”; e.g. lung, liver, parotid. The utility of mean dose as a predictive metric is puzzling, as an underlying biologic basis is challenging to define. So, why does this work? Superficially, in a group of patients treated in a relatively-uniform manner (e.g. with respect to dose and beam arrangement), mean dose may simply be highly correlated with other potentially “more-biologically-logical” parameters such as threshold-based metrics (e.g. V≥x, the fraction of organ receiving a dose ≥ x). We herein present an alternative explanation for the association between mean dose and clinical outcomes based on both (a) the dose-response relationship between local dose and local function, and (b) basic geometric principles for modern RT beams.

      The dose-response relationship between local dose and local function

      For parallel-like organs, it is reasonable to consider that each region functions relatively independently, and that changes in global organ function (that manifest as clinical toxicity) might be expected to reflect the sum of regional effects (often termed ‘integrated response’) (6);
      Sumregionaleffects=D=0D=DmaxVD*EffectD


      Where VD = the fractional volume at dose D, and EffectD = degree of local effect at dose D.
      If the dose-response function for regional injury (i.e. EffectD) is ≈ linear within the clinically-relevant dose range, it is reasonable to suppose that the sum of regional effects might be fairly well correlated with mean dose. Inherent to this argument is that volume a surrogate for function (i.e. that different regions of equal size carry equal functional burdens). We recognize that this overall model is certainly simplistic, but we believe it is a useful construct within which to discuss these issues.
      Using lung as an example, the regional function dose response with conventional fractionation, (assessed using perfusion single photon emission tomography) is roughly linear in the 10-55 Gy range (Figure 1a); i.e. doses received by much of the incidentally-irradiated lung (7, 8). A similar linear response was observed following lung stereotactic body radiotherapy (SBRT) (9). The same is seen in other organs. For parotid, the regional dose response function (assessed using metabolic clearance of 11C-methonine) is noisier, but is approximately linear from 5-40 Gy (Figure 1b) (10). While the deep parotid is often close to the target (and thus receives ≈ target doses of 60-70 Gy), much of the parotid receives lower doses. For liver, regional portal vein perfusion one month post-RT declined linearly with local dose (11).
      Figure 1:
      Figure 1Dose response curves for regional function. Left: reductions in regional lung perfusion, data from Duke and Netherlands Cancer Institute (based on 70 and 50 patients, respectively, with non-small cell lung cancer [NSCLCa]; adapted from Fried et al. (6). Each data point is the weighted average from multiple regions from multiple patients. Right: reductions in voxel-specific parotid function estimated by 11C-methionine clearance via PET, Buus, et al. (10). For both images, the solid line is added to illustrate that the data resembles a linear function. The curved line in the right- is from the Buus publication. Adapted with permission from references 6 and 10.
      Thus, from a physiologic perspective, mean dose is perhaps predictive due to the approximate linear nature of the dose response function for regional injury, and that much of the incidental dose is in this ‘region of linearity’. In other words, mean dose is a reasonable surrogate for the integrated response.
      Nevertheless, mean dose is also suboptimal since it ignores where dose is delivered (thus, failing to consider possible spatial functional heterogeneities). For example, it has been argued that lung toxicity is more common in patients receiving RT for lower lobe tumors (12). This argument, however, applies both to mean dose and Vx metrics (discussed below). Mean dose also ignores the general shape of the regional dose response functions that appear to have a threshold (below which there is no injury), and a plateau (above which there is no further injury). Reducing regional doses within regions already at doses < threshold, or within the plateau region, will alter mean dose, but not effect the integrated response. To lower the integrated response, at least some of dose reduction has to move regions that are on the plateau or linear regions to lesser doses along the linear or pre-threshold regions. Nevertheless, given the uncertainties in these models, and imprecision of our normal tissue toxicity assessments, the ability to predict clinical outcomes based on mean dose vs. the integrated response are likely similar. The general concept of the threshold and plateau has implications for how one considers competing treatment plans (e.g. with 3D, vs IMRT vs proton-based).
      In the QUANTEC, HyTEC and PENTEC reviews, threshold metrics (e.g. V≥x, percent organ receiving dose ≥ x) are predictive in several parallel-like organs (e.g. 1-4). From a physiologic perspective, this makes sense if we believe there is essentially a ‘step function’ for regional injury (i.e. a steep dose response; regional function ceases above some ‘regionally-toxic’ dose), and one wants to keep some critical organ volume (or fractional volume) below this ‘regionally-toxic’ dose. Given what we know about the often-shallow nature of regional dose response functions (7-10), V≥x metrics are perhaps not logical. Further, the threshold doses sometimes considered for parallel-like organs (e.g. V≥20 for lung) are not consistent with the whole organ tolerance data (e.g. pulmonary risks following fractionated 20 Gy whole lung RT appear low; see Fig 3 in Lung QUANTEC review). Nevertheless, since V≥x metrics are often correlated with metrics such as mean dose (1, 2), they may remain clinically useful. However, this correlation is likely technique-dependent, and with evolving and more-varied treatment techniques, V≥x metrics may not be generalizable or transferrable, and mean dose might be more predictive than V≥x metrics. Conversely, V≥x metrics might be more predictive than mean dose if, for example, the dose response curve for regional dysfunction is steep, and/or the incidental doses received by a meaningful fraction of the organ is outside of the ‘region of linearly’ in the regional dose-response function. And, in these cases, mean dose might be predictive for outcomes due to its correlation with V≥x.

      Basic geometric principles for modern RT beams

      Interestingly, the degree of correlation between the mean dose and any V≥x appears to be related to be number of treatment beam orientations used. This concept is illustrated in the idealized 2D representation in Figure 2, using area, rather than volume. As the number of treatment beam orientations increases, the normal tissue area receiving low doses increases (the absolute area depends on the organ size; ie, R in Figure). Conversely, as the number of beam orientations increases, the area of normal tissue receiving high doses (e.g. prescription dose) will decrease (e.g. area of the polygon formed by the intersection of multiple beams decreases). As the number of beam orientations increases to a high number, the area of the surrounding polygon approaches the target area, thus reducing the normal tissue area receiving the full prescription dose (this is typical with, and is indeed an essence of, SBRT/SRS).
      Figure 2:
      Figure 2The idealized 2D situation of a small circular tumor in a larger circular organ at risk is considered, and ignoring effects related to beam divergence and beam attenuation. (a) As the number of treatment beam orientations increases, the area of normal tissue receiving a low dose increases, and the absolute area depends on the size of the normal organ (e.g. R). As the number of treatment beam orientations increases, the area of normal tissue receiving the full prescription dose decreases approximately as shown (e.g. assuming full dose delivered to the area of the regular polygon formed by the intersection of multiple beams surrounding a circular target). (b) For an arc rotation, the dose to a point at distance L from the target center (DL) / dose to the target dose (Dr) = θ / π. Since sin (θ/2) = r/L; θ = 2 arcsin (DL/Dr), and DL/Dr = 2/π * arcsin (r/L). (c) For an arc rotation, the isodose lines are circular with the relative doses at variable values of L, and the area receiving doses ≥ DL, computed as shown.
      However, the situation at intermediate doses is more complex. For example, consider the area receiving ≥1/12th, ≥1/6th, ≥1/3rd of a 60 Gy prescription dose (analogous to V≥5, V≥10, or V≥20 for a lung target receiving 60 Gy). The best way to reduce the area receiving 1/12th, 1/6th, 1/3rd of the prescription dose is to use >12, >6, and >3 beam orientations, respectively. (This is true since the area within the entrance and exit beam path is typically much larger than the area within the regions closer to the target where the treatment beams intersect, since the targets are usually small relative to total organ volume). Once these threshold numbers of beam orientations are used, intermediate doses will still be delivered in areas where the beams overlap. The sizes of these areas of overlap (again analogous to V≥5, V≥10, or V≥20) are highly dependent on beam orientations and can fluctuate widely with beam number (see Appendix 1 for examples).
      Nevertheless, with large numbers of beam orientations, these intermediate V≥x values typically will stabilize. For example, a full arc rotation in the simple 2D case leads to circular isodose lines, with doses (relative to the target dose) and radii (relative to the target radius) following a simple relationship (Figures 2 b, c). This can provide planners with some a priori estimate of the relative dose to nearby normal structures based on its distance from the target center relative to the target radius, for a simple unoptimized/unmodulated arcs. Since mean dose is essentially constant irrespective of beams and technique (as suggested by the cartoon on the top of Figure 2A, and formally demonstrated by others (13-15), and further discussed in Appendix 1), a stabilizing value for Area≥x (i.e. area at dose ≥X) makes Area≥x and mean dose highly correlated. Thus, with the large number of beam orientations commonly used with SBRT/SRS, V≥x and mean dose will tend to be highly correlated with each other.
      This idealized 2D representation can be readily expanded to a similarly-idealized 3D representation of a cylindrical target (analogous to a lung lesion with respiratory excursion superiorly-inferiorly) within a cylindrical lung, treated with a set of beams orthogonal to the long axis of the cylinder (analogous to an axial arc-based treatment); Figure 3. Interestingly, using a set of reasonable assumptions applied to the idealized model, one can compute V≥20 and mean lung doses for axial treatments of small lung nodules that are similar to clinical data in patients treated using ‘axial-like’ beams (Figure 3 and Appendix 2). We recognize that the assumption to ignore divergence and attenuation are not ideal, but these effects are modest.
      Figure 3:
      Figure 3(a) On the left, an idealized 3D representation of a cylindrical target within a cylindrical normal tissue being treated with an axial arc. The percent of normal tissue volume at dose ≥ DL can be computed as shown (essentially rearranging the equations in the prior figures, derivation in appendix). On the right, the mean dose to the normal organ can be estimated as shown (for the setting of a single treatment beam orientation). The mean dose is relatively-stable irrespective of the number of beam orientation, as suggested by the cartoon on the top of a and formally demonstrated by others (13-15). (b and c) The solid lines are from the idealized 3D model-based calculations of the predicted mean lung dose (panel b) and V20 (panel c) for targets with variable radii from 0.75-2.5 cm. These are highly correlated with the clinical data from one of the author's centers (dots in panels b and c). The fit between the idealized model and the clinical data are clearly imperfect. Nevertheless, it is interesting that a simple model can provide reasonable estimates of clinical data. Further details are provided and discussed in Appendix 2.
      Figure 3:
      Figure 3(a) On the left, an idealized 3D representation of a cylindrical target within a cylindrical normal tissue being treated with an axial arc. The percent of normal tissue volume at dose ≥ DL can be computed as shown (essentially rearranging the equations in the prior figures, derivation in appendix). On the right, the mean dose to the normal organ can be estimated as shown (for the setting of a single treatment beam orientation). The mean dose is relatively-stable irrespective of the number of beam orientation, as suggested by the cartoon on the top of a and formally demonstrated by others (13-15). (b and c) The solid lines are from the idealized 3D model-based calculations of the predicted mean lung dose (panel b) and V20 (panel c) for targets with variable radii from 0.75-2.5 cm. These are highly correlated with the clinical data from one of the author's centers (dots in panels b and c). The fit between the idealized model and the clinical data are clearly imperfect. Nevertheless, it is interesting that a simple model can provide reasonable estimates of clinical data. Further details are provided and discussed in Appendix 2.
      In summary, the limited dose-response data for regional injury for some organs, such as lung and parotid, provide some physiologic rationale that may explain the predictive value of mean dose. However, depending on the steepness of the dose-response function, and the incidental doses delivered to the organ at risk, V≥x values may be physiologically rational as well. Further, there appear to be basic geometric principles that may drive a strong correlation between mean dose and V≥x. Thus, from a pragmatic clinical-utility perspective, both mean dose and V≥x can both be useful. Additional work is needed to better define dose-response functions for regional injury that may help to define optimal dosimetric predictors for global toxicity in parallel-like organs. However, the geometric realities predict high correlations between many of the commonly-used dosimetric parameters (that are indeed seen in clinical data), and these correlations challenge our ability to assess their relative utility in predicting normal tissue injury.

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      Declaration of interests

      The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
      The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:
      Ellen Yorke reports financial support was provided by National Cancer Institute.

      Acknowledgements

      As this work grew out of the HyTEC initiative, we acknowledge the support of the AAPM for the HyTEC project.

      Appendix. Supplementary materials