^{1,a)}, Ingrid Spadinger

^{1}, Rosey Rasoda

^{2}, W. James Morris

^{3}and Septimiu Salcudean

^{4}

### Abstract

**Purpose:**

In postimplant dosimetry for prostate brachytherapy,dose is commonly calculated using the TG-43 1D formalism, because seed orientations are difficult to determine from CTimages, the current standard for the procedure. However, the orientation of stranded seeds soon after implantation is predictable, as these seeds tend to maintain their relative spacing, and orient themselves along the implant trajectory. The aim of this study was to develop a method for determining seed orientations from reconstructed strand trajectories, and to use this information to investigate the dosimetric impact of applying the TG-43 2D formalism to clinical postimplant analysis.

**Methods:**

Using in-house software, the preplan to postimplant seed correspondence was determined for a cohort of 30 patients during routine day-0 CT-based postimplant dosimetry. All patients were implanted with stranded-seed trains. Spline curves were fit to each set of seeds composing a strand, with the requirement that the distance along the spline between seeds be equal to the seed spacing within the strand. The orientations of the seeds were estimated by the tangents to the spline at each seed centroid. Dose distributions were then determined using the 1D and 2D TG-43 formalisms. These were compared using the TG-137 recommended dose metrics for the prostate, prostatic urethra, and rectum.

**Results:**

Seven hundred and sixty one strands were analyzed in total. Defining the*z*-axis to be cranial-positive and the *x*-axis to be left-lateral positive in the CT coordinate system, the average seed had an inclination of 21° ± 10° and an azimuth of −81° ± 57°. These values correspond to the average strand rising anteriorly from apex to base, approximately parallel to the midsagittal plane. Clinically minor but statistically significant differences in dose metrics were noted. Compared to the 2D calculation, the 1D calculation underestimated prostate V100 by 1.1% and D90 by 2.3 Gy, while overestimating V150 and V200 by 1.6% and 1.3%, respectively. Urethral and rectal dose quantifiers tended to be underestimated by the 1D calculation. The most pronounced differences were in the urethral D30 and rectal D2cc, which rose by 3.8 and 1.9 Gy, respectively, using the 2D calculation. The total volume of the 100% isodose region as a percentage of the prostate volume was found to increase by 0.4%.

**Conclusions:**

Stranded seeds in the supine patient are not oriented in a uniformly random manner, nor are they aligned along the axis of the CT scanner. Instead, this study identified a consistent anterior pitch that is likely attributable to differences in patient pose between implant and CTimaging. The angle of the ultrasound probe with respect to the patient during implant may have also been a contributing factor. The dose metrics derived using the 1D formalism were found to be within 2%, on average, of those derived using the 2D formalism. For greater accuracy, 2D dosimetry can be pursued using the strand-fitting method described in this work. If a 1D representation is used, integrating over the empirically determined seed orientation density reported here may be more appropriate than assuming that seed inclinations are distributed uniformly.

I. INTRODUCTION

II. METHODS

II.A. Study cohort

II.B. Plan reconstruction

II.C. Strand fitting

II.D. Effect of seed localization uncertainty on fit

II.E. Spline sensitivity analysis

II.F. Phantom evaluation

II.G. Dose calculations using the 2D formalism

II.H. Evaluation

II.H.1. 2D anisotropy on day-0 dosimetry

III. RESULTS

III.A. Phantom validation

III.B. Sensitivity simulation

III.C. Strand orientation

III.D. Dosimetric impact of computing seed orientation

IV. DISCUSSION

V. CONCLUSIONS

### Key Topics

- Dosimetry
- 42.0
- Anisotropy
- 18.0
- Computed tomography
- 17.0
- Medical imaging
- 8.0
- Brachytherapy
- 7.0

## Figures

Incorporating uncertainty in the determined seed orientation. The bold vector represents the nominal fitted orientation of the seed which defines the *z*-axis in the TG-43 coordinate system. The coordinates of a dose point in this frame are (*r*,θ,φ). The error in the seed orientation relative to the nominal orientation is described by inclination α and azimuth β. The separation angle γ is the angle between the seed and the measurement point.

Incorporating uncertainty in the determined seed orientation. The bold vector represents the nominal fitted orientation of the seed which defines the *z*-axis in the TG-43 coordinate system. The coordinates of a dose point in this frame are (*r*,θ,φ). The error in the seed orientation relative to the nominal orientation is described by inclination α and azimuth β. The separation angle γ is the angle between the seed and the measurement point.

Phantom validation of the fitting procedure. Strands were embedded into grooves based on (a) cubic and (b) quadratic functions, which were etched into sheets of solid water. The phantom was imaged and the seeds segmented and fit using the spline algorithm. For both cases, the RMS error was <0.3, the maximum absolute error was 0.5 mm, and the maximum error in the curve tangent (seed orientation) was 2°.

Phantom validation of the fitting procedure. Strands were embedded into grooves based on (a) cubic and (b) quadratic functions, which were etched into sheets of solid water. The phantom was imaged and the seeds segmented and fit using the spline algorithm. For both cases, the RMS error was <0.3, the maximum absolute error was 0.5 mm, and the maximum error in the curve tangent (seed orientation) was 2°.

The sensitivity simulation-derived probability density of the seed inclination error α for four different levels of applied noise in (a) central and (b) terminal seeds. Noise levels (see Table I) are in increasing order of magnitude.

The sensitivity simulation-derived probability density of the seed inclination error α for four different levels of applied noise in (a) central and (b) terminal seeds. Noise levels (see Table I) are in increasing order of magnitude.

The sensitivity simulation-derived probability density of the seed azimuth error β for four different levels of applied noise in (a) central and (b) terminal seeds. Noise levels (see Table I) are in increasing order of magnitude.

The sensitivity simulation-derived probability density of the seed azimuth error β for four different levels of applied noise in (a) central and (b) terminal seeds. Noise levels (see Table I) are in increasing order of magnitude.

The probability densities of the fitted seed inclination (in deg^{−1}) plotted for measurement points every 10° of inclination for (a) central seeds and (b) terminal seeds. Multiple densities are overlaid on a single plot for compactness; each is centered at the measurement inclination (θ) and falls to zero on either side. The dependence of these plots on the azimuth φ of the measurement point can be seen when comparing the densities of points at and using the probability density of the fitted seed azimuth derived from sensitivity-simulations. Making the assumption that *P*(β) is uniform splits the difference in the extremes and eliminates the dependence of the measurement azimuth [in the seed (TG-43) frame].

The probability densities of the fitted seed inclination (in deg^{−1}) plotted for measurement points every 10° of inclination for (a) central seeds and (b) terminal seeds. Multiple densities are overlaid on a single plot for compactness; each is centered at the measurement inclination (θ) and falls to zero on either side. The dependence of these plots on the azimuth φ of the measurement point can be seen when comparing the densities of points at and using the probability density of the fitted seed azimuth derived from sensitivity-simulations. Making the assumption that *P*(β) is uniform splits the difference in the extremes and eliminates the dependence of the measurement azimuth [in the seed (TG-43) frame].

Probability density (in deg^{−1}) of the (a) inclination and (b) azimuth for stranded seeds in the CT coordinate system. Histogram bars are the raw counts in bins of 5°, and solid curves represent the orientation probability density *P* _{SO}. The probability density assumed by the 1D formalism is shown for comparison. (c) In this image, *P* _{SO} is projected onto the unit sphere using bright intensity to highlight regions of high density. The grid interval is 10°, and the directions of the CT axes are labeled (A—anterior, P—posterior, R—right, L—left, I—inferior, S—superior). Most seeds exhibit an approximately 20° pitch from the coronal plane toward the anterior of the patient.

Probability density (in deg^{−1}) of the (a) inclination and (b) azimuth for stranded seeds in the CT coordinate system. Histogram bars are the raw counts in bins of 5°, and solid curves represent the orientation probability density *P* _{SO}. The probability density assumed by the 1D formalism is shown for comparison. (c) In this image, *P* _{SO} is projected onto the unit sphere using bright intensity to highlight regions of high density. The grid interval is 10°, and the directions of the CT axes are labeled (A—anterior, P—posterior, R—right, L—left, I—inferior, S—superior). Most seeds exhibit an approximately 20° pitch from the coronal plane toward the anterior of the patient.

(a) Volumes with an absolute difference between 1D and 2D-weighted dose distributions of greater than 7.2 Gy (5% of prescription) (left) and 14.4 Gy (10% of prescription) (right). (b) 1D (left) and 2D (right) prescription isodose surfaces for one patient. The 1D formalism tends to overestimate dose margins at the superior and inferior aspects, and underestimate dose margins at the lateral boundaries and in the medial regions.

(a) Volumes with an absolute difference between 1D and 2D-weighted dose distributions of greater than 7.2 Gy (5% of prescription) (left) and 14.4 Gy (10% of prescription) (right). (b) 1D (left) and 2D (right) prescription isodose surfaces for one patient. The 1D formalism tends to overestimate dose margins at the superior and inferior aspects, and underestimate dose margins at the lateral boundaries and in the medial regions.

## Tables

Noise level parameters, expressed as the standard deviations in each CT axis for zero-mean Gaussian noise.

Noise level parameters, expressed as the standard deviations in each CT axis for zero-mean Gaussian noise.

Errors in sensitivity analysis. A test-set of five reconstructed strands had simulated noise applied and the effects on the computed position and orientation of their seeds were studied. Fitted error in seed position and polar orientation (α) are reported for each noise level. Values are means, plus/minus one standard deviation for the orientations. Maximum values are displayed in parentheses.

Errors in sensitivity analysis. A test-set of five reconstructed strands had simulated noise applied and the effects on the computed position and orientation of their seeds were studied. Fitted error in seed position and polar orientation (α) are reported for each noise level. Values are means, plus/minus one standard deviation for the orientations. Maximum values are displayed in parentheses.

Cohort means and patient-controlled differences in selected dose metrics, comparing the 1D and 2D-weighted dose-calculations methods. Listed p-values are for *t*-tests of the respective mean paired difference from zero.

Cohort means and patient-controlled differences in selected dose metrics, comparing the 1D and 2D-weighted dose-calculations methods. Listed p-values are for *t*-tests of the respective mean paired difference from zero.

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