Specific absorbed fractions and radionuclide S-values for tumors of varying size and composition

Abstract

Accurate estimates of tumor absorbed dose are essential for the evaluation of treatment efficacy in radiopharmaceutical cancer therapy. Although tumor dosimetry via the MIRD schema has been previously investigated, prior studies have been limited to the consideration of soft-tissue tumors. In the present study, specific absorbed fractions (SAFs) for monoenergetic photons, electrons, and alpha particles in tumors of varying compositions were computed using Monte Carlo simulations in MCNPX after which self-irradiation S-values for 22 radionuclides (along with 14 additional alpha-emitter progeny) were generated for tumors of both varying size and tissue composition. The tumors were modeled as spheres with radii ranging from 0.10 cm to 6.0 cm and with compositions varying from 100% soft tissue (ST) to 100% mineral bone (MB). The energies of the photons and electrons were varied on a logarithm energy grid from 10 keV to 10 MeV. The energies of alpha particles were varied along a linear energy grid from 0.5 MeV to 12 MeV. In all cases, a homogenous activity distribution was assumed throughout the tumor volume. Furthermore, to assess the effect of tumor shape, several ellipsoidal tumors of different compositions were modeled and absorbed fractions were computed for monoenergetic electrons and photons. S-values were then generated using detailed decay data from the 2008 MIRD Monograph on Radionuclide Data and Decay Schemes. Our study results demonstrate that a soft-tissue model yields relative errors of 25% and 71% in the absorbed fraction assigned to uniform sources of 1.5 MeV electrons and 100 keV photons, respectively, localized within a 1 cm diameter tumor of MB. The data further show that absorbed fractions for moderate ellipsoids can be well approximated by a spherical shape of equal mass within a relative error of < 8%. S-values for 22 radionuclides (and their daughter progeny) were computed with results demonstrating how relative errors in SAFs could propagate to relative errors in tumor dose estimates as high as 86%. A comprehensive data set of radionuclide S-values by tumor size and tissue composition is provided for application of the MIRD schema for tumor dosimetry in radiopharmaceutical therapy.

1. Introduction

Radiopharmaceutical therapy (RPT) is one of the more rapidly expanding fields of cancer treatment. Its growth can be attributed to both the development of new targeting strategies for agent localization (e.g. prostate-specific membrane antigen, PSMA) (Vahidfar et al 2019), as well as the shift in focus from high-energy beta-particle emitters (e.g. ⁹⁰Y) to more therapeutically advantageous lower-energy beta emitters (e.g. ¹⁷⁷Lu) (Zakaly et al 2020) and a variety of alpha-particle emitters (e.g. ²²³Ra, ²¹²Bi, and ²²⁵Ac) (Sgouros 2020). In most forms of radiotherapy, the treatment objective is to deliver a prescribed absorbed (or equieffective) dose to a targeted tumor site, while minimizing energy deposition to critical organs and tissues within or adjacent to the treatment field (Bentzen et al 2012). In contrast, RPT is typically administered for widely disseminated disease, and thus the objective of dosimetry-based treatment-planning is typically not driven by achievement of a desired tumor dose, but is structured to ensure doses to normal organs approach but stay below thresholds for organ toxicity and functional impairment (ICRP 2019). Tumor doses in RPT are thus maximized indirectly through optimized timing and adjustment of administered activity, as guided by patient-specific biokinetic assessment. Retrospective assessment of tumor dosimetry in RPT may still be performed, however, to monitor treatment response and to quantify therapeutic index (Goetz et al 2020). This is typically performed through the selection of one or more ‘representative’ tumor sites amenable to quantitative imaging (Ljungberg and Gleisner 2018).

Estimates of tumor dose in RPT may be computed using radiation transport simulation or dose point-kernel convolution within voxelized models of the tumor(s) and surrounding anatomy originating from the CT portion of the patient’s PET/CT or SPECT/CT hybrid images. Tumor heterogeneity is thus modeled via the range of CT numbers within the region of interest defining the tumor site. An alternative approach to tumor dosimetry in RPT is the application of the MIRD schema if suitable radionuclide S-values are available. The MIRD schema states that the absorbed dose $D\left(r_T\right)$ to target region $r_T$ from radiation emissions from a source region $r_s$ may be calculated as

$$D\left(r_T\right)=\sum _{r_s}\tilde{A}\left(r_s\right)S\left(r_T\leftarrow r_s\right)$$ (1)

where $\tilde{A}\left(r_s\right)$ is the time-integrated activity (total number of decays) of the radionuclide within the source region, and $S\left(r_T\leftarrow r_s\right)$ is the mean absorbed dose to the target region per nuclear decay in the source region. The S-value is then computed as

$$S\left(r_T\leftarrow r_s\right)=\sum _i{E}_i{Y}_i\mathrm{\Phi }\left(r_T\leftarrow r_s,E_i\right)$$ (2)

where $E_i$ and $Y_i$ are energies and yields of the $i^{\text{th}}$ nuclear transformation of the radionuclide, respectively, and $\mathrm{\Phi }\left(r_T\leftarrow r_s,E_i,t\right)$ is the specific absorbed fraction (SAF) for a radiation particle of energy $E_i$ for a given source-target combination (Bolch et al 2009). The SAF is further defined as the ratio of the absorbed fraction $\varphi \left(r_T\leftarrow r_s,E_i\right)$ (fraction of particle energy $E_i$ emitted within the source region $r_s$ that is deposited in the target region $r_T$) and the target region mass $m\left(r_T\right)$:

$$SAF=\mathrm{\Phi }\left(r_T\leftarrow r_s,E_i\right)=\frac{\varphi \left(r_T\leftarrow r_s,E_i\right)}{m\left(r_T\right)}.$$ (3)

Of interest in the present study are values of SAF and associated radionuclide S-values for model tumors that are both the radiopharmaceutical source as well as the dosimetric target (e.g. S-values for tumor self-dose).

Siegel and Stabin (1994) first investigated this subject by computing absorbed fractions for electrons homogenously distributed in ST spheres. Both the electron energy and the sphere size were varied, and the absorbed fractions were computed using a spherical extension of a scaled absorbed dose distribution for point sources developed by the MIRD Committee (M J 1968, Berger 1971, Ellet and Humes 1971). These absorbed fractions were then used to calculate S-values for various radionuclides. The paper demonstrated the need to correct for the traditional assumption of unity in the electron self-dose absorbed fraction (Siegel and Stabin 1994).

Stabin and Konijnenberg (2000) later reevaluated the data within Siegel and Stabin (1994) by calculating absorbed fractions using Monte Carlo methods via two separate radiation transport codes: EGS4 and MCNP4B. Their investigation used similar methods as before, varying sphere size and particle energy, with the exception that absorbed fractions were computed for photons as well as for electrons. The results from both codes were compared to one another, and to values published in MIRD Pamphlet No. 3 (Brownell et al 1968). Recommended energy-dependent values of electron and photon absorbed fraction were then suggested by the authors. These values were later adopted within the OLINDA/EXM v1 dosimetry code for reporting tumor self-dose (Stabin et al 2005).

Amato et al (2009a, 2009b, 2011) investigated absorbed fractions for both photons and electrons emitted within ellipsoidal volumes. These papers used methods similar to those of Stabin and Konijnenberg (2000). Absorbed fractions were determined for both photons and electrons using Monte Carlo simulations in GEANT4 for tumors composed of soft tissue (ST). These investigations were unique as they mainly focused on determining absorbed fractions for ellipsoidal tumor models. The authors reported empirical formulae to predict the absorbed fraction as a function of a generalized radius $\rho$, defined as 3 V S⁻¹ where V and S are the model tumor volume and surface, respectively.

In this present study, we provide radionuclide S-values for 22 radionuclides (plus 14 additional radionuclides within the alpha-emitter decay series) of potential application in RPT. As with previous studies, we consider variations in tumor size, and explore issues of dose accuracy with regard to an assumption of a spherical tumor shape. However, our study extends previous studies in three respects. First, we consider tumor tissue compositions other than generic soft tissue, thus allowing for applications to the treatment of tumors with varying degrees of bone mineralization. Second, we include radiation transport simulations for alpha particles, in addition to those for monoenergetic photons and electrons. S-values for alpha-emitters and their progeny are thus reported here. Third, we report S-values by five categories of radiation emissions: photons, beta particles, electrons, alpha particles, and alpha recoil ions. This approach allows one to weight the dose contributions by radiation category by an appropriate relative biological effectiveness (RBE). The study data—both SAFs and radionuclide S-values—are provided as a series of electronic datafiles for ease of incorporation within any dosimetry software that applies the MIRD schema.

2. Methods

2.1. Comprehensive absorbed fraction data set

Tumors were modeled as tissue spheres of 15 sizes, with radii ranging from 1 mm to 6 cm. Although the work by Stabin and Konijnenberg (2000) modeled tumors with radii as large as 11.3 cm (6 kg), this investigation focused on tumors of sizes more clinically relevant to RPT. Furthermore, the material composition of each spherical tumor was further varied to compute absorbed fractions for different clinically relevant tissues, ranging from pure ST to pure mineral bone (MB). Five homogeneous material variations were created to span this range: 100% ST, 75% ST/25% bone, 50% ST/50% bone, 25% ST/75% bone, and 100% MB. The elemental compositions for ST and MB were taken from table A1 of ICRU Report 46 (ICRU 1992), with ST modeled as Adult ICRU-44 (male) Average ST (mass density of 1.03 g cm⁻³) and MB as Adult Cortical Bone (mass density of 1.92 g cm⁻³). The elemental compositions and mass densities of these tissues and their mixtures are given in table 1. The variations in ST and MB percentages thus allow for applications to osteoblastic or osteoclastic metastatic disease. Note that, although the tumor composition was varied, all tumors were modeled within a semi-infinite ST medium, in accordance with the related studies.

Table 1.

Elemental composition of the five tumor compositions.

	Elemental mass fraction (%)											Density
Tumor composition	H	C	N	O	Na	Mg	P	S	Cl	K	Ca	(g cm−3)
Soft tissue (ST)	10.50	25.60	2.70	60.20	0.10	0.00	0.20	0.30	0.20	0.20	0.00	1.03
75% ST	8.73	23.08	3.08	56.03	0.10	0.04	2.73	0.30	0.14	0.14	5.63	1.25
50% ST	6.95	20.55	3.45	51.85	0.10	0.10	5.25	0.30	0.10	0.10	11.25	1.48
25% ST	5.18	18.03	3.83	47.68	0.10	0.15	7.78	0.30	0.04	0.04	16.87	1.70
Mineral bone (MB)	3.40	15.50	4.20	43.50	0.10	0.20	10.30	0.30	0.00	0.00	22.50	1.92

With 75 different tumor models (15 sizes and five compositions), SAFs (SAFs) were computed for monoenergetic photons, electrons, and alpha particles. A total of 25 different energies were considered for both photons and electrons across a logarithmic energy grid spanning 10 keV to 10 MeV, while 24 alpha particle energies were considered along a linear energy grid from 0.5 MeV to 12.0 MeV. The lower alpha energies were used to ensure low-energy convergence to SAF-limiting values of inverse tumor mass. This monoenergetic and systematic approach allows for rapid interpolation across the decay spectrum of many therapy radionuclides of interest.

Radiation particle transport was performed on the HiPerGator supercomputer at the University of Florida, using MCNPX version 2.7.0 (Pelowitz 2011). Over 50 000 histories were run for all simulations to achieve statistical uncertainties in the absorbed fraction below 1%. The absorbed dose to each tumor was tallied in these simulations, and then divided by both the initial emission energy and tumor mass to yield the SAF. Furthermore, all particle interactions were simulated, including bremsstrahlung and pair production. Stabin and Konijnenberg (2000) had previously verified the use of the MCNP code for similar dosimetry purposes.

To benchmark the new data set, absorbed fractions were compared to those reported in recent publications. Absorbed fractions for 100 keV and 400 keV electrons and absorbed fractions for 140 keV and 364 keV photons were compared to data published by Amato et al for soft-tissue tumors with radii ranging from 0.13 cm up to 2.88 cm (Amato et al 2009a, 2011). Similarly, absorbed fractions were also compared to those reported by Stabin and Konijnenberg (2000) for 200 keV and 700 keV electrons and 100 keV and 364 keV photons for soft-tissue tumors between 0.1 g and 500 g.

2.2. S-value calculations

The next step in evaluating tumor dosimetry was to compute S-values to better assess the effect of tumor composition on dosimetry calculations. S-values were computed using the monoenergetic SAF data set for 22 radionuclides actively considered in RPT, across all tumor compositions and sizes. Radionuclide decay data on emission energies and yields were taken from the MIRD Monograph on Radionuclide Data and Decay Schemes (Eckerman and Endo 2008) as given in the MIRD-07.RAD and MIRD-07.BET datafiles. To improve the accuracy of our computations at the smallest tumor sizes, the enhanced decay data for Auger and Coster–Kronig electron energy spectra given in MIRD-07.ACK were used in lieu of their abbreviated energy spectra of MIRD-07.RAD file. Table 2 lists these 22 radionuclides as grouped into alpha-particle emitters, beta-particle emitters, and Auger-electron emitters. For alpha-emitters, principal therapy radiations are those alpha particles of highest yield across the decay series. Principal imaging photons include all x-ray and γ-rays of highest yield, and are indicated by either the parent or progeny radionuclide. For beta-emitters, the maximum and average beta energy are given. For Auger-electron emitters, principal therapy radiations are given separately for Auger electrons and internal conversion electrons to include their energy range and cumulative particle yield.

Table 2.

Radionuclides considered in this study for S-value calculation. Continuous Slowing-Dose Approximation (CSDA) ranges for alpha particles were taken from the NIST ASTAR database for liquid water,¹ while CSDA ranges for electrons were taken from the NIST ESTAR database for ICRU Report 46 ST.² Ranges listed for Auger and conversion electrons are given at energies of highest yield as indicated. Data on radiation energies and decay yields were taken from the Eckerman and Endo (2008).

Radionuclide	Physical half-life	Principal therapy radiations	RCSDA in tissue	Principal imaging photons
Alpha-emitters
211At	7.21 h	5.87 MeV α (211At)	48 μm	211At: 77 keV (11.6%) and 93 keV (2.0%) x-rays
		7.45 MeV α (211Po)	70 μm	211Po: 570 keV (0.54%) and 898 keV (0.56%) γ-rays 207Bi: 73 keV (21.8%), 75 keV (36.6%) x-rays, 590 keV (97.8%) γ-ray
212Bi	1.01 h	6.06 MeV α (212Bi)	50 μm	212Bi: 727 keV (6.6%) γ-ray
		8.79 MeV α (212Po)	91 μm	208Tl: 583 keV (84.5%) and 860 keV (12.4%) γ-rays
213Bi	45.6 m	5.87 MeV α (213Bi)	48 μm	213Bi: 440 keV (26.1%) γ-ray
		8.38 MeV α (213Po)	85 μm	209Tl: 117 keV (84.3%) and 465 keV (96.9%) γ-rays
223Ra	11.4 d	5.61 MeV α (223Ra)	45 μm	223Ra: 144 keV (3.3%), 154 keV (5.7%), 269 keV (14%), 324 keV (4.0%) γ-rays
		5.72 MeV α (223Ra)	46 μm	219Rn: 271 keV (10.8%) and 402 keV (6.6%) γ-rays
		6.82 MeV α (219Rn)	61 μm	211Pb: 405 keV (3.8%) and 427 keV (1.8%) γ-rays
		7.39 MeV α (215Po)	69 μm	211Bi: 351 keV (12.9%) γ-ray
		6.62 MeV α (211Bi)	58 μm	211Po: 570 keV (0.54%) and 898 keV (0.56%) γ-rays
		7.45 MeV α (211Po)	70 μm
225Ac	10.0 d	5.73 MeV α (225Ac)	46 μm	225Ac: 86 keV (1.9%) x-ray and 157 keV (0.36%) γ-ray
		5.79 MeV α (225Ac)	47 μm	221Fr: 218 keV (11.6%) γ-ray
		5.83 MeV α (225Ac)	48 μm	213Bi: 440 keV (26.1%) γ-ray
		6.13 MeV α (221Fr)	51 μm	209Tl: 117 keV (84.3%) and 465 keV (96.9%) γ-rays
		6.34 MeV α (221Fr)	54 μm
		7.07 MeV α (217At)	64 μm
		5.87 MeV α (213Bi)	48 μm
		8.38 MeV α (213Po)	85 μm
227Th	18.7 d	5.70 MeV α (227Th)	46 μm	227Th: 211 keV (1.3%), 236 keV (12.9%), 256 keV (7.0%), 286 keV (1.7%) γ-rays
		5.71 MeV α (227Th)	46 μm	227Th: 290 keV (1.9%), 300 keV (2.2%), 330 keV (2.9%) γ-rays
		5.76 MeV α (227Th)	47 μm	[plus, 223Ra series x-rays and γ-rays]
		5.98 MeV α (227Th)	49 μm
		6.04 MeV α (227Th) [plus 223Ra α-particles]	50 μm
Beta-emitters
89Sr	50.5 d	β-(Emax = 1490 keV)	7.1 mm	Bremsstrahlung x-rays
		(Eave = 583 keV)	2.2 mm
90Y	2.67 d	β-(Emax = 2280 keV)	11 mm	Bremsstrahlung x-rays
		(Eave = 934 keV)	4.0 mm	511 keV (0.0032%) annihilation photons
124I	4.18 d	β-(Emax = 610 keV)	23 mm	511 keV (45.7%) annihilation photons
		(Eave = 188 keV)	0.41 mm	603 keV (62.9%) and 723 keV (10.4%) γ-rays
131I	8.02 d	β-(Emax = 610 keV)	23 mm	80 keV (2.6%), 284 keV (6.1%), 364 keV (81.7%)
		(Eave = 182 keV)	0.39 mm	637 keV (7.2%), 723 keV (1.8%) γ-rays
153Sm	46.5 h	β-(Emax = 705 keV)	2.8 mm	69 keV (4.8%) and 103 keV (29.8%) γ-rays
		(Eave = 225 keV)	0.54 mm
166Ho	26.8 h	β-(Emax = 1854 keV)	9.0 mm	80.6 keV (6.7%) and 1379 keV (0.93%) γ-rays
		(Eave = 672 keV)	2.6 mm
177Lu	6.65 d	β-(Emax = 500 keV)	1.8 mm	113 keV (6.4%) and 208 keV (11.0%) γ-rays
		(Eave = 133 keV)	0.23 mm
186Re	3.72 d	β-(Emax = 1070 keV)	4.8 mm	137 keV (9.4%) γ-ray
		(Eave = 323 keV)	0.94 mm
188Re	17.0 h	β (Emax = 2120 keV)	10 mm	155 keV (15.6%), 478 keV (1.1%), 633 (1.4%) γ-rays
		(Eave = 765 keV)	3.1 mm
Auger electron-emitters
103Pd	17.0 d	AE (2–22 keV) (7.44 nt−1) from 103Pd	0.02 μm (0.26 keV AE)	20 keV (58%) 103Pd x-rays
		AE (16–40 keV) (0.99 nt−1) from 103mRh CE (None)		20 keV (6%) 103mRh x-rays
111In	2.80 d	AE (40 eV-26 keV) (7.43 nt−1)	0.02 μm (0.35 keV AE)	171 keV (90.6%) and 245 keV (94.1%) γ-rays
		CE (144–245 keV) (0.16 nt−1)	520 μm (219 keV CE)
117mSn	13.8 d	AE (10 eV-28 keV) (14.2 nt−1)	0.02 μm (0.4 keV AE)	156 keV (2.1%) and 159 keV (86.4%) γ-rays
		CE (126–313 keV) (1.15 nt−1)	215 μm (127 keV CE)
123I	13.3 h	AE (20 eV-30 keV) (13.7 nt−1)	0.02 μm (0.45 keV AE)	159 keV (83.3%) and 529 keV (1.4%) γ-rays
		CE (127–1068 keV) (0.16 nt−1)	215 μm (154 keV CE)
125I	59.4 d	AE (20 eV-30 keV) (23.0 nt−1)	0.02 μm (0.45 keV AE)	27 keV (116%) and 31 keV (20.4%) x-rays
		CE (3.7–36 keV) (0.945 nt−1)	18.2 μm (30.6 keV CE)	35.5 keV (6.68%) γ-ray
193mPt	4.33 d	AE (40 eV-74 keV) (27.4 nt−1)	0.016 μm (70 eV AE)	65–76 keV (13.8%) x-rays
		CE (1–135 keV) (2.99 nt−1)	2.5 μm (10 keV CE)	135.5 keV (0.11%) γ-ray
195mPt	4.02 d	AE (40 eV-74 keV) (36.6 nt−1)	0.015 μm (60 eV AE)	65.3 keV (22.7%) and 67.1 keV (38.8%) x-rays
		CE (5.9–239 keV) (2.78 nt−1)	8.8 μm (20.3 keV CE)	30.9 keV (2.3%), 98.9 keV (11.4%), 129.7 keV (2.8%) γ-rays

All S-value calculations were performed using a MATLAB^™ script with SAF interpolation across particle energies using piecewise cubic hermite interpolation polynomials (or PCHIP). S-values were computed for five different radiation categories: (1) photons, (2) beta particles, (3) electrons, (4) alpha particles, and (5) alpha recoil particles. For the fifth category, the SAF for alpha recoils was taken to be that for a 2 MeV alpha particle (an approach previously adopted by the ICRP) (ICRP 2016). For each radionuclide and tumor composition combination, the total cumulative S-value data was fit to a power-law model across all tumor sizes as per equation (4),

$$S\left(r,RN,TC\right)=A{r}^B,$$ (4)

where $\mathrm{S}$ is the S-value (unit: Gy Bq⁻¹ s⁻¹) for a particular tumor radius $r$ (unit: cm), radionuclide $RN$, and tumor composition $TC$. For the alpha-particle emitters, separate fits were made to each radionuclide’s low-LET radiation emissions (photons, electrons, and beta particles) and to their high-LET radiation emissions (alpha particles and recoil ions), so that RBE values may be applied as appropriate.

These S-values were further benchmarked against three datasets from previous publications and software codes. First, S-values were taken for nine radionuclides (At-211, Ra-223, Ac-225, I-131, Sm-153, Lu-177, In-111, I-123 and Pt-193m) in soft-tissue spheres of 1 mm and 5 mm radius, and compared to corresponding S-values given by the MIRDcell V2.0 software (Vaziri et al 2014). MIRDcell provides soft-tissue S-values for spherically shaped sources (cells) using analytic absorbed fraction calculations and the 2008 MIRD decay scheme data applied in the present investigation (Eckerman and Endo 2008). Second, S-values for five radionuclides (Lu-177, I-131, Sm-153, Re-186, and Y-90) were compared between data of the present study and those given by the OLINDA/EXM v1 code (Stabin et al 2005) for tumor masses of 1 g, 10 g, and 100 g. Third, comparisons of Sm-153 S-values were made between the work of the current study and those reported in Senthamizhchelvan et al (2012) for the 3D-RD software package for soft-tissue tumors ranging in size from 0.6 cm to 5.0 cm in radius. This later study reported good agreement between tumor doses reported by 3D-RD using patient-specific imaging data, and the model spherical tumors of the OLINDA/EXM v1 code for the bone-seeking radiopharmaceutical Sm-153 EDTMP.

2.3. Absorbed fractions for ellipsoids

The effect of tumor shape on absorbed fraction was also investigated. Although no tumor is perfectly spherical, ellipsoidal tumors are often modeled as spheres of equal volume or mass. Absorbed fractions are thus typically provided for spheres instead of ellipsoids as using spheres greatly reduces the number of simulations, simplifying data reporting, and improving data accessibility. Amato et al (Amato et al 2009a, 2009b, 2011) took a different approach by computing absorbed fractions for various ellipsoids and providing empirical fits to calculate absorbed fractions as a function of a generalized radius for any ellipsoid. The studies by Amato et al were limited because they only addressed ST tumors and did not provide a quantitative analysis of the error caused by approximating an ellipsoid as a sphere of equal volume. Thus, in the present work, a study was performed to determine the magnitude of the error introduced when approximating ellipsoidal tumors as having an equivalent spherical volume.

In this analysis, absorbed fractions were computed for four different ellipsoidal tumors: two oblate and two prolate. Each of these tumors had different degrees of ellipticity, $e$, which is defined as

$$e\equiv \left\{\begin{array}{l}\sqrt{\frac{a^2-c^2}{a^2}},c<a\\ \sqrt{\frac{c^2-a^2}{c^2}},c>a\end{array}\right.$$ (5)

where $c$ and $a$ are the equatorial and polar radius, respectively.³ The ellipticity of the model tumors was varied from 0.89 ($a/c=2.2$) to 0.98 ($a/c=5.5$) to assess even the most extreme cases. Furthermore, the material composition of each of these tumors was additionally varied to include tumors of 100% ST, a 50/50 mixture of ST and MB, and 100% MB. Finally, radiation transport was run using MCNPX for both photon and electron sources, with energies ranging from 10 keV to 2 MeV, and the absorbed fractions were compared to those of spherical tumors of equal volume. Alpha particles were not simulated as tumor shape has minimal effect for particles with very short ranges. Figure 1 shows a 3D schematic images of a spherical tumor, and the four additional ellipsoidal tumors of equivalent volume considered in this sensitivity study.

Figure 1.

Schematic images of the tumor models adopted in the shape sensitivity study.

3. Results

3.1. Comprehensive SAF data set

The SAFs for monoenergetic photons, electrons, and alpha particles emitted in tumors of varying size and tissue composition are reported in supplementary data (A)–(C), available online at https://stacks.iop.org/PMB/65/235015/mmedia, respectively. The number of particles simulated in each Monte Carlo simulation was sufficient to maintain absorbed fraction uncertainties below 1% for all particles.

Below are two figures that highlight notable features. Figures 2(A)–(B) plot the absorbed fraction and SAF, respectively, versus tumor radius for 1.5 MeV electrons emitted within each of the five tumor tissue compositions. Figures 3(A)–(B) similarly give these values for 100 keV photons sources emitted within tumors of varying size and compositions. Comparisons of the absorbed fraction dataset of the present study to those published by Amato et al (Amato et al 2009a, 2011) are reported in tables 3 and 4 for electrons and photons, respectively, and are in agreement to within 0.5% for electrons and to within 4.8% for photons. A second comparison to the absorbed fractions reported by Stabin and Konijnenberg (2000) is shown in tables 5 and 6 for electrons and photons, respectively. Similarly, the electron absorbed fractions agree to within 1.1% and the photons values are to within 5.9%.

Figure 2.

(A) Absorbed fractions and (B) SAFs for 1.5 MeV electrons emitted uniformly within tumors of varying size and tissue composition.

Figure 3.

(A) Absorbed fractions and (B) SAFs for 100 keV photons emitted uniformly within tumors of varying size and tissue composition.

Table 3.

Absorbed fractions reported for electrons in soft tissue tumors of varying size. Percent differences between the present study (PS) and the Amato et al (2011) comparative study (CS) are given in the last column.

	Absorbed fractions				Percent difference (%)
	Present study		Amato et al (2011)		(CS—PS)/PS
Tumor radius (cm)	100 keV	400 keV	100 keV	400 keV	100 keV	400 keV
0.13	9.57E-01	6.41E-01	9.57E-01	6.42E-01	0.0%	0.1%
0.29	9.81E-01	8.39E-01	9.81E-01	8.34E-01	0.0%	−0.5%
0.62	9.92E-01	9.24E-01	9.91E-01	9.21E-01	−0.1%	−0.3%
1.34	9.96E-01	9.62E-01	9.96E-01	9.63E-01	0.0%	0.1%
2.12	9.97E-01	9.76E-01	9.97E-01	9.76E-01	0.0%	0.0%
2.88	9.98E-01	9.83E-01	9.98E-01	9.82E-01	0.0%	−0.1%

Table 4.

Absorbed fractions reported for photons in ST tumors of varying size. Percent differences between the present study (PS) and the Amato et al (2009a) comparative study (CS) are given in the last column.

	Absorbed fractions				Percent difference (%)
	Present study		Amato et al (2009a)		(CS—PS)/PS
Tumor radius (cm)	140 keV	364 keV	140 keV	364 keV	140 keV	364 keV
0.62	1.33E-02	1.54E-02	1.30E-02	1.50E-02	−2.3%	−2.7%
0.78	1.67E-02	1.92E-02	1.60E-02	1.90E-02	−4.2%	−1.2%
1.13	2.44E-02	2.78E-02	2.40E-02	2.70E-02	−1.5%	−2.8%
1.34	2.92E-02	3.28E-02	2.80E-02	3.20E-02	−4.1%	−2.4%
1.68	3.73E-02	4.12E-02	3.60E-02	4.10E-02	−3.4%	−0.5%
2.43	5.57E-02	6.03E-02	5.30E-02	5.90E-02	−4.8%	−2.1%

Table 5.

Absorbed fractions reported for electrons in ST tumors of varying size. Percent differences between the present study (PS) and the Stabin and Konijnenberg (2000) comparative study (CS) are given in the last column.

		Absorbed fractions				Percent difference (%)
		Present study		Stabin and konijnenberg (2000)		(CS—PS)/PS
Tumor radius (cm)	Tumor mass (g)	200 keV	700 keV	200 keV	700 keV	200 keV	700 keV
0.29	0.1	9.42E-01	6.64E-01	9.37E-01	6.56E-01	−0.5%	−1.1%
0.61	1	9.72E-01	8.38E-01	9.71E-01	8.32E-01	−0.1%	−0.7%
1.32	10	9.87E-01	9.23E-01	9.87E-01	9.20E-01	0.0%	−0.4%
2.85	100	9.94E-01	9.64E-01	9.93E-01	9.62E-01	−0.1%	−0.2%
4.88	500	9.96E-01	9.78E-01	9.96E-01	9.76E-01	0.0%	−0.2%

Table 6.

Absorbed fractions reported for photons in ST tumors varying size. Percent differences between the present study (PS) and the Stabin and Konijnenberg (2000) comparative study (CS) are given in the last column.

		Absorbed fractions				Percent difference (%)
		Present study		Stabin and Konijnenberg (2000)		(CS—PS)/PS
Tumor radius (cm)	Tumor mass (g)	100 keV	364 keV	100 keV	364 keV	100 keV	364 keV
0.61	1	1.27E-02	1.53E-02	1.20E-02	1.50E-02	−5.4%	−1.8%
1.32	10	2.84E-02	3.24E-02	2.70E-02	3.20E-02	−4.8%	−1.2%
2.85	100	6.69E-02	7.07E-02	6.30E-02	6.90E-02	−5.9%	−2.5%
4.11	300	1.03E-01	1.03E-01	1.00E-01	1.02E-01	−2.6%	−1.2%
4.88	500	1.25E-01	1.22E-01	1.21E-01	1.21E-01	−3.5%	−1.1%

3.2. S-value calculations

S-values were calculated using an in-house MATLAB^™ script according to equation (2) for the 22 radionuclides listed in table 2. The data were fit to a power regression model across various tumor sizes for all radionuclide and tumor composition combinations, according to equation (4). All fits yielded a coefficient of determination (R²) greater than 0.997. The resulting fitting coefficients are listed in tables 7A and 7B for alpha-emitters and table 7C for beta-particle and Auger-electron emitters, so as to facilitate future calculations. For the alpha-particle emitters, separate fitting parameters are provided for the high-LET (table 7A) and low-LET (table 7B) emissions of their decay scheme. Final radionuclide S values are given in supplementary data D for the alpha-particle emitters (including all daughter progeny) and in supplementary data E for the beta-particle and Auger-electron emitters.

Table 7A.

Parameters for S-value power-law regression for all tumor compositions for alpha-particle emitters and their progeny (high-LET radiations). A (Gy Bq⁻¹ s⁻¹) B (cm⁻¹).

Radionuclide	Coefficient	Mineral bone	25% ST	50% ST	75% ST	ST
Alpha-emitters—High-LET radiations (alphas and alpha recoils)
At-211	A	4.970E-11	5.630E-11	6.470E-11	7.620E-11	9.260E-11
	B	−2.999E+00	−2.998E+00	−2.998E+00	−2.998E+00	−2.998E+00
(Po-211)	A	1.507E-10	1.704E-10	1.960E-10	2.307E-10	2.803E-10
	B	−2.997E+00	−2.997E+00	−2.997E+00	−2.996E+00	−2.996E+00
(Bi-207)	A	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00
	B	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00
Bi-212	A	4.410E-11	4.990E-11	5.740E-11	6.750E-11	8.210E-11
	B	−2.998E+00	−2.998E+00	−2.998E+00	−2.998E+00	−2.998E+00
(Po-212)	A	1.776E-10	2.008E-10	2.310E-10	2.718E-10	3.302E-10
	B	−2.996E+00	−2.996E+00	−2.995E+00	−2.995E+00	−2.994E+00
(Tl-208)	A	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00
	B	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00
Bi-213	A	2.300E-12	2.600E-12	3.000E-12	3.600E-12	4.300E-12
	B	−2.999E+00	−2.998E+00	−2.998E+00	−2.998E+00	−2.998E+00
(Po-213)	A	1.694E-10	1.915E-10	2.203E-10	2.593E-10	3.150E-10
	B	−2.996E+00	−2.996E+00	−2.996E+00	−2.995E+00	−2.994E+00
(Tl-209)	A	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00
	B	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00
(Pb-209)	A	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00
	B	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00
Ra-223	A	1.148E-10	1.298E-10	1.494E-10	1.759E-10	2.138E-10
	B	−2.999E+00	−2.999E+00	−2.998E+00	−2.998E+00	−2.998E+00
(Rn-219)	A	1.367E-10	1.546E-10	1.779E-10	2.094E-10	2.545E-10
	B	−2.998E+00	−2.998E+00	−2.997E+00	−2.997E+00	−2.997E+00
(Po-215)	A	1.495E-10	1.690E-10	1.945E-10	2.289E-10	2.781E-10
	B	−2.997E+00	−2.997E+00	−2.997E+00	−2.996E+00	−2.996E+00
(Pb-211)	A	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00
	B	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00
(Bi-211)	A	1.327E-10	1.500E-10	1.726E-10	2.032E-10	2.470E-10
	B	−2.998E+00	−2.998E+00	−2.998E+00	−2.997E+00	−2.997E+00
(Tl-207)	A	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00
	B	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00
(Po-211)	A	1.507E-10	1.704E-10	1.960E-10	2.307E-10	2.803E-10
	B	−2.997E+00	−2.997E+00	−2.997E+00	−2.996E+00	−2.996E+00
Ac-225	A	1.171E-10	1.325E-10	1.524E-10	1.795E-10	2.182E-10
	B	−2.999E+00	−2.998E+00	−2.998E+00	−2.998E+00	−2.998E+00
(Fr-221)	A	1.276E-10	1.443E-10	1.661E-10	1.955E-10	2.376E-10
	B	−2.998E+00	−2.998E+00	−2.998E+00	−2.998E+00	−2.997E+00
(At-217)	A	1.431E-10	1.618E-10	1.861E-10	2.191E-10	2.663E-10
	B	−2.998E+00	−2.997E+00	−2.997E+00	−2.997E+00	−2.996E+00
see Bi-213 decay series above
Th-227	A	1.190E-10	1.346E-10	1.549E-10	1.823E-10	2.216E-10
	B	−2.999E+00	−2.998E+00	−2.998E+00	−2.998E+00	−2.998E+00
see Ra-223 decay series above

Table 7B.

Parameters for S-value power-law regression for all tumor compositions for alpha-particle emitters and their progeny (low-LET radiations). A (Gy Bq⁻¹ s⁻¹) B (cm⁻¹).

Radionuclide	Coefficient	Mineral bone	25% ST	50% ST	75% ST	ST
Alpha-emitters—Low-LET radiations (photons, beta particles, electrons)
At-211	A	3.000E-13	3.000E-13	3.000E-13	3.000E-13	3.000E-13
	B	−2.694E+00	−2.718E+00	−2.748E+00	−2.784E+00	−2.819E+00
(Po-211)	A	1.000E-14	1.000E-14	1.000E-14	1.000E-14	1.000E-14
	B	−2.316E+00	−2.325E+00	−2.325E+00	−2.329E+00	−2.330E+00
(Bi-207)	A	3.800E-12	4.100E-12	4.300E-12	4.700E-12	5.100E-12
	B	−2.508E+00	−2.509E+00	−2.508E+00	−2.505E+00	−2.495E+00
Bi-212	A	8.200E-12	1.150E-11	1.010E-11	1.150E-11	1.320E-11
	B	−2.785E+00	−2.715E+00	−2.743E+00	−2.715E+00	−2.677E+00
(Po-212)	A	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00
	B	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00
(Tl-208)	A	1.380E-11	1.500E-11	1.650E-11	1.840E-11	2.090E-11
	B	−2.687E+00	−2.684E+00	−2.678E+00	−2.670E+00	−2.654E+00
Bi-213	A	8.100E-12	9.000E-12	1.020E-11	1.170E-12	1.380E-11
	B	−2.877E+00	−2.867E+00	−2.853E+00	−2.836E+00	−2.810E+00
(Po-213)	A	6.000E-17	6.000E-17	7.000E-17	7.000E-17	8.000E-17
	B	−2.386E+00	−2.390E+00	−2.393E+00	2.392E+00	−2.388E+00
(Tl-209)	A	1.410E-11	1.540E-11	1.700E-11	1.900E-11	2.160E-11
	B	−2.713E+00	−2.706E+00	−2.696E+00	−2.682E+00	−2.660E+00
(Pb-209)	A	3.800E-12	4.300E-12	4.900E-12	5.700E-12	6.900E-12
	B	−2.964E+00	−2.959E+00	−2.953E+00	−2.946E+00	−2.935E+00
Ra-223	A	1.900E-12	2.100E-12	2.400E-12	2.700E-12	3.200E-12
	B	−2.871E+00	−2.883E+00	−2.896E+00	−2.910E+00	−2.922E+00
(Rn-219)	A	2.000E-13	2.000E-13	3.000E-13	3.000E-13	3.000E-13
	B	−2.685E+00	−2.703E+00	−2.721E+00	−2.743E+00	−2.764E+00
(Po-215)	A	3.000E-16	3.000E-16	4.000E-16	4.000E-16	4.000E-16
	B	−2.449E+00	−2.464E+00	−2.482E+00	−2.502E+00	−2.521E+00
(Pb-211)	A	8.100E-12	9.100E-12	1.020E-11	1.180E-11	1.380E-11
	B	−2.879E+00	−2.866E+00	—2.850E+00	−2.830E+00	−2.802E+00
(Bi-211)	A	3.000E-13	3.000E-13	3.000E-13	4.000E-13	4.000E-13
	B	−2.770E+00	−2.782E+00	−2.794E+00	−2.809E+00	−2.820E+00
(Tl-207)	A	8.700E-12	9.700E-12	1.090E-11	1.250E-11	1.470E-11
	B	−2.876E+00	−2.862E+00	−2.844E+00	−2.821E+00	−2.790E+00
(Po-211)	A	1.000E-14	1.000E-14	1.000E-14	1.000E-14	1.000E-14
	B	−2.316E+00	−2.325E+00	−2.325E+00	−2.329E+00	−2.330E+00
Ac-225	A	6.000E-13	7.000E-13	7.000E-13	9.000E-13	1.000E-12
	B	−2.948E+00	−2.952E+00	−2.957E+00	−2.960E+00	−2.959E+00
(Fr-221)	A	2.000E-13	3.000E-13	3.000E-13	3.000E-13	4.000E-13
	B	−2.818E+00	−2.832E+00	−2.847E+00	−2.865E+00	−2.880E+00
(At-217)	A	2.000E-15	2.000E-15	3.000E-15	3.000E-15	3.000E-15
	B	−2.831E+00	−2.843E+00	—2.856E+00	−2.870E+00	—2.884E+00
see Bi-213 decay series above
Th-227	A	1.900E-12	2.100E-12	2.400E-12	2.700E-12	3.200E-12
	B	−2.896E+00	−2.904E+00	−2.911E+00	−2.918E+00	−2.923E+00
see Ra-223 decay series above

Table 7C.

Parameters for S-value power-law regression for all tumor compositions for beta-particle and Auger-electron emitters. A (Gy Bq⁻¹ s⁻¹) B (cm⁻¹).

Radionuclide	Coefficient	Mineral bone	25% ST	50% ST	75% ST	ST
Beta-emitters
Sr-89	A	9.900E-12	1.100E-11	1.240E-11	1.420E-11	1.660E-11
	B	−2.851E+00	−2.834E+00	−2.813E+00	−2.787E+00	−2.752E+00
Y-90	A	1.420E-11	1.560E-11	1.740E-11	1.960E-11	2.240E-11
	B	−2.753E+00	−2.729E+00	−2.701E+00	−2.666E+00	−2.622E+00
I-124	A	4.600E-12	5.000E-12	5.400E-12	5.900E-12	6.500E-12
	B	−2.618E+00	−2.610E+00	−2.599E+00	−2.584E+00	−2.565E+00
I-131	A	4.300E-12	4.800E-12	5.400E-12	6.200E-12	7.300E-12
	B	−2.869E+00	−2.873E+00	−2.877E+00	−2.881E+00	−2.883E+00
Sm-153	A	5.500E-12	6.200E-12	7.000E-12	8.100E-12	9.600E-12
	B	−2.930E+00	−2.928E+00	−2.927E+00	−2.927E+00	−2.925E+00
Ho-166	A	1.170E-11	1.300E-11	1.460E-11	1.660E-11	1.940E-11
	B	−2.830E+00	−2.813E+00	−2.793E+00	−2.767E+00	−2.734E+00
Lu-177	A	3.000E-12	3.400E-12	3.800E-12	4.500E-12	5.400E-12
	B	−2.960E+00	−2.960E+00	−2.959E+00	−2.958E+00	−2.955E+00
Re-186	A	6.300E-12	7.000E-12	8.000E-12	9.200E-12	1.090E-11
	B	−2.919E+00	−2.911E+00	−2.900E+00	−2.887E+00	—2.867E+00
Re-188	A	1.270E-11	1.400E-11	1.570E-11	1.780E-11	2.060E-11
	B	−2.798E+00	−2.779E+00	−2.755E+00	−2.727E+00	−2.689E+00
Auger electron-emitters
Pd-103/Rh-103m	A	1.100E-12	1.300E-12	1.400E-12	1.600E-12	1.800E-12
	B	−2.959E+00	−2.950E+00	−2.941E+00	−2.933E+00	−2.935E+00
In-111	A	1.500E-12	1.600E-12	1.700E-12	1.900E-12	2.000E-12
	B	−2.645E+00	−2.655E+00	−2.666E+00	−2.683E+00	−2.712E+00
Sn-117m	A	3.700E-12	4.100E-12	4.700E-12	5.400E-12	6.300E-12
	B	−2.917E+00	−2.921E+00	−2.926E+00	−2.931E+00	−2.941E+00
I-123	A	1.100E-12	1.200E-12	1.300E-12	1.400E-12	1.500E-12
	B	−2.723E+00	−2.727E+00	−2.735E+00	−2.751E+00	−2.791E+00
I-125	A	9.000E-13	1.000E-12	1.000E-12	1.100E-12	1.100E-12
	B	−2.804E+00	−2.781E+00	−2.762E+00	−2.756E+00	−2.800E+00
Pt-193m	A	2.800E-12	3.200E-12	3.700E-12	4.300E-12	5.200E-12
	B	−2.981E+00	−2.982E+00	−2.983E+00	−2.983E+00	−2.982E+00
Pt-195m	A	4.100E-12	4.600E-12	5.200E-12	6.000E-12	7.100E-12
	B	−2.944E+00	−2.950E+00	−2.956E+00	−2.964E+00	−2.970E+00

The effect of tumor composition was evaluated by comparing the absorbed dose for a given tumor size across multiple radionuclides and tissue types using data of the present study. S-values for a 0.5-cm radius tumor composed of 100% MB, 50/50 mixture of MB and ST, and 100% ST are listed in table 8 for several radionuclides. To further assess the significance of tissue composition on tumor dosimetry, the percent difference between ST and MB S-values are shown in the last column of table 8. S-values in tumors of pure MB were shown to be consistently lower than their corresponding values in ST, due to similar energy deposition in a higher density volume and thus higher mass. The single exception is shown for the Auger-electron emitter I-125 (a percent difference of only −1%), which is driven primarily by the photon component of the S-value. The S-values listed in table 8 are total S-values, but the electron component for I-125 follows a similar trend to that seen for the other radionuclides. The difference in photon S-value contribution is due to the non-linear nature of photon attenuation coefficients. For all radionuclides considered in table 8, average percent differences are shown to be −45% for the six alpha-particle emitters, −40% for the eight beta-particle emitters, and −31% for the seven Auger-electron emitters.

Table 8.

S-values for 22 radionuclides within tumors of 0.5 cm radius with varying material composition are reported. The 50% ST corresponds to a tumor composed of 50% Soft Tissue (ST) and 50% bone.

Radionuclide	Mineral bone (Gy Bq−1 s−1)	50% ST (Gy Bq−1 s−1)	ST (Gy Bq−1 s−1)	Percent difference (MB-ST)/ST (%)
Alpha-emitters
At-211	4.00E-10	5.20E-10	7.44E-10	−46%
Bi-212	4.18E-10	5.40E-10	7.60E-10	−45%
Bi-213	8.29E-11	1.05E-10	1.45E-10	−43%
Ra-223	9.32E-10	1.21E-09	1.73E-09	−46%
Ac-225	9.42E-10	1.23E-09	1.75E-09	−46%
Th-227	9.66E-10	1.26E-09	1.80E-09	−46%
Beta-emitters
Sr-89	8.05E-11	1.01E-10	1.34E-10	−40%
Y-90	1.15E-10	1.40E-10	1.75E-10	−34%
I-124	3.05E-11	3.63E-11	4.50E-11	−32%
I-131	3.11E-11	3.97E-11	5.54E-11	−44%
Sm-153	4.30E-11	5.46E-11	7.60E-11	−43%
Ho-166	9.44E-11	1.17E-10	1.54E-10	−39%
Lu-177	2.34E-11	3.02E-11	4.27E-11	−45%
Re-186	5.01E-11	6.38E-11	8.81E-11	−43%
Re-188	1.02E-10	1.26E-10	1.62E-10	−37%
Auger electron-emitters
Pd-103	8.97E-12	1.10E-11	1.39E-11	−35%
In-111	9.08E-12	1.05E-11	1.23E-11	−26%
Sn-117m	2.78E-11	3.51E-11	4.83E-11	−42%
I-123	7.26E-12	8.13E-12	9.50E-12	−24%
I-125	6.86E-12	7.04E-12	6.96E-12	−1%
Pt-193m	2.24E-11	2.91E-11	4.13E-11	−46%
Pt-195m	3.13E-11	4.01E-11	5.63E-11	−44%

In table 9, S-values from the MIRDcell v2.0 software are compared to values given in the current study for selected alpha-emitters, beta-emitters, and Auger-emitters for spherical ST tumors of 1- and 5-mm radius. For the lower-range emitters—alpha and Auger emitters—percent differences are shown range from between 0% and 2%. For the three beta-emitters, percent differences in radionuclide S-value are 3.0% to 3.8% at 5-mm radius and between −10.9% to −8.6% for the smaller sphere size of 1 mm radius. These discrepancies can be attributed to either (1) differences in transport method or (2) differences in decay scheme energy interpolation. In table 10, percent differences in radionuclide S-values for tumor masses of 1 g, 10 g, and 100 g between those of the OLINDA/EXM v1 code and those of the present study range from a minimum of 0.1% to a maximum of −3.0%. The one exception is that for Y-90 at the smallest tumor mass of 1 g, where a percent difference of 9.5% is seen. Here, both approaches use Monte Carlo radiation transport simulation to assess photon and electron absorbed fractions, but different decay scheme data (and potentially different energy interpolation schemes) were used, as the Stabin and Konijnenberg (2000) study predates the release of the 2008 MIRD Monograph (Eckerman and Endo 2008). In table 11, we report percent differences in Sm-153 S-values given in the 3D-RD study of Senthamizhchelvan et al (2012) and the current investigation. Here, percent differences range from a low of 0% (0.6-cm radius tumor) to a high of 0.51% (5.0-cm radius tumor). We further note that the Senthamizhchelvan et al study provides curve fits to the Sm-153 S-values generated by the OLINDA/EXM software, but the study does not explicitly report their own calculations. They do discuss, however, the relative strong agreement in Sm-153 S-values between those given by the 3D-RD code and those reported by the OLINDA/EXM v1 software.

Table 9.

S-values are reported for 1 mm and 5 mm radii ST tumors using both the in-house MATLAB^™ script and the MIRDcell V2.0 software. Percent differences between the two results are given in the last two columns.

	S-value (Gy Bq−1 s−1)		S-value (Gy Bq−1 s−1)		Percent difference (%)
	Present study		MIRDCell v2.0		(CS—PS)/PS
Radionuclides	1 mm	5 mm	1 mm	5 mm	1 mm	5 mm
At-211	9.20E-08	7.44E-10	9.20E-08	7.49E-10	0.0%	0.7%
Ra-223	2.15E-07	1.73E-09	2.15E-07	1.75E-09	0.1%	0.9%
Ac-225	2.17E-07	1.75E-09	2.18E-07	1.77E-09	0.4%	0.9%
I-131	5.84E-09	5.39E-11	5.21E-09	5.58E-11	−10.8%	3.4%
Sm-153	8.05E-09	7.54E-11	7.18E-09	7.83E-11	−10.9%	3.8%
Lu-177	4.96E-09	4.25E-11	4.53E-09	4.38E-11	−8.6%	3.0%
In-111	1.14E-09	1.01E-11	1.13E-09	1.03E-11	−0.9%	2.0%
I-123	9.78E-10	8.26E-12	9.82E-10	8.42E-12	0.4%	1.9%
Pt-193m	4.79E-09	4.04E-11	4.82E-09	4.12E-11	0.6%	2.0%

Table 10.

S-values reported various ST tumors using the MATLAB^™ script of the present study and as reported by the OLINDA/EXM v1 software. Percent differences between the two results are given in the last column.

	S-values (Gy Bq−1 s−1)			S-values (Gy Bq−1 s−1)			Percent difference (%)
	Present study			OLINDA/EXM v1			(OLINDA/EXM v1—PS)/PS
Radionuclide	1 g	10 g	100 g	1 g	10 g	100 g	1 g	10 g	100 g
Lu-177	2.34E-11	2.43E-12	2.41E-13	2.33E-11	2.36E-12	2.39E-13	−0.5%	−3.0%	−0.9%
I-131	3.06E-11	3.30E-12	3.50E-13	3.09E-11	3.24E-12	3.49E-13	1.1%	−1.8%	−0.2%
Sm-153	4.18E-11	4.41E-12	4.41E-13	4.19E-11	4.29E-12	4.37E-13	0.1%	−2.6%	−0.9%
Re-186	4.91E-11	5.31E-12	5.32E-13	5.05E-11	5.23E-12	5.33E-13	2.8%	−1.5%	0.2%
Y-90	1.04E-10	1.30E-11	1.39E-12	1.14E-10	1.33E-11	1.41E-12	9.5%	2.2%	1.2%

Table 11.

Comparison of Sm-153 S-values calculated in the present study (PS) with those reported by Senthamizhchelvan et al (2012) in our comparative study (CS) for ST tumors of varying sizes.

Tumor radius (cm)	Tumor mass (g)	S-value comparative study (Gy Bq−1 s−1)	S-value present study (Gy Bq−1 s−1)	Percent difference (CS—PS)/PS (%)
0.6	0.93	4.45E-11	4.45E-11	0.00%
0.8	2.21	1.90E-11	1.90E-11	−0.01%
1.0	4.31	9.84E-12	9.84E-12	−0.01%
1.5	14.56	2.96E-12	2.96E-12	0.04%
2.0	34.52	1.26E-12	1.26E-12	−0.03%
3.0	116.49	3.79E-13	3.78E-13	0.33%
4.0	276.13	1.62E-13	1.61E-13	0.45%
5.0	539.31	8.38E-14	8.33E-14	0.51%

3.3. Absorbed fractions for ellipsoids

The absorbed fractions for monoenergetic photons and electrons emitted in four different elliptically shaped tumors of varying composition were determined. Over 75 000 particles were simulated to reduce relative errors in the absorbed fraction to less than 1%. Percent differences between the absorbed fractions in spherically modeled tumors and absorbed fractions in elliptically shaped tumors were calculated and averaged across all particle energies. Tables 12 and 13 show the average percent error due to spherical tumor modeling for three tissue compositions for electrons and photons, respectively. To visualize this data further, figures 4 and 5 plot percent error versus tumor ellipticity for electrons and photons, respectively, across the three tissue compositions.

Table 12.

Average percent errors for electron absorbed fractions due to an assumed spherical tumor shape assessed across tumor three tissue compositions.

	Percent error (%) for electron sources
Ellipticity	Soft tissue	50% ST	Mineral bone
0.891	0.8	0.5	0.3
0.909	0.8	0.6	0.5
0.958	2.1	1.4	1.1
0.983	6.1	4.2	3.2

Table 13.

Average percent errors due to spherical models of spheroids for photon absorbed fractions across three tissue compositions.

	Percent error (%) for photon sources
Ellipticity	Soft tissue	50% ST	Mineral bone
0.891	3.0	1.8	2.2
0.909	4.9	4.2	4.7
0.958	11.5	10.5	10.4
0.983	21.7	21.1	21.6

Figure 4.

Percent errors due to spherical modeling of spheroids are plotted versus ellipticity across three tissue compositions for electrons.

Figure 5.

Percent errors due to spherical modeling of spheroids are plotted versus ellipticity across three tissue compositions for photons.

4. Discussion

4.1. Comprehensive absorbed fraction data set

In this study, we provide SAFs for monoenergetic photons, electrons, and alpha particles in spherical tumors of varying compositions as needed for tumor dosimetry via the MIRD schema. The ranges of particle energy and tumor radii are large enough to provide a clinically relevant and comprehensive database for tumor dosimetry. The data set is reported for monoenergetic particles to allow S-value calculations for any radionuclide. Ultimately, a simple table look-up and PCHIP interpolation scheme is best suited for S-value calculations.

Overall, the data support the hypothesis that tissue composition should be considered when performing dosimetry for tumor self-irradiation. Figure 2(A) shows large differences in absorbed fraction for a tumor radius between 0.5 cm and 2.5 cm. In fact, the absorbed fraction for a soft-tissue model for 1.5 MeV electrons emitted in a 1 cm diameter bony tumor would have a 25% relative error. This error is particularly important as a 1 cm diameter tumor is quite common in many cancers. In fact, the TNM staging system for non-small lung cancer classifies a T1 stage as a tumor with a diameter less than 3 cm and a T2 stage as a tumor with diameter between 3 cm and 7 cm (Kalemkerian 2011). These sizes fall within the previously mentioned range where large differences in absorbed fractions are observed, and would thus require attention to the tumor composition. Figure 3(A) makes a similar argument for photons. There is a wide spread in absorbed fractions across tissue compositions at radii greater than 1 cm. A soft-tissue model for 100 keV photons emitted in a 1 cm diameter bone tumor would have a 71% relative error. On the other hand, extreme cases where particle energy is very low and tumor size is large, or vice versa, would minimize the differences in absorbed fractions. Nevertheless, tissue composition is often important for clinically relevant cases.

In addition, benchmark comparisons of the new soft-tissue absorbed fractions with previous studies validate the reported values. Tables 3 and 5 show excellent agreement for electron absorbed fractions with the two other data sets, within 1.3% overall. Stabin and Konijnenberg found differences up to 5% for electron absorbed fractions when comparing identical simulations using different Monte Carlo codes. For photons, they found percent differences ranging from 20–40% between Monte Carlo codes for most energies and tumor sizes (Stabin and Konijnenberg 2000). The present study was within 7.1% of both studies for photon absorbed fractions, averaging a 2.9% difference with the Amato et al study (table 4) and 3.9% difference with the Stabin and Konijnenberg study (table 6). Differences in absorbed fractions are attributed to several factors including elemental tissue composition, tissue density, and the particle transport cross sections within the Monte Carlo codes. The absorbed fractions of this study are based on the elemental composition of ST from ICRU Report 46 (ICRU 1992) (density of 1.03 g cm⁻³), while Stabin and Konijnenberg employed the same unit density tissue composition used in the work by Siegel and Stabin in 1994. Percent differences for photons were negative, indicating that the present study had slightly higher absorbed fractions than the others due to the increased density of the ST models, as well as the other previously mentioned factors. Overall, both photon and electron ST absorbed fractions display minor differences with the other two studies and are well within acceptable ranges.

4.2. S-value calculations

S-values were successfully calculated for the 22 radionuclides in table 2 and power-law regressions were performed and reported in tables 7A–7C to facilitate S-value calculations. The data from table 8 demonstrates how errors introduced in selecting the incorrect SAFs can propagate when performing tumor dosimetry. The percent error caused by modeling tumors as ST spheres increases with increasing tumor MB content. Furthermore, the S-values calculated in table 8 are in units of Gy Bq⁻¹ s⁻¹, meaning that tumor dose can be directly calculated by simply multiplying the S-values by the time-integrated activity. Given this direct relationship between S-value and absorbed dose, the dosimetry for a bony tumor treated with ¹⁵³Sm can in error by as much as 77%.

The data from table 9 compares S-value computed using the in-house MATLAB^™ script with the SAF data set of this study to those available in MIRDcell v2.0. Percent differences in S-values ranged from −10.9% to + 3.8% across the 9 radionuclides and two tumor sizes considered, but this result was expected, as there are several differences in methodology. For one, MIRDcell uses an analytic method to calculate absorbed fractions, whereas the present study uses Monte Carlo simulations. MIRDcell is also designed to estimate self-dose S-values for cells on the order of microns, yet the smallest sphere in the new dataset has a 1 mm radius. In table 10, S-values from the current study were in agreement with those reported by the OLIND/EXM v1 code to within −3.0% to + 2.8% for the radionuclide considered across tumor masses of 1 g, 10 g, and 100 g. The one exception was a + 9.5% difference in Y-90 S values for the smallest tumor size (1 g). Upon further investigation, it is evident that this particular discrepancy can be attributed to a difference in computational methodology. Tumor self-dose S-values in OLINDA/EXM v1 appear to have been computed using only the mean beta-particle energy, while the present study considers the full beta energy spectrum (5000 energy bins) in the computation of tumor self-dose⁴. This difference—mean energy versus full energy spectrum—can lead to increasing discrepancies in the self-dose S-value as the tumor size decreases. Table 11 similarly compares Sm-153 S-values of this study with those given in Senthamizhchelvan et al (2012), which are actually curve fits to the Sm-153 S-values reported by the OLINDA/EXM software. Senthamizhchelvan et al claim to be within 4% of the OLINDA/EXM curve-fit, but they do not explicitly state their calculated S-values. Still, the present study is in strong agreement with the OLINDA/EXM curve-fit, with an average percent difference of just 0.16%. Any differences between the S-values of this study and those from OLINDA/EXM are attributed to differences in choice of Monte Carlo software and decay scheme data. The in-house MATLAB^™ script uses radionuclide decay schemes published by the MIRD Committee in 2008 (Eckerman and Endo 2008), while OLINDA/EXM uses decay schemes published prior to 2002 by the Brookhaven National Laboratory (Stabin and da Luz 2002). These direct comparisons serve as benchmarks to validate the accuracy of the present methodology and to further emphasize the necessity of accounting for tumor composition.

4.3. Mass scaling of tumor self-dose S values

Given radionuclide S values that differ by both tumor size and tissue composition, it is of interest to explore the degree to which a simple mass ratio correction may be applied to soft-tissue S-values to approximate their values for tumors of MB composition. Table 14 thus shows MB S-values for three radionuclides and five tumor sizes. Corresponding ST tumor S-values were then scaled by the ratio of ST-to-MB masses to approximate S-values for tumors composed of 100% MB. As shown, this mass scaling approach underestimates tumor S values of MB up to 40.4% for Y-90 and In-111, but the effect is significantly less for Ra-223. These results thus confirm the importance of having an S-value database for tumor self-dose that explicitly considers variations in tumor tissue composition, although alpha S-values may be approximated using mass scaling.

Table 14.

Comparison of tumor S values between those computed for 100% MB and those for 100% ST but mass-corrected by the ratio of ST to MB masses for equivalent tumor volumes.

	S-value (Gy Bq−1 s−1)
	Ra-223			Y-90			In-111
Tumor radius (cm)	MB S-value	Mass-corrected ST S-value	% Diff.	MB S-value	Mass-corrected ST S-value	% Diff.	MB S-value	Mass-corrected ST S-value	% Diff.
0.1	1.16E-07	1.15E-07	−0.5	5.30E-09	3.16E-09	−40.4	7.95E-10	6.43E-10	−19.1
0.2	1.45E-08	1.45E-08	−0.3	1.17E-09	7.58E-10	−35.3	1.15E-10	8.94E-11	−22.2
0.5	9.32E-10	9.31E-10	−0.1	1.15E-10	9.38E-11	−18.5	9.08E-12	6.58E-12	−27.5
1.0	1.17E-10	1.17E-10	−0.2	1.64E-11	1.50E-11	−8.8	1.38E-12	9.64E-13	−29.9
2.0	1.46E-11	1.46E-11	−0.2	2.18E-12	2.09E-12	−4.1	2.26E-13	1.52E-13	−32.6

4.4. Absorbed fractions for ellipsoids

As shown in figures 4 and 5, the percent error in estimating an ellipsoidal shaped tumor as a sphere increases with increasing ellipticity. For electrons, this error ranges from less than 1% to about 6%, and is most prominent in ST tumors. Figure 4 exhibits a linear-quadratic behavior, where the errors are kept below 2% for an ellipticity up to ~ 0.95, but increases sharply thereafter. Even then, the percent errors in estimating an ellipsoidal shaped tumor as a sphere are still about an order of magnitude smaller than the potential errors introduced when not taking tissue composition into consideration. Also, keep in mind that in the most extreme case, which had a 6.1% error, this was for an ellipsoid with an $a/c$ ratio of 5.5. This corresponds to a tumor shape with a very high degree of ellipticity in which the tumor would resemble more of a disk than a sphere. In such cases, the user would be advised to reconsider the appropriateness of using data for spherically shaped tumor masses.

Photons exhibited a similar trend with increasing error with increasing ellipticity. The percent errors are much higher though, ranging from about 2% up to 22%. The shape of the curve in figure 5 has a similar linear-quadratic shape, although the change in slope about an ellipticity of 0.92 is not as drastic as it was for electrons at 0.95. Furthermore, the percent error appears to be greatest for ST tumors, but the effect of tissue composition is not as prominent as was seen for electrons. When using spheres to estimate absorbed fractions for photons, the ellipticity should be kept below 0.92 in order to avoid dosimetry errors in excess of 5%.

4.5. Limitations

There are several major limitations that must be considered when using the data from this study. First, the MIRD schema assumes that the radionuclide is uniformly distributed within the tumor volume. This assumption is in accordance with all previously mentioned investigations, but it known that the blood flow to the center of tumors can be restricted as the tumor enlarges. Microvascular changes throughout the tumor lead to a non-uniform, volumetric radionuclide uptake, which deviates from the present models. Investigations such as those by Howell et al have shown that dose-rates vary by about a factor of 2 even for uniform distributions of radionuclides in the tumor due to edge effects, and vary by much more depending on the nonuniform distribution of the radionuclide, the range of the emitted radiations, and the tumor size (Howell et al 1989). Nevertheless, these distributions of radioactivity are difficult to estimate in clinical practice and this issue is thus beyond the scope of the present investigation. Second, the results herein are strictly for self-dose to the tumor, where r_T = r_s. Other studies have shown that the tumor-dose contribution due to cross-irradiation from other sources can be significant, especially for radionuclides that emit higher energy gamma-rays, such as I-131. Howard et al found that the tumor-dose estimate by OLINDA/EXM underestimated the total tumor-dose by up to 31% due to a lack of cross-dose consideration (Howard et al 2011), and Grimes and Cellar similarly found that only tumor-self dose underestimated the total tumor dose by up to 22% (Grimes and Celler 2014). A full consideration of tumor dose from non-tumor radionuclide sources thus requires radiation transport within an image-based or phantom-based model of the patient. Finally, although the material composition of the spherical tumors was varied from 100% ST to 100% MB, all tumors were surrounded by a semi-infinite volume of ST. This was selected in order to be consistent with previous studies. A subsequent investigation is needed to assess the effects of varying the surrounding medium, such as either lung or skeletal tissues. Changes in absorbed fractions could possibly be seen due to increases in backscatter from the denser surrounding medium, but a detailed analysis is necessary to quantify this effect.

5. Summary and clinical implementation

This investigation confirmed the significance of considering tumor composition when performing dosimetry for tumor self-irradiation. A comprehensive database of SAFs for monoenergetic photons, electrons, and alpha particles in tumors of varying compositions and sizes was created. Although the data set is large, it is readily accessible for applications of tumor dosimetry via any spreadsheet or other software. Furthermore, comparing the absorbed fractions across the five material compositions demonstrated an absorbed fraction dependence on tissue composition, particularly in the ranges that are clinically relevant.

S-values for 22 radionuclides (and 14 additional alpha-emitter decay progeny) frequently used for internal radiotherapy were also calculated for tumors of varying compositions. The results demonstrated how errors in SAFs could propagate into tumor dosimetry estimates. Choosing to use conventional absorbed fractions for ST tumors instead of bone can lead to relative dose errors as high as 86%. Comparisons in S-value between those from the MIRDcell v2.0 software and those calculated using the new database reveal differences of between 0.7% and 2%. Comparisons with other previously published soft-tissue S-values confirm the accuracy of the present methodology and highlight the need for a more comprehensive data set that considers tumor composition. This work also evaluated the validity of using spheres to approximate ellipsoidal tumors. As long as the spherical tumor volume and composition is equal to that of the ellipse, this approximation is adequate when the tumor ellipticity is below 0.95 for electrons and 0.92 for photons. An investigation similar to that by Amato et al (Amato et al 2009a, 2011) for the four other tissue compositions would help reduce the error when estimating absorbed fraction for ellipses of all sizes.

In order to implement this new dataset in the clinic, one must first determine the size and composition of the targeted tumor. Both parameters can be estimated from a CT image of the patient, which is commonly taken for most cancer patients. The size of the tumor can be estimated from a CT by measuring the radius or diameter of the tumor from a cross-sectional slice. This would serve as a rough estimate and would assume spherical geometry, but it has been shown that ellipsoids are well approximated as spheres. An estimate of the tumor composition is more challenging, but feasible using a Hounsfield Unit (HU) calibration. One would need to estimate the HU of a pure soft-tissue tumor, which would most likely range between 0–100 HU, and the HU of a pure mineral-bone tumor, which most likely ranges between 900–1000 HU. A simple linear fit would thus indicate that the HU for a 75%/25%, 50%/50%, and 25%/75% soft-tissue/mineral-bone tumor would be approximately 250 HU, 500 HU, and 750 HU, respectively. Regions-of-interest in a patient’s CT image can then be taken of a particular tumor to estimate the soft-tissue/mineral-bone content. The proposed method is a rough estimate assuming a 0 HU ST tumor and a 100 HU mineral-bone tumor, but future studies can collect data on various tumor HUs to create a more accurate fit. Alternatively, a commercially available tissue characterization CT phantom, such as the Gammex Multi-Energy CT Phantom, can be used instead of patient images. These phantoms provide multiple rods of tissue equivalent materials that can be used for image calibration including cortical bone, inner-bone, varying levels of calcium, and soft-tissue inserts. Ultimately, the use of CT in combination with the S-value dataset of this study can significantly improve tumor dose estimates and patient outcomes for treatments using RPT.

Supplementary Material

Supplementary material for this article is available online