PET for bone imaging

The two image in the top row (the image on the left hand side is plotted on log scale and on the right hand side is plotted on linear scale) show the output of the spectral analysis showing its frequencies components grouped around three clusters, referred to as high, intermediate and low frequencies, supporting the assumption of three compartments in the Hawkins model corresponding to plasma, bone ECF and bone mineral compartment respectively. The image at the bottom row shows the IRF plotted using the frequency components obtained previously.

The standardized uptake value (SUV) is defined as tissue concentration (KBq/ml) divided by activity injected normalized for body weight.[35]

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,[36] cheap, and quicker to perform, making it more attractive for clinical use. It has been used in diagnosing and assessing the efficacy of therapy.[37][38] 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.[32]

The SUV has emerged as a clinically useful, albeit controversial, semi-quantitative tool in PET analysis.[39] Standardizing imaging protocols and measuring the SUV at the same time post-injection of the radiotracer, is necessary to obtain a correct SUV[40] because imaging before the uptake plateau introduces unpredictable errors of up to 50% with SUVs.[41] 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.[42][43] 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.[44][45]

The method was first described by Cunningham & Jones[46] 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.[47] The method was used by numerous other studies.[48][49][50] 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.

A diagrammatic view of the process of kinetic modelling using Hawkins model used to calculate the rate of bone metabolism at a skeletal site. C_p refers to the plasma concentration of the tracer, C_e refers to the tracer concentration in ECF compartment, C_b refers to the concentration of tracer in bone mineral compartment, M1 refers to mass of tracer in the C_e compartment, M2 refers to the mass of tracer in the C_b compartment, C_T is the total mass in the C_e+C_b, PVE refers to the partial volume correction, FA refers to the femoral artery, ROI refers to region of the interest, B-Exp refers to the bi-exponential, .

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.[22] 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.[51]

Patlak method[52] 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. It is crucial to note that 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 ${\displaystyle Y=m*X+c}$

Patlak analysis where a linear regression is fitted between the data on y- and x-axis to obtain the estimates of the Ki, which is the slope of the fitted regression line.

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.[52]

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.[53] showed in 2012 that it is possible to measure K_i values in bones using static [¹⁸F]NaF PET scans. Blake et al.[32] 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,[52] the semi-population based arterial input function,[54] 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.