Water Phantom Characterization of a Novel Optical Fiber Sensor for LDR Brachytherapy

This work considers the feasibility of using a novel optical fiber-based sensor, employing a terbium-doped gadolinium oxysulfide (Gd2O2S:Tb) inorganic scintillator, as a real-time in vivo dosimetry solution for applications in low-dose-rate (LDR) prostate brachytherapy (BT). This study specifically considers the influence of scintillator geometry (hemisphere tip versus cylindrical cavity), polymethyl methacrylate (PMMA) fiber core diameter (0.5 versus 1.0 mm), and sensor housing material (stainless steel versus plastic) on the measured scintillation signal. Characterization measurements were performed using a silicon photon-multiplier (SiPM) detector and a commercial water phantom system, integrated with custom 3-D printed components to allow for precise positioning of the LDR BT radiation source with respect to the optical fiber sensor (OFS). Significant differences in the rate of fall-off in the scintillation signal, with distance from the source, were observed between the different scintillator geometries considered. The hemisphere tip geometry was shown to be the most accurate, tracking with the expected fall-off in dose-rate, within measurement uncertainty. Reducing the fiber core diameter from 1.0 to 0.5 mm resulted in a sixfold reduction in the detected scintillation signal. A further 57% reduction was observed when housing the 0.5-mm fiber within a stainless steel LDR BT needle applicator. Initial results demonstrate the feasibility of employing an OFS, for applications in LDR BT, given the excellent agreement of measurements with theoretical expectations. Furthermore, a calibration process has been described for converting the detected scintillation signal into absorbed dose/dose rate, using our water phantom-based experimental setup.

some form of RT, after their cancer diagnosis [1], [2], [3]. RT can be divided into external beam RT (EBRT), referring to the use of a linear accelerator to deliver radiation from outside the patient's body, or brachytherapy (BT), referring to the use of radioactive sources that are implanted within, or in close proximity to, the critical target. BT can be divided into categories based on the dose rate of the employed radioactive source, with low-dose-rate (LDR) and high-doserate (HDR) BT employing sources with dose rates of <2 and >12 Gy/h, respectively [4]. BT treatments can also be categorized based on the implantation technique employed (i.e., permanent implantation versus temporary implantation). This work focuses on LDR prostate BT, employing a permanent implantation technique, and iodine 125 ( 125 I) as the LDR BT source [5], [6]. Due to the steep dose gradients associated with LDR BT sources, precise positioning of the source is crucial to ensure both target coverage and minimization of the dose to uninvolved tissues and organs at risk (e.g., bladder, urethra, and rectum). Currently, source positioning during treatment delivery relies on transrectal ultrasound imaging (TRUS), which can be challenging due to the low echogenic properties of radioactive 125 I sources (typically referred to as seeds) [7]. The final verification of source positions is therefore typically performed using a post-implantation computed tomography (CT) image and/or magnetic resonance imaging (MRI) [8]. Post-implantation CT images are typically acquired on the day of treatment as an initial indication of implant quality, and 30 days after treatment, at which point swelling of the prostate has subsided providing a better estimate of relevant dosimetric metrics, in terms of target coverage (D90, V100, V150, and so on) and OAR constraints (D2cc Rectum, D30 Urethra, and so on) [6].
Post-implantation MRI imaging, in combination with CT imaging, has also been considered as a method for improving soft tissue delineation [9]. Post-implantation CT/MRI imaging, however, does not represent real-time independent monitoring of the actual dose delivered to the patient and, as such, creates an environment where errors may go undetected during implantation. Furthermore, any issues that are identified using post-implantation CT/MRI may be difficult for clinicians to address, if at all possible.
The Horizon 2020 funded ORIGIN Project (Grant Agreement ID: 871324) aims to overcome these challenges, by providing a real-time in vivo dosimetry solution for applications in LDR prostate BT. This will be achieved through the development of a novel optical fiber-based sensor system, employing a terbium-doped gadolinium oxysulfide (Gd 2 O 2 S:Tb) inorganic scintillator. Optical fiber-based sensors have the small size required for insertion into BT needle applicators/catheters, allowing for potential in vivo measurements during a procedure. Previous work by Woulfe et al. [10] has reported initial results demonstrating the potential of Gd 2 O 2 S:Tb as a scintillator for applications in LDR prostate BT.
This study considers the characterization of the optical fiber sensor (OFS) design previously described [10], as well as additional sensor designs varying both the scintillator geometry at the tip of the sensor and the polymethyl methacrylate (PMMA) fiber core diameter. Characterization was performed using a commercial water phantom system integrated with custom 3-D printed components, allowing for a more precise evaluation of sensor performance. Therriault-Proulx et al. [11] have previously considered validation of a plastic scintillation detector for applications in LDR BT using a water phantom setup, with a reported accuracy of ±1.0 mm. This work, however, considers a water phantom system employing a mechanical stage with an improved accuracy of ±0.1 mm and provides a detailed description of the integrated novel 3-D printed components. Finally, the influence of housing the sensor in a clinically relevant stainless steel (CP Medical Inc., GA, USA) LDR BT needle applicator was considered. To the best of the authors' knowledge, we are the first group to characterize inorganic scintillation detectors, for applications in LDR BT, in this way.

II. METHODS AND MATERIALS
A. Optical Fiber Sensors 1) Scintillator Powder: Gd 2 O 2 S:Tb powder was considered as the scintillator in this work. It was obtained from Phosphor Technology Ltd., U.K. (Type UKL65/F-R1), has a density of ∼7.34 g/cm 3 and a median particle size of 3.5 μm [12].
When exposed to ionizing radiation, Gd 2 O 2 S:Tb produces scintillation light, characterized by a primary peak wavelength of 545 nm and a decay time of ∼600 μs [13], implying that each primary gamma interaction within the scintillator powder produces a series of single photons, requiring the use of detectors with single-photon detection capabilities.
2) Scintillator Geometry: Two scintillator geometries were considered to investigate their influence on measurement accuracy: 1) a cylindrical cavity and 2) a hemisphere tip (see Fig. 1(a) and (b), respectively). For the cylindrical cavity, precision drilling of the PMMA fiber core tip was performed to create a 0.7-mm-diameter cavity; two different cavity depths of 5 and 7 mm were considered. The cavities were filled with scintillator powder and sealed at the tip with Henkel Loctite Hysol medical device epoxy. A hemisphere tip geometry was also considered, where the diameter of the hemisphere was chosen to be 1.0 mm. The tip itself consisted of a mixture of scintillator powder and Norland optical adhesive (NOA) 61 epoxy (3:2 ratio). The hemisphere tip was affixed to the PMMA fiber core and cured using an externally applied ultraviolet (UV) lamp.
3) PMMA Fiber Core Diameter: Measurements were performed with sensors fabricated using a PMMA fiber core diameter of 1.0 mm and a total outer diameter of 2.2 mm, including the protective jacket (Mitsubishi Chemical Group, Super Eska, SH4001). Measurements were also performed with a sensor fabricated using a PMMA fiber core diameter of 0.5 mm (hemisphere tip scintillator geometry only) and a total outer diameter of 1.0 mm, including the protective jacket (Asahi Kasei, DC-500). With a total outer diameter of 1.0 mm, this sensor has the potential to be inserted into the clinically employed 18 gauge needle applicator, which has an inner diameter of ∼1.0 mm.

B. Water Phantom System
All of the measurement data presented in this study were acquired using the PTW MP3-XS water phantom system (PTW, Freiburg, Germany). The use of a water phantom, filled with sterile water, allows for clinically relevant measurements to be obtained in a standardized, reproducible, and dosimetrically accurate environment. A key feature of the water phantom system is the 3-D stainless steel mechanical stage capable of moving in increments of 0.1 mm, therefore allowing for high-resolution sampling of the steep dose gradients associated with BT sources. The accuracy of the mechanical stage is also defined as being ±0.1 mm by the manufacturer [14]. The water phantom system also has an accompanying control software, MEPHYSTO mc 2 , which allows the user to design custom "task lists," enabling the design of measurement schemes, automating data acquisition.
To achieve the goal of employing the water phantom system described, for LDR BT measurements, a method for positioning the radiation source relative to the OFS had to be developed. This was achieved through the design of custom 3-D printed components, which were integrated with the PTW water phantom system (see Fig. 2). PolyJet 1 material (model 1 Trademarked. MED610), which is a polymer-like material with a density of 1.17-1.18 g/cm 3 , was employed for all 3-D printing. The 3-D printer employed was the Stratasys Objet Connex500 with a printing accuracy of ∼64 μm.
Three main components were printed: 1) a support frame; 2) a source holder; and 3) a sensor holder. The support frame consists of a T-junction top support, which slots onto the top of the water phantom. A 10-cm-long "guide tube" extends from the support frame into the water phantom; this feature was designed to reduce setup uncertainty by constraining the alignment of the source holder, helping to ensure that the source is positioned in a parallel orientation. The T-junction design also ensures that the source holder is delivered to the same point within the water phantom for each setup. A 3-D printed source holder was also designed for housing a plastic needle (obtained from Eckert & Ziegler Bebig GmbH, Berlin, Germany, Product Ref: LLA200-KB) within which the 125 I seed is placed. The 3-D printed source holder component once again acts to constrain the alignment of the plastic needle, through which the source is inserted, and in this way ensures that the source is positioned parallel to the sensor. The cuboid design of both the guide tube and the source holder component eliminates rotational variations between experiments (i.e., the source holder does not rotate within the guide tube). The plastic needle protrudes (∼10 mm) from the bottom of the 3-D printed cuboid section to allow for more precise positioning and alignment of the 125 I source with the sensor. Finally, the 3-D printed OFS holder is designed such that it slots into the PTW TRUFIX "Markus Electron Chamber" holder. It can therefore be mounted on the mechanical stage of the water phantom, allowing for precise measurements to be performed. Once again, the guide tube design feature is employed to enable accurate setup and positioning of the sensor with respect to the source. Variations of this component were designed to allow for the sensor to be housed within a plastic needle (obtained from Eckert & Ziegler Bebig GmbH, Product Ref: LLA250-K15) or a stainless-steel BT needle (obtained from CP Medical Inc., Product Ref: CPPS-MN-1821-5), with only the diameter of the 3-D printed channel changing to match that of the specific needle employed. The specific needle housing the sensor protrudes (∼10 mm) from the guide tube component to allow for more precise positioning and alignment of the sensor with the 125 I source.

C. Photon Counting Detector System
The photon counting detector system employed in this work is provided as part of the CAEN SP5600E Educational Photon Kit [15]. The components employed were the CAEN SP5600 power supply and amplification unit (PSAU), the CAEN DT5720A desktop digitizer (required only for activation of the CAEN control software), and a Hamamatsu silicon photomultiplier (SiPM) [model S13360-1350CS]. This SiPM has an active area of 1.3 × 1.3 mm 2 and a pixel pitch of 50 μm. The SiPM holder, which mounts onto the front end of the PSAU, has an embedded temperature feedback sensor allowing for gain compensation. The OFS is interfaced to the SiPM via an FC terminated connection. Furthermore, the PSAU and digitizer are interfaced to the laptop via USB 2.0 connections,  allowing for readout and analysis of measurement data. Fig. 3 shows the full schematic of the experimental setup employed, incorporating the water phantom system, the custom 3-D printed components, and the photon counting detector system.

D. Measurement Procedure
For the water phantom setup shown in Figs. 2 and 3, the 125 I seed is surrounded with ≥10 cm of sterile water in all directions, ensuring consistent and accurate scattering conditions. Both the 125 I seed and sensor were also ≥10 cm from the metal components of the water phantom system at all times. Initially, the sensor and the radiation source are positioned parallel to one another, with their centers aligned, defining a preliminary "null point" (i.e., the origin of the coordinate system). This in turn defines the position of the sensor relative to the 125 I seed (note: the sensor is positioned 10 mm from the source when defining the "null point"). This null point definition is then confirmed radiologically, that is to say, when the sensor and the source are correctly aligned, the scintillation signal detected at equal distances, on opposing sides of the source, should be equivalent. This is due to the cylindrically symmetrical dose distribution about the source and symmetrical geometries of the scintillators (cylindrical cavities and hemisphere tip). Fine-tuning of the null point, in 0.1-mm increments, is performed until this condition is met, within measurement uncertainty.
Within the CAEN Control Software, the user defines a number of data acquisition parameters (see Table I), including the SiPM bias voltage (defined by Hamamatsu as the breakdown voltage +3 V), amplifier gain, threshold voltage above which an "event" is counted, and various counting parameters [e.g., the gate width and the number of data points averaged when calculating the photon counting rate (PCR)]. The measurement acquisition process has been automated through the use of the MEPHYSTO mc 2 control software, allowing the user to define the position and duration of each measurement. The dark count rate (DCR), which is the signal generated within the SiPM in the absence of a scintillation signal, was averaged over 30 s and performed off-axis > 10 cm from the source, an AgX100 125 I seed, provided by Theragenics Corporation (GA, USA), with an activity of approximately 0.336 mCi. The PCR, which is the signal generated within the SiPM in excess of the DCR, due to the presence of a scintillation signal, was similarly averaged over 30 s. For all of the data presented in this study, consistent ambient conditions (e.g., lighting and room temperature) were maintained. Note that measurements were performed in a dark room, with the sensors housed in needles that were opaque to visible light.
Measurement files generated using the CAEN detector system were analyzed using Microsoft Excel. Data analysis involved: 1) segmentation of the raw dataset file consisting of thousands of raw count values, so as to assign each data point to the relevant measurement position; 2) subtraction of the mean DCR value from the raw counts to provide a measure of the PCR; and 3) calculation of the mean PCR signal and the standard deviation of the PCR signal, at each position.

E. Error Analysis
The random noise component, of the overall measurement uncertainty, is defined as the standard error of the mean measurement signal (σμ i ). Multiply σμ i by 3 to give an approximate coverage of 99.7%.
There is, however, also a positional uncertainty component, which is given by the mechanical stage accuracy, defined by the manufacturer as ±0.1 mm. To understand how this positional uncertainty manifests as an uncertainty in the mean measurement signal, at a particular distance from the 125 I source, we must first understand how the expected dose rate varies as a function of distance from the 125 I source. If we begin by considering the American Association of Physicists in Medicine (AAPM) Task Group No. 43 Report [16], [17], the dose rate can be calculated at some point of interest P(r, θ), where r is the distance from the center of the source to the point of interest and θ is the angle with respect to the long axis of the sourcė Here, S K and are constants, which describe the air-kerma strength of the source and the dose rate constant, respectively. F(r, θ) describes the anisotropy function, which is equal to unity in this work since all measurements were performed at θ = π/2 radians, with respect to the source. G(r, θ) is the geometry factor and g(r ) is the radial dose function. For the purpose of this work, the constants S K and , and the function F(r, θ), do not influence the rate at which the calculated dose rate varies with distance from the source. We, therefore, focus on G(r, θ) and g(r ). Employing the line source approximation, the geometry factor is calculated as where L is the source active length (3.5 mm for the AgX100 125 I seed) and β is the angle subtended by the source with respect to the point of interest. The radial dose function is described by a modified polynomial [18] g(r ) = a 0 r −2 + a 1 r −1 + a 2 + a 3 r + a 4 r 2 + a 5 r 3 · e −a 6 r .
(3) Fit parameters were calculated by the Carleton Laboratory for Radiotherapy Physics (CLRP) using the egs_brachy Monte Carlo software, as part of their AAPM Task Group No. 43 parameter database (for the AgX100 125 I seed) [19].
Using (2) and (3), we can calculate the values for G(r, θ) and g(r ) at any nominal distance, and we can also calculate the influence of a ±0.1-mm shift at any nominal distance. Furthermore, since G(r, θ) and g(r ) are the two parameters, which influence the fall-off in the dose rate as a function of distance from the source (in this work), dividing normalized PCR measurement data by G(r, θ)/G(r 0 , θ 0 ), for example, should yield a result, which scales with g(r ) or vice versa. In this way, a comparison between theoretical expectation and measured PCR data can be performed. Fig. 4 shows a relative comparison of the PCR fall-off along the transverse axis as a function of distance from the 125 I source, over the distance range of 5-30 mm, for each of the scintillator geometries considered in this work. The "expected" fall-off in the dose rate as a function of distance from the source is characterized by G(r, θ) and g(r ) as previously discussed. Dividing normalized data in Fig. 4 by the value for G(r, θ)/G(r 0 , θ 0 ) at each measurement distance yields the results shown in Fig. 6 for each scintillator geometry. Note that the error bars on the measurement data in Figs. 4-7 represent three times the standard error of the mean (σμ i ) measurement signal at each distance. Fig. 5 shows the measured PCR, using a hemisphere tip geometry (1.0-mm-diameter core fiber), as a function of distance from the source. The DCR at the employed discriminator threshold voltage of −15 mV was ∼63 kHz. The PCR signal  5 mm from the source was ∼279 kHz (in excess of the DCR), falling to ∼4 kHz (in excess of the DCR) 30 mm from the source. Agreement with theoretical expectation [i.e., the product of G(r, θ)/G(r 0 , θ 0 ) and g(r )] is clearly evident, across the entire range of distances considered, within measurement uncertainty. Note that the error bars on the theoretical data represent the calculated impact of a ±0.1-mm shift at each distance. The influence of moving from a 1.0-mm-diameter core fiber (in a plastic needle) to 0.5-mm-diameter core fiber (in plastic and stainless-steel needles) was also considered. Ratios of the measured PCRs (mean, minimum, and maximum) across the entire measurement range are provided in Table II. Moving from a 1.0-mm-diameter core fiber to 0.5-mm-diameter core fiber (both in plastic needles) results in an approximate sixfold reduction in the PCR. A further ∼57% reduction in the PCR is observed when housing the 0.5-mm-diameter fiber in a stainless steel LDR BT needle applicator. It is worth noting that a hemisphere tip geometry (diameter 1.0 mm) was employed for both the 1.0-mm fiber core diameter and the 0.5-mm fiber core diameter and that the DCR remained at ∼63 kHz for the three datasets shown. Normalizing the measurement data and dividing by the value for G(r, θ)/G(r 0 , θ 0 ) at each measurement distance yields the results provided in Fig. 7, for each fiber diameter/needle configuration considered.

IV. DISCUSSION
Water phantom measurements presented in this work were obtained over the clinically relevant distance range of 5-30 mm. The minimum distance of 5 mm was chosen since this distance describes the spacing on the LDR BT template grid. The maximum distance of 30 mm was chosen based on average dimensions of the prostate gland of ∼60 mm, as obtained from an in-house clinical audit. Assuming a distribution of sensor locations within the prostate and indeed within the urethra, sensitivity over a range of 30 mm was deemed sufficient to ensure coverage of the entire prostate.
Considering Fig. 4, differences in the relative response for each of the scintillator geometries considered are observed. A relative comparison was performed since each of the sensors considered in this work was fabricated manually. Therefore, differences in the raw scintillation signal generated by each sensor may be observed, which are not due to the scintillator geometry employed. For example, slight variations in the FC connection from one sensor to the next, the use of pure scintillator powder in the cylindrical cavities versus a powder-NOA 61 epoxy mix in the hemisphere tip, the distribution of dopant sites within the scintillator volume, and so on. However, the relative change in the response as a function of distance from the source should be independent of the raw scintillation signal and in this way should allow for a comparison of the scintillator geometries considered. The observed differences in Fig. 4 are particularly evident when the normalized data are divided by G(r, θ)/G(r 0 , θ 0 ) and compared to g(r ), as shown in Fig. 6. The hemisphere tip geometry provides the best agreement with theoretical expectation, agreeing with g(r ) within uncertainty. For both of the cylindrical cavity geometries considered, a relative under-response is observed at distances closer than 10 mm (normalization distance) from the 125 I source, while an overresponse is observed at distances greater than 10 mm. It is also noted that the observed deviation from theoretical expectation increases with increasing depth of the cavity. This finding can, in part, be explained by the steep dose gradient around the 125 I source, that is to say, due to variation in the dose rate to the scintillator across its length, theoretical data (calculations) may need to be integrated over the length of the scintillator, as previously described by Therriault-Proulx et al. [11] for their 5-mm-long plastic scintillation detector. However, there is an additional consideration for inorganic scintillation detectors (such as Gd 2 O 2 S:Tb); the scintillator material is not transparent with respect to the scintillation light it produces. Therefore, as you move along the cylindrical volume of the cavity (away from fiber), less scintillation light is coupled into the fiber, reducing the light collection efficiency [20]. Furthermore, for the cavity geometries considered, although they are cylindrically symmetrical, eliminating any angular  dependence for the measurements performed in this work, they are likely to exhibit an angular dependence in the longitudinal plane [20], which could possibly limit future use as an in vivo dosimeter. Based on these findings, it was decided to focus on the hemisphere tip scintillator geometry for the remainder of this work.
Referring to Fig. 5, agreement between measured PCR data and theoretical expectation, within uncertainty, is evident across the 5-30-mm distance range. This finding indicates that an "energy correction" to account for changes in the response of the sensor as a function of distance from the source, due to changes in the energy spectra and the high-density nontissue equivalent inorganic scintillator material, may not be required for applications in LDR prostate BT. Furthermore, previous Monte Carlo modeling studies have independently reported that changes in the energy spectra of 125 I with distance are small, particularly over the distance range considered in this work [21], [22]. Based on these findings, we believe that the conversion from PCR to absorbed dose will require only a single calibration coefficient. Calculation of this calibration coefficient could be achieved in the water phantom using an AgX100 125 I seed "calibrated" by performing a measurement using a radiation dosimeter with a traceable calibration to a "primary standard." The PCR could be measured at a reference distance of 10 mm from the source on the transverse axis, in the water phantom; using the known dose rate from the calibrated seed, a calibration coefficient could be calculated with units of absorbed dose per unit frequency (e.g., cGy/kHz). A distance of 10 mm from the 125 I source for the purposes of calibration strikes a balance between the influence of the positional and noise components of the measurement uncertainty. It is also consistent with the definition of the "reference position" from the AAPM Task Group No. 43 Report [16], [17], defined as lying on the transverse bisector of the source at a distance of 1 cm.
The influence of the fiber core diameter on the measured PCR was also considered, with a sixfold reduction observed for the 0.5-mm fiber core compared to the 1.0-mm fiber core, when both are housed within a plastic needle. Taking the 0.5-mm fiber core and then placing it within the clinically relevant stainless steel LDR BT needle applicator resulted in a further ∼57% reduction in the PCR, due to attenuation of the low-energy photons associated with 125 I (maximum energy ∼35.5 keV). A result of the observed reduction in the scintillation signal for the 0.5-mm fiber core diameter was a relative increase in the magnitude of the noise component of the uncertainty (error bars), as shown in Fig. 7. This is due to the fluctuations in the DCR dominating the fluctuations in the measurement signal (DCR + PCR), as the PCR reduces. There is the additional possibility for transient fluctuations in the measurement signal to negatively impact measurement accuracy, particularly as the PCR reduces. It is worth highlighting, however, that the data obtained using the 0.5-mm fiber core diameter, shown in Fig. 7, remained in agreement with theoretical expectations within uncertainty, for distances of less than or equal to ≈25 mm. The observed reductions in PCR may necessitate the use of larger integration times during data acquisition, to achieve acceptable levels of statistical uncertainty. Future work could investigate if further optimization of the scintillator geometry, optical coupling, and/or SiPM characteristics (e.g., thermo-electric cooling) could be employed to offset the observed reductions in PCR.
These results indicate that the rate of fall-off in the PCR is dependent on the characteristics of the radiation source and the geometry of the scintillator (see Figs. 4 and 6). The use of the 0.5-mm fiber would have the advantage of allowing more fibers to occupy the same space (e.g., within a needle/catheter) and it would also allow for jacketed fibers to be inserted into the needle applicator, which would reduce the impact of ambient or parasitic light during a clinical procedure. Data obtained in this work demonstrate that the experimental design employed, including the water phantom system and 3-D printed components, has the potential to be employed for the characterization and calibration of OFSs, within different BT needles/applicators.

V. CONCLUSION
A custom water-phantom system-based experimental design is presented in this work and employed to obtain precise OFS measurements, for applications in LDR prostate BT. Results obtained demonstrate that the rate at which the scintillation signal falls off with distance from the clinical radiation source (AgX100 125 I seed), specifically for the hemisphere tip geometry, agrees with theoretical expectation for the fall-off in dose rate. Furthermore, water phantom measurements indicate that an "energy correction" to account for both changes in the energy spectra as a function of distance from the radiation source and the high-density nontissue equivalent inorganic scintillator material may not be required for applications in LDR prostate BT. Based on these findings, we believe that the conversion from scintillation signal to absorbed dose requires only a single calibration coefficient, and a calibration process has been described. The influence of employing a 0.5-mmdiameter fiber on the scintillation signal was also considered, with and without incorporating a clinically relevant LDR BT stainless steel needle applicator. While these configurations pose challenges due to the reduced signal, initial results demonstrate that they are feasible, given the continued agreement with theoretical expectations.