Efficient ¹⁶⁹Yb high‐dose‐rate brachytherapy source production using reactivation

Abstract

Purpose

To present and quantify the effectiveness of a method for the efficient production of ¹⁶⁹Yb high‐dose‐rate brachytherapy sources with 27 Ci activity upon clinical delivery, which have about the same dose rate in water at 1 cm from the source center as 10 Ci ¹⁹²Ir sources.

Materials

A theoretical framework for ¹⁶⁹Yb source activation and reactivation using thermal neutrons in a research reactor and ¹⁶⁸Yb‐Yb₂O₃ precursor is derived and benchmarked against published data. The model is dependent primarily on precursor ¹⁶⁸Yb enrichment percentage, active source volume of the active element, and average thermal neutron flux within the active source.

Results

Efficiency gains in ¹⁶⁹Yb source production are achievable through reactivation, and the gains increase with active source volume. For an average thermal neutron flux within the active source of 1 × 10¹⁴ n cm⁻² s⁻¹, increasing the active source volume from 1 to 3 mm³ decreased reactor‐days needed to generate one clinic‐year of ¹⁶⁹Yb from 256 days yr⁻¹ to 59 days yr⁻¹, and 82%‐enriched precursor dropped from 80 mg yr⁻¹ to 21 mg yr⁻¹. A resource reduction of 74%–77% is predicted for an active source volume increase from 1 to 3 mm³.

Conclusions

Dramatic cost savings are achievable in ¹⁶⁹Yb source production costs through reactivation if active sources larger than 1 mm³ are used.

Short abstract

1. Introduction

Tumor dose conformity can be substantially improved relative to conventional high‐dose‐rate brachytherapy (HDR‐BT) through the use of applicators with partially shielded radiation sources such as those used in rotating shield brachytherapy (RSBT)^{1, 2, 3, 4, 5, 6, 7} or direction‐modulated brachytherapy (DMBT).^{8, 9, 10, 11, 12, 13, 14} Through partial shielding and controlling the direction of the radiation emission from implanted intracavitary and/or interstitial applicators, the RSBT and DMBT approaches provide the practical capability to realize the dose conformity benefits initially identified by Ebert.^{15, 16} These benefits could include the removal or reduction of supplemental interstitial needles needed to obtain high‐risk clinical target volume coverage for cervical cancer patients or for achieving urethral avoidance and/or dose escalation for prostate cancer patients. ¹⁶⁹Yb has the potential to provide superior dose conformity with RSBT or DMBT relative to ¹⁹²Ir, the conventional HDR‐BT isotope. This is because ¹⁶⁹Yb emits photons with an average photon energy of 93 keV, substantially lower than the 380 keV of ¹⁹²Ir, enabling more effective partial shielding within intracavitary and interstitial applicators. Also, ¹⁶⁹Yb has a high enough specific activity to enable dose rates that match those of 10 Ci of ¹⁹²Ir in a source that is small enough to fit in commercially available afterloaders, a critical capability for any isotope to be used for HDR‐BT.

¹⁶⁹Yb sources have been considered for brachytherapy before,^{17, 18, 19, 20} and a Food and Drug Administration approved ¹⁶⁹Yb radiation source is currently available from the Source Production and Equipment Company (SPEC, St. Rose, LA).²¹ A major impediment to the clinical adoption of ¹⁶⁹Yb as an HDR‐BT source is cost. ¹⁶⁹Yb is generated by irradiating a precursor material containing ¹⁶⁸Yb, such as 82% enriched ¹⁶⁸Yb‐Yb₂O₃, with thermal neutrons in a nuclear reactor. Naturally occurring Yb contains only 0.13% ¹⁶⁸Yb,²² thus, the precursor needs to be enriched by two orders of magnitude. For an active source with a volume small enough for HDR‐BT of 1–4 mm³, it is desirable for the active source to contain enough ¹⁶⁸Yb such that, when activated, the active source has a high enough ¹⁶⁹Yb activity to match the ¹⁹²Ir‐based HDR‐BT dose rate. The cost of 82%‐enriched ¹⁶⁸Yb‐Yb₂O₃ is currently $692 mg⁻¹. The competing isotope, ¹⁹²Ir, has a negligible precursor cost due to the high abundance of ¹⁹¹Ir (37.3%) in naturally occurring iridium. During the activation process, ¹⁶⁹Yb is simultaneously produced by ¹⁶⁸Yb neutron absorption and lost to radioactive decay and ¹⁶⁹Yb neutron absorption, and a maximum ¹⁶⁹Yb activity exists for any combination of starting ¹⁶⁸Yb mass in the precursor and thermal neutron flux within the active source. The ¹⁶⁹Yb activity asymptotically approaches its maximum with thermal neutron irradiation time, and, when the activation process stops at the maximum ¹⁶⁹Yb activity, a substantial amount of ¹⁶⁸Yb still exists within the active source.

In this work, it will be demonstrated that the physical properties of ¹⁶⁸Yb are such that a 1 mm³ active source of 82%‐enriched ¹⁶⁸Yb, irradiated with a realistic thermal neutron flux of 1 × 10¹⁴ n cm⁻² s⁻¹, is barely capable of producing the ¹⁶⁹Yb activity needed to match the dose rate in water at 1 cm off‐axis from a 10 Ci ¹⁹²Ir source. This is of major practical concern since typical HDR‐BT sources have volumes of about 1 mm³. For such a source, the activation process is highly asymptotic with time and therefore inefficient. After the ¹⁶⁹Yb activity drops to the minimum clinically acceptable activity, the residual ¹⁶⁸Yb is wasted since its quantity is insufficient for reactivation back to the desired initial ¹⁶⁹Yb activity. It will be demonstrated that ¹⁶⁹Yb sources larger than 1 mm³ can be generated much more efficiently based on the principles of (a) minimizing reactor time by avoiding asymptotic activation and (b) minimizing expensive precursor waste by ensuring enough ¹⁶⁸Yb is initially present to enable effective reactivation. Reactivation as a means of improving efficiency of ¹⁶⁹Yb has been previously described in concept,²³ and in this work, we quantitatively describe the benefits of the approach.

2. Materials and Methods

2.A. ¹⁶⁹Yb activity range

The desired dose rate for a fresh ¹⁶⁹Yb source in water at 1 cm off‐axis from the source is assumed to be the same as that of a 10 Ci ¹⁹²Ir source. According to Angelopoulos et al.,²⁴ a 10 Ci (370 GBq) ¹⁹²Ir Varian VariSource (Varian Medical Systems, Palo Alto, CA) has an air kerma strength (S_k ) of 10.28 × 10⁻⁸ U Bq⁻¹, a dose rate constant of 1.101 cGy h⁻¹ U⁻¹, and therefore, a dose rate 1 cm lateral to the source in water of 4.184 × 10⁴ cGy h⁻¹. To obtain the dose rate per unit activity in water at 1 cm from a ¹⁶⁹Yb source, the air kerma strength per unit activity (S_k /A) for such sources was obtained from two references: Das et al.,²⁵ with S_k /A of 0.042 μGy m² MBq⁻¹ h⁻¹, which is equal to 1.554 cGy cm² h⁻¹ mCi⁻¹ [U mCi⁻¹] and Reynoso et al., with S_k /A of 1.15 cGy cm² h⁻¹ mCi⁻¹ [U mCi⁻¹]. Multiple authors^{17, 21} reported a dose rate constant for ¹⁶⁹Yb sources of 1.19 cGy h⁻¹ U⁻¹, and the dose rate per activity value for the ¹⁶⁹Yb source from Das et al. is therefore 1.84 cGy h⁻¹ mCi⁻¹ and that from Reynoso et al. (2017) is 1.37 cGy h⁻¹ mCi⁻¹. The average of these numbers is 1.605 cGy h⁻¹ mCi⁻¹, and, dividing that number by the dose rate in water at 1 cm from the 10 Ci ¹⁹²Ir source, one finds that a 26 Ci ¹⁶⁹Yb source is needed to match the 10 Ci ¹⁹²Ir dose rate. Adding a 4% safety margin, we obtain the 27 Ci quantity for the ¹⁶⁹Yb activity required to match the dose rate at 1 cm in water for a 10 Ci ¹⁹²Ir HDR BT source.

It is conservatively assumed that the minimum useful clinical dose rate is that which provides that same dose rate in water at 1 cm lateral to the source as a ¹⁹²Ir source used for HDR‐BT just prior to a typical source change. Such an ¹⁹²Ir source typically has an initial activity of 10 Ci and is replaced after 90 days. With a half‐life of 73.83 days, the ¹⁹²Ir activity at the time of replacement under these assumptions would be 4.30 Ci. The ¹⁶⁹Yb activity that would match the dose rate of a ¹⁹²Ir at the time of replacement is thus (27 Ci) (4.3/10) = 11.6 Ci, which, for an initial clinical ¹⁶⁹Yb activity of 27 Ci, would occur after 39 days of clinical use given the 32‐day half‐life of ¹⁶⁹Yb.

2.B. Active source geometry

The ¹⁶⁹Yb active source is assumed to have a diameter with a range of 0.60–0.69 mm and a length ranging from 3.5 to 10.5 mm. This corresponds to a volume range of 1–4 mm³, and these active sources have the potential to fit inside lengthened capsules corresponding to the Varian GammaMed Plus, Classic Nucletron (Elekta, Stockholm, Sweden), Nucletron Microselectron v2, and Nucletron Flexisource, all of which are shown in detail by Rasmussen et al.²⁶ The practical density of the precursor assumed to be 82%‐enriched ¹⁶⁸Yb‐Yb₂O₃ throughout this work was assumed to be 8.15 mg mm⁻¹ although other densities are possible.

2.C. Thermal neutron irradiation and source delivery

¹⁶⁹Yb can be produced by irradiating the precursor in a research nuclear reactor, with or without the presence of existing ¹⁶⁹Yb within the active source. The precursor can be in many forms including a pellet,²³ glass,²³ or ceramic.^{23, 27} A wide range of thermal neutron fluxes is available at various research reactors, and examples of maximum available thermal neutron fluxes are 6.0 × 10¹³ n cm⁻² s⁻¹ at the Massachusetts Institute of Technology Research Reactor (MITR‐II), 6.0 × 10¹⁴ n cm⁻² s⁻¹ at the University of Missouri Research Reactor (MURR), and 5 × 10¹⁵ n cm⁻² s⁻¹ for the SM‐3 reactor at the State Scientific Center — Research Institute of Atomic Reactors (RIAR) in Dimitrovgrad, Russia.²⁸ Throughout this work, a default average thermal neutron flux within the active source of 1 × 10¹⁴ n cm⁻² s⁻¹ will be used in activation calculations, which is realistically obtainable at multiple research reactors, including MURR.

With the goal of delivering 27 Ci of ¹⁶⁹Yb to clinics following source activation, the time needed to allow undesirable radioactive impurities to decay away, to transport the radiation source wire to the receiving clinic, install it in the respective afterloader, and complete the associated quality assurance is assumed to be five calendar days. Under this assumption, the ¹⁶⁹Yb activity at the end of activation in the reactor is 30 Ci, such that the amount of activity lost by source decay during transportation to the clinic is accounted for and 27 Ci of ¹⁶⁹Yb is present at the time of delivery to the clinic.

2.D. ¹⁶⁹Yb source activation

To demonstrate how precursor quantity in the active source element (active source volume), thermal neutron irradiation time, and source activity is quantitatively related, equations for radioactive source activation are derived. The equations will be used to calculate the amount of ¹⁶⁸Yb precursor material and nuclear reactor time needed to produce one clinic‐year of ¹⁶⁹Yb as a function of active source volume. One clinic‐year of ¹⁶⁹Yb is defined as the ¹⁶⁹Yb activity needed to support one afterloader for 1 year, assuming the ¹⁶⁹Yb source activity is 27 Ci on delivery and 11.6 Ci on removal, corresponding to 39 days of clinical usage per source and 9.35 source exchanges per year on average.

Consider three isotopes of a given element, indexed by 1, 2, and 3. Isotope 1 is stable, has mass number M − 1, and becomes the active source isotope after absorbing a thermal neutron. Examples of Isotope 1 are ¹⁹¹Ir and ¹⁶⁸Yb, and their relevant physical properties are listed in Table 1. Isotope 2 is the gamma ray emitting (therapeutic) active source isotope with mass number M, a half‐life of t _1/2 [s], a decay constant of λ ₂ = ln(2)/t _1/2 [s⁻¹], and is the result of thermal neutron absorption by isotope 1 with cross section σ ₁ [cm²]. Examples of Isotope 2 are ¹⁹²Ir and ¹⁶⁹Yb. Isotope 3 is stable, has mass number M + 1, and is the result of thermal neutron absorption by isotope 2 with cross section σ ₂ [cm²]. Examples of isotope 3 are ¹⁹³Ir and ¹⁷⁰Yb.

Table 1.

Isotope properties for Ir and Yb

Isotope	σ [cm2]	t 1/2 [days]	λ [s−1]	$\overline{E}$ [keV]	Natural abundance (%)	Enriched abundance
191Ir	954 × 10−24	Stable	0	N/A	37.3	N/A
192Ir	1420 × 10−24 [30]	73.8	1.1 × 10−7	360	0	0%
193Ir	111 × 10−24	Stable	0	N/A	62.7	N/A
168Yb	2300 × 10−24	Stable	0	N/A	0.13	82%
169Yb	3600 × 10−24	32	2.5 × 10−7	93	0	0%
170Yb	10 × 10−24	Stable	0	N/A	3	0%

Definitions: σ = thermal neutron absorption cross section; t _1/2 = half‐life, λ = ln(2)/t _1/2 = decay constant, $\overline{E}$ = average emitted photon energy. σ‐values are from Mughabghab²⁹ (via the IAEA Live Chart of Nuclides) or Abdel‐Rahman and Podgorsak,³⁰ which was used for the ¹⁹²Ir thermal neutron absorption cross section in order to ensure consistency of parameters for benchmarking the model against the literature.^{22, 30}

Given an average thermal neutron flux within the active source of φ [n cm⁻² s⁻¹], including the effects of thermal neutron attenuation, changes with time, t, of the numbers of the atoms, N_m (t) of each isotope, m, (m = 1,...,3), for a given element are described by the following three differential equations:

$$\frac{d{N}_1(t)}{\mathit{dt}}=-\phi {\sigma }_1{N}_1(t),$$ (1)

$$\begin{array}{cc}\frac{d{N}_2(t)}{\mathit{dt}}=\hfill & \phi {\sigma }_1{N}_1(t)-{\lambda }_2{N}_2(t)-\phi {\sigma }_2{N}_2(t)\hfill \\ =\hfill & \phi {\sigma }_1{N}_1(t)-\left({\lambda }_2+\phi {\sigma }_2\right)N_2(t),\hfill \end{array}$$ (2)

and

$$\frac{d{N}_3(t)}{\mathit{dt}}=\phi {\sigma }_2{N}_2(t),$$ (3)

with the boundary conditions:

$$N_1\left(0\right)=N_1^0,N_2\left(0\right)=N_2^0,\text{and}{N}_3\left(0\right)=N_3^0,$$ (4)

which are the respective isotope counts at time t = 0. For the case of ¹⁶⁹Yb production, Eq. (2) accounts for ¹⁶⁸Yb conversion to ¹⁶⁹Yb by thermal neutron absorption with a rate constant of $\phi {\sigma }_1$ [s⁻¹], ¹⁶⁹Yb loss by radioactive decay to ¹⁶⁹Tm with a 32‐day half‐life and a rate constant of λ ₂ [s⁻¹], and ¹⁶⁹Yb loss by thermal absorption to become ¹⁷⁰Yb at a rate constant of $\phi {\sigma }_2$ [s^‐1]. Eqs. (1)–(3) have a similar form to those solved by Bateman,³¹ which can be solved by taking the Laplace transform of the variables, yielding the following equations for the isotopic quantities:

$$N_1(t)=N_1^0{e}^{-\phi {\sigma }_{1}t},$$ (5)

$$\begin{array}{cc}{N}_2(t)=\hfill & \frac{\phi {\sigma }_1}{\phi {\sigma }_2+{\lambda }_2-\phi {\sigma }_1}{N}_1^0\left[e^{-\phi {\sigma }_{1}t}-e^{-\left(\phi {\sigma }_2+{\lambda }_2\right)t}\right]\hfill \\ +N_2^0{e}^{-\left(\phi {\sigma }_2+{\lambda }_2\right)t},\hfill \end{array}$$ (6)

and

$$N_3(t)=\frac{\phi {\sigma }_2}{\phi {\sigma }_3-\phi {\sigma }_2}{N}_2^0\left(e^{-\phi {\sigma }_{2}t}-e^{-\phi {\sigma }_{3}t}\right)+N_3^0{e}^{-\phi {\sigma }_{3}t}.$$ (7)

The second term on the right‐hand side of Eq. (6) accounts for existing therapeutic isotopes (¹⁹²Ir or ¹⁶⁹Yb) in the active source, which is nonzero for the case in which the source is being reactivated. For the case of $N_2^0=0$, which occurs during initial activation when there is no therapeutic isotope (¹⁹²Ir or ¹⁶⁹Yb) activity, Eq. (6) reduces to Eq. (32) as derived by Abdel‐Rahman and Podgorsak.³⁰ Isotope 3 may be capable of undergoing further thermal neutron absorptions, and Eq. (7) accounts for isotope 3 (¹⁷⁰Yb or ¹⁹³Ir) generation and loss by radioactive decay.

By the definition of radioactivity, A_m (t), the activity of isotope m at time t can be calculated as:

$$A_m(t)={\lambda }_m{N}_m(t),$$ (8)

where λ_m is the decay constant for isotope m.

Eq. (6) can be differentiated with respect to time, set equal to zero, and solved for t_s , the saturation time when the maximum activity of isotope 2 (¹⁹²Ir or ¹⁶⁹Yb) occurs, obtaining the following:

$$\begin{array}{c}{t}_s=-\frac{1}{\phi {\sigma }_2+{\lambda }_2-\phi {\sigma }_1}\\ \times ln\left[\frac{\phi {\sigma }_1}{\phi {\sigma }_2+{\lambda }_2}·\frac{\phi {\sigma }_1{N}_1^0}{\phi {\sigma }_1{N}_1^0-\left(\phi {\sigma }_2+{\lambda }_2-\phi {\sigma }_1\right)N_2^0}\right].\\ \end{array}$$ (9)

According to Eq. (8), the activity of isotope 2 at saturation, $A_2^{\text{sat}}$, is:

$$A_2^{\text{sat}}={\lambda }_2{N}_2\left(t_s\right),$$ (10)

and a general expression for the number of therapeutic isotopes (¹⁹²Ir or ¹⁶⁹Yb) in the source at the time of saturation can be obtained by substituting Eq. (9) into Eq. (6) to obtain:

$$\begin{array}{c}{N}_2\left(t_s\right)\\ =\frac{\phi {\sigma }_1{N}_1^0}{\phi {\sigma }_2+{\lambda }_2-\phi {\sigma }_1}\\ \times {\left[\frac{\phi {\sigma }_1}{\phi {\sigma }_2+{\lambda }_2}·\frac{\phi {\sigma }_1{N}_1^0}{\phi {\sigma }_1{N}_1^0-\left(\phi {\sigma }_2+{\lambda }_2-\phi {\sigma }_1\right)N_2^0}\right]}^{\frac{\phi {\sigma }_1}{\phi {\sigma }_2+{\lambda }_2-\phi {\sigma }_1}}\\ +\left(N_2^0-\frac{\phi {\sigma }_1{N}_1^0}{\phi {\sigma }_2+{\lambda }_2-\phi {\sigma }_1}\right)\\ \times {\left[\frac{\phi {\sigma }_1}{\phi {\sigma }_2+{\lambda }_2}·\frac{\phi {\sigma }_1{N}_1^0}{\phi {\sigma }_1{N}_1^0-\left(\phi {\sigma }_2+{\lambda }_2-\phi {\sigma }_1\right)N_2^0}\right]}^{\frac{\phi {\sigma }_2+{\lambda }_2}{\phi {\sigma }_2+{\lambda }_2-\phi {\sigma }_1}}.\\ \end{array}$$ (11)

For the case in which the initial therapeutic isotope activity is zero, $N_2^0=0$ and Eq. (11) reduces to:

$$\begin{array}{c}{N}_2\left(t_s\right)=\frac{\phi {\sigma }_1}{\phi {\sigma }_2+{\lambda }_2-\phi {\sigma }_1}{N}_1^0\left[{\left(\frac{\phi {\sigma }_1}{\phi {\sigma }_2+{\lambda }_2}\right)}^{\frac{\phi {\sigma }_1}{\phi {\sigma }_2+{\lambda }_2-\phi {\sigma }_1}}\right.\\ \left.-{\left(\frac{\phi {\sigma }_1}{\phi {\sigma }_2+{\lambda }_2}\right)}^{\frac{\phi {\sigma }_2+{\lambda }_2}{\phi {\sigma }_2+{\lambda }_2-\phi {\sigma }_1}}\right].\\ \end{array}$$ (12)

2.E. Precursor overhead

Precursor overhead is the minimum precursor mass needed to generate a therapeutic radiation source with the desired activity at a given average thermal neutron flux within the active source. Precursor overhead is a relevant quantity as it enables one to determine the minimum cost of the precursor needed to activate or reactivate a radiation source. An equivalent definition of precursor overhead is the precursor mass that yields the desired therapeutic radiation source activity at saturation. Precursor overhead can be calculated by solving Eq. (11) for $N_1^0=N_1^{min}$, the minimum ¹⁶⁸Yb isotope count at the start of thermal neutron irradiation, when all other quantities are known, and then applying the appropriate scaling factors to obtain precursor (¹⁶⁸Yb‐Yb₂O₃) mass.

The precursor overhead for a realistic neutron flux of 1 × 10¹⁴ n cm⁻² s⁻¹ was calculated for a 30 Ci ¹⁶⁹Yb source with (a) a starting ¹⁶⁹Yb activity of zero ($N_2^0=0$), and (b) a starting ¹⁶⁹Yb activity of 10.4 Ci ($N_2^0=10.4\text{Ci}/{\lambda }_2$), which would be the ¹⁶⁹Yb activity remaining after an 11.6 Ci source is transferred back to the reactor from the clinic after 5 days of transport.

To determine the impact of thermal neutron flux on precursor overhead, an infinite thermal neutron flux was also considered. Eq. (11) can be simplified to determine this by taking the limit as φ becomes infinite, obtaining an expression relating $N_2^{min}$, the minimum number of ¹⁶⁹Yb atoms needed to obtain the desired activity, to $N_1^{min}$:

$$\begin{array}{c}{N}_2^{min}=N_2\left(t_s\right)\\ =N_1^{min}\frac{{\sigma }_1}{{\sigma }_2-{\sigma }_1}{\left[\frac{{\sigma }_1}{{\sigma }_2}·\frac{{\sigma }_1{N}_1^0}{{\sigma }_1{N}_1^0-\left({\sigma }_2-{\sigma }_1\right)N_2^0}\right]}^{\frac{{\sigma }_1}{{\sigma }_2-{\sigma }_1}}\\ +\left(N_2^0-\frac{{\sigma }_1{N}_1^0}{{\sigma }_2-{\sigma }_1}\right){\left[\frac{{\sigma }_1}{{\sigma }_2}·\frac{{\sigma }_1{N}_1^0}{{\sigma }_1{N}_1^0-\left({\sigma }_2-{\sigma }_1\right)N_2^0}\right]}^{\frac{{\sigma }_2}{{\sigma }_2+{\sigma }_1}}.\\ \end{array}$$ (13)

Eq. (13) simplifies to the following for the case of an active source that does not contain ¹⁶⁹Yb, thus $N_2^0=0$:

$$N_2^{min}=N_2\left(t_s\right)=N_1^{min}\frac{{\sigma }_1}{{\sigma }_2-{\sigma }_1}\left[{\left(\frac{{\sigma }_1}{{\sigma }_2}\right)}^{\frac{{\sigma }_1}{{\sigma }_2-{\sigma }_1}}-{\left(\frac{{\sigma }_1}{{\sigma }_2}\right)}^{\frac{{\sigma }_2}{{\sigma }_2-{\sigma }_1}}\right].$$ (14)

2.F. Activation and reactivation

Activation and reactivation sequences were simulated using Eq. (6). To quantify the relative costs required to generate ¹⁶⁹Yb sources with the claimed active volumes of 1–4 mm³, calculations were performed to determine the number of reactor days at an average thermal neutron flux within the active source of 1 × 10¹⁴ n cm⁻² s⁻¹ and precursor mass needed to generate one clinic‐year of ¹⁶⁹Yb over a 1–4 mm³ range of active source volumes. For comparison, the number of reactor days needed to generate one clinic‐year of ¹⁹²Ir was calculated at an average neutron flux within the active source of 1 × 10¹⁴ n cm⁻² s⁻¹. The ¹⁹²Ir source was assumed to be a Varian GammaMedplus source with an active source diameter of 0.70 mm and length of 3.50 mm as described by Rasmussen et al.,²⁶ and clinical source lifetime was set to 90 days. Active source density was 22.56 mg mm⁻³, isotopic abundances were those listed in Table 1, and activity on clinic delivery was 10 Ci. Days between removal from the reactor, transport, and clinical installation for the ¹⁹²Ir source were assumed to be five calendar days, consistent with the ¹⁶⁹Yb source as defined in Section II.C.

3. Results

3.A. Model benchmarking

The model for source activation was benchmarked against published literature for ¹⁹²Ir activation to assess its accuracy and determine if it can be applied confidently for modeling ¹⁶⁹Yb generation. Results are shown in Table 2, which indicates that the calculation model was accurate to within 3.3% of reference values from two previous works: Abdel‐Rahman and Podgorsak³⁰ and the IAEA.²² The model is thus sufficiently accurate for ¹⁹²Ir activation calculations to justify its application to ¹⁶⁹Yb production.

Table 2.

Benchmarking results for activation calculations, which were performed for ¹⁹²Ir since references exist

$\phi$ (n cm−2s−1)	t (days)			$\text{SA}\left[\frac{\text{Ci}}{\text{g Ir}}\right],{}^{\ast }\left[\frac{\text{Ci}}{\mathrm{g}{}^{191}\text{Ir}}\right]$			Ref.
	Ref.	Calc.	Diff. (%)	Ref.	Calc.	Diff. (%)
1 × 1012	7		0	1.85	1.91	3.2	[22]
1 × 1012	28		0	6.73	6.94	3.1	[22]
1 × 1012	504 (Sat.)		0	28.9	28.5	−1.4	[22]
1 × 1013	258	261	1	579*	580*	0.2	[30]
2 × 1014	42.8	41.4	−3.3	2250*	2274*	1	[30]

Definitions: φ = average thermal neutron flux inside active source; t = time in reactor; SA = specific activity at time t; Ref. = reference citation or quantity; Calc. = calculated value; Diff. = percentage difference between Calc. and Ref., Sat. = saturation.

For numbers with * next to them, the units are [Ci / (g¹⁹¹Ir)]

3.B. Precursor overhead

To obtain a 30 Ci ¹⁶⁹Yb source with an average thermal neutron flux within the active source of 1 × 10¹⁴ n cm⁻² s⁻¹, one obtains a precursor overhead of 8.48 mg of 82%‐enriched ¹⁶⁸Yb‐Yb₂O₃ when the starting ¹⁶⁹Yb activity is zero and 7.8 mg when the starting ¹⁶⁹Yb activity is 10.4 Ci. For a precursor density of 8.15 mg mm⁻³, this corresponds to 1.04 mm³ of active source volume when starting with 0 Ci of ¹⁶⁹Yb, and the cost would be $5868 per physical source. At an infinite thermal neutron flux and all else equal, precursor overhead is 6.1 mg, corresponding a 0.76 mm³ active source volume and a cost of $4221 per physical source. Going from an infinite neutron flux down to an average thermal neutron flux within the active source of 1 × 10¹⁴ n cm⁻² s⁻¹ thus reduces precursor overhead cost by $1647, or 28%.

3.C. Activation of ¹⁶⁸Yb

Activation and precursor burnup curves for ¹⁶⁹Yb are shown in Fig. 1 for precursor volumes of 1, 1.25, 2, and 4 mm³. In Fig. 1(a), the activation and precursor burnup curves are shown for a full year of precursor irradiation by thermal neutrons at an average thermal neutron flux within the active source of 1 × 10¹⁴ n cm⁻² s⁻¹, indicating saturation occurs at approximately 30 days after the start of activation. Following one year of irradiation by thermal neutrons, nearly all ¹⁶⁸Yb atoms in the precursor have been activated regardless of the active source volume, corresponding to an activation of the following precursor (¹⁶⁸Yb‐Yb₂O₃) masses: 8.15, 10.19, 16.30, and 32.60 mg for 1, 1.25, 2, and 4 mm³ precursor volumes, respectively. Focusing on the first 30 days in the activation process, Fig. 1(b) indicates the activation times needed to achieve the goal ¹⁶⁹Yb activity at the end of activation of 30 Ci.

Figure 1.

Simulated activation and precursor burnup curves for four volumes of 82% enriched ¹⁶⁸Yb‐Yb₂O₃ ranging from 1 to 4 mm³, with a density of 8.15 mg mm⁻³, in a nuclear reactor with an average thermal neutron flux within the active source of 1 × 10¹⁴ n cm⁻² s⁻¹. Curves for (a) 1 year of continuous activation and (b) 30 days of continuous activation are shown. [Color figure can be viewed at wileyonlinelibrary.com]

3.D. Activation and reactivation

As shown in Fig. 1(a), the 2 mm³ and 4 mm^{3 169}Yb sources have clearly not been activated to their full potential after reaching 30 Ci, and substantial useful precursor remains in the active source volume. These sources can be reactivated after clinical usage; however, reactivation of the 1 mm³ and 1.25 mm³ sources to 30 Ci is infeasible under the assumptions herein since the amount of remaining precursor after the first activation, 5.8 mg and 7.2 mg, respectively, is below the required precursor overhead for reactivation of 7.8 mg in each case (Section III.B).

Reactivation curves for ¹⁶⁹Yb sources with active volumes of 1 mm³ and 3 mm³ are shown in Fig. 2(a), along with plots in Fig. 2(b) indicating clinic‐years and precursor burnup with reactor months at an average thermal neutron flux within the active source of 1 × 10¹⁴ n cm⁻² s⁻¹. For 1 mm³ vs 3 mm^{3 169}Yb active sources, 80 mg yr⁻¹ vs 21 mg yr⁻¹ or precursor per clinic‐year is needed, corresponding to a precursor cost of $55 360 yr⁻¹ vs $14 532 yr⁻¹, respectively. Reactor‐days per clinic‐year for 1 mm³ vs 3 mm³ active ¹⁶⁹Yb sources are reduced from 256 days yr⁻¹ to 59 days yr⁻¹. Increasing active source volume from 1 to 3 mm³ corresponds to a total reduction in ¹⁶⁹Yb annual precursor and reactor time costs of 74% and 77%, respectively.

Figure 2.

(a) Activity–time curves for ¹⁶⁹Yb sources with active source volumes of 1 mm³ and 3 mm³. A 1 mm³ source can be activated once before exhausting its ¹⁶⁸Yb precursor supply below the threshold needed for effective reactivation (7.8 mg in this example), and it will never meet the target activity of 30 Ci of ¹⁶⁹Yb (Fig. 1(b)). A 3 mm³ source, however, can be activated and reactivated at total of ten times. (b) Clinic‐years obtained and precursor burnup with reactor‐months assuming an average thermal neutron flux within the active source of 1 × 10¹⁴ n cm⁻² s⁻¹, for 1 mm³ and 3 mm³ active sources. For 1 mm³ sources, 256 reactor‐days and 80 mg per year of 82%‐enriched ¹⁶⁸Yb‐Yb₂O₃ precursor are needed to generate one clinic‐year of ¹⁶⁹Yb. For 3 mm³ sources, only 59 reactor‐days and 21 mg of precursor are needed per clinic‐year. [Color figure can be viewed at wileyonlinelibrary.com]

Resource consumption results for ¹⁶⁹Yb generation are shown in Fig. 3 for two precursor enrichment levels of 82% and 88%. Whether 82% enriched or 88% enriched ¹⁶⁸Yb‐Yb₂O₃ is used to generate the ¹⁶⁹Yb sources, the precursor mass and the number of reactor days needed per clinic‐year drop by 26%–31% when the active source volume increases from 2 to 3 mm³. The drop is another 11%–21% between 3 mm³ and 4 mm³ active source volumes at both enrichment levels.

Figure 3.

Resource consumption needed for one clinical‐year ¹⁶⁹Yb source generation. Average thermal neutron flux within the active radiation source was assumed to be 1 × 10¹⁴ n cm⁻² s⁻¹.

For comparison to the ¹⁶⁹Yb‐values, the number of reactor‐days needed to generate one clinic‐year of ¹⁹²Ir at an average thermal neutron flux within the active source of 1 × 10¹⁴ n cm⁻² s⁻¹ was calculated to be 64 days, which is 5 days longer than needed for the reactivated 3 mm^{3 169}Yb source generated with 82%‐enriched ¹⁶⁸Yb‐Yb₂O₃.

4. Discussion

4.A. Activation efficiency

Two primary principles dictate activation efficiency: the quantity of ¹⁶⁸Yb isotopes in the active source and activation linearity. At the start of the activation process, assuming no ¹⁶⁹Yb is already present in the active source, N ₁(t) >> N ₂(t) (¹⁶⁸Yb isotopes >> ¹⁶⁹Yb isotopes), and, as indicated by Eq. (2), the rate of change in the number of ¹⁶⁹Yb isotopes, dN ₂(t)/dt, is nearly a linear function of N ₁(t) (¹⁶⁸Yb isotope count) since the first term on the right‐hand side dominates. This means that the time efficiency of the activation process increases with the number of ¹⁶⁸Yb atoms present, as there are more targets that absorb thermal neutrons inside the active source. As the ¹⁶⁸Yb in the active source is converted to ¹⁶⁹Yb, the number of ¹⁶⁸Yb isotopes (targets) diminishes as described by Eq. (5), reducing the rate of ¹⁶⁹Yb generation and therefore activation efficiency. The activation curve will still remain approximately linear if sufficient ¹⁶⁸Yb remains in the active source, as shown for the 2 mm³ and 4 mm³ sources in Fig. 1(b), indicating activation times of 3–7 days. The activation curve will become asymptotic as the quantity of remaining ¹⁶⁸Yb approaches the minimum required to generate a 30 Ci source, as shown in Fig. 1(b) for the 1 mm³ and 1.25 mm³ sources. This is because ¹⁶⁹Yb generation is being limited by dwindling ¹⁶⁸Yb availability, and is also competing with ¹⁶⁹Yb losses due to radioactive decay and thermal neutron absorption by the ¹⁶⁹Yb isotopes. The asymptotic activation effect substantially decreases activation efficiency below that for linear activation, with 15–30 day activation times for the 1 mm³ and 1.25 mm³ sources, and, for the case of the 1 mm³ active source, the goal activity of 30 Ci is never reached. This is made clear in Fig. 1(b), where activation of the 1 mm³ source is clearly an asymptotic process and activation of the 3 mm³ source is clearly a linear process.

4.B. Minimizing precursor waste

Minimizing precursor waste is achieved by delaying the step in which the source is discarded as much as possible through reactivation. As shown in Fig. 2(b), for a 1 mm³ source, 5.8 mg of precursor is wasted after each source usage due to the inability to reactivate the source. For the 3 mm³ source, however, unactivated precursor is not discarded until after ten activations, and approximately the same amount of precursor is discarded as for a 1 mm³ active source, drastically reducing precursor waste per activation.

4.C. Accounting for radioactive impurities

The primary radioactive impurities of concern in a ¹⁶⁹Yb source are ¹⁷⁵Yb and ¹⁷⁷Yb, which have half‐lives of 105 h and 1.9 h, respectively.²² The radioactive impurities are produced during thermal neutron activation of the precursor due to the presence of isotopic impurities of ¹⁷⁴Yb and ¹⁷⁶Yb. The expected ¹⁷⁵Yb activity following 12 days of neutron irradiation of 17.1% enriched ¹⁶⁸Yb‐Yb₂O₃ at 1 × 10¹⁴ n cm⁻² s⁻¹ is 0.2% of the ¹⁶⁹Yb activity, and 10 days of cooling are recommended prior for decay of radionuclidic impurities.²² In the cited study, the ratio of ¹⁶⁸Yb to ¹⁷⁴Yb in the initial sample was 17.1/15.5 = 1.10. With 82% enriched ¹⁶⁸Yb‐Yb₂O₃ precursor, approximate ¹⁷⁴Yb and ¹⁷⁶Yb abundances are 3% and 1%, respectively, and the resulting ¹⁷⁵Yb impurity following activation would be (0.2%) (1.10) (3/82) = 0.008%. This is a ¹⁷⁵Yb‐to‐¹⁶⁹Yb activity reduction by a factor of 1/25 relative to that of the 17.1% enriched ¹⁶⁸Yb‐Yb₂O₃ precursor, which is equivalent to 20 days of cooling given the 105 h (4.38 days) half‐life of ¹⁷⁵Yb. This is effectively 10 days beyond the IAEA recommended 10 days of cooling for the ¹⁶⁹Yb source with 0.2% activity ¹⁷⁵Yb impurity.²² In the cited study, the activity of the lower half‐life isotope, ¹⁷⁵Yb, was considered acceptable after 10 days of cooling when the ratio of ¹⁷⁴Yb to ¹⁷⁶Yb in the initial sample was 15.5%/4.5% = 3.4. As the ratio of ¹⁷⁴Yb to ¹⁷⁶Yb in the 82%‐enriched ¹⁶⁸Yb‐Yb₂O₃ precursor is 3 — lower than that from the cited study — the produced ¹⁷⁵Yb from the 82%‐enriched precursor would be sufficiently low immediately after activation to be acceptable without additional cooling. The reduction in ¹⁷⁵Yb obtained by using 82% or higher enriched precursor is therefore sufficient to enable prompt shipment of the ¹⁶⁹Yb source following activation; thus, a 5‐day period between the removal of the ¹⁶⁹Yb source from the nuclear reactor and installation in a clinical afterloader is a realistic time frame.

4.D. Thermal neutron attenuation within the active source volume

An active source volume containing precursor material and some existing quantity of the therapeutic isotope will attenuate the thermal neutrons impinging upon it; thus, the thermal neutron flux at the center of the active source will be lower than that at the surface. This attenuation is implicitly accounted for in the calculations presented in the current work since the quantity, φ, is the average thermal neutron flux inside the active radiation source, thus including the effects of neutron attenuation in the source. The thermal neutron flux that would need to be delivered in the volume in which the active source will be placed within the reactor, that is, the advertised reactor neutron flux, ${\phi }_0$, would thus need to be greater than the average thermal neutron flux inside the active radiation source, φ.

Gaining a thorough understanding of the distribution of thermal neutrons within an active radiation source placed inside a research reactor, and the change in the distribution over time as isotopes are converted, could be accomplish using Monte Carlo simulations. Here, we present a simple analytical thermal neutron attenuation calculation in which it is demonstrated that Yb precursor material is substantially less attenuating than Ir precursor material of the same geometric size. The thermal neutron flux at depth [cm] can be calculated as:

$$\phi (x)={\phi }_0{e}^{-\mu x},$$ (15)

where ${\phi }_0$ is the flux of the thermal neutron beam at the surface of the active source and the attenuation coefficient, μ, [cm⁻¹] is obtained as³²:

$$\mu =\frac{\sigma N_A\rho }{m_a},$$ (16)

where N_A is Avogadro's number (6.023 × 10²³ g mol⁻¹), ρ is the density of the medium [mg mm⁻³], m_a is the atomic mass of the medium [g mol⁻¹], σ is the thermal neutron absorption cross section for the active source [cm²], which is a linear combination of σ ₁, σ ₂, and σ ₃, weighted by the relative abundances of isotopes 1 (¹⁹¹Ir, ¹⁶⁸Yb), 2 (¹⁹²Ir, ¹⁶⁹Yb), and 3 (¹⁹³Ir, ¹⁷⁰Yb) from Table 1. In this analysis, σ will be set equal to σ ₁, corresponding to the case of an active source that is pure unactivated precursor material. Attenuation coefficients for Ir and Yb precursor materials are calculated using ρ‐values of 22.56 g cm⁻³ and 8.15 g cm⁻³, respectively, and m_a ‐values of 192.2 and 173, respectively, and are 88.1 cm⁻¹ and 53 cm⁻¹, respectively. Assuming that the precursor material has a diameter of 0.6 mm, the distance on an axial cross section between the surface of the active source and the center is 0.3 mm, and the thermal neutron transmission between those points for Ir and Yb active sources would be 7% and 20%, respectively. These relative attenuation values will shift throughout the activation process as the respective quantities of ¹⁶⁸Yb and ¹⁶⁹Yb change with time.

Based on these attenuation calculations, we conclude that, if the geometric and physical properties of Ir and Yb active source precursors are set such that the total thermal neutron irradiation time needed to generate one clinic‐year of ¹⁹²Ir and ¹⁶⁹Yb is equal according to the activation model of Eq. (5), Eq. (6), and Eq. (7), then, once thermal neutron attenuation is fully accounted for, the total actual neutron irradiation time needed to generate ¹⁶⁹Yb will be less than for ¹⁹²Ir.

4.E. Mechanical considerations on the impact of active source volume

Brachytherapy applicators in current clinical use for HDR‐BT have been, in general, developed for use with active radiation sources of 3–5 mm²⁶ length, thus shorter than the maximum proposed ¹⁶⁹Yb source length of 10.5 mm examined in the current work. A mechanical concern with active radiation sources of greater length is that their capability for use in conventional applicators may be mechanically limited due to their inability to navigate curves in existing applicators which were designed for use with shorter active sources. For example, a conventional tandem and ring applicator used for cervical cancer, such as the 3D Interstitial Ring Applicator from Varian, has a ring diameter of 30 mm and a channel diameter that supports a 3.5 mm long source, which is likely too small to support a 10.5 mm radiation source without substantial modification of the channel diameter. As the primary purpose of the proposed ¹⁶⁹Yb source is to deliver brachytherapy treatments using applicators containing partial radiation shields, the applicators will, by default, need to be redesigned and reapproved to make clinical adoption possible. Accommodating a ¹⁶⁹Yb source that is 10.5 mm in length, for example, would be relatively straightforward in a tandem and ovoid applicator designed for cervical cancer by adjusting the curvature of the ovoid applicator channels. For tandem and ring applicators used for cervical cancer brachytherapy, however, more significant changes would be required to support a larger source, such as increasing the size of both the ring and the source channel. It is possible, due to the capability to deliver tumor‐conformal doses using RSBT or DMBT, that a ring applicator, and possibly ovoids as well, may become obviated and can likely be modified or discarded pending treatment planning analyses. In addition, the tandem applicator geometry will need to be reevaluated, possibly by widening the “elbow” through which the source passes on the way to the distal end, to enable a longer radiation source to pass through a 45° curve.

Applicators for other sites such as the breast, which can be treated effectively with a multicurved channel applicator such as SAVI (Cianna Medical, Aliso Viejo, CA), may also need to be redesigned. Such an applicator can be replaced with an RSBT‐type applicator, which may deliver a dose distribution that is clinically equivalent to the SAVI applicator but with a lower curvature in the channel or channels used for the delivery in support of the longer radiation source. The cylindrical applicators used to treat endometrial and vaginal cancer tend to have straight channels and can be used with a ¹⁶⁹Yb source that is 10.5 cm long, for example, with little to no modification. It may be that such patients would benefit from a DMBT or RSBT applicator as well, which has been considered previously.^{33, 34}

4.F. Other additional costs of ¹⁶⁹ Yb usage beyond ¹⁹²Ir

In addition to source production and applicator redesign costs, the shorter half‐life of ¹⁶⁹Yb relative to ¹⁹²Ir would add cost regarding the more frequent source exchanges and additional quality assurance necessary at each source change. As with the advent of intensity‐modulated radiation therapy, which added quality assurance costs beyond 3D conformal radiation therapy when introduced to radiotherapy clinics, the rationale for adding such costs must be weighed relative to the expected clinical benefits associated with them.

5. Conclusion

Dramatic savings are achievable in ¹⁶⁹Yb source production costs through reactivation if active sources larger than 1 mm³ are used.

Conflict of Interest

RTF is the founder of pxAlpha, LLC, which is developing a rotating shield brachytherapy delivery device.

Flynn RT, Adams QE, Hopfensperger KM, Wu X, Xu W, Kim Y. Efficient ¹⁶⁹Yb high‐dose‐rate brachytherapy source production using reactivation. Med. Phys.. 2019;46:2935–2943. 10.1002/mp.13563