A novel calibration for L-shell x-ray fluorescence measurements of bone lead concentration using the strontium Kβ/Kα ratio

Abstract

Objective.

Lead (Pb) is a well-known toxic element. In vivo bone Pb concentration measurement is a long-term exposure metric complementary to blood Pb concentration measurement which is a metric of recent exposure. In vivo human tibia bone Pb measurements using Pb K-shell or L-shell x-ray fluorescence (KXRF or LXRF) emissions were developed in the 1980s. KXRF bone Pb measurements using Cd-109 gamma rays and coherent-to-fluorescence ratio to account for differences between phantom and in vivo measurements, was employed in human studies. Bone Pb LXRF method employed x-ray tubes. However, calibration procedures using ultrasound measurements of the soft tissue thickness (STT) proved inaccurate.

Approach.

In this study, bone and soft tissue (ST) phantoms simulated in vivo bone Pb measurements. Seven plaster-of-Paris cylindrical bone phantoms containing 1.01 mg g⁻¹ of strontium (Sr) were doped with Pb in 0, 8, 16, 29, 44, 59, and 74 μg g⁻¹ concentrations. Polyoxymethylene (POM), resin, and wax were each used to fabricate four ST phantoms in the approximate 1–4 mm thickness range. Pb LXRF measurements were performed using a previously developed optimal grazing incidence position method.

Main results.

Linear attenuation coefficients measurements of ST materials indicated that POM and resin mimicked well attenuation of Pb x-rays in skin and adipose tissue, respectively. POM and resin data indicated a bone Pb detection limit of 20 μg g⁻¹ for a 2 mm STT. Derived relationships between the Pb concentration, Pb LXRF and Sr K_β/K_α ratio data did not require STT knowledge. Applied to POM and resin data, the new calibration method yielded unbiased results.

Significance.

In vivo bone Pb measurements in children were suggested following considerations of radiation dose, STT, detectability and distribution of Pb and Sr in bone. This research meets with the concerns regarding the negative effects of low levels of Pb exposure on neurodevelopment of children.

1. Introduction

Lead (Pb) is a well-known toxic element. Its negative effects on human health have triggered sustained research efforts in several areas related to human health and environment over the past several decades. The result was an impressive body of scientific literature which led to strict industrial and environmental regulatory policies worldwide aimed at reducing the overall Pb exposure.

While adverse health effects of Pb were known for centuries, scientific studies conducted in the past decades revealed that exposure to this element is detrimental in several ways with demonstrated negative impact on the function of several organs and systems: brain, central nervous system, cardiovascular system, kidneys, reproductive system, and bone (Goyer 1993, Gidlow 2004, Flora et al 2012). Research studies have also identified several molecular mechanisms of Pb toxicity as summarized in several relatively recent review papers (Verstraete et al 2008, Yedjou et al 2010, Singh et al 2018).

Recently published results from population surveys in several parts of the world indicate that regulatory policies aimed at mitigating the human Pb exposure were effective. Thus, the study by McNeill et al (2018) conducted between 2009 and 2011 in Toronto, Canada found that tibia bone Pb levels in adults were about half of those corresponding to the same age group measured 17 years prior. In United States, the 2007–2010 National Health and Nutrition Examination Survey (NHANES) reported a geometric mean blood Pb level (BLL) of 1.3 μg dl⁻¹ in young children (< 6 years) (Wheeler and Brown 2013), which was a 90% decrease from the 12.8 μg dl⁻¹ estimate of the 1976–1980 NHANES II (Pirkle et al 1994). Using NHANES data between 1999 and 2014, Tsoi et al (2016) reported a continual decrease in general population BLLs in the United States. In South Korea, a general population survey from 2008 to 2017 also found that the geometric mean BLL in adults decreased from 2.37 to 1.46 μg dl⁻¹ and was linked to a 61% decrease of local atmospheric Pb content (Ahn et al 2019).

Despite the public health success in reduction of Pb exposure, two concerns remain: (i) developing countries and (ii) demonstrated negative effects of low BLLs on cognitive development of children. In 2012, the Centers for Disease Control and Prevention (CDC) of the United States, following the recommendations of the Advisory Committee on Childhood Lead Poisoning Prevention, set a new reference BLL value of 5 μg dl⁻¹. The CDC response came following the mounting evidence of associations between low BLLs and developmental and cognitive problems in children (Needleman et al 1990, Schwartz 1994, Lanphear et al 2000, 2005, Canfield et al 2003, Bellinger 2008). Based on a survey between 2010 and 2014 which included 69 children and adolescents in Uruguay, Pascale et al (2016) identified e-waste recycling as a new source of Pb exposure. The negative health impact on children exposed to Pb and other heavy metals as a consequence of e-waste recycling was also identified as a global environmental health issue in the review article by Zeng et al (2016). Informal used lead-acid battery recycling which occurs in an estimated 90 developing countries was also shown to be a source of Pb exposure in the study by Ericson et al (2016). Past and current industrial growth in many parts of the world appears to increase the risks of Pb exposure, particularly in children. Following a national BLL survey of over 30 000 children (<7 years old) conducted in China between 2013 and 2015, Li et al (2020) reported a geometric mean BLL of 2.67 μg dl⁻¹ with 8.6% of subjects exceeding 5 μg dl⁻¹. A survey of children less than 6 years of age conducted in the United States between 2009 and 2015, also found BLLs over 5 μg dl⁻¹ in about 3% of all cases (McClure et al 2016). A detailed analysis of the impact of Pb-contaminated soils from Pb-based additives in gasoline on children BLLs in 8 urban areas in California was reported by Mielke et al (2010). Using existing industrial and environmental data, the authors estimated that over 190 000 tons of Pb were released as atmospheric particles in these urban areas between 1950 and 1982. Presented evidence led the authors to state that ‘lead in soil is at least co-equal to Pb-based paint as an explanation for the urban pattern of Pb exposure of children’.

In summary, recent population surveys provided evidence of continuing decreasing of Pb exposure. Worldwide, however, concerns remain. Neurodevelopmental deficits associated with low BLLs in children pose new challenges in efficiently addressing both assessment and mitigation measures. While BLL measurement is a standard clinical metric of human Pb exposure, in vivo bone Pb concentration measurement is also an established biometric tool (Chettle 2005). Mean life of blood Pb following the exposure is relatively short on the order of 30 d while the mean life of bone Pb is on the time scale of 10–30 years (Rabinowitz 1991). Bone Pb, therefore, reflects the long-term Pb exposure of individuals and it represents about 95% of the adult human Pb burden (Barry 1975). In combination with blood Pb measurements, in vivo bone Pb measurements of exposed and unexposed populations at time intervals longer than a decade, depicted a complex picture of Pb kinetics in the human body which depends on age, exposure level and duration (Brito et al 2008, Behinaein et al 2012, 2017). Barbosa et al (2005) also argued that bone Pb measurements can distinguish the exogenous and endogenous (Pb turnover from bone to blood) components of exposure, thus, giving a more precise clinical significance of blood Pb data.

In vivo bone Pb concentration XRF measurements were pioneered in Sweden in the 1970s (Ahlgren et al 1976, Ahlgren and Mattson 1979). Initial method of Ahlgren and Mattson (1979) measured Pb concentration in finger bone and relied on a ⁵⁷Co gamma-ray excitation of the K-shell Pb electrons (KXRF) and subsequent detection of the Pb characteristic x-ray photons. In early 1980s, first Laird et al (1982), and then, Sommervaille et al (1985), improved the in vivo KXRF tibia bone Pb concentration measurement method. Method’s sensitivity was increased by adopting the 88.0 keV photon emissions of the ¹⁰⁹Cd gamma rays to excite the bone Pb in an approximately 180° backscatter geometry selected to increase the energy separation between the Compton peak and the Pb KXRF peaks. A calibration method based on the measured ratio between the number of counts of the coherent peak and the cumulative number of counts of the observed Pb KXRF peaks was also developed to account for experimental variations of the soft tissue (ST) x-ray attenuation, bone size, and overall excitation-detection geometry. In the same period, alternative L-shell XRF (LXRF) in vivo bone Pb concentration measurement studies were conducted, first using the excitation of gamma-ray emissions of the ¹²⁵I (Wielopolski et al 1983), and later, the partially plane polarized x-ray photons emitted by an x-ray tube with a silver target (Wielopolski et al 1989). The calibration method involved ultrasound measurements of the ST thickness (STT) overlying the bone and a generic ST attenuation coefficient at the 10.5 and 12.6 keV photon energies of the two Pb Lα and Pb Lβ peaks. Although additional LXRF studies followed, the ¹⁰⁹Cd KXRF method was widely adopted for in vivo bone Pb measurements (Chettle 2005).

The calibration methodology in the early LXRF studies was closely scrutinized by Todd (2002a). Todd pointed out that the precision of LXRF Pb concentration measurements is highly dependent on the selection of the linear attenuation coefficient of the ST overlying the bone, concluding that ‘… attenuation in LXRF is probably intractable because the attenuation coefficient(s) are not, and cannot, be known’. Todd (2002a) also estimated that the ultrasound measurements of the STT may introduce additional significant relative measurement uncertainties of up to 46%. In a separate publication (Todd 2002b), improvements of the LXRF system were investigated using a molybdenum (Mo) target x-ray tube and various combinations of filters in a polarization by reflection experimental setup. After finding the optimal LXRF experimental setup, the Pb Lα calibration line data from 15 min irradiations of Pb-doped plaster-of-Paris (poP) bone phantoms and polystyrene ST phantoms, gave Pb detection limit (DL) values in the range from to 2.7 to 23.1 μg g⁻¹ corresponding to a range of skin thickness equivalent range of 0.0–1.7 mm. However, translation of the LXRF phantom method developed by Todd (2002b) to Pb bone concentration measurements in human cadaver tibiae produced inaccurate results when compared to atomic absorption spectrometry measurements (Todd et al 2002).

Despite issues surrounding the calibration and DLs of the LXRF method, the approach, if improved, still holds promise for large population bone Pb surveys, particularly when considering the benefits of portable XRF devices. Studies over the past decade indicated that commercial portable XRF devices can measure low concentrations (<1 μg g⁻¹) of trace elements such as arsenic (As), selenium (Se), zinc (Zn), or chromium (Cr) in human skin and nails (Fleming and Gherase 2007, Gherase et al 2010a, Roy et al 2010, Fleming and Ware 2017, Fleming et al 2017). An initial portable XRF study by Fleming et al (2011) on circular Pb-doped poP bone phantoms found Pb DLs too large for in vivo applications. The portable LXRF study of Nie et al (2011) on Pb-doped poP bone phantoms reported a Pb DL of 29 μg g⁻¹ for 5 mm of Lucite layer thickness. The authors also demonstrated a nonlinear relationship between the measured Compton scattered photon count rate and the Lucite layer thickness. The similar portable LXRF approach was later employed by Specht et al (2014) who derived a Pb calibration method based on a spectral background subtraction and empirical relationships between the backscatter peak of bremsstrahlung x-rays and the phantom STT. Using goat and cadaver bone samples with added Lucite layers of 1, 2, and 3 mm thickness, the authors demonstrated that the novel calibration method yielded superior correlations to the KXRF measurements than the traditional methods relying on peak fitting and calibration line data. The portable LXRF and background subtraction method were applied in recently published bone Pb studies (Specht et al 2016, 2019).

In brief, portable XRF devices demonstrated capabilities for detection and quantification of trace elements content of the superficial human tissues. These devices are currently designed as commercial tools intended for industrial and agricultural applications and readily applicable to specific areas of investigation such as forensic science or archeology. Additional research is required to optimize portable XRF devices for medical applications. Towards this goal, our research group developed a bone Pb LXRF measurement which demonstrated improved Pb detectability by employing an optimal grazing incidence position (OGIP) measurement method and a table-top XRF experimental platform consisting of an integrated x-ray tube and polycapillary x-ray lens (PXL) and an x-ray detector (Gherase and Al-Hamdani 2018a). A later investigation (Gherase and Al-Hamdani 2018b) significantly reduced the OGIP irradiation time (more than threefold) and demonstrated its reproducibility. Pb DL of 5 μg g⁻¹ for three 5 min irradiations and 1 mm polyoxymethylene (POM) phantom was obtained (Gherase and Al-Hamdani 2018a). This bone Pb DL value is in within the roughly 1–6 μg g⁻¹ average bone Pb concentration range measured by a KXRF system in a cohort of the general population in Canada (McNeill et al 2018).

While bone Pb detectability is essential, it is not sufficient for developing a robust in vivo bone Pb LXRF measurement method. Accurate in vivo bone Pb concentration measurement depends on establishing a correct relationship between the spectrometric LXRF measurements (i.e. the net areas of the observed Pb Lα and Pb Lβ peaks) and bone Pb concentration (i.e. calibration method). For a given set of experimental parameters: excitation-detection geometry, performance parameters of x-ray beam and x-ray detector, similar bone phantoms doped with known Pb concentrations, ST phantom material and thickness, the abovementioned relationship is linear and it is known as a calibration line. However, in vivo conditions cannot be easily replicated by phantom-based experiments since the x-ray attenuation of the bone and overlying ST is not a priori known. The past LXRF calibration relied on ultrasound STT measurements and on an average value of the ST linear attenuation coefficient. Despite careful carrying out of experiments and a detailed data analysis, the calibration method based on phantom calibration line data was demonstrated not be reliable (Todd et al 2002). On the other hand, the KXRF calibration uses the ratio between the Pb KXRF signal and the coherent peak of the 88.0 keV incident gamma rays which was demonstrated to be independent of detection geometry, bone size and shape, and ST x-ray attenuation (Sommervaille et al 1985). KXRF calibration method, which is also known as coherent normalization, was recently translated to measurements of bone gadolinium (Gd) (Keldani et al 2017) and bone lanthanum (La) (Nguyen et al 2020). Notably, however, coherent normalization was not able to correct for the ST attenuation in the KXRF finger strontium (Sr) measurements as reported by Zamburlini et al (2008).

In this paper, a novel bone Pb LXRF calibration method based on Sr K_β/K_α ratio measurements was demonstrated using a microbeam experimental platform and the OGIP measurement method (Gherase and Al-Hamdani 2018b). In vivo bone Pb LXRF measurements were simulated using bone and ST phantoms. Seven poP cylindrical bone phantoms containing Sr with a measured concentration of (1.01 ± 0.07) mg g⁻¹ (section 3.1) were doped with Pb in 0, 8, 16, 29, 44, 59, and 74 μg g⁻¹ concentrations. Three sets of four cylindrical shell ST phantoms each were made out of polyoxymethylene (POM), resin, and wax with approximate thickness in the 1–4 mm range. Linear attenuation measurements results included in section 3.2 indicated that POM and resin mimicked well the average attenuation of Pb x-rays in skin and adipose tissue, respectively. POM and resin calibration line data presented in section 3.3 was used to estimate a bone Pb DL of about 20 μg g⁻¹ for a 2 mm STT. Relationships between spectrometric measurements and bone Pb concentration were derived in section 2.6 and replaced the past calibration based on STT data. Two nondimensional parameters a_α and a_β linked the Sr K_β/K_α ratio to calibration line slope data. The values of these two parameters were different for the three ST materials. However, numerically, differences were small. A weighted average value was employed to compute the Pb concentrations using data from the three ST phantom materials. The procedure produced unbiased Pb concentration estimates using the POM and resin data and a relatively small bias was observed from the wax data. Further considerations of the Pb and Sr in vivo detectability and distribution, radiation dose, and some instrumental considerations were included in section 4. The experimental results and discussions included in this study indicated bone Pb measurements in children as a possible application.

2. Materials and methods

2.1. Bone and ST phantoms

Bone phantoms doped with Pb were made of calcium sulfate hemihydrate (CaSO₄ · 1/2H₂O) which is also known as poP. The poP in powder form (Sigma-Aldrich Co., St. Louis, MO) was mixed with distilled water and doped with known quantities of lead (Pb) using pipette-measured volumes of Pb standard atomic absorption solution (Sigma-Aldrich Co., St. Louis, MO). The chemical form of Pb was not known and the manufacturer provided a Pb concentration of (1000 ± 4) mg l⁻¹. The solvent was a diluted water-based nitric acid (HNO₃) solution (2% w/w). The viscous mixture was then poured into aluminum cylindrical molds. Following solidification, the bone phantoms were rigid cylinders with an approximate diameter of 29 mm. Their masses were measured and the final Pb concentration was calculated based on the initial Pb standard solution volume measurement. Seven bone phantoms with varying Pb concentrations of 0, 8, 16, 29, 44, 59, and 74 μg g⁻¹ were made. The resulting poP density for each phantom was also calculated based on measured masses and volumes. The poP density values ranged between 0.99 and 1.06 g cm⁻³. The average and standard deviation of the poP density values of the seven bone phantoms were 1.05 g cm⁻³ and 0.05 g cm⁻³, respectively.

The poP material contained Sr in a concentration of (1.01 ± 0.07) mg of Sr per gram of solidified poP. The Sr concentration was evaluated and the results were summarized in section 3.1. The molar mass ratio between calcium (Ca) and solidified poP or calcium sulfate hydrate (CaSO₄ · 2H₂O) is 0.232 78 and the above Sr concentration can be converted to a (4.3 ± 0.3) mg of Sr per gram of Ca. Although Sr was not a trace element of interest in this study, Sr K_α (14.1 keV) and Sr K_β (15.8 keV) peaks were used in the calibration method that determined Pb concentration. Therefore, Sr concentration determination was important, particularly when compared to human tibia bone Sr levels estimated to be in the range from 0.1 to 0.3 mg Sr per gram of Ca (Pejović-Milić et al 2004). The Sr concentration was determined using poP Sr-doped bone phantoms. These were made employing the same procedure as for poP Pb-doped bone phantoms. The main difference was that the poP and water mixture was doped with known quantities of strontium (Sr) using pipette-measured volumes of Sr standard atomic absorption solution (Sigma-Aldrich Co., St. Louis, MO, US). The solvent was a diluted water-based nitric acid (HNO₃) solution (2% w/w). The chemical form of Sr in the standard solution was not known and its concentration provided by the manufacturer was (1000 ± 4) mg l⁻¹. Four Sr-doped poP bone phantoms were made with 203, 426, 660, and 991 μg g⁻¹ added Sr concentration.

ST phantoms consisted of cylindrical shells of 30 mm inner diameter and varying 1, 2, 3, and 4 mm nominal thickness. Three different materials were used: polyoxymethylene (POM) (United States Plastic Co., Lima, OH, US), fiberglass resin (3 M Co., St. Paul, MN, US), and paraffin wax (Sigma-Aldrich Co., St. Louis, MO, US). Due to differences in mechanical properties, different procedures were employed to make the cylindrical shell ST phantoms for each material. Thus, the POM ST phantoms were machined out of a larger cylindrical rod. The resin and wax ST phantoms were made using a custom-made aluminum mold which consisted in four cylindrical cups of 31, 33, 35, and 37 mm diameter and 6 cm height. In the center of each of the four cylindrical cups, a solid aluminum cylinder with a 29 mm diameter and 5 cm in height was placed and secured to the bottom of each cup using a screw. The liquid resin was mixed with a hardener solution using about 10 droplets of hardener per 30 ml of resin. The resin mixture was poured in the mold which was pre-coated with non-stick oil to prevent adherence to the mold following hardening. The resin cylindrical shells solidified after several hours. The irregular blocks of paraffin wax were slowly melted in a metallic cylindrical can using the heat generated by a commercial propane-fueled torch (Coleman Inc., Columbus, OH, US). The resulting liquid wax was poured into the aluminum mold in a similar manner to the resin. The melted wax cooled down to room temperature and solidified after several minutes. The samples were then carefully extracted using a utility knife. The cylindrical shells made of the three materials were later cut into smaller sections as can be seen in the photograph of figure 1. This procedure led to an easier and better mounting of the ST phantom on the cylindrical bone phantom. The bone and ST phantom assembly was secured with an elastic band that did not interfere with the XRF experiments.

Figure 1.

Digital photograph of the ST phantoms composed of cylindrical shell sections made of POM (a), wax (b), and resin (c).

For POM and paraffin wax the chemical formula was taken from technical data provided by the manufacturer. For the fiberglass resin, the elemental composition was not known, the approximated bulk elemental composition provided in table 1 was extracted from the paper of Gawdzik et al (2001). The density ρ is also given in table 1 and was calculated as the ratio of direct mass measurements using a digital balance (LW Measurements LLC, Santa Rosa, CA, US) and water displacement volume measurements using calibrated glass cylinders (Corning Co., Corning, NY, US). The thickness of each ST phantom was measured at the location of the x-ray beam incidence using a micrometer (Anytime Tools Inc., Granada Hills, CA, US). Thickness measurements are included in table 2. The x-ray beam incidence location was identified by the intersection of two perpendicular lines (horizontal and vertical) marked on each phantom as can be seen in the digital photograph of figure 1.

Table 1.

Density and elemental composition of the three ST phantom materials. The number in the round parentheses is the uncertainty in the last one or two significant figures of the corresponding value.

Material	Density (g cm−3)	Chemical formula/elemental composition
POM	1.40(6)	(CH O2 )n
Resin	1.08(9)	5%H, 60%C, 35%O
Wax	1.07(8)	C30H62

Table 2.

Thickness measurements of the ST phantoms. The number in the round parentheses is the uncertainty in the last one or two significant figures of the corresponding value.

Material	Thickness (mm)
POM	1.10(5)	1.89(5)	3.23(3)	4.36(3)
Resin	1.10(5)	2.18(5)	3.05(3)	4.19(3)
Wax	1.47(8)	2.04(8)	3.48(8)	4.01(5)

2.2. XRF experimental setup

The experimental equipment was presented in our previous publications (Gherase and Al-Hamdani 2018a, 2018b). For clarity, the experimental equipment and setup are presented in this subsection. The microbeam XRF equipment used consisted of: (i) an integrated x-ray tube and PXL (Polycapillary X-beam Powerflux model, X-ray Optical Systems, East Greenbush, NY, US), (ii) silicon-drift x-ray detector with integrated pulse-height analyzer (X-123 SDD model, Amptek Inc., Bedford, MA, US), (iii) XY modular motorized linear positioning stage unit (Newport Co., Irvine, CA, US).

A schematic of the experimental setup is provided in figure 2. The continuous emission x-ray tube was air-cooled and its target was made of tungsten (W). The built-in PXL was 100 mm in length and 10 mm outer diameter. The x-ray tube voltage and current could be varied in 0.1 kV and 1 μA increments, respectively. The x-ray tube and PXL unit was also equipped with a filter wheel placed in front of the PXL. For XRF experiments a 1.8 mm Al filter was used to reduce the negative effect of the W L-shell characteristic x-rays. Their maximum values of 50 kV and 1 mA were used during the XRF measurements. The circular active area of the detector was 25 mm² (or 5.6 mm diameter) and 500 μm thickness and the window consisted of a 12.5 μm thick beryllium (Be) sheet. Counting rate capability of the detector provided by the manufacturer was 10⁵ counts/s. The distance between the central axis of the detector and the PXL was set and kept constant at 15 mm to allow free motion of the bone, ST, and x-ray detector assembly relative to the microbeam. A 20 mm long aluminum (Al) collimator was attached to the end of the x-ray detector to reduce x-ray scatter.

Figure 2.

Schematic drawing of the XRF experimental setup.

The x-ray detector was mounted on an Al plate attached to the XY positioning stage unit. The plate also served as the support of the bone and ST which were placed standing in front of the Al collimator with no space in between. This setup allowed simultaneous precise positioning of the two phantoms and x-ray detector assembly relative to the direction of the microbeam to establish the OGIP as presented in the next section 2.3. The entire XRF setup was placed on an x-ray shield consisting of a 56 cm × 62 cm and 6.35 mm thickness Al plate. The plate was covered by a rectangular-shaped stainless steel cover with a slightly smaller base area and 46 cm in height which could be opened to allow operating access to the XRF setup. The cover was manually opened after the microbeam was turned off. The entire XRF experimental setup was mounted on an optical table (Newport, Irving, CA, US) which also served as mechanical support for the x-ray tube power supply and the laptop computer that controlled the x-ray tube, x-ray detector, and positioning unit.

The characteristics of the microbeam shaped by the PXL were presented in detail in a previous publication (Gherase and Vargas 2017). A few results of the x-ray beam geometry are provided. The focal distance of the PXL was 4 mm. The microbeam full-width at half maximum (FWHM) at the focal point was measured to be roughly in the 15–30 μm range as measured in the 8–30 keV photon energy range. FWHM at the focal point provided by the manufacturer was 24.8 μm at 9.67 keV (W L_β1). The microbeam divergence downstream from the focal point was measured to be about 76 mrad. Using the schematic of the beam geometry shown in figure 3, the FWHM value at distance d from the focal point can be estimated as follows:

$$\text{FWHM}(\text{mm})=0.0248+2\cdot 0.076\cdot d(\text{mm}).$$ (1)

At the aforementioned x-ray incidence distance of 15 mm from the PXL, d = 15 − 4 = 11 mm. Employing equation (1), the microbeam FWHM was calculated to be 1.7 mm.

Figure 3.

Schematic representation of the microbeam geometry.

2.3. Optimal grazing-incidence position XRF measurements

The OGIP was obtained by sequences of 10 s x-ray spectra acquisitions at positions separated by equal 0.5 mm steps bringing the bone phantoms closer to the microbeam. The initial position was randomly selected such that the microbeam was incident on the ST phantom, but not on the bone phantom. The OGIP corresponded to the maximum of the a convolution function between a Gaussian and exponential functions which was fitted to the Sr K_α peak area versus position data obtained after the sequence of 10 s x-ray spectra acqusitions (Gherase and Al-Hamdani 2018b). The Sr K_α peak was the most intense of the characteristic x-rays of both Sr and Pb, and, therefore, naturally selected to determine the optimal XRF geometry. A schematic of the OGIP experimental procedure is given in figure 4. The number of 0.5 mm steps varied between 7 and 12 with the least number of steps corresponding to the bare bone XRF experiments.

Figure 4.

Schematic of the experimental setup of the OGIP method employed in all XRF measurements. The dotted lines indicate sequential positions of the microbeam relative to the bone and ST assembly separated by equal 0.5 mm steps.

X-ray spectra from Pb-doped poP bone phantoms were acquired using the OGIP method and analyzed to yield the calibration lines data. At the optimal position three x-ray spectra were acquired in 300 s each. Each peak area was measured using data analysis procedures presented in section 2.5. The final peak area result and its error were calculated as the weighted average and error (statistical weight was computed as the inverse of the squared error). The same procedure was applied using the Sr-doped bone phantoms to yield the Sr calibration lines data with the difference that the three trials were acquired in shorter 120 s time intervals.

2.4. Linear attenuation coefficient measurements

The schematic of the experimental setup used for the linear attenuation coefficient measurements is shown in figure 5.

Figure 5.

Schematic of linear attenuation coefficients experimental setup.

Samples were positioned at the focal point of the PXL where the microbeam FWHM was about 25 μm. Due to shielding, the maximum attainable distance between the measured sample and the Si detector was 170 mm. Therefore, the solid angle subtended by the Si detector of 25 mm² area was calculated to be only about 0.9 × 10⁻³ steradians which reduced the amount of forward scattered photons reaching the detector. The FWHM of the microbeam at the position of the x-ray detector, without an attenuating sample, was calculated to be 25.9 mm which was larger than the 5.6 mm diameter of the detector element. Hence, the detector positioned as in figure 1, only captured a fraction of the microbeam. The linear attenuation coefficient was measured using the relative decrease of the x-ray signal at a given photon energy with increasing thickness of the sample. Keeping constant the initial alignment was essential, therefore, the x-ray detector position was secured by its firm magnetic base holder.

The linear attenuation coefficient as a function of photon energyE, μ(E), and layer thickness t of the three ST phantom materials were used to interpret Pb calibration line data and measured Pb Lβ/Lα and Sr Kβ/Kα ratios presented in sections 3.3 and 3.4, respectively. The samples consisted of all the ST phantoms described in the previous section 2.1 and pure Al layers of varying thickness. The Al layers were made by superimposing square Al sheets (1 cm × 1 cm) of the same 0.005 inches (or 0.127 mm) thickness. Thus, four thickness values of 0.127, 0.254, 0.381, and 0.508 mm were used by incremental overlapping of Al square sheets. The square sheets were cut out of a single sheet of 99.99% pure Al (Sigma-Aldrich Co., St. Louis, MO, US). Al was used a reference material to validate the experimental methods employed in these measurements. The measured linear attenuation coefficients at various photon energies (~9–22 keV range) for all four materials were compared with values calculated as follows. The total mass attenuation coefficient (μ/ρ)(E) was calculated for the four materials using the XCOM photon cross section database (Berger et al 2010) and their elemental composition or chemical formula provided in table 1. The linear attenuation coefficient μ(E) for each material was then calculated as the product of (μ/ρ)(E) and density ρ taken from table 1. The density of Al used in calculations was 2.7 g cm⁻³.

For each sample, three x-ray spectra were acquired, each in 15 s. The x-ray detector dead time percentage was below 1% by operating the x-ray tube at a lower current of 0.05 mA (or 50 μA). The voltage was maintained at its maximum value of 50 kV and no additional beam filter was used. This setting allowed measurement of the linear attenuation coefficient at the energies of the observed W LXRF emissions: 9.65 keV (L_β1 and L_β6), 9.93 keV (L_β2, L_β3, and L_β15), 11.27 keV (L_β17 and L_γ1), 11.58 keV (L_γ2 and L_γ3). The significantly lower intensity peaks at 8.38 keV (L_α1 and L_α2) and 12.0 keV were not used in the analysis. X-ray photon energies of atomic transitions were taken from the compilation data tables of Deslattes et al (2003). The procedure employed for fitting the Gaussian peaks is described in the following section 2.5. The peak area was then proportional to the number of photons attenuated at each peak center energy. For other energies the following procedure was employed. The larger photon energy range from 12 to 21 keV was divided in 10 intervals of 1 keV each. The arithmetic average of the number of counts recorded in the channels included in each 1 keV interval was taken as representing the measured number of photons at the center of the interval. Thus, the average number of counts in the 12–13 keV interval represented the number of photons at the 12.5 keV and so on. The error was calculated as the sample standard deviation. For each interval, the final measured value and uncertainty corresponded to the weighted average and weighted error, respectively. Statistical weight was computed as the inverse error squared; the weighted average was performed over the measured values from the three trials. At each photon energy E, the linear attenuation coefficient μ(E) was calculated as the positive slope of the line fitted to the ln [S(0)/S(t)] versus t data. S(0) and S(t) represented the measured spectrometric quantity corresponding to no attenuation (t = 0) and to attenuation from sample of thickness t, respectively. The two measured spectrometric quantities were the peak area for W LXRF lines and the weighted average number of counts as described above. The equations for best fit line slope and uncertainty were taken from the textbook by Taylor (1997) and were implemented in a template using the Excel software (Microsoft Co., Redmond, WA, US). Samples of these computations are shown in the two plots of figure 6.

Figure 6.

(a) Plot of an x-ray spectrum sample obtained using the linear attenuation coefficient measurement procedures described in the text. The spectrum corresponds to POM ST phantom of 1.1 mm thickness. Inset plot shows six spectrometric quantities denoted by S as described in the text The red curves indicated the fitted W LXRF peaks and the vertical red lines indicate the photon energies. (b) Sample plot of the best fit line and the corresponding data yielding the linear attenuation coefficient of the POM material at 13.5 keV photon energy.

2.5. Data analysis

All XRF peaks were fitted using the built-in nonlinear curve fitting tool in the OriginPro 2020 data analysis and plotting software (OriginLab, Northampton, MA, US). Custom fitting functions were written using the Origin nonlinear fitting tool. The fitting functions f (x) were written as the sum of a background represented by a first or second order polynomial function P (x) and one or two Gaussian functions G(x; x_i, w_i) characterized by peak area A_i , center x_i and width w_i. An example of f (x) function including P (x) as a first order polynomial and a single Gaussian function G(x; x₁, w₁) is given in the following equation:

$$f(x)=P(x)+G\left(x;x_1,w_1\right)=a+bx+\frac{A_1}{w_1\sqrt{\pi }}\text{exp}\left[-{\left(\frac{x-x_1}{w_1}\right)}^2\right].$$ (2)

The nonlinear fitting was performed using the statistical weighting option. Thus, the statistical weight of a y-axis value y, was computed as 1/y as predicted by the Poisson statistics governing the number of counts recorded in individual channels of the x-ray detector and multi-channel analyzer unit. The goodness-of-fit of multiple peak fittings for a data set was done by monitoring the reduced chi-squared (χ²/n) values. The chi-squared test was performed to determine if χ²/n were significantly larger than unity. The test was performed using Excel’s CHISQ.DIST.RT function which computes the right-tailed probability of the chi-squared distribution. Test results below 5% indicated a χ²/n value significantly larger than unity.

Samples of peak fittings (red lines) are shown in the x-ray spectra plots of figure 7. The x-ray spectra were taken using the 29 mg g⁻¹ Pb concentration poP bone phantom in two cases: (i) no overlying ST phantom (i.e. bare bone), (ii) 2.04 mm overlying wax layer. The inset plots enclosed in (b) and (c) show the Pb Lα and Pb Lβ peaks. Peak area (A) values, their uncertainties, reduced chi-squared values and p values in round parentheses are also provided for each fitted function. The maximum lower index of the A parameter indicates the number of Gaussian peaks included in the fitting function. The x-ray spectrum of figure 7(a) plot also indicates an escape peak at the 1.95 keV energy. This corresponds to the difference between the average Ca K_α peak energy of 3.69 keV (see table 3) and the Si K_α peak energy of 1.74 keV (Deslattes et al 2003). The effect of x-ray attenuation by the 2.04 mm wax ST phantom can be noticed by comparing the Pb and Sr peak areas from plots of figures 7(b) and (c). The XRF peaks of calcium (Ca), sulfur (S), and phosphorus (P) contained in the poP bone phantoms were only observed in the bare bone experiments. The observed Cu K_α and Cu K_β peaks were constant and were attributed to excitation by scattered x-rays of the copper (Cu) contained within the Al collimator and metallic cylinder enclosing of the x-ray detector window and stainless steel shield.

Figure 7.

Sample of peak fittings (red lines) corresponding to x-ray spectra from the 29 μg g⁻¹ Pb concentration poP bone phantom in two experimental conditions: (i) no overlying ST shown in plots (a) and (b) and (ii) overlying wax ST phantom of 2.04 mm thickness shown in plot (c).

Table 3.

XRF energy values and atomic transitions for elements contained in the poP Pb-doped bone phantoms. The numbers in the round parentheses represent the uncertainties in the last two significant figures of the photon energy value.

Element	Z	Siegbhan notation	IUPAC notation	Photon energy (keV)	Method	Peak center (keV)
S	16	KL2	Kα2	2.306 700(38)	Experimental	2.3119(11)
		KL3	Kα1	2.307 885(34)
		KM2	Kβ3	2.469 73(62)	Theoretical	2.475(5)
		KM3	Kβ1	2.467 53(72)
Ca	20	KL2	Kα2	3.688 128(49)	Experimental	3.6915(5)
		KL3	Kα1	3.691 719(49)
		KM2	Kβ3	4.014 32(59)	Theoretical	4.0136(13)
		KM3	Kβ1	4.014 68(58)
Sr	38	KL2	Kα2	14.098 03(24)	Experimental	14.1059(13)
		KL3	Kα1	14.165 20(24)
		KM2	Kβ3	15.825 17(90)		15.799(5)
		KM3	Kβ1	15.835 89(60)
		KM4	K${\beta }_5^{\text{II}}$	15.970 20(77)	Theoretical
		KM5	K${\beta }_5^{\text{I}}$	15.971 90(76)
Pb	82	L3M4	Lα2	10.449 59(65)	Experimental	10.506(10)
		L3M5	Lα1	10.551 60(27)
		L3N4	Lβ15	12.6012(13)		12.58(4)
		L3N5	Lβ2	12.6228(13)
		L2M4	Lβ1	12.613 80(57)

The XRF photon energies for elements contained in the Pb-doped poP bone phantoms were taken from the compilation tables of Deslattes et al (2003) and summarized in table 3. The last column shows the peak center parameter values from the peak fittings shown in the spectra of figures 7(a) and (b) (i.e. bare bone experiments). Due to energy resolution limitations of the x-ray detector, XRF emissions separated in energy by less than 150 eV were not spectroscopically resolved and appeared as a single peak. The Gaussian peak width (w) value obtained from the peak fitting procedures measures the broadening effect of the convolution of the individual atomic XRF emissions with the energy resolution function of the x-ray detector which increases with the absorbed photon energy (Papp et al 2005).

The measured values of K_β/K_α ratio of S, Ca and Sr and L_β/L_α ratio of Pb were also taken from the literature and grouped in table 4. The value of measured K_β/K_α ratios and Pb L_β/L_α ratio, denoted byr_m, was corrected for the varying detection efficiency (ε) at the XRF photon energies as follows:

$$r_{\text{c}}=r_{\text{m}}\frac{\epsilon \left(E_{\alpha }\right)}{\epsilon \left(E_{\beta }\right)}.$$ (3)

In equation (3), r_c represents the corrected ratio, r_m is the measured ratio. Notations ε(E_α) and ε(E_β) represent the detection efficiency values at the average XRF energy values E_α and E_β, respectively.

Table 4.

Measured values for K_β/K_α ratio of S, Ca and Sr and L_β/L_α ratio of Pb. The number in the round parentheses is the uncertainty in the last one or two significant figures of the corresponding value.

Element	Kβ/Kα ratio	References
Ca	0.123(6)	Ertuğral et al (2007)
Sr	0.181(9)
	Lβ/Lα ratio
Pb	1.04(9)	Garg et al (1984)

The quantum detection efficiency as a function of photon energy E was denoted by ε(E) and was calculated based on the x-ray attenuation properties of the Si detector element (i.e. absorption) and Be window (i.e. transmission). The two parts of the x-ray detector were characterized by t_Be = 12.5 μm and t_Si = 0.5mm thickness, and ρ_Be = 1.85 g cm⁻³ and ρ_Si = 2.392 g cm⁻³ density, respectively. The final expression for ε(E) is given by

$$\epsilon (E)=\text{exp}\left[-{(\mu /\rho )}_{\text{Be}}(E){\rho }_{\text{Be}}{t}_{\text{Be}}\right]\left\{1-\text{exp}\left[-{(\mu /\rho )}_{\text{Si}}(E){\rho }_{\text{Si}}{t}_{\text{Si}}\right]\right\}.$$ (4)

The values of the mass attenuation coefficients at different photon energies for Be and Si, (μ/ρ)_Be(E) and (μ/ρ)_Si(E) were taken from the XCOM database (Berger et al 2010).Employing equation (4), values of ε(E) at eight XRF photon energy (E) values were computed and are provided in table 5.

Table 5.

Values of ε(E) at eight XRF photon energy values (E).

XRF peak	E(keV)	ε(E)
S Kα	2.3	0.893 822
S Kβ	2.5	0.916 935
Ca Kα	3.7	0.974 726
Ca Kβ	4.0	0.980 116
Pb Lα	10.5	0.968 953
Pb Lβ	12.6	0.871 973
Sr Kα	14.1	0.772 182
Sr Kβ	15.8	0.653 613

Given the corrected r_c value of measured K_β/K_α or L_β/L_α ratio r_m, one can compute the average x-ray attenuation length $\overline{L}$ by writing down the relationship between r_c and the known atomic ratio denoted by r_a:

$$r_c=r_a\frac{\text{exp}\left[-\mu \left(E_{\beta }\right)\overline{L}\right]}{\text{exp}\left[-\mu \left(E_{\alpha }\right)\overline{L}\right]}=r_a\text{exp}\left\{\left[\mu \left(E_{\alpha }\right)-\mu \left(E_{\beta }\right)\right]\overline{L}\right\}.$$ (5)

Excluding the shell and subshell edges, the linear attenuation coefficient μ(E) is amonotonic decreasing function with photon energy E. Since E_β > E_α, it follows that μ(E_α) > μ(E_β) and, further, that r_c > r_a. In other words, the observed K_β/K_α or L_β/L_α ratio is larger than its atomic value due to a larger x-ray attenuation of E_α photons than that of E_β photons.Using notation Δμ = μ(E_α) − μ(E_β) and equation (3), the average x-ray attenuation length $\overline{L}$ is given by

$$\overline{L}=\frac{1}{\Delta \mu }\text{ln}\left\{\left(r_m/r_a\right)\left[\epsilon \left(E_{\alpha }\right)/\epsilon \left(E_{\beta }\right)\right]\right\}.$$ (6)

Three important notes can be made. First, equations (5) and (6) require a homogeneous medium and knowledge of the linear attenuation coefficients difference Δμ. Extensions of these equations to inhomogeneous media or layers of different material require knowledge of the average Δμ over all possible photon paths. Second, the average length determined using equation (6) is independent of the energy spectrum of the incident photons; x-ray attenuation of E_α and E_β photons occurs after excitation. Finally, the $\overline{L}$ value is independent of the concentration of the element which emits the XRF photons. The relative uncertainty on $\overline{L}$ measurement, however, depends on the element concentration. A higher concentration will yield a lower relative error on the r_m measurement which, in turn, will result in a lower relative error on $\overline{L}$ estimate.

As in previous studies (Fleming et al 2011, Gherase and Al-Hamdani 2018a), the Pb DL was calculated by plugging the slope of the calibration line (s) and the uncertainty of the zero Pb concentration peak area estimate (σ₀) in the following equation:

$$\text{DL}=3{\sigma }_0/\text{s}.$$ (7)

2.6. Calibration method

Given a known calibration line of slope s and null y-axis intercept, the unknown bone Pb concentration, c, can be calculated from its measured peak area A as follows:

$$c=A/s.$$ (8)

The Pb LXRF emissions are observed as two peaks Pb L_α (10.5 keV) and Pb L_β (12.6 keV) as shown in the x-ray spectra of figures 7(b) and (c), yielding two peak area measurements A_α and A_β. Therefore, equation (8) yields two estimates of the bone Pb concentration c_α and c_β:

$$c_{\alpha }=A_{\alpha }/s_{\alpha },$$ (9a)

$$c_{\beta }=A_{\beta }/s_{\beta }.$$ (9b)

For an in vivo bone Pb measurement, the s_α and s_β values are not known. Translation from bare bone phantom calibration lines slope values denoted by s_α0 and s_β0 to adequate values for in vivo bone Pb measurements denoted by s_α1 and s_β1 requires corrections accounting for several factors. These include: (i) x-ray attenuation of incident and emergent LXRF photons by the overlying ST, (ii) differences in the x-ray attenuation properties between bone phantom material and human cortical bone, (iii) geometrical differences between bare bone phantom and in vivo bone Pb measurements. In this study, s_αt and s_βt will denote the Pb calibration lines corresponding to poP bone phantoms with overlying ST phantoms made of the same material and thickness t. The following relationships between the bare bone calibration line slopes s_α0 and s_β0 and s_αt and s_βt were found:

$$s_{\alpha t}=s_{\alpha 0}{\left[r_{m,0}^{\text{Sr}}/r_{m,t}^{\text{Sr}}\right]}^{a_{\alpha }},$$ (10a)

$$s_{\beta t}=s_{\beta 0}{\left[r_{m,0}^{\text{Sr}}/r_{m,t}^{\text{Sr}}\right]}^{a_{\beta }}.$$ (10b)

In equations (10a) and (10b), $r_{m,0}^{\text{Sr}}$ and $r_{m,t}^{\text{Sr}}$ Sr K_β/K_α ratio measured for poP bare bone phantoms and ST phantom layer of thickness t, respectively. The measured Sr K_β/K_α ratio was selected having much lower relative uncertainty than the Pb L_β/L_α ratio. A treatment of the measured Pb L_β/L_α ratio and its relationship to overlying ST attenuation was included in the synchrotron-based study of Gherase et al (2017). The a_α and a_β are nondimensional parameters that will contain the ST material and excitation-detection geometry dependences as it will be shown below.

Taking into account the x-ray attenuation in bone and ST and employing equation (6) twice, the ratio $r_{m,0}^{\text{Sr}}/r_{m,t}^{\text{Sr}}$ from equations (10a) and (10b) can be rewritten as

$$\frac{r_{m,0}^{\text{Sr}}}{r_{m,t}^{\text{Sr}}}=\frac{r_{m,0}^{\text{Sr}}}{r_a^{\text{Sr}}}\cdot \frac{r_a^{\text{Sr}}}{r_{m,t}^{\text{Sr}}}=\left[\frac{\epsilon \left({\text{E}}_{\text{Sr\hspace{0.17em}}K\alpha }\right)}{\epsilon \left({\text{E}}_{\text{Sr\hspace{0.17em}}K\beta }\right)}\cdot \text{exp}\left(\Delta {\mu }_{\text{b}}^{\text{Sr}}{\overline{t}}_{\text{b},0}\right)\right]\times \left[\frac{\epsilon \left({\text{E}}_{\text{Sr\hspace{0.17em}}K\beta }\right)}{\epsilon \left({\text{E}}_{\text{Sr\hspace{0.17em}}K\alpha }\right)}\cdot \text{exp}\left(-\Delta {\mu }_{\text{b}}^{\text{Sr}}{\overline{t}}_{\text{b},\text{t}}\right)\text{exp}\left(-\Delta {\mu }_{\text{st}}^{\text{Sr}}{\overline{t}}_{\text{st}}\right)\right].$$ (11)

In equation (11), ${\overline{t}}_{\text{b},0}$, ${\overline{t}}_{\text{b},\text{t}}$ and ${\overline{t}}_{\text{st}}$ represent the average path lengths of Sr x-rays in bone with no ST (i.e. bare bone), in bone with overlying ST of thickness t, and in ST, respectively. Quantities $\Delta {\mu }_{\text{b}}^{\text{Sr}}$ and $\Delta {\mu }_{\text{st}}^{\text{Sr}}$ denote the differences between the linear attenuation coefficients at Sr K_α energy of 14.1 keV and Sr K_α energy of 15.8 keV in bone (b) and ST (st), respectively. In equation (11), the detector efficiency values at Sr KXRF energies cancel, and the right-hand side is

$$\frac{r_{m,t}^{\text{Sr}}}{r_{m,0}^{\text{Sr}}}=\text{exp}\left[\Delta {\mu }_{\text{st}}^{\text{Sr}}{\overline{t}}_{\text{st}}+\Delta {\mu }_{\text{b}}^{\text{Sr}}\left({\overline{t}}_{\text{b},\text{t}}-{\overline{t}}_{\text{b},0}\right)\right].$$ (12)

From equation (12) the average photon path length of emergent XRF photons ${\overline{t}}_{\text{st}}$, is given by

$${\overline{t}}_{\text{st}}=\left\{\text{ln}\left[r_{m,t}^{\text{Sr}}/r_{m,0}^{\text{Sr}}\right]-\Delta {\mu }_{\text{b}}^{\text{Sr}}\left({\overline{t}}_{\text{b},\text{t}}-{\overline{t}}_{\text{b},0}\right)\right\}/\Delta {\mu }_{\text{st}}^{\text{Sr}}.$$ (13)

However, the relationships between s_αt and s_α0 and between s_βt and s_β0 imply the x-ray attenuation of Pb L-shell x-rays as well as that of incident excitation photons:

$$s_{\alpha t}=s_{\alpha 0}\left\{{\int }_E\text{g}(E)\text{exp}\left[-{\mu }_{\text{st}}(E){\overline{L}}_{\text{st}}\right]dE\right\}\text{exp}\left[-{\mu }_{\text{st}}\left(E_{\text{Pb\hspace{0.17em}}L\alpha }\right){\overline{t}}_{\text{st}}\right],$$ (14a)

$$s_{\beta t}=s_{\beta 0}\left\{{\int }_E\text{g}(E)\text{exp}\left[-{\mu }_{\text{st}}(E){\overline{L}}_{\text{st}}\right]dE\right\}\text{exp}\left[-{\mu }_{\text{st}}\left(E_{\text{Pb\hspace{0.17em}}L\beta }\right){\overline{t}}_{\text{st}}\right].$$ (14b)

In equations (14a) and (14b), ${\overline{L}}_{\text{st}}$ represents the average path length in ST of the incident photons. The integration of the exponential attenuation function is performed over all photon energies: (i) larger than the Pb L₃ edge energy (~13.04 keV) in equation (14a) and (ii) larger than Pb L₂ edge energy (~15.20 keV) in equation (14b). The edge energies of Pb L subshells were taken from Deslattes et al (2003). The function g(E) represents the energy spectrum of the incident x-ray beam and it is also required to keep the integral value nondimensional. In the schematic of figure 4,t_b ≪ t_st. It follows that

$$L_{\text{st}}\cong \sqrt{{\left(D_{\text{b}}/2+t_{\text{st}}\right)}^2-{\left(D_{\text{b}}/2\right)}^2}=\sqrt{{\left(t_{\text{st}}\right)}^2+D_{\text{b}}{t}_{\text{st}}}.$$ (15)

For a noncircular bone cross section and constant overlying ST layer thickness, preserving the ortoghonal OGIP excitation-detection geometry implies that ${\overline{L}}_{\text{st}}>{\overline{t}}_{\text{st}}$. Therefore, in general,

$${\overline{L}}_{\text{st}}=n{\overline{t}}_{\text{st}}$$ (16)

In equation (16), the added nondimensional parameter is above unity (n > 1) to preserve the inequality written above. Using equation (16), equations (14a) and (14b) can be simplified to

$$s_{\alpha t}=s_{\alpha 0}\text{\hspace{0.17em}exp}\left\{-\left[n{\mu }_{\text{st}}\left({\overline{E}}_{\alpha }\right)+{\mu }_{\text{st}}\left(E_{\text{Pb\hspace{0.17em}}L\alpha }\right)\right]{\overline{t}}_{\text{st}}\right\},$$ (17a)

$$s_{\beta t}=s_{\beta 0}\text{\hspace{0.17em}exp}\left\{-\left[n{\mu }_{\text{st}}\left({\overline{E}}_{\beta }\right)+{\mu }_{\text{st}}\left(E_{\text{Pb\hspace{0.17em}}L\beta }\right)\right]{\overline{t}}_{\text{st}}\right\}.$$ (17b)

In equations (17a)) and (17b), ${\overline{E}}_{\alpha }$ and ${\overline{E}}_{\beta }$ are the average incident photon energies for which $\text{exp}\left[-{\mu }_{\text{st}}\left({\overline{E}}_{\alpha }\right){\overline{t}}_{\text{st}}\right]$ and $\text{exp}\left[-{\mu }_{\text{st}}\left({\overline{E}}_{\beta }\right){\overline{t}}_{\text{st}}\right]$ are equal to the value of the integral from equations (14a) and (14b), respectively. Neglecting the term $\Delta {\mu }_{\text{b}}^{\text{Sr}}\left({\overline{t}}_{\text{b},\text{t}}-{\overline{t}}_{\text{b},0}\right)$ in equation (13) and plugging the ${\overline{t}}_{\text{st}}$ result in equations (17a) and (17b), the initial equations (10a) and (10b) are obtained. The nondimensional parameters a_α and a_β are given by

$$a_{\alpha }=\frac{n{\mu }_{\text{st}}\left({\overline{E}}_{\alpha }\right)+{\mu }_{\text{st}}\left(E_{\text{Pb\hspace{0.17em}}L\alpha }\right)}{\Delta {\mu }_{\text{st}}^{\text{Sr}}},$$ (18a)

$$a_{\beta }=\frac{n{\mu }_{\text{st}}\left({\overline{E}}_{\beta }\right)+{\mu }_{\text{st}}\left(E_{\text{Pb }L\beta }\right)}{\Delta {\mu }_{\text{st}}^{\text{Sr}}}.$$ (18b)

In section 3.5 of this paper, parameters a_α and a_β will be determined by fitting equations (10a) and (10b) to the experimental data. Further discussions were included in sections 3.5 and 4.1.

Plugging the slope values computed using equations (10a) and (10b) in equations (9a) and (9b) gives two estimates of the bone Pb concentration c_α and c_β:

$$c_{\alpha }=A_{\alpha }/s_{\alpha t}=\left(A_{\alpha }/s_{\alpha 0}\right){\left[r_{m,t}^{\text{Sr}}/r_{m,0}^{\text{Sr}}\right]}^{a_{\alpha }},$$ (19a)

$$c_{\beta }=A_{\beta }/s_{\beta t}=\left(A_{\beta }/s_{\beta 0}\right){\left[r_{m,t}^{\text{Sr}}/r_{m,0}^{\text{Sr}}\right]}^{a_{\beta }}.$$ (19b)

Equations (19a) and (19b) contain spectrometric quantities, no additional measurements are required. The bare bone calibration line slope values s_α0 and s_β0 include the x-ray attenuation properties of bone tissue and Pb excitability. The x-ray attenuation properties of the ST and geometrical parameters are contained within the values of the a_α and a_β parameters. Error propagation can be carried out in equations (19a) and (19b) to compute the errors on Pb concentration estimates $\delta c_{\alpha }$ and$\delta c_{\beta }$. The expressions of $\delta c_{\alpha }$ and $\delta c_{\beta }$ are as follows:

$$\delta c_{\alpha }=c_{\alpha }\sqrt{{\left(\frac{\delta A_{\alpha }}{A_{\alpha }}\right)}^2+{\left(\frac{\delta s_{\alpha 0}}{s_{\alpha 0}}\right)}^2+{\left(a_{\alpha }\right)}^2\left[{\left(\frac{\delta r_{m,t}^{\text{Sr}}}{r_{m,t}^{\text{Sr}}}\right)}^2+{\left(\frac{\delta r_{m,0}^{\text{Sr}}}{r_{m,0}^{\text{Sr}}}\right)}^2\right]+{\left[\text{ln}\left(a_{\alpha }\right)\delta a_{\alpha }\right]}^2},$$ (20a)

$$\delta c_{\beta }=c_{\beta }\sqrt{{\left(\frac{\delta A_{\beta }}{A_{\beta }}\right)}^2+{\left(\frac{{\delta }_{\beta 0}}{s_{\beta 0}}\right)}^2+{\left(a_{\beta }\right)}^2\left[{\left(\frac{\delta r_{m,t}^{\text{Sr}}}{r_{m,t}^{\text{Sr}}}\right)}^2+{\left(\frac{\delta r_{m,0}^{\text{Sr}}}{r_{m,0}^{\text{Sr}}}\right)}^2\right]+{\left[\text{ln}\left(a_{\beta }\right)\delta a_{\beta }\right]}^2}.$$ (20b)

The final Pb concentration estimate c and its error δc can be computed as the weighted average:

$$c=\frac{c_{\alpha }{\left(\delta c_{\alpha }\right)}^{-2}+c_{\beta }{\left(\delta c_{\beta }\right)}^{-2}}{{\left(\delta c_{\alpha }\right)}^{-2}+{\left(\delta c_{\beta }\right)}^{-2}},$$ (21a)

$$\delta c={\left[{\left(\delta c_{\alpha }\right)}^{-2}+{\left(\delta c_{\beta }\right)}^{-2}\right]}^{-1/2}.$$ (21b)

3. Results

3.1. Sr concentration measurement

The results of the Sr calibration measurements are given in the two plots of figure 8 shown below. The best fit lines and the corresponding fitting parameters are also provided in the two plots. In both cases, the peak area error bars appeared to be smaller than the observed variations above and below the fitted line. The variations are significant as indicated by the reduced chi-squared values significantly larger than unity (p < 0.05). Two possible explanations can be brought forward. First, Sr KXRF peak area measurements correspond to much larger Sr concentrations than the added values in the plots, hence, small relative uncertainties. Second, the uncertainties on the prepared Sr concentrations were not taken into account and could be responsible for the observed variations.

Figure 8.

Two plots of the peak area measurements versus added Sr concentration corresponding to Sr K_α peak (a) and Sr K_β peak (b).

The intrinsic Sr concentration c_Sr of the commercial poP bone phantoms can be determined using the values of the y-axis intercept (b) and slope (a) obtained following the analytical linear fit:

$$c_{\text{Sr}}=b/a.$$ (22)

Thus, based on the two calibration line data, two values of the intrinsic Sr concentration were calculated as (1.01 ± 0.07) mg g⁻¹ (Sr K_α peak data) and (1.0 ± 0.2) mg g⁻¹ (Sr K_β peak data). The weighted average of these two estimates is (1.01 ± 0.07) mg g⁻¹.

As indicated in section 2.1, the weighted average Sr concentration of (1.01 ± 0.07) mg of Sr per gram of solidified poP can be converted to (4.3 ± 0.3) mg per gram of Ca concentration. Similar Sr contamination levels in commercial poP in the range of 4 mg Sr g⁻¹ Ca to 6 mg Sr g⁻¹ Ca were also measured by Pejović-Milić et al (2004).

DL values for Sr can also be calculated using equation (7) in which σ₀ value was the Sr peak area uncertainty (in counts · keV units) corresponding to zero added Sr concentration: 0.74 for Sr K_α and 0.48 for Sr K_β. Thus, Sr DL values of (39.6 ± 0.2) μg g⁻¹ and (0.15 ± 0.02) mg g⁻¹ were computed corresponding to Sr K_α and Sr K_β data, respectively. Conversion to per gram of Ca concentration values yields, in the same order, (170.1 ± 0.9) μg Sr g⁻¹ Ca and (0.64 ± 0.09) mg Sr g⁻¹ Ca. These are similar to the 0.11 mg Sr g⁻¹ Ca to 0.75 mg Sr g⁻¹ Ca range of DL values reported by Pejović-Milić et al (2004). However, the DL values reported here were obtained from three trials of 120 s each (360 s total time), while Pejović-Milić et al (2004) used 1800 s real time acquisitions. The more recent study of Da Silva et al (2013) detailed the chemical preparation of pure hydroxyapatite (the main component of biological bone mineral) bone phantoms (i.e. levels of Sr, Pb and other metals were below DLs) and, using a commercial XRF spectrometer and 1200 s live time acquisition, a limit of detection value of 0.4 μg Sr g⁻¹ Ca was reported.

3.2. Linear attenuation coefficients

The measurements of the linear attenuation coefficients (μ) of the three ST phantom materials (POM, resin, and wax) and aluminum (Al) as reference material were plotted against photon energy in the 9–22 keV range. The plots are included in figure 9 shown below. For comparison, the red color data points indicate the linear attenuation coefficients values calculated as the product between the XCOM database mass attenuation coefficients and density. The measured density values and elemental composition or chemical formula for the three ST materials were taken from table 1. The error on these data points was computed as propagated error from density measurements and a 1% relative error on XCOM mass attenuation coefficients as estimated by Hubbell in his review of photon interaction cross section data (Hubbell 1999).

Figure 9.

Plots of the measured linear attenuation coefficients (μ) of four materials: aluminum (Al), polyoxymethylene (POM), resin, and wax as a function of photon energy in the 9–22 keV range. The red data points correspond to values calculated using the XCOM database; the corresponding relative errors were below 2%.

The measured linear attenuation coefficients as a function of photon energy data was described well by an ad hoc empirical three-parameter exponentially decreasing function. The fitted parameters and their errors resulting from the nonlinear fitting routines are provided in the plots of figure 9. The percentage differences between the measured and XCOM-computed values in the 10–20 keV photon energy range are shown in the plot of figure 10. At the boundaries of the energy range, small relative differences at or below 5% can be observed for Al and resin. For POM and wax materials, the XCOM linear attenuation coefficients were between 10% and 35% larger than their measured counterparts at all photon energies. These systematic differences were likely due to an overestimation of the measured density (higher XCOM μ values) and thickness (lower measured μ values). The density overestimation likely resulted from an underestimate of the measured water volume in the water displacement method: the volume was read off the graduated cylinder without accounting for the edge volume due to the wetting phenomenon. For wax, due to fragility of cylindrical shell wax phantoms, volume and mass measurements of the the unmelted large wax blocks were taken to estimate its density. In the phantom-making process consisting of melting, mold pouring, and solidification, a lower density of the wax phantom could have resulted from the trapping of microscopic air bubbles. Measurements of the circular phantom layer by the parallel-plate jaws configuration of the micrometer was the source of thickness overestimated measurements, an effect likely more pronounced for the more fragile wax phantoms. The same systematic errors occurred in the density and thickness measurements of the resin phantoms. However, the bulk elemental composition of the resin provided in table 1 from section 2.1 was taken from the literature (Gawdzik et al 2001). A commercial product intended for repairs of various household items, the resin presumably contained contaminants consisting of chemical elements with a larger atomic number than that of the predominant carbon (Z = 6). Speculatively, the authors assumed that the higher average atomic number of the resin resulted in higher values of the measured linear attenuation coefficients, thus, counteracting the systematic effects of overestimated density and thickness discussed above. Therefore, as can be noticed in the plot of figure 10, the relative differences between the XCOM and measured linear attenuation coefficients of resin resemble those of reference material Al.

Figure 10.

Plot of the percentage differences between the measured and XCOM-computed linear attenuation coefficients versus photon energy for the four ST phantom materials and aluminum.

The peak of the differences between measured and XCOM calculated μ values observed in the plot of figure 10 for all four materials can be explained as follows. As detailed in section 2.4, the linear attenuation coefficients measurements at photon energies above 12 keV were based on the average number of counts estimated at the middle of a photon energy range of roughly 1 keV width. The energy broadening effect of the counting x-ray detectors with multi-channel analyzers is well known. For any given channel (or photon energy following the energy calibration procedure), the recorded number of counts included contributions from photons of lower and higher energy. If the broadening effect would be constant across all photon energies, the data analysis procedure employed in this study should result in an unbiased linear attenuation coefficient estimate. However, the energy broadening effect increases with detected photon energy (Papp et al 2005). Therefore, the number of counts associated with the middle of a certain 1 keV range includes more contributions from higher energy photons than from the lower energy side. Consequently, the measured linear attenuation coefficient (μ) will be systematically underestimated (μ decreases with photon energy E). In the plot of figure 9, one can observe that the systematic effect is more pronounced at the photon energies from the middle of the broad bremsstrahlung peak, or 15–17 keV range as seen in the x-ray spectrum plot of figure 6(a), and less so at the photon energies at the edges. No further analysis of this effect and the systematic uncertainties discussed above was performed.

Despite the spectrometric-based systematic error discussed above, it was estimated that the linear attenuation coefficient measurements were slightly more accurate than those computed using the XCOM database. The estimation particularly takes into account that any systematic errors of the density measurements and elemental compositions were not investigated. The linear attenuation coefficients of the three ST materials at the Pb and Sr XRF energies were computed using the three-parameter fitted exponential decay function.

The error on each linear attenuation coefficient value was computed using general error propagation equation (Taylor 1997) in the exponential decay function μ = μ(E). The errors on the three fitted parameters were assumed to be statistically independent. The results are grouped in table 6. For comparison, the linear attenuation coefficients of human skin, adipose tissue, poP, tibia cortical bone are also provided and were calculated using the XCOM database. Average density and elemental compositions of human skin and adipose tissues were taken from ICRU-44 (ICRU Report 44 1989) compilation. Chemical formula of calcium sulfate hydrate (CaSO₄ · 2H₂O) and measured average density of solidified poP of (1.05 ± 0.05) g cm⁻³ were used to compute the poP linear attenuation coefficients. The elemental composition of the adult human cortical bone and its density of 1.90 g cm⁻³ were taken from Zhou et al (2009) paper. It can be noted that the linear attenuation coefficients of the ICRU-44 skin and adipose tissue were approximated well by POM and resin materials, respectively. The linear attenuation coefficient of wax was significantly lower than that of skin or adipose tissue. The calculated linear attenuation coefficients of poP were about 34% lower than those of adult human cortical bone. Therefore, for more accurate Pb concentration measurements, future investigations might consider other bone substitute materials to produce the bare bone Pb calibration line data. A good bone phantom material could be the hyroxyapatite phantom solution of Da Silva et al (2013) which would increase the bone phantom density and allow Pb doping through the addition of the standard Pb solution.

Table 6.

Linear attenuation coefficients of the three ST phantom materials, skin, adipose tissue, and poP in units of mm⁻¹ evaluated at four photon energies. The numbers in the round parentheses are the uncertainties in the last one or two significant figures of the corresponding value.

Photon energy (keV)	XRF peak	POM	Resin	Wax	Skin (ICRU-44)	Adipose tissue (ICRU-44)	Plaster-of-Paris (poP)	Human cortical bone
10.5	Pb Lα	0.43(2)	0.30(4)	0.14(4)	0.468(5)	0.270(3)	3.10(17)	4.69(5)
12.6	Pb Lβ	0.243(10)	0.17(2)	0.083(17)	0.277(3)	0.162(2)	1.85(9)	2.79(3)
14.1	Sr Kα	0.171(6)	0.121(13)	0.061(10)	0.202(2)	0.1204(12)	1.34(6)	2.03(2)
15.8	Sr Kβ	0.123(4)	0.089(8)	0.047(6)	0.149(15)	0.0906(9)	0.97(5)	1.469(15)

3.3. Pb calibration lines

Samples of the Pb L_α and Pb L_β calibration lines obtained using POM ST material and employing the OGIP XRF measurement method are shown in the two plots of figure 11. Details of the Pb L_α and Pb L_β calibration lines for all three ST phantom materials were summarized in table 6. A careful analysis of the calibration lines data reveals the following features. Seven Pb L_α calibration lines have an y-axis intercept value significantly larger than zero (i.e. value is more than twice larger than its error). Five Pb L_α calibration lines also resulted from a linear fit for which the reduced chis-squared value was significantly larger than unity (p < 0.05). The same features were not observed for Pb L_β calibration line data. These observations indicated a possible contamination of the Pb L_α peak area data. A plausible explanation related to the known Pb contamination of the PXL glass material is included in section 4.3.

Figure 11.

Samples of Pb L_α and Pb L_β calibration lines corresponding to cylindrical Pb-doped poP bone phantoms with overlapping cylindrical shell section of POM ST phantoms of 0, 1, and 4 mm approximate thickness. Slope values of the best fit lines going through origin are provided. See text for further details.

The Pb L_α and Pb L_β slope values dependence on the ST thickness (t_st) denoted by s_αt and s_αt followed the attenuation model described by equations (14a) and (14b) in section 2.6. In the plots of figure 12 shown below, the slope values were normalized to the value of the bare bone slope values denoted by s_α0 and s_β0. As shown in the plots, data was fitted by a double exponential function as in equations (14a) and (14b). The following steps were taken for the numerical application. The values of the linear attenuation coefficients in units of mm⁻¹ corresponding to the Pb L-shell x-rays, μ_st (E_{Pb Lα}) and μ_st (E_{Pb Lβ}), were taken from table 6 of section 3.2. The integral describing the ST attenuation of the incident photons over their energy spectrum in equations (14a) and (14b) was replaced by an effective linear attenuation coefficient denoted by μ_eff. The ST attenuation length of the incident x-rays denoted by L_st was computed as a function of ST thickness values t_st using the approximate relationship of equation (15). The values of μ_eff obtained from the fitting procedures were included in the six plots of figure 12. The weighted average μ_eff values of the two estimates corresponding to the Pb L_α and Pb L_β data (left and right plots in figure 9) for POM, resin, and wax, in units of mm⁻¹, were: 0.073 ± 0.007, 0.065 ± 0.012, and 0.055 ± 0.003, respectively. Using the fitted curves of linear attenuation coefficients versus photon energy of figure 9, the incident photon energy values corresponding to μ_eff values of POM, resin, and wax, were: 20.9 keV, 18.5 keV, and 14.7 keV, respectively. The noticeable photon energy increase with increasing linear attenuation coefficient of the ST phantom material μ_st is due to the x-ray beam ‘hardening’ effect: a higher μ_st and/or ST thickness results in an increase of the average energy of the incident photons which can reach the poP bone Pb and excite its L-shell electrons.

Figure 12.

Plots of the Pb L_α and Pb L_β calibration lines slope values s_αt and s_βt dependence on ST thickness (t_st). In the plots the slope values were normalized to the values of the bare bone phantom calibration lines s₀.

It was also found that the background levels were strongly correlated with the uncertainty of the zero Pb concentration peak area estimate (σ₀). The σ₀ values were used to compute the Pb DL values using equation (7) and were included in table 8. Thus, the Pb L_β slope value of (0.012 ± 0.003) keV μg⁻¹ g corresponding to the strongest attenuating 4.36 mm POM phantom is larger than the Pb L_β slope value of (0.002 ± 0.003) keV μg⁻¹ g (i.e. zero slope) of the less attenuating 4.19 mm resin phantom. The numerical σ₀ values from table 8, in the same order, are 0.138 and 0.218, indicating a substantially larger background in the x-ray spectrum of the resin phantom compared to that of the POM phantom.

Table 8.

Pb detection limit (DL) values for bare bone and ST phantom calibration line data.

ST	Thickness t (mm)	Pb Lα		Pb Lβ
		σ0 (counts keV)	DL (μg g−1)	σ0 (counts keV)	DL (μg g−1)
	0.00	0.068	2.76(8)	0.179	7.3(5)
POM	1.10(5)	0.105	9.0(7)	0.281	20(2)
	1.89(5)	0.123	24(5)	0.297	39(3)
	3.23(3)	0.090	35(6)	0.187	35(3)
	4.36(3)	0.079	40(18)	0.138	35(3)
Resin	1.10(5)	0.088	5.5(3)	0.281	16.2(1.5)
	2.18(5)	0.174	15(2)	0.265	26(2)
	3.05(3)	0.114	27(5)	0.250	38(8)
	4.19(3)	0.092	45(17)	0.161	54(14)
Wax	1.47(8)	0.108	7.0(9)	0.302	19.6(9)
	2.04(8)	0.123	10.4(5)	0.286	34(9)
	3.48(8)	0.149	17(2)	0.328	30(3)
	4.01(5)	0.107	15(2)	0.262	35(6)

3.4. Analysis of Ca K_β/K_α, Sr K_β/K_α and Pb L_β/L_α ratio measurements

Equation (6) can be applied to the Ca K_β/K_α, Sr K_β/K_α and Pb L_β/L_α ratio measurements from the x-ray spectra acquired in the bare bone OGIP experiments. Thus, one can estimate the average thickness ${\overline{t}}_{\text{b},0}$ of x-ray attenuation within the poP bone phantom. Figure 13 shows the linear fit of the ln (r_c/r_a) versus Δμ for the ratio measurements aforementioned. The slope of the best fit line is the ${\overline{t}}_{\text{b},0}$ estimate. All quantities were defined in section 2.5 and detection efficiencies values were taken from table 5. Linear attenuation coefficients of poP bone phantoms were calculated using the XCOM database (Berger et al 2010), calcium sulfate hydrate chemical formula (CaSO₄ · H₂O) and measured average poP density of 1.05 g cm⁻³. The estimate of the average x-ray attenuation thickness in the bone phantom was (44 ± 9) μm. Estimates of ${\overline{t}}_{\text{b},0}$ can also be computed for each of the Ca K_β/K_α, Sr K_β/K_α and Pb L_β/L_α ratio measurements. The following ${\overline{t}}_{\text{b},0}$ values, in μm, were: (44 ± 9), (−20 ± 80), and (230 ± 120). The weighted average and error was (44 ± 9) μm which is identical with that of the linear fit method as expected.

Figure 13.

Plot of the ln (r_c/r_a) from Ca K_β/K_α, Sr K_β/K_α and Pb L_β/L_α ratio measurements (r_m) versus linear attenuation difference Δμ. See text for further details and notations.

Different x-ray attenuation of 14.1 keV and 15.8 keV Sr K-shell x-rays in bone and ST phantoms can be expressed by equation (12). The $\text{ln}\left(r_{m,t}^{\text{Sr}}/r_{m,0}^{\text{Sr}}\right)$ quantity was calculated as the weighted average over all seven measurements at a given ST thickness (for each Pb concentration). The plots of these quantity against the ST thickness (t_st) for each phantom material were included in figure 14 shown below.

Figure 14.

Plots of the $\text{ln}\left(r_{m,t}^{Sr}/r_{m,0}^{Sr}\right)$ quantity versus ST thickness (t_st) for POM, resin, and wax.

The differences in the linear attenuation coefficients $\Delta {\mu }_{\text{st}}^{\text{Sr}}$ and $\Delta {\mu }_{\text{b}}^{\text{Sr}}$ corresponding to ST and poP bone phantoms can be calculated based on the data from table 6 and XCOM database, respectively. These are: (0.048 ± 0.007) mm⁻¹ for POM, (0.032 ± 0.015) mm⁻¹ for resin, (0.014 ± 0.012) mm⁻¹ for wax, and (0.44 ± 0.02) mm⁻¹ for poP. From a comparison of the quantities plotted in figure 14 with equation (12), one can deduce that the ST thickness (t_st) corresponds to the average photon attenuation thickness $\left({\overline{t}}_{\text{st}}\right)$, the slope of the best line fit (m) is the ST linear attenuation coefficient difference $\Delta {\mu }_{\text{st}}^{\text{Sr}}$, and the intercept corresponds to the poP bone phantom attenuation factor $\Delta {\mu }_{\text{b}}^{\text{Sr}}\left({\overline{t}}_{\text{b},\text{t}}-{\overline{t}}_{\text{b},0}\right)$. However, the slope of the best fit line m is significantly larger than $\Delta {\mu }_{\text{st}}^{\text{Sr}}$ for POM and resin phantom materials computed from the corresponding values of table 6. This result implies that the average ST attenuation thickness, ${\overline{t}}_{\text{st}}$, is about twice larger than the ST layer thickness, t_st. The observations can be qualitatively explained by considering the spatial extent of the incident x-ray beam, not just its main direction. After a closer visual inspection of the experimental schematic provided in figure 2, one can notice that Sr x-rays following the excitation-emission-detection processes chain can be triggered by primary and scattered photons on the side of the poP bone closer to the x-ray beam incidence. In our approximate estimation, these Sr x-rays will represent an important contribution added to those originating from the small poP bone volume assumed by the OGIP geometry or the size of the ${\overline{t}}_{\text{b},0}$ estimate. The same analysis holds for the wax ST phantom material as well, but the difference between slope and $\Delta {\mu }_{\text{st}}^{\text{Sr}}$ is hindered by the large measurement-driven uncertainties. Data plot and analysis for wax shows a non-zero y-axis intercept. Given the previously determined (44 ± 9) μm estimate of ${\overline{t}}_{\text{b},0}$ the average x-ray attenuation thickness in bone corresponding to wax data, ${\overline{t}}_{\text{b},t}$, was estimated to be (0.23 ± 0.07) mm. For POM and resin materials, the data is consistent with a null y-axis intercept. A simplified interpretation of equation (12) leads to ${\overline{t}}_{\text{b},\text{t}}\approx {\overline{t}}_{\text{b},0}$. A more unifiying interpretation of these results for all three ST materials is that ${\overline{t}}_{\text{b},\text{t}}$ increases with ST thickness t_st, an effect more pronounced for the stronger attenuating POM and resin materials and less so for the wax material. This effect can be explained by the beam ‘hardening’ process which also explained the photon energy of the effective linear attenuation coefficient (μ_eff) discussed in the previous section 3.3. A stronger attenuation by the ST of the incident beam will have higher energy primary x-ray photons being able to excite Sr and Pb atoms located deeper within the bone. Hence, the increase of ${\overline{t}}_{\text{b},\text{t}}$ with ST thickness t_st. As a secondary consequence, the t_b-dependent part of the $\Delta {\mu }_{\text{b}}^{\text{Sr}}{\overline{t}}_{\text{b},\text{t}}$ attenuation factor could be added to the $\Delta {\mu }_{\text{st}}^{\text{Sr}}{\overline{t}}_{\text{st}}$ term in equation (12). This operationwill supplement the interpretation of the slope values being larger than the measured $\Delta {\mu }_{\text{st}}^{\text{Sr}}$ for POM and resin.

3.5. Pb concentration calculations

Equations (10a) and (10b) derived in section 2.6 were applied to the Pb and Sr peak area measurements. The results were summarized in the plots of figure 15. The slope values of the best fit lines provided in the plots of figure 15 correspond to the nondimensional quantities a_α and a_β from equations (10a) and (10b). The values of a_α and a_β were also included in table 9. The last column provides the $\Delta {\mu }_{\text{st}}^{\text{Sr}}$ value for each ST material; $\Delta {\mu }_{\text{st}}^{\text{Sr}}$ was the difference between the linear attenuation coefficients of the ST evaluated at the Sr K_α and Sr K_β energies of 14.1 keV and 15.8 keV, respectively. The linear attenuation coefficient values of the three ST materials were taken from table 6. It can be noticed that the values of a_α and a_β increase with decreasing $\Delta {\mu }_{\text{st}}^{\text{Sr}}$. This result is consistent with the inverse proportionality relationships between these quantities from equations (18a) and (18b). The measured a_α, a_β, $\Delta {\mu }_{\text{st}}^{\text{Sr}}$ values of wax are smaller than twice their uncertainty values. This observation combined with the results of the bare bone analysis in previous section 3.4 points to the following limits of applicability of the calibration method developed in this study. The applicability relies on the condition: Δμt ⩾ 1, where Δμ is the linear attenuation coefficient difference evaluated at the two XRF emissions of a given element and attenuating material and t is the average thickness of the attenuating material.

Figure 15.

Log–log plots of the spectrometric quantities as defined in the text. Top row: plots corresponding to the Pb Lα data for the three ST phantom materials; bottom row: plots corresponding to the Pb Lβ data for the three ST phantom materials.

Table 9.

Values of a_α, a_β, $\Delta {\mu }_{\text{st}}^{\text{St}}$ measured values for the three ST phantom materials. The errors in the last one or two significant figures were provided in the round parentheses.

Material	aα	aβ	$\Delta {\mu }_{\text{st}}^{\text{St}}$ (mm−1)
POM	6.5(2)	4.7(3)	0.048(7)
Resin	7.9(1.4)	6.8(8)	0.032(15)
Wax	7.5(6)	7.9(7)	0.014(12)
Weighted average	6.6(2)	5.4(3)	0.038(6)

The last row of table provides the weighted average values of a_α and a_β across the three ST phantom materials. These values were used to determine the Pb concentrations (c_Pb) and their uncertainties (δc_Pb) using equations (19a) through (21b). Quantities from equations (19a) and (19b) were calculated using the weighted average Pb and Sr peak areas from each three trial set corresponding to one particular ST material, thickness, and Pb concentration. Thus, 84 estimates of the Pb concentration were computed using an Excel template. Apart from weighted average values of a_α and a_β, the only a priori knowledge serving as input in all calculations were: (i) the Pb L_α and Pb L_β bare bone calibration lines slopes s_α0 and s_β0 respectively, and (ii) $r_{m,0}^{\text{Sr}}$ which was the measured Sr K_β/K_α ratio corresponding to the bare bone phantom experiments. The values of the s_α0 and s_β0 were (0.073 ± 0.002) keV μg⁻¹ g and (0.073 ± 0.005) keV μg⁻¹ g as extracted from table 7 and was $r_{m,0}^{\text{Sr}}$ 0.1664 ± 0.0015. The value of $r_{m,0}^{\text{Sr}}$ is smaller than its literature atomic value of 0.181 ± 0.009 because the detector efficiency difference was not accounted for. The corresponding corrected value was 0.200 ± 0.002 which is larger than its atomic value as expected. For data sets in which the positive y-axis intercept values of the Pb calibration lines were significant (i.e. value was larger than twice its error), the Pb peak areas were corrected by subtracting the value of the y-axis intercept. The physical origin of this positive y-axis intercept values was discussed in section 4.3.

Table 7.

Calibration lines data for poP bare bone (t = 0 mm) and poP bone with overlapping POM, resin and wax ST phantoms of four different thickness values. The number in the round parentheses is the uncertainty in the last one or two significant figures of the corresponding value.

ST	Thickness t (mm)	Pb Lα			Pb Lβ
		Slope(s)(keV μg−1g)	y-intercept(keV)	R2	Slope(s)(keV μg−1g)	y-intercept(keV)	R2
	0.00	0.073(2)	0.09(7)	0.996	0.073(5)	−0.1(2)	0.975
POM	1.10(5)	0.035(3)	0.14(11)	0.973	0.042(5)	−0.1(2)	0.932
	1.89(5)	0.015(3)	0.33(12)	0.839	0.023(2)	−0.03(8)	0.962
	3.23(3)	0.0076(13)	0.25(6)	0.868	0.016(2)	0.20(13)	0.914
	4.36(3)	0.006(3)	0.29(11)	0.494	0.0119(13)	0.03(5)	0.945
Resin	1.10(5)	0.048(2)	−0.14(9)	0.989	0.052(5)	−0.2(2)	0.958
	2.18(5)	0.036(4)	0.03(18)	0.932	0.030(2)	0.14(10)	0.968
	3.05(3)	0.013(2)	0.12(10)	0.857	0.020(4)	−0.02(18)	0.811
	4.19(3)	0.006(2)	0.11(10)	0.583	0.009(2)	0.14(10)	0.738
Wax	1.47(8)	0.046(6)	0.0(2)	0.923	0.046(2)	−0.15(9)	0.989
	2.04(8)	0.0355(16)	0.11(7)	0.989	0.025(6)	0.1(3)	0.760
	3.48(8)	0.026(3)	0.30(13)	0.933	0.033(4)	0.02(15)	0.945
	4.01(5)	0.021(3)	0.16(13)	0.898	0.022(4)	0.05(16)	0.872

The plots of the calculated Pb concentration versus the known bone phantom Pb concentration were grouped in figures 16–18 shown below and corresponded to the POM, resin, and wax ST phantoms, respectively. The relative errors on the bone Pb concentration estimates were, in general, large with many values exceeding 20%. These results indicating a poor precision are a direct consequence of the large relative errors associated with the the Pb L_α and Pb L_β peak area measurements which were also around or above 20%.

Figure 16.

Plot of the calculated versus known bone phantom Pb concentrations for the POM ST phantoms. The diagonal line represents indicates the perfect agreement.

Figure 18.

Plot of the calculated versus known bone phantom Pb concentrations for the wax ST phantoms. The diagonal line represents indicates the perfect agreement.

Statistical bias was verified by plotting the differences between the estimated and known Pb concentration values divided by the error on the estimated Pb concentration. The three plots corresponding to the three ST materials were included in figure 19. The data set number of the horizontal axes corresponds to data obtained from experiments in the order of increasing poP bone Pb concentration and from lowest to the largest ST phantom thickness. Thus, data sets designated by numbers 2 and 18 in the POM data plot corresponded to data obtained from experiments characterized by: (i) 1.10 mm POM thickness (see tables 2) and 8 μg g⁻¹ poP bone Pb concentration and (ii) 3.23 mm POM thickness and 44 μg g⁻¹ poP bone Pb concentration, respectively.

Figure 19.

Plot of the differences between estimated and known Pb concentration values divided by the error on the Pb concentration estimate.

A close inspection of the plots of figure 19 shows that most of the differences were found to be approximately equally distributed around zero with most of the differences being within the interval centered on zero and bounded by twice the error on the estimate or the 95% confidence interval assuming Gaussian statistics. In the POM and wax plots, only one and two data points, respectively, were slightly outside the 95% Gaussian confidence interval.

4. Discussion

4.1. Calibration methods in bone Pb LXRF studies

The main advantage of the calibration method presented in this paper is that bone Pb concentration is determined using several spectrometric measurements that were defined in sections 2.5 and 2.6. The method eliminates the need for STT measurements. Although past calibration procedures employed in past LXRF bone Pb concentrations varied, a common feature is the measurement of STT using clinical ultrasound equipment and procedures. Thus, Wielopolski et al (1989) measured the net counts of the observed Pb L_α for intact amputated human tibia bone and bare bone following STT measurement and its subsequent removal. The study measured nine bone samples with measured STT in the 3–8 mm range and Pb concentration of the bare bones was measured using a flameless atomic absorption spectroscopy (AAS) method. Assuming that a single exponential attenuation described the decrease of the measured Pb L_α net counts, the measurements gave an effective linear attenuation coefficient of (0.48 ± 0.05) mm⁻¹. This value is comparable with the linear attenuation coefficients values of POM and ICRU-44 skin from table 6. Wielopolski et al (1989) did not include measurement uncertainties on their STT measurements which would have likely increased the overall uncertainties. A separate issue noted by Todd (2002a) is the report of AAS Pb concentrations measurements as μg of Pb per gram of wet bone by Wielopolski et al (1989). This comes in contradiction with known sample preparation in AAS analytical measurements procedures whereas sample water content is removed.

Todd et al (2002) selected a phantom-based method to estimate the Pb concentration in nine human cadaver tibia bones. Following the US measurements of the overlying ST thickness, the authors calculated the equivalent polystyrene thickness by taking into account the attenuation differences between the polystyrene and the skin and adipose tissues separately for Pb L_α and Pb L_β x-rays. The corresponding slope values were then determined using an interpolation of the slope data from poP bone and polystyrene ST phantoms measurements reported in a separate publication (Todd 2002b). The following observations can be made. First, the equivalent thickness calculations only accounted for the attenuation of Pb x-rays and ignored the attenuation of incident x-rays. As shown in section 3.3, the attenuation of incident polyenergetic x-rays is required to correctly describe the calibration line slope dependence on the ST thickness. Second, the procedure did not account for differences in the x-ray scattering background between in vivo skin and adipose tissues on one side, and polystyrene, on the other. These differences were likely large given that that polystyrene is significantly less attenuating than both skin and adipose tissues at the two Pb x-ray energies. The linear attenuation coefficients of polystyrene computed using (C₈H₈)n chemical formula, mass density of 1.05 g cm⁻³, and XCOM database were 0.203 mm⁻¹ and 0.124 mm⁻¹ at the 10.5 keV and 12.6 keV energies. By comparison, these values are about 55% and 24% lower than the corresponding values for skin and adipose tissue, respectively. Third, the procedure could not account for x-ray attenuation differences between the average and the individual skin and adipose tissues overlying the human the cadaver bone samples. The effect of inherent variability on bone Pb LXRF signal was analyzed in detail in the Todd (2002a) paper.

An interesting spectrometric calibration method for bone Pb LXRF measurements performed with portable XRF devices was developed more recently by Nie and her collaborators (Nie et al 2011, Specht et al 2014). After demonstrating a monotonic increase of the spectral background with increasing STT, Nie et al (2011) computed the unknown Pb concentration of a bone sample from its measured net count for one of the two Pb peaks and the corresponding measured net count rate of a bone sample with known Pb concentration (i.e. reference sample). The net count rates were determined by subtracting the background count rate which was, in turn, related to the larger Compton peak count rate which intrinsically included the STT information. By using the background subtraction method, the authors circumvented traditional peak fitting to determine the bone Pb concentration. Later, the background subtraction method and two other variations were compared to the results given by separate applications of the ¹⁰⁹Cd KXRF method and the traditional peak fitting LXRF method on Pb-doped poP phantoms (0–100 μg g⁻¹ Pb concentration range), four goat bone samples, and ten human cadaver bones out of which three had intact overlying ST.

The following critical comments can be made regarding the methods employed by Specht et al (2014). The authors claim that ‘… MC (Monte Carlo) simulations were performed to test the differences between poP and bone and Lucite and ST in terms of Pb over Compton signal. No significant differences in XRF spectra were found between poP with Lucite and bone with ST.’ It is difficult to deduce from these statements what was actually tested and how. No further results or details were provided. In the traditional peak fitting methodology, Specht et al (2014) stated that Pb concentration in the goat and cadaver bones was determined by the net Pb counts corrected for Lucite or ST attenuation. One can only speculate of how these corrections were done. In the case of overlying Lucite, one might assume that the net Pb counts were related to calibration line data for the corresponding Lucite thickness. However, no such calibration data was provided. In the case of overlying ST, an equivalent Lucite thickness method adapted from Todd et al (2002) could have been applied, but, no further clarifications were given. Lucite is also known as polymethyl methacrylate (PMMA). Using PMMA chemical formula, (C₅O₂H₈)_n, its mass density of 1.17 g cm⁻³ (ICRU-44), and XCOM database, the linear attenuation coefficients at the two Pb x-rays energies were computed to be 0.341 and 0.204 mm⁻¹. These values fall in between the linear attenuation coefficients of adipose tissue and skin of table 6, arguably a better choice to mimic the STs overlying the tibia bone than the polystyrene used by Todd et al (2002). Specht et al (2014) assessed the agreement between the KXRF and LXRF Pb concentration outcomes using the R² metric, also known as the coefficient of determination, which measures the likelihood of a linear relationship between the outcomes of two measured quantities. As discussed by Altman and Bland (1983), regression lines and associated metrics are not the best assessment tools for agreement. A high R² value indicates linearity, but does not measure or indicates bias or other trends in the difference between two data sets subject of an agreement test.

Despite criticism, the relationship between the large scattered peak and the ST thickness demonstrated by Nie et al (2011) remains a viable alternative as a spectrometric-based calibration method for in vivo bone Pb concentration measurements. However, further work is required to adjust the method to the expected in vivo variations of the ST and bone x-ray attenuation and geometry.

In summary, POM and resin ST phantoms used in this study were better skin and adipose tissue phantoms than the polystyrene used by Todd et al (2002) or Lucite used by Specht et al (2014). The measured Sr K_β/K_α ratio was demonstrated to be a reliable metric of the ST x-ray attenuation. Nondimensional parameters a_α and a_β connecting the Sr K_β/K_α ratio measurements to the Pb calibration lines did not vary significantly amongst the three ST materials. Average values of parameters a_α and a_β yielded unbiased Pb concentration values for POM and resin materials. Therefore, these values are expected to yield accurate results in future investigations involving biological tissues.

4.2. In vivo bone Pb measurements considerations

The radiation dose to the ST and bone were not measured or estimated in this study. The dose to the 3.23 mm POM ST phantom was approximately estimated to be 0.5 mGy for a 180 s irradiation (Gherase and Al-Hamdani 2018a). This result translates to a 1.0 mGy dose (or 1.0 mSv equivalent dose) delivered to the POM phantom in the 360 s total irradiation time employed in this study: 300 s for bone Pb measurements and 60 s as average time dedicated to measurements establishing the OGIP. The 1.0 mSv equivalent dose is significantly lower than the 22.9 mSv equivalent dose to skin computed using 0.5 h of irradiation and the 45.83 mSv h⁻¹ equivalent dose to skin rate reported in the bone Pb KXRF dosimetry study of Nie et al (2007). The total body effective doses for three age groups: 5 year old, 10 year old, and adults reported by Nie et al (2007) were 8.45/9.37 μSv (female/male), 4.20 μSv, and 0.26 μSv, respectively. The effective doses associated with the bone Pb LXRF method applied in this study can be speculated to be also lower than its KXRF counterparts by taking into account the lower irradiation time, and the microbeam’s confined tissue irradiation. The dose to the radiosensitive bone marrow is also a concern, particularly in children. The OGIP method combined with the small x-ray beam size would mitigate the radiation dose to the bone marrow. A separate dosimetric study using suitable hollow bone phantoms will quantitatively support this statement.

A critical component of in vivo applications is Pb detectability assessed by the DL estimate. Clinical applications of in vivo bone Pb LXRF measurements require a DL below 10 μg g⁻¹, a capability achieved by existent KXRF systems. DL values above 20 μg g⁻¹ are considered high for clinical applications, particularly if population assessments of low levels of Pb exposure are targeted. Strict application of this criterion to the measured DL values summarized in table 8 limits bone Pb LXRF measurements to individuals whose STT is around or below 2 mm regardless of the precise division in skin and adipose tissue layers. The ranges of thickness measurements of the ST overlying the adult human tibia bone reported in past investigations were: 3–8 mm (Wielopolski et al 1989), 2.6–8.3 mm (Todd et al 2002), ~2.5–8 mm (Pejović-Milić et al 2002), and 0.4–4.1 mm (Nie et al 2011). The analysis above appears to indicate that applicability of in vivo bone Pb LXRF measurements could be restricted to a small segment of the general population. The authors could not find reported measurements of ST thickness overlying the tibia bone in children. However, thickness measurements of children’s skin of legs and arms below 2 mm were reported (Waller and Maibach 2005). Relatively recent studies also indicated an overall skin thickness increase with age at various anatomical locations (Derraik et al 2014, Van Mulder et al 2020). Combined with a predicted lower effective dose, the STT data point to a more plausible in vivo bone Pb LXRF in children.

Another factor that might play a role in the transition to in vivo bone Pb measurements is the expected physiological bone Sr concentration. The poP bone phantom Sr concentration was measured to be (1.01 ± 0.07) mg of Sr per gram of solidified poP or (4.3 ±0.3) mg of Sr per gram of Ca. By comparison, physiological bone Sr concentrations are in the 0.1 to 0.3 mg g⁻¹ Ca range (Pejović-Milić et al 2004), therefore, poP bone phantom Sr levels were more than one order of magnitude larger than the expected physiological bone levels. Although the measured values of Sr K_β/K_α ratio do not depend on Sr concentration, their relative uncertainties will increase with decreasing Sr concentration, thus, lowering the precision of bone Pb concentration measurements. In this context, it is important to estimate Sr DL which was not determined from direct measurements. From the data of the three 300 s OGIP spectra, the weighted average area of the Sr K_α peak, measured in units of counts · keV, decreased from 278.6±0.6 for bare bone samples to 117.2±0.5 for overlying POM phantom of 1.89 mm thickness. Hence, the Sr K_α peak signal decreased by a factor equal to 117/279 ≈ 0.42. The Sr K_α peak slope of 0.056±0.005 in counts · keV peak area units divided by μg g⁻¹ Pb concentration units was derived from the linear fit of figure 8(a) using three trials of 120 s data. The Sr K_α peak slope for 300 s OGIP trials can then be computed to be 0.140±0.012 assuming an increase by a factor equal to 300/120 = 2.5. The product of the two factors gives a slope value of 0.059±0.005. The average uncertainty of the Sr K_α peak from the 1.89 mm POM data was about 1.3 counts · keV, about 10 times larger than the σ₀ values in table 8, an expected result due to the much higher photon scattering background under the Sr K_α peak. Applying equation (7), one obtains an approximate Sr DL of 65 μg g⁻¹. Using the mass concentration of 0.221 82 g of Ca per gram of cortical bone from Zhou et al (2009) paper, Sr DL can be converted to 0.29 mg Sr g⁻¹ Ca. This result places Sr DL close to the upper bound of the 0.1–0.3 mg g⁻¹ Ca range extracted from Pejović-Milić et al (2004). However, this range refers to average bone tissue Sr concentrations. A micro-particle induced x-ray emission (micro-PIXE) method was employed by Zamburlini et al (2009) to investigate the Sr distribution within the 2.5 mm thickness outer layer of cortical bone samples extracted from five ex vivo human cadaver fingers. The reported average Sr concentrations were above our 65 μg g⁻¹ estimate and the average Sr concentration of one bone sample was around 85 μg g⁻¹. These results suggests that, for an overlying ST thickness not exceeding 2 mm, the current bone Pb measurement and calibration method are applicable, but require future experimental demonstrations on a bone sample.

Physiological distribution of Sr and Pb within the cortical bone tissue can also be taken into account. In the micro-XRF studies of Zoeger et al (2005) and Bellis et al (2009), Pb was observed to be distributed mostly within the outer regions (within ~1 mm) of the human cortical bone samples. The synchrotron-based microscopic XRF investigations of Pemmer et al (2013) indicated that Sr and Pb concentrations in bone structural units (i.e. osteons and bone packets) increased with the degree of mineralization. The mean-free-path (MFP) of Pb x-rays in human cortical bone were computed as the inverse of the linear attenuation values from the last column of table 6. The MFP values were below 0.4 mm indicating that superficial Pb distribution in cortical bone tissues favors its detectability. Microscopic Sr distribution along bone depth axis was studied by Zamburlini et al (2009) by employing a micro-PIXE method. Bone slices were sampled across the 3 MeV proton microbeam (5 μm spot size) at selected points about 100 μm apart within the ~2.5 mm superficial layer of cortical bone. The study concluded that Sr was uniformly distributed in cortical bone. Co-localization of Sr and Pb in cortical bone can aid optimal sensitivity when applying the OGIP method to in vivo conditions. Given that lower energy Pb x-rays are more attenuated than the higher energy Sr x-rays, the Pb detectability might be improved by finding an OGIP method for Pb detection which might not coincide with that of optimal Sr detection. Such adjustment could be achieved by using animal bone samples with a Pb concentration higher than the normal physiological level. An example is the study of Specht et al (2014) that used Pb-doped goat bones.

4.3. Instrumental considerations

While the microbeam used in this study minimized the x-ray scatter, it was not optimized for bone Pb measurements. The focal distance of the PXL equal to 4 mm was too small to co-localize the beam focal spot with the poP bone incidence. The bone incidence was at a point located 11 mm from the tip of the PXL. At that location, in air, the beam FWHM was evaluated to be about 1.7 mm. By comparison, the FWHM of a beam generated by a portable XRF instrument was measured to be 7 mm by employing a thin copper wire scanning technique (Gherase et al 2010b). The overlying ST phantom certainly enlarged the beam incident on the bone due to Compton and elastic scattering of x-ray photons. A smaller size of x-ray beam incident on the bone can reduce the x-ray scatter background in the acquired x-ray spectra improving Pb detection. On the other hand, a much smaller x-ray beam sampling the nonuniform distribution of Pb with bone depth could potentially complicate the interpretation of in vivo results.

The x-ray beam size also bears on the overall size of the XRF instrument. The 10 cm long PXL attached to the x-ray tube significantly increases the size of the units and would make the design of a portable unit difficult. The size of PXLs can be adjusted to accommodate desirable beam output properties such as focal length, beam size or transmission at certain photon energies as reviewed by MacDonald (2010). The use of miniature x-ray tube combined with a shorter PXL could result in a larger x-ray beam size, but gaining much reduced size and weight of the overall instrument. In current designs of portable XRF instruments, the x-ray lens and x-ray detector are placed in a vacuum-tight chamber made of Al and sealed by a thin Kapton or Be window. The experimental setup used in study could be replicated by a more compact XRF device if the detector and x-ray tube and x-ray lens were separated in two units such that the x-ray detector window was parallel to the direction of the x-ray beam. The detector unit could also be attached to a hard plastic brace that would shape the overlying ST in a circular segment for a better resemblance with the phantom experiments. Adjustable distance between the x-ray detector and the x-ray beam in steps of 0.1 mm combined with an automated prebuilt sequential spectral acquisition could provide data required to obtain the optimal beam-detector distance.

Optimization of x-ray beam excitation can also be considered, although significant improvements in Pb and Sr detectability are not expected from such efforts. Higher Al filtration could reduce the scatter background under the Pb L_β which could improve Pb detection for higher ST attenuation. On the other hand, increased beam filtration decreases incident photon count rate. Pb detectability (expressed as calibration line slope) was investigated by Gherase et al (2017) using plane polarized synchrotron mononergetic beams of 15.8, 16.6 and 17.5 keV photon energy intended to optimize LXRF Pb excitation. The photon energies also mimicked the K-shell XRF emissions of three elements as potential candidates for a dedicated x-ray tube target material: zirconium (Zr), niobium (Nb), and molybdenum (Mo), respectively. The lowest 15.8 keV photon energy was found to be optimal for Pb detection in bare bone phantom experiments since it was the closest to the Pb L₂ and L₃ subshell binding energies of 15.2 keV and 13.0 keV, respectively. However, Pb detectability differences among the three photon energies significantly decreased when a 3.1 mm resin ST phantom covered the Pb-doped bone phantoms. XRF signal gain provided by enhanced x-ray absorption of Pb atoms was lost due to a higher x-ray attenuation in the resin layer and an elevated background under the Pb LXRF peaks due to the Compton peak being closer to the Pb x-rays. In this study, beam hardening of the polyenergetic incident x-rays by the overlying ST combined with the excitation-detection geometry was beneficial: increased absorption of lower energy incident x-rays by thickest ST phantom layers resulted in a decrease of the spectral background under the 10.5 and 12.6 keV Pb peaks as demonstrated by the σ₀ values provided in table 8.

5. Conclusions

A new calibration method for LXRF bone Pb concentration measurements was developed using bone and ST phantoms that simulated in vivo human tibia bone Pb measurements. Seven cylindrical poP bone phantoms containing 1.01 mg g⁻¹ of Sr were doped with Pb in the 0–74 μg g⁻¹ range. POM, resin, and wax were used to make three sets of four cylindrical shell-shaped ST phantoms of thickness in the approximate 1–4 mm range. Linear attenuation coefficients measurements demonstrated that POM and resin mimicked well the attenuation of Pb x-rays in skin and adipose tissue, respectively. Calibration line data for POM and resin was used to calculate a bone Pb DL of about 20 μg g⁻¹ for a STT of 2 mm. Calibration based on STT knowledge was replaced by relationships between Pb concentration and spectrometric data. The new method yielded unbiased results using POM and resin data, thus promising more accurate bone Pb concentrations from in vivo measurements than those reported in the past. Considerations of radiation dose, STT, and detectability and distribution of Pb and Sr, indicated bone Pb LXRF measurements in children as a clinical application. This research meets with the current health concerns regarding the negative effects of low levels of Pb exposure on the neurodevelopment of children.

Figure 17.

Plot of the calculated versus known bone phantom Pb concentrations for the resin ST phantoms. The diagonal line represents indicates the perfect agreement.