^{1}, Pavel Stavrev

^{2}, Nadia Stavreva

^{2}and B. Gino Fallone

^{3,a)}

### Abstract

This paper outlines a theoretical approach to the problem of estimating and choosing dose-volume constraints. Following this approach, a method of choosing dose-volume constraints based on biological criteria is proposed. This method is called “reverse normal tissue complication probability (NTCP) mapping into dose-volume space” and may be used as a general guidance to the problem of dose-volume constraint estimation. Dose-volume histograms (DVHs) are randomly simulated, and those resulting in clinically acceptable levels of complication, such as NTCP of , are selected and averaged producing a mean DVH that is proven to result in the same level of NTCP. The points from the averaged DVH are proposed to serve as physical dose-volume constraints. The population-based critical volume and Lyman NTCP models with parameter sets taken from literature sources were used for the NTCP estimation. The impact of the prescribed value of the maximum dose to the organ,, on the averaged DVH and the dose-volume constraint points is investigated. Constraint points for 16 organs are calculated. The impact of the number of constraints to be fulfilled based on the likelihood that a DVH satisfying them will result in an acceptable NTCP is also investigated. It is theoretically proven that the radiation treatmentoptimization based on physical objective functions can sufficiently well restrict the dose to the organs at risk, resulting in sufficiently low NTCP values through the employment of several appropriate dose-volume constraints. At the same time, the pure physical approach to optimization is self-restrictive due to the preassignment of acceptable NTCP levels thus excluding possible better solutions to the problem.

This research was supported by the Alberta Foundation for Medical Research, the Alberta Cancer Board, and the Translational Research Training in Cancer program, as well as the Alberta Cancer Board Research Initiative Program Grant RI-218/20624. We would like to thank Dr. Rufus Scrimger for discussions of clinical maximum dose range values for the 16 organs investigated in this work.

I. INTRODUCTION

II. BACKGROUND

II.A. Some definitions

II.B. NTCP models

II.B.1. The Lyman (Sigmoidal dose response) NTCP model

II.B.2. Critical volume population model

II.C. Model parameters

III. METHOD

III.A. Reverse mapping of NTCP onto DVH space—A theoretical approach for dose-volume constraint estimation

III.A.1. Generation of random DVHs

III.A.2. Scaling the random DVHs

III.A.3. Probability that a DVH, with a user-specified NTCP, passes through a given point in the dose-volume histogram space

III.A.4. Constraint point estimation

III.B. Reverse mapping of NTCP onto DVH space—A theoretical approach to the investigation of the radiobiological impact of a set of dose-volume constraints

IV. RESULTS

V. DISCUSSION

V.A. Suitability of the calculated constraint points for inverse planning

V.B. Random integral DVHs and dose scaling

V.C. Comparison with other methods of dose-volume constraint determination

V.D. Physical versus radiobiological optimization

VI. CONCLUSIONS

### Key Topics

- Dosimetry
- 121.0
- Anatomy
- 61.0
- Radiation treatment
- 15.0
- Cancer
- 9.0
- Medical treatment planning
- 8.0

## Figures

A sample of DVHs with (gray dotted curves) calculated according to the Lyman NTCP model (a) and the CV population NTCP model (b) for the end-point heart pericarditis. Also shown in each of these subplots are the average of all DVHs with along with the constraint points interpolated from this curve (solid black curve with diamond points). The Emami constraints for the heart are shown for comparison as circles, along with the 5% iso-NTCP envelope that passes near to the Emami points. The DVH point probabilities (curves in the plane) calculated for heart are shown for the Lyman model (c) and the CV population model (d). The average of DVHs with is shown (dark curve in plane) for comparison, along with the 67% confidence limits (dashed curves in plane) that were calculated by means of the DVH point probabilities. Panels (e) and (f) show the most probable DVH curves for the heart, calculated from the DVH point probabilities in (c) and (d). The Lyman model was used for (e), and the Lyman NTCP of the most probable DVH is shown in the upper right corner. For (f), the CV population model results are shown, along with the CV population NTCP for this curve.

A sample of DVHs with (gray dotted curves) calculated according to the Lyman NTCP model (a) and the CV population NTCP model (b) for the end-point heart pericarditis. Also shown in each of these subplots are the average of all DVHs with along with the constraint points interpolated from this curve (solid black curve with diamond points). The Emami constraints for the heart are shown for comparison as circles, along with the 5% iso-NTCP envelope that passes near to the Emami points. The DVH point probabilities (curves in the plane) calculated for heart are shown for the Lyman model (c) and the CV population model (d). The average of DVHs with is shown (dark curve in plane) for comparison, along with the 67% confidence limits (dashed curves in plane) that were calculated by means of the DVH point probabilities. Panels (e) and (f) show the most probable DVH curves for the heart, calculated from the DVH point probabilities in (c) and (d). The Lyman model was used for (e), and the Lyman NTCP of the most probable DVH is shown in the upper right corner. For (f), the CV population model results are shown, along with the CV population NTCP for this curve.

Plot of the average DVH for each of six NTCP intervals for the end-point heart pericarditis. Averages were calculated based on the Lyman NTCP model (a) and the CV population model (b). From the lowest volume to the highest volume curves, the intervals are (first solid line), [10%, 20%] (first dotted line), [20%, 30%] (dash-dotted line), [50%, 60%] (dashed line), [70%, 80%] (second solid line), and [90%, 100%] (second dotted line).

Plot of the average DVH for each of six NTCP intervals for the end-point heart pericarditis. Averages were calculated based on the Lyman NTCP model (a) and the CV population model (b). From the lowest volume to the highest volume curves, the intervals are (first solid line), [10%, 20%] (first dotted line), [20%, 30%] (dash-dotted line), [50%, 60%] (dashed line), [70%, 80%] (second solid line), and [90%, 100%] (second dotted line).

A subset of DVHs that have a CV population NTCP of for the end-point lung pneumonitis (gray dashed curves), the average of these DVHs, and the interpolated constraint points (solid curve with black diamonds). Each subplot shows these curves for a different DVH dose scaling: (a) , (b) , (c) , and (d) . In each subplot, the 5% iso-NTCP envelope is shown along with the Emami points for lung pneumonitis. The NTCPs are those of the average DVHs.

A subset of DVHs that have a CV population NTCP of for the end-point lung pneumonitis (gray dashed curves), the average of these DVHs, and the interpolated constraint points (solid curve with black diamonds). Each subplot shows these curves for a different DVH dose scaling: (a) , (b) , (c) , and (d) . In each subplot, the 5% iso-NTCP envelope is shown along with the Emami points for lung pneumonitis. The NTCPs are those of the average DVHs.

Lyman (left) and CV population (right) NTCP probability distributions for the end-point brain necrosis for the given sets of calculated dose-volume constraint points. A total of DVHs were simulated in order to build these distributions, and of those, the ones that passed within a vicinity of were deemed to satisfy the constraints. Shown in each subplot are two additional quantities: the probability that a DVH that satisfies the given constraint(s) will have a NTCP that is greater than 5.5% and the 95% confidence intervals (CIs) for the distributions. In Figs. 4(a) and 4(b), the NTCP distributions for the Emami , constraint point is shown for comparison (black line).

Lyman (left) and CV population (right) NTCP probability distributions for the end-point brain necrosis for the given sets of calculated dose-volume constraint points. A total of DVHs were simulated in order to build these distributions, and of those, the ones that passed within a vicinity of were deemed to satisfy the constraints. Shown in each subplot are two additional quantities: the probability that a DVH that satisfies the given constraint(s) will have a NTCP that is greater than 5.5% and the 95% confidence intervals (CIs) for the distributions. In Figs. 4(a) and 4(b), the NTCP distributions for the Emami , constraint point is shown for comparison (black line).

Averaged DVH (dotted line) and dose-volume constraint estimates (diamonds) for the rectum, calculated by means of the reverse mapping method. The Lyman NTCP model with the Burman parameters was used to obtain these constraint points. Shown for comparison are the averaged DVH for nonbleeders from Jackson *et al.* (Ref. 60) (upper solid curve) and the lower limit of its 67% confidence range (lower solid curve). The Lyman NTCP of the nonbleeder DVH is given in the upper right corner of this plot.

Averaged DVH (dotted line) and dose-volume constraint estimates (diamonds) for the rectum, calculated by means of the reverse mapping method. The Lyman NTCP model with the Burman parameters was used to obtain these constraint points. Shown for comparison are the averaged DVH for nonbleeders from Jackson *et al.* (Ref. 60) (upper solid curve) and the lower limit of its 67% confidence range (lower solid curve). The Lyman NTCP of the nonbleeder DVH is given in the upper right corner of this plot.

## Tables

Estimates for the clinical maximum dose range to 16 critical structures (, ) that typically occur during the listed treatments (values based on treatments given at the Cross Cancer Institute). Also shown are values for the maximum dose range ( and ) calculated according to both the Lyman and CV population models. The parameters and are used to scale randomly generated DVHs appropriately to calculate constraint points using the reverse mapping method. The following abbreviations are used: CNS—central nervous system; PTV—planning target volume; H&N—head and neck.

Estimates for the clinical maximum dose range to 16 critical structures (, ) that typically occur during the listed treatments (values based on treatments given at the Cross Cancer Institute). Also shown are values for the maximum dose range ( and ) calculated according to both the Lyman and CV population models. The parameters and are used to scale randomly generated DVHs appropriately to calculate constraint points using the reverse mapping method. The following abbreviations are used: CNS—central nervous system; PTV—planning target volume; H&N—head and neck.

Constraint points interpolated from the average of DVHs with a Lyman NTCP of for 16 organs. For each of the relative volumes shown across the top of the chart, the interpolated dose in grays is given for each organ if this value is nonzero. The far right column in the table shows the fraction of randomly generated DVHs, out of a total of , that have a NTCP of , .

Constraint points interpolated from the average of DVHs with a Lyman NTCP of for 16 organs. For each of the relative volumes shown across the top of the chart, the interpolated dose in grays is given for each organ if this value is nonzero. The far right column in the table shows the fraction of randomly generated DVHs, out of a total of , that have a NTCP of , .

Same as Table II, but calculated using the CV population NTCP model.

Same as Table II, but calculated using the CV population NTCP model.

This table illustrates the effect of range on calculated constraint points. Dose-volume constraint points were calculated for the lungs with the random DVHs scaled according to ranges of , , , and , using both the Lyman and CV population models.

This table illustrates the effect of range on calculated constraint points. Dose-volume constraint points were calculated for the lungs with the random DVHs scaled according to ranges of , , , and , using both the Lyman and CV population models.

This table shows three quantities calculated for the end-point brain necrosis for the given sets of constraints. Calculations were done with both the Lyman and CV population NTCP models, and for each constraint volume given, a dose value from either Table II (for the Lyman model analysis) or Table III (for the CV population model analysis) was selected (except for the Emami constraint point, for which the dose was ). The first quantity in this table is , which represents the probability that a DVH with passes within the -vicinity of the chosen constraint(s). The value is the probability that a DVH that satisfies the chosen constraint(s) has a NTCP that is greater than 5.5%. Finally, this table shows the NTCP range in which 95% of the DVHs satisfying the constraint(s) result.

This table shows three quantities calculated for the end-point brain necrosis for the given sets of constraints. Calculations were done with both the Lyman and CV population NTCP models, and for each constraint volume given, a dose value from either Table II (for the Lyman model analysis) or Table III (for the CV population model analysis) was selected (except for the Emami constraint point, for which the dose was ). The first quantity in this table is , which represents the probability that a DVH with passes within the -vicinity of the chosen constraint(s). The value is the probability that a DVH that satisfies the chosen constraint(s) has a NTCP that is greater than 5.5%. Finally, this table shows the NTCP range in which 95% of the DVHs satisfying the constraint(s) result.

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