PET for bone imaging

Positron emission tomography for bone imaging, as an in vivo tracer technique, allows the measurement of the regional concentration of radioactivity proportional to the image pixel values averaged over a region of interest (ROI) in bones. Positron emission tomography is a functional imaging technique that uses [¹⁸F]NaF radiotracer to visualise and quantify regional bone metabolism and blood flow. [¹⁸F]NaF has been used for imaging bones for the last 60 years. This article focuses on the pharmacokinetics of [¹⁸F]NaF in bones, and various semi-quantitative and quantitative methods for quantifying regional bone metabolism using [¹⁸F]NaF PET images.

The measurement of regional bone metabolism is critical to understand the pathophysiology of metabolic bone diseases.

• Bone biopsy is considered the gold standard to quantify bone turnover; however, it is invasive, complex and costly to perform and subject to significant measurement errors.^[2]
• Measurements of serum or urine biomarkers of bone turnover are simple, cheap, quick, and non-invasive in measuring changes in bone metabolism, but only provide information on the global skeleton.^[3]
• The functional imaging technique of dynamic [¹⁸F]NaF PET scans can quantify regional bone turnover at specific sites of clinical importance such as the lumbar spine and hip^[4] and has been validated by comparison with the gold standard of bone biopsy.^[5][6][7]

The chemically stable anion of Fluorine-18-Fluoride is a bone-seeking radiotracer in skeletal imaging. [¹⁸F]NaF has an affinity to deposit at areas where the bone is newly mineralizing.^{[6][9][10][11][12]} Many studies have [¹⁸F]NaF PET to measure bone metabolism at the hip,^[4] lumbar spine, and humerus.^[13] [¹⁸F]NaF is taken-up in an exponential manner representing the equilibration of tracer with the extracellular and cellular fluid spaces with a half-life of 0.4 hours, and with kidneys with a half-life of 2.4 hours.^[14] The single passage extraction of [¹⁸F]NaF in bone is 100%.^[15] After an hour, only 10% of the injected activity remains in the blood.^[16]

¹⁸F- ions are considered to occupy extracellular fluid spaces because, firstly, they equilibrate with transcellular fluid spaces and secondly, they are not entirely extracellular ions.^[17][18][19] Fluoride undergoes equilibrium with hydrogen fluoride, which has a high permeability allowing fluoride to cross the plasma blood membrane.^[20] The fluoride circulation in red blood cells accounts for 30%.^[21] However, it is freely available to the bone surface for uptake because the equilibrium between erythrocytes and plasma is much faster than the capillary transit time. This is supported by studies reporting 100% single-passage extraction of whole-blood ¹⁸F- ion by bone^[15] and the rapid release of ¹⁸F- ions from erythrocytes with a rate constant of 0.3 per second.^[22]

[¹⁸F]NaF is also taken-up by immature erythrocytes in the bone marrow,^[23] which plays a role in fluoride kinetics.^[24] The plasma protein binding of [¹⁸F]NaF is negligible.^[25] [¹⁸F]NaF renal clearance is affected by diet^[26] and pH level,^[27] due to its re-absorption in the nephron, which is mediated by hydrogen fluoride.^[28] However, large differences in urine flow rate^[21] are avoided for controlled experiments by keeping patents well hydrated.^[23]

The exchangeable pool and the size of the metabolically active surfaces in bones determines the amount of tracer accumulated or exchanged^[29] with bone extracellular fluid,^[30] chemisorption onto hydroxyapatite crystals to form fluorapatite,^[16][31][11] as shown in Equation-1:^[32][33]

${\displaystyle Ca_{10}(PO_{4})_{6}(OH)_{2}+2F-=>Ca_{10}(PO_{4})_{6}F_{2}+2.OH-}$ Equation-1

Fluoride ions from the crystalline matrix of bone are released when the bone is remodelled, thus providing a measure of the rate of bone metabolism.^[34][35][36]

In the context of PET bone imaging using [18F]NaF, the estimated radiation dose to patients is approximately 4.3 millisieverts (mSv) for a typical injected activity of 250 megabecquerels (MBq). This estimate aligns with international radiation protection standards. However, the actual dose delivered to a patient depends on several variables, including how different organs absorb the tracer, the biological clearance rates, and anatomical differences such as organ size and position. These estimates are typically based on reference models of average adult males and females. It's important to note that additional radiation exposure may occur from associated imaging procedures, such as the CT scan that often accompanies PET in combined PET/CT scans. The contribution from the CT scan varies depending on the scan settings, including tube current and exposure time. Recent studies have demonstrated that lower doses of [18F]NaF, such as 90 MBq, may still provide adequate image quality, especially in targeted regions like the lumbar spine. Research is also ongoing into ways to reduce dosage even further, particularly in preclinical studies. However, radiation dosimetry is a complex field and not the primary focus here.

Before the scan, patients are advised to stay relaxed and well-hydrated to promote consistent renal clearance of the tracer. They are typically asked to empty their bladder just before imaging to reduce variability in tracer uptake. During the scan, the patient lies in a supine position so that the target region (such as the hip or spine) is fully within the scanner’s field of view. The imaging session begins with a CT scan, followed by the injection of [18F]NaF. The tracer is administered intravenously within seconds of starting the scan, followed by a saline flush. Imaging data are collected continuously, and blood samples are taken at specific time points to measure tracer concentration in both whole blood and plasma. These measurements are necessary for quantifying tracer kinetics. Blood samples are processed to separate plasma from cells, and radioactivity is measured. Accurate measurements require weighing the tubes before and after sample collection to calculate the actual volume based on fluid density. Tracer concentrations in plasma and whole blood are compared over time to adjust the estimated arterial input function (AIF), which is used in modeling how the tracer behaves in the body.

To ensure consistent analysis, all tracer concentration values—whether from imaging or blood samples—are corrected to a single reference time point, typically the time of injection. The residual tracer left in the syringe after injection is also measured and adjusted for radioactive decay to calculate the actual amount of tracer delivered to the patient. [18F]NaF does not undergo metabolism or significant protein binding, so the tracer concentration in plasma reflects the parent compound directly. This simplifies the calculation of tracer kinetics.

The PET data are processed using various frame durations to accurately capture both the early rapid changes and the later slower changes in tracer distribution. Corrections for physical factors such as scatter, random coincidences, dead time, and attenuation are applied during reconstruction. Quantitative analysis can be performed using either two-dimensional (2D) or three-dimensional (3D) image reconstruction methods. Each has strengths and limitations. 3D reconstruction generally provides better visualization due to higher sensitivity but may introduce variability in measured activity across the field of view. In contrast, 2D reconstruction offers more uniform sensitivity, which may make it more reliable for quantitative analysis. Thus, the choice of reconstruction method can impact the accuracy of tracer uptake measurements.

The standardized uptake value (SUV) is defined as tissue concentration (KBq/ml) divided by activity injected normalized for body weight.^[38]

The SUV measured from the large ROI smooths out the noise and, therefore, more appropriate in [¹⁸F]NaF bone studies as the radiotracer is fairly uniformly taken up throughout the bone. The measurement of SUV is easy,^[39] cheap, and quicker to perform, making it more attractive for clinical use. It has been used in diagnosing and assessing the efficacy of therapy.^[40][41] SUV can be measured at a single site, or the whole skeleton using a series of static scans and restricted by the small field-of-view of the PET scanner.^[34]

The SUV has emerged as a clinically useful, albeit controversial, semi-quantitative tool in PET analysis.^[42] Standardizing imaging protocols and measuring the SUV at the same time post-injection of the radiotracer, is necessary to obtain a correct SUV^[43] because imaging before the uptake plateau introduces unpredictable errors of up to 50% with SUVs.^[44] Noise, image resolution, and reconstruction do affect the accuracy of SUVs, but correction with phantom can minimize these differences when comparing SUVs for multi-centre clinical trials.^[45][46] SUV may lack sensitivity in measuring response to treatment as it is a simple measure of tracer uptake in bone, which is affected by the tracer uptake in other competing tissues and organs in addition to the target ROI.^[47][48]

The quantification of dynamic PET studies to measure Ki requires the measurement of the skeletal time-activity curves (TAC) from the region of interest (ROI) and the arterial input function (AIF), which can be measured in various different ways. However, the most common is to correct the image-based blood time-activity curves using several venous blood samples taken at discrete time points while the patient is scanned. The calculation of rate constants or K_i requires three steps:^[4]

• Measurement of the arterial input function (AIF), which acts as the first input to the mathematical model of tracer distribution.
• Measurement of the time-activity curve (TAC) within the skeletal region of interest, which acts as the second input to the mathematical model of tracer distribution.
• Kinetic modelling of AIF and TAC using mathematical modelling to obtain net plasma clearance (K_i) to the bone mineral.

The method was first described by Cunningham & Jones^[49] in 1993 for the analysis of dynamic PET data obtained in the brain. It assumes that the tissue impulse response function (IRF) can be described as a combination of many exponentials. Since A tissue TAC can be expressed as a convolution of measured arterial input function with IRF, C_bone(t) can be expressed as:

${\displaystyle C_{bone}(t)=\sum _{k=1}^{n}\alpha _{i}.{\bigl (}C_{plasma}(t)\otimes exp(-\beta _{i}.t){\bigr )}}$

where, ${\displaystyle \otimes }$ is a convolution operator, C_bone(t) is the bone tissue activity concentration of tracer (in units: MBq/ml) over a period of time t, C_plasma(t) is the plasma concentration of tracer (in units: MBq/ml) over a period of time t, IRF(t) is equal to the sum of exponentials, β values are fixed between 0.0001 sec⁻¹ and 0.1 sec⁻¹ in intervals of 0.0001, n is the number of α components that resulted from the analysis and β₁, β₂,..., β_n corresponds to the respective α₁, α₂,..., α_n components from the resulted spectrum. The values of α are then estimated from the analysis by fitting multi-exponential to the IRF. The intercept of the linear fit to the slow component of this exponential curve is considered the plasma clearance (K_i) to the bone mineral.

The method was first described by Williams et al. in the clinical context.^[50] The method was used by numerous other studies.^[51][52][53] This is perhaps the simplest of all the mathematical methods for the calculation of K_i but the one most sensitive to noise present in the data. A tissue TAC is modelled as a convolution of measured arterial input function with IRF, the estimates for IRF are obtained iteratively to minimise the differences between the left- and right-hand side of the following Equation:

${\displaystyle C_{bone}(t)=C_{plasma}(t)\otimes IRF(t)}$

where, ${\displaystyle \otimes }$ is a convolution operator, C_bone(t) is the bone tissue activity concentration of tracer (in units: MBq/ml) over a period of time t, C_plasma(t) is the plasma concentration of tracer (in units: MBq/ml) over a period of time t, and IRF(t) is the impulse response of the system (i.e., a tissue in this case). The K_i is obtained from the IRF in a similar fashion to that obtained for the spectral analysis, as shown in the figure.

The measurement of Ki from dynamic PET scans require tracer kinetic modelling to obtain the model parameters describing the biological processes in bone, as described by Hawkins et al.^[24] Since this model has two tissue compartments, it is sometimes called a two-tissue compartmental model. Various different versions of this model exist; however, the most fundamental approach is considered here with two tissue compartments and four tracer-exchange parameters. The whole kinetic modelling process using Hawkins model can be summed up in a single image as seen on the right-hand-side. The following differential equations are solved to obtain the rate constants:

${\displaystyle {\operatorname {d} \!C_{e}(t) \over \operatorname {d} \!t}=K_{1}*C_{p}(t)-(k_{2}+k_{3})*C_{e}(t)+k_{4}*C_{b}(t)}$

${\displaystyle {\operatorname {d} \!C_{b}(t) \over \operatorname {d} \!t}=k_{3}*C_{e}(t)-k_{4}*C_{b}(t)}$

The rate constant K₁ (in units: ml/min/ml) describes the unidirectional clearance of fluoride from plasma to the whole of the bone tissue, k₂ (in units: min⁻¹) describes the reverse transport of fluoride from the ECF compartment to plasma, k₃ and k₄ (in units min⁻¹) describe the forward and backward transportation of fluoride from the bone mineral compartment.

K_i represents the net plasma clearance to bone mineral only. K_i is a function of both K₁, reflecting bone blood flow, and the fraction of the tracer that undergoes specific binding to the bone mineral k₃ / (k₂ + k₃). Therefore, ${\displaystyle K_{i}=\left({\frac {K_{1}*k_{3}}{k_{2}+k_{3}}}\right)}$

Hawkins et al. found that the inclusion of an additional parameter called fractional blood volume (BV), representing the vascular tissue spaces within the ROI, improved the data fitting problem, although this improvement was not statistically significant.^[54]

Patlak method^[55] is based on the assumption that the backflow of tracer from bone mineral to bone ECF is zero (i.e., k₄=0). The calculation of K_i using Patlak method is simpler than using non-linear regression (NLR) fitting the arterial input function and the tissue time-activity curve data to the Hawkins model. The Patlak method can only measure bone plasma clearance (K_i), and cannot measure the individual kinetic parameters, K₁, k₂, k₃, or k₄.

The concentration of tracer in tissue region-of-interest can be represented as a sum of concentration in bone ECF and the bone mineral. It can be mathematically represented as

${\displaystyle {\frac {C_{bone}(T)}{C_{plasma}(T)}}=K_{i}*{\frac {\int \limits _{0}^{T}C_{plasma}(t)dt}{C_{plasma}(T)}}+V_{o}}$

where, within the tissue region-of-interest from the PET image, C_bone(T) is the bone tissue activity concentration of tracer (in units: MBq/ml) at any time T, C_plasma(T) is the plasma concentration of tracer (in units: MBq/ml) at time T, V_o is the fraction of the ROI occupied by the ECF compartment, and ${\displaystyle \int \limits _{0}^{T}C_{plasma}(t)dt}$ is the area under the plasma curve is the net tracer delivery to the tissue region of interest (in units: MBq.Sec/ml) over time T. The Patlak equation is a linear equation of the form ${\displaystyle Y=m*X+c}$

Therefore, linear regression is fitted to the data plotted on Y- and X-axis between 4–60 minutes to obtain m and c values, where m is the slope of the regression line representing K_i and c is the Y-intercept of the regression line representing V_o.^[55]

The calculation of Ki using arterial input function, time-activity curve, and Hawkins model was limited to a small skeletal region covered by the narrow field-of-view of the PET scanner while acquiring a dynamic scan. However, Siddique et al.^[57] showed in 2012 that it is possible to measure K_i values in bones using static [¹⁸F]NaF PET scans. Blake et al.^[34] later showed in 2019 that the K_i obtained using the Siddique–Blake method has precision errors of less than 10%. The Siddique–Blake approach is based on the combination of the Patlak method,^[55] the semi-population based arterial input function,^[58] and the information that V_o does not significantly change post-treatment. This method uses the information that a linear regression line can be plotted using the data from a minimum of two time-points, to obtain m and c as explained in the Patlak method. However, if V_o is known or fixed, only one single static PET image is required to obtain the second time-point to measure m, representing the K_i value. This method should be applied with great caution to other clinical areas where these assumptions may not hold true.

The most fundamental difference between SUV and K_i values is that SUV is a simple measure of uptake, which is normalized to body weight and injected activity. The SUV does not take into consideration the tracer delivery to the local region of interest from where the measurements are obtained, therefore, affected by the physiological process consuming [¹⁸F]NaF elsewhere in the body. On the other hand, K_i measures the plasma clearance to bone mineral, taking into account the tracer uptake elsewhere in the body affecting the delivery of tracer to the region of interest from where the measurements are obtained. The difference in the measurement of K_i and SUV in bone tissue using [¹⁸F]NaF are explained in more detail by Blake et al.^[36]

It is critical to note that most of the methods for calculating K_i require dynamic PET scanning over an hour, except, the Siddique–Blake methods. Dynamic scanning is complicated and costly. However, the calculation of SUV requires a single static PET scan performed approximately 45–60 minutes post-tracer injection at any region imaged within the skeleton.

Many researchers have shown a high correlation between SUV and K_i values at various skeletal sites.^[59][60][61] However, SUV and K_i methods can contradict for measuring response to treatment.^[48] Since SUV has not been validated against the histomorphometry, its usefulness in bone studies measuring response to treatment and disease progression is uncertain.

An additional advantage of using the metabolic flux parameter Ki from dynamic PET imaging, rather than relying solely on standardized uptake values (SUV) from static scans, is its greater sensitivity to treatment-related changes in bone turnover. In one study, postmenopausal women with low bone mineral density at the spine or hip were treated for six months with teriparatide, a parathyroid hormone analog that stimulates bone formation. Tracer uptake in the lumbar spine was assessed both before and after treatment using two approaches: a kinetic modeling method to derive Ki, and static PET imaging to calculate SUV. The average increase in Ki was 23.8%, compared to only a 3.0% increase in SUV. This discrepancy was attributed to a decrease in plasma tracer concentration following treatment, as a greater proportion of the tracer was absorbed by cortical bone in the peripheral skeleton. Since SUV does not account for changes in tracer input function, Ki provides a more reliable measure of treatment response in such contexts. A follow-up study using similar methods found that treatment effects varied by skeletal site: after 12 weeks of teriparatide, Ki increased by 50.7% at the femoral shaft and 17.8% at the lumbar spine. These imaging results were consistent with changes observed in biochemical markers of bone turnover, which also indicated a peak treatment response around 12 weeks. As a result, it is generally recommended to perform PET imaging assessments of therapeutic efficacy no earlier than 12 weeks after initiating bone-forming treatments.^[62]

^[63]

In [¹⁸F]sodium fluoride PET imaging, the parameter Ki quantifies the net influx rate of fluoride tracer from plasma into the bone mineral compartment and is widely used as a biomarker of bone turnover. It reflects the efficiency with which the tracer is delivered to bone tissue and irreversibly incorporated into the mineral phase. Mathematically, Ki is defined by the equation

Ki = K₁ × k₃ / (k₂ + k₃),

where K₁ is the rate of delivery of the tracer from plasma to bone, k₂ represents the rate at which the tracer returns from bone to plasma, and k₃ corresponds to the rate of tracer binding to bone mineral. This expression captures the proportion of delivered tracer that becomes fixed in the bone matrix, providing a mechanistic representation of local skeletal metabolic activity.

The units of Ki are typically expressed as mL/min/mL, which may be interpreted as the volume of plasma cleared of tracer per minute for each milliliter of bone tissue. For instance, a Ki value of 0.01 mL/min/mL means that each milliliter of bone tissue incorporates fluoride tracer at a rate equivalent to that contained in 0.01 milliliters of plasma every minute. This gives Ki a direct physiological meaning: it measures how much tracer is being deposited in the bone over time relative to the available tracer in circulation. In practical terms, higher Ki values indicate greater bone metabolic activity, which may be due to increased bone formation, remodeling, or both.

Ki is considered more reliable than the standardized uptake value (SUV) when evaluating dynamic changes in bone turnover, particularly in response to treatment. SUV measurements are influenced by changes in systemic tracer distribution and plasma concentration, which can lead to misleading results in cases where treatment alters the overall biodistribution of the tracer. In contrast, Ki accounts for changes in plasma tracer concentration, making it more accurate for detecting true biological changes at the imaging site. This is particularly important in diseases or therapies that affect the skeleton broadly, such as metastatic bone disease, Paget's disease, or systemic osteoporosis treatments.

Clinically, Ki has been shown to increase following anabolic therapies such as teriparatide, reflecting enhanced bone formation, and to decrease with antiresorptive agents like bisphosphonates, which suppress bone turnover. Because Ki captures site-specific remodeling activity, it is valuable for monitoring regional treatment responses, assessing bone quality, and stratifying fracture risk. Its quantitative nature and strong correlation with histomorphometric bone formation rates make Ki a central parameter in the noninvasive assessment of metabolic bone diseases using PET imaging.

In dynamic [¹⁸F]NaF PET imaging, the parameter K₁ quantifies the rate at which fluoride tracer is transferred from the blood plasma into the bone’s extracellular fluid space, reflecting the local blood flow to bone tissue. It is expressed in units of mL/min/mL, indicating how much plasma is cleared of tracer per minute per milliliter of bone tissue. This metric is especially important in assessing regional bone perfusion, which plays a critical role in bone metabolism and remodeling.

The relationship between K₁ and actual blood flow is defined by the equation

K₁ = E × F × (1 − PVC),

where E is the extraction efficiency of the tracer during a single capillary pass, F is the local blood flow, and PVC represents the plasma volume correction accounting for red blood cell volume. The extraction efficiency E itself is derived from the formula

E = 1 − exp(−P×S/F),

where P denotes capillary permeability to the tracer, S is the surface area of the capillary bed, and F is again the blood flow. In bone, fluoride has a high extraction efficiency due to its small molecular size and ability to rapidly diffuse into surrounding tissues, allowing K₁ to closely approximate actual blood flow in low-flow conditions.

Empirical studies have shown that K₁ values below approximately 0.16 mL/min/mL correspond well with bone blood flow values measured by [¹⁵O]H₂O PET, the gold standard for perfusion imaging. However, at higher flow rates, the limited diffusivity of fluoride leads to an underestimation of actual perfusion, as the tracer cannot fully equilibrate with the bone compartment within the short capillary transit time. Despite this limitation, measured K₁ values at metabolically active skeletal sites, such as the lumbar spine, are generally within this reliable range in both healthy individuals and those with bone diseases, supporting its utility as a biomarker for bone perfusion.

K₁ is physiologically meaningful because bone blood flow is intricately linked to bone remodeling. Adequate perfusion supports osteoblast and osteoclast function and regulates the delivery of nutrients and removal of waste products in bone tissue. With aging, reductions in bone blood supply contribute to increased bone loss and the progression of osteoporosis, particularly in trabecular-rich regions. Studies have demonstrated correlations between K₁ and calcium uptake rates, as well as new bone formation in patients with osteoporosis. Although the precise mechanisms connecting perfusion and bone metabolism remain under investigation, impaired vascular regulation—such as reduced nitric oxide or prostaglandin PGI₂ signaling—is believed to play a role in age-related declines in bone blood flow.

Regional differences in K₁ also help explain site-specific variations in bone metabolic activity. For instance, the spine tends to show higher K₁ values compared to the proximal femur or humerus, aligning with the higher metabolic activity and richer red marrow content of the vertebral bones. In contrast, peripheral bones often show reduced K₁ values, partly due to marrow composition changes with age, including the replacement of red marrow with fatty yellow marrow. These findings underscore the role of vascular supply in maintaining bone health and highlight K₁ as a key parameter in evaluating site-specific bone perfusion and metabolism, particularly in aging and disease contexts.

In dynamic [¹⁸F]NaF PET imaging, the parameters k₂ and k₃ represent the rates at which the sodium fluoride tracer moves between the extravascular extracellular fluid (ECF) compartment and other compartments of the bone. Specifically, k₂ describes the rate of efflux, or movement of tracer from the ECF back into the plasma, while k₃ reflects the forward flux of tracer from the ECF into the bone mineral matrix, where it binds to hydroxyapatite crystals. Although the exact physiological interpretation of k₂ and k₃ remains partially unclear, they collectively contribute to understanding the efficiency of fluoride tracer extraction and its incorporation into bone tissue.

These parameters are essential components of compartmental kinetic models used to estimate the net influx rate constant (Ki), which is derived from the equation Ki = K₁ × k₃ / (k₂ + k₃). Thus, k₂ and k₃ play key roles in governing how the tracer is distributed and retained within bone, particularly in the context of metabolic bone diseases and treatment response.

In clinical studies, antiresorptive treatments—such as bisphosphonates—have been shown to increase k₂, likely reflecting reduced tracer retention within bone tissue due to suppressed bone remodeling. Conversely, in high-turnover bone diseases such as Paget’s disease, pathological changes in bone structure significantly alter these kinetic parameters. In Pagetic bone, the extracellular fluid space is often enlarged and structurally complex, contributing to an atypical tracer distribution pattern. Clinical PET studies have observed elevated k₃ values—indicating increased tracer uptake into mineralizing surfaces—alongside decreased k₂ values, suggesting slower return of the tracer to the bloodstream. These patterns reflect the increased bone turnover and altered vascular and cellular environment characteristic of Paget’s disease.

The interpretation of k₂ and k₃ is also influenced by changes in bone architecture. For instance, in pathological bone, marrow space may be replaced with fibrotic tissue, limiting the volume and permeability of the ECF compartment and affecting tracer availability. Since k₂ quantifies how quickly the tracer leaves the ECF, and k₃ measures how efficiently it binds to mineral, the balance between these rates can indicate how tracer is partitioned between temporary residence in fluid and permanent incorporation into bone.

Overall, k₂ and k₃ offer important insight into the microenvironmental dynamics of bone tissue, including cellular activity, vascular function, and mineralization processes. Their values can change significantly in response to both therapeutic interventions and disease-related remodeling, making them useful indicators in the study of bone metabolism under both normal and pathological conditions. Although less frequently discussed than K₁ or Ki, these parameters help refine our understanding of how the tracer is handled within the bone, enhancing the utility of dynamic PET in clinical and research applications.

The parameter k₄ represents the backward rate constant describing the movement of the [¹⁸F]fluoride tracer from the bone mineral compartment back into the extravascular extracellular fluid (ECF). This rate characterizes the reversibility of tracer binding to bone mineral surfaces. Although fluoride ions bind to hydroxyapatite crystals within the bone matrix, not all binding interactions are permanent; some are relatively weak and reversible. This is evidenced by the fact that multiple studies have consistently reported nonzero values for k₄, indicating that a fraction of the tracer can dissociate from the mineralized matrix and re-enter the surrounding fluid compartment. Typically, measured k₄ values are low, around 0.01 min⁻¹, which corresponds to a tracer half-life in the bone mineral compartment of approximately 70 minutes.

Because of the small magnitude of k₄, certain analytical methods—such as the Patlak graphical approach—simplify the model by assuming k₄ = 0. While this assumption facilitates data analysis, it can lead to a systematic underestimation of the Ki parameter by as much as 25%, since Ki reflects the net influx of tracer into bone mineral and disregarding k₄ overlooks the reversible component of binding. Therefore, even though k₄ is small, it can significantly impact the accuracy of bone metabolic flux measurements in dynamic PET studies. Clinical data suggest that k₄ does not vary significantly between skeletal regions such as the lumbar spine and the hip, making it a relatively stable parameter across different bone sites.

From a modeling perspective, k₄ serves a critical role in distinguishing strongly bound tracer (which remains fixed in mineralized bone) from weakly bound or exchangeable tracer that may leave the mineral surface over time. Its value provides insight into the dynamic equilibrium between tracer deposition and release at the bone surface, adding temporal resolution to the interpretation of bone turnover and fluoride retention. As such, accurate estimation of k₄ is essential for refining Ki measurements, especially in advanced kinetic models that go beyond simplified assumptions. In summary, although k₄ is often small, its inclusion in compartmental models enhances the physiological accuracy of PET-based assessments of bone metabolism and tracer kinetics in both clinical and research contexts.

The K1/k2 ratio represents the effective volume of distribution for tracer within the extracellular fluid space. This parameter quantifies the fraction of the skeletal volume of interest that comprises the bone extracellular fluid compartment, based on the assumption of passive fluoride diffusion between plasma and extracellular fluid. This assumption appears reasonable given fluoride's low atomic weight and diffusion properties. The fluoride-18 ion demonstrates the ability to bind with hydrogen atoms, forming electrically neutral hydrogen fluoride, a small and highly diffusible molecule capable of crossing cell membranes. The sodium fluoride tracer within the bone extracellular fluid compartment may exhibit reduced availability for inter-compartmental exchange due to binding interactions with marrow spaces, which limits access to bone mineral surfaces. Empirical measurements indicate K1/k2 values corresponding to approximately 48% of volume of interest in vertebral bone and 34% in humeral bone. These findings align with earlier estimates showing 10% for bone extracellular fluid space alone and 24% for bone marrow extracellular fluid in tibial samples. These observations support the concept that extracellular fluid spaces are more extensive in trabecular-rich bone containing greater marrow content compared to cortical bone. Clinical observations reveal three-fold lower Ki values at the hip compared to the lumbar spine. Beyond differences in K1 related to marrow fat composition variations, the significantly larger K1/k2 values in vertebrae compared to hip may contribute to this difference. The lumbar spine contains more functioning red marrow, providing a greater volume fraction of bone region accessible to fluoride ions, whereas hip bone marrow is predominantly fatty. This suggests a substantially smaller extracellular fluid space at the hip, containing relatively limited tracer available for bone mineral uptake, potentially contributing to lower Ki values at this site. Treatment studies demonstrate decreased K1/k2 ratios at the lumbar spine in osteoporotic subjects following antiresorptive therapy, suggesting reduced bone extracellular fluid space post-treatment. This finding may reflect increased bone mineral density at the site, potentially due to remodeling space filling. Conversely, K1/k2 parameters show significant elevation in Pagetic bone, indicating enlarged bone and non-bone extracellular fluid spaces, which combined with increased delivery and clearance to total bone tissue, contributes to overall enhanced clearance of sodium fluoride tracer to the bone mineral compartment.

The k3/(k2 + k3) ratio quantifies the proportion of tracer transported via osseous blood flow to the extracellular fluid compartment that subsequently undergoes binding within the bone mineral matrix, mathematically equivalent to Ki/K1. Research has demonstrated comparable values for this ratio across lumbar spine and hip regions in healthy postmenopausal populations. Treatment response studies reveal distinct patterns in k3/(k2 + k3) modulation depending on therapeutic mechanism. Anabolic agents like teriparatide elevate this ratio, while antiresorptive compounds such as risedronate reduce it, thereby altering the fraction of tracer achieving specific mineral binding. These differential responses indicate the ratio's potential utility as a treatment efficacy biomarker. The k3/(k2 + k3) parameter demonstrates superior treatment-response-to-precision-error characteristics compared to other Hawkins model components. Given that this ratio directly reflects the tracer fraction undergoing mineral binding and consequently represents treatment effects on bone formation processes, monitoring k3/(k2 + k3) changes emerges as a highly effective approach for non-invasive assessment of bone formation rate alterations through dynamic fluoride-18 sodium fluoride positron emission tomography protocols. This parameter's specificity for mineral-bound tracer, combined with its robust statistical properties and direct correlation with therapeutic outcomes, positions it as a primary endpoint alongside Ki for evaluating bone formation dynamics in clinical and research applications.

Monitoring treatment response in bone diseases requires methods that can detect changes quickly and accurately. Biochemical markers found in blood and urine offer one approach for measuring bone metabolism changes within weeks of starting treatment. These markers fall into two categories: those indicating bone breakdown and those showing bone formation activity.

For bone breakdown, doctors commonly measure serum carboxy-terminal collagen crosslinks and N-terminal telopeptide. For bone formation, they track serum bone-specific alkaline phosphatase and serum procollagen 1 N-terminal propeptide. When patients begin treatment with bone-preserving medications like alendronic acid, the breakdown markers drop rapidly within the first month. However, formation markers take longer to change, typically showing decreases only after 3 to 6 months due to the natural bone renewal cycle.

Sodium fluoride PET scanning provides another method for tracking treatment effects by measuring changes in bone metabolism parameters. Research has examined how different osteoporosis treatments affect these measurements in postmenopausal women. In studies with the bone-preserving drug risedronate, the Ki parameter decreased by approximately 18% after six months of treatment, matching similar decreases seen in blood markers of bone formation. The k3/(k2 + k3) ratio also dropped by about 18%, while k2 values increased significantly.

Conversely, treatment with bone-building medications like teriparatide produced opposite effects. The Ki parameter increased by nearly 24% after six months, with the k3/(k2 + k3) ratio showing similar increases. These changes demonstrate that sodium fluoride PET can effectively detect both bone-preserving and bone-building treatment responses.

The timing for measuring treatment response using 18F-NaF PET is particularly advantageous, with quantitative sodium fluoride PET-CT capable of detecting changes in regional bone formation within approximately 12 weeks after treatment initiation.^[67] This represents a significant advantage over bone mineral density measurements by DXA scanning, which typically require much longer periods to show detectable changes. The ability to assess treatment effectiveness within three months allows clinicians to make earlier decisions about continuing, modifying, or changing therapeutic approaches.

An important finding from these studies involves the comparison between Ki measurements and standardized uptake values. While Ki showed significant changes with teriparatide treatment at the spine, standardized uptake values remained essentially unchanged at this location. However, standardized uptake values did increase significantly at hip locations. This difference highlights why Ki measurements provide more accurate assessments of local bone changes, especially when treatments affect the entire skeleton or when patients have widespread bone disease. In such cases, the total amount of tracer gets distributed across many active bone sites, making standardized uptake values less reliable indicators of true regional changes in bone metabolism.

The measurement accuracy of individual parameters within the Hawkins model, expressed through coefficient of variation values, along with therapeutic response shown as percentage changes from initial measurements to six months following bone anabolic agent teriparatide treatment, were reported in clinical research. Parameters demonstrating substantial therapeutic response combined with minimal measurement error represent the most reliable indicators for assessing treatment effectiveness and need fewer study participants to achieve statistically meaningful differences in clinical trials.

In the referenced teriparatide investigation, only Ki and k3/(k2 + k3) possessed adequate measurement reliability and therapeutic response to provide clinical value. The other parameters (K1, k2, k3, k4, and Fbv) showed measurement errors of approximately 30% or higher, rendering them insufficiently reliable for most research applications.

The measurement precision data reveals significant variability across different parameters. K1 demonstrated a coefficient of variation of 36%, while k2 showed the poorest precision at 52%. Parameter k3 exhibited 28% variation, k4 showed 33%, and Fbv displayed 55% coefficient of variation. In contrast, the k3/(k2 + k3) ratio achieved much better precision at 19%, and Ki demonstrated the best measurement reliability with only 15% coefficient of variation. For comparison, standardized uptake value measurements showed 11% coefficient of variation.^[69]

These precision characteristics explain why Ki and k3/(k2 + k3) emerge as the most practical parameters for clinical monitoring. Their superior measurement consistency, combined with meaningful changes in response to treatment, makes them ideal candidates for tracking therapeutic effectiveness in bone metabolism studies. The high variability in other parameters limits their utility despite their potential physiological significance, highlighting the importance of balancing theoretical relevance with practical measurement capabilities in clinical applications.

> Osteoporosis: [18F]NaF PET has demonstrated utility in evaluating treatment efficacy in osteoporosis by quantifying regional bone turnover. Frost et al. (2013) showed that it provides sensitive detection of treatment response to bone-active agents, especially at the hip, offering a noninvasive biomarker for monitoring therapeutic impact.^[70]

> Paget’s Disease: In Paget’s disease, [18F]NaF PET is effective for assessing disease activity and monitoring response to therapy. Installé et al. (2005) found that fluoride uptake patterns correlate with disease extent and therapeutic changes, making it a reliable imaging tool for disease management.^[71]

> [18F]NaF PET provides insight into altered bone metabolism in CKD-MBD, where standard imaging may fall short. Feng et al. (2021) highlighted its role in evaluating early kinetic changes, enabling noninvasive assessment of skeletal complications in CKD patients.^[72]

> In patients with breast cancer, [18F]NaF PET can assess skeletal metastases with high sensitivity. Azad et al. (2019) showed that measuring fluoride metabolic flux offers superior assessment of treatment response compared to conventional SUV measurements, improving clinical decision-making.^[73]

> [18F]NaF PET/CT aids in evaluating disease activity in rheumatoid arthritis by detecting bone remodeling and joint involvement. Park et al. (2021) demonstrated its utility in estimating inflammatory burden and tracking treatment response, particularly when traditional markers are inconclusive.^[74]

> [18F]NaF PET is valuable in assessing fracture healing, including challenging cases like atypical femoral shaft fractures. It helps distinguish between viable and nonviable bone, enabling early detection of impaired healing and guiding orthopedic management.^[75]

> Joshi et al. (2014) demonstrated that [18F]NaF PET can identify active, high-risk coronary plaques that are prone to rupture — outperforming traditional imaging by detecting microcalcification activity.^[76]

> Syed et al. (2022) showed that [18F]NaF PET/CT detects disease activity in acute aortic syndromes, providing critical insight into vascular wall remodeling and potential instability in real time.^[77]

> Fibrodysplasia Ossificans Progressiva (FOP): [18F]NaF PET/CT can detect early heterotopic ossification in FOP. Eekhoff et al. (2018) found it effective in identifying subclinical disease activity before structural changes appear on conventional imaging.^[78]

> Osteonecrosis of the Jaw (ONJ): In patients receiving bisphosphonates or antiresorptive therapies, [18F]NaF PET helps detect osteonecrosis of the jaw early. Raje et al. (2008) showed that it offers valuable metabolic and structural information, aiding in the diagnosis and monitoring of ONJ in cancer patients.^[79]

• Bone
• Positron emission tomography
• Time-activity curve
• Arterial input function
• Medical Imaging
• Radiology
• Molecular Imaging
• Medical Imaging
• Bone scintigraphy