Multiparametric Cardiac ¹⁸F-FDG PET in Humans: Pilot Comparison of FDG Delivery Rate with ⁸²Rb Myocardial Blood Flow

Fourteen patients with ischemic heart disease (IHD) or suspected cardiac sarcoidosis were referred for PET myocardial viability or inflammation assessment by PET and enrolled into this study after giving informed consent. The study is approved by Institutional Review Board at the University of California, Davis. Each patient underwent a dynamic ⁸²Rb-PET/CT scan first and then a dynamic FDG-PET/CT scan, both scans operated on a GE Discovery ST PET/CT scanner in two-dimensional mode. The time between the Rb scan and FDG scan ranges from a few minutes to one hour, depending on the oral glucose loading procedure.

For dynamic Rb-PET imaging, patients received approximately 30 mCi ⁸²Rb-chloride with a bolus injection. A low-dose transmission CT scan was performed at the beginning of PET scan to provide CT images for PET attenuation correction. The dynamic scan lasted for nine minutes. The acquired raw data were binned into 16 dynamic frames: 12 $\times$ 10 s, 2 $\times$ 30 s, 1 $\times$ 60 s, 1 $\times$ 300 s. Dynamic Rb-PET images were reconstructed using the standard ordered-subset expectation-maximization (OSEM) algorithm. All data corrections, including normalization, dead-time correction, attenuation correction, scatter correction, and random correction, were included in the reconstruction process.

For dynamic FDG-PET imaging, patients received approximately 20 mCi ¹⁸F-FDG with a bolus injection. Data acquisition was commenced right after the FDG injection and lasted for 60 minutes. The acquired raw data were then binned into a total of 49 dynamic frames: 30 $\times$ 10 s, 10 $\times$ 60 s and 9 $\times$ 300 s. Other processing was the same as for the Rb-PET scans.

An ellipsoidal region of interest (ROI) was manually placed in the left ventricle (LV) to extract an image-derived input function $C_{\mathrm{IDIF}}(t)$. An additional 17 ROIs were placed within the 17 segments of myocardium according to the AHA-17 standard [35]. These segment ROIs were combined into a global myocardial ROI and used to extract a global myocardial TAC $C_{T}(t)$ using ROI mean. Due to high noise of the dynamic data in individual segments, the analysis of this study was focused on global myocardial quantification. Based on the analysis in our previous work [31], a reversible two-tissue compartmental model [23] was used to model the one-hour dynamic FDG-PET data:

	$\displaystyle\frac{dC_{1}(t)}{dt}$	$\displaystyle=$	$\displaystyle K_{1}C_{p}(t)-(k_{2}+k_{3})C_{1}(t)+k_{4}C_{2}(t),$		(1)
	$\displaystyle\frac{dC_{2}(t)}{dt}$	$\displaystyle=$	$\displaystyle k_{3}C_{1}(t)-k_{4}C_{2}(t),$		(2)

where $C_{p}(t)$ is the FDG concentration in the plasma, $C_{1}(t)$ is the concentration of free FDG and $C_{2}(t)$ denotes the activity concentration of metabolized tracer in the myocardium tissue space. $K_{1}$ is the tracer delivery rate from the blood space to the tissue space with the unit mL/min/cm³[36]; $k_{2}$ (/min) is the rate constant of tracer exiting the tissue space; $k_{3}$ (/min) is phosphorylation rate; $k_{4}$ (/min) is the dephosphorylation rate. With $v_{b}$ denoting the fractional blood volume and $C_{wb}(t)$ denoting the activity in the whole blood, the total activity that is measured by PET is described by

$$C_{T}(t)=(1-v_{b})[C_{1}(t)+C_{2}(t)]+v_{b}C_{wb}(t).$$

(3)

The unknown kinetic parameters ($v_{b}$, $K_{1}$, $k_{2}$, $k_{3}$, $k_{4}$) were estimated using nonlinear least-square curve fitting. Their initial values were set to (0.1,0.1,0.1,0.1,0.001). The lower bounds were zero and the upper bounds were set to (1.0,2.0,2.0,1.0,0.1) [31].

In theory the IDIF represents the activity in the whole blood, i.e., $C_{\mathrm{IDIF}}(t)=C_{wb}(t)$. The plasma input function $C_{p}(t)$ is related to $C_{wb}(t)$ following the model

$$C_{p}(t)=\mathrm{PBR}(t)\cdot C_{wb}(t),$$

(4)

where $\mathrm{PBR}(t)$ denotes the plasma-to-blood ratio (PBR) function. In this work, we first used $\mathrm{PBR}(t)=1$ by assuming the difference between $C_{p}(t)$ and $C_{wb}(t)$ is small [32]. Unless specified otherwise, the main results of this paper were obtained with $\mathrm{PBR}(t)=1$. In addition, we investigated the effect of PBR correction using a recent nonlinear model [33],

$$\mathrm{PBR}(t)=1/[0.97-0.06\exp(-0.085t)].$$

(5)

We describe the relation between FDG $K_{1}$ and the blood glucose level $C_{\mathrm{glu}}$ using a linearized approximation,

$$K_{1}\triangleq f(C_{\mathrm{glu}})\approx f(C_{\mathrm{glu},0})+\dot{f}(C_{\mathrm{glu},0})\cdot\left(C_{\mathrm{glu}}-C_{\mathrm{glu},0}\right),$$

(6)

where $f(\cdot)$ is a nonlinear function of $C_{\mathrm{glu}}$ determined from the Renkin-Crone (RC) model and Michaelis-Menten model [23], see Eq. (19) in the Appendix. $C_{\mathrm{glu},0}$ denotes a reference blood glucose level and $\dot{f}(\cdot)$ is the first-order derivative of $f$. Based on the derivations in the Appendix I, we have

$$\dot{f}(C_{\mathrm{glu},0})<0.$$

(7)

This suggests FDG $K_{1}$ is inversely correlated with the blood glucose level,

$$K_{1}\approx-s\cdot C_{\mathrm{glu}}+int,$$

(8)

where $s\triangleq-\dot{f}(C_{\mathrm{glu},0})$ denotes the absolute slope ($s>0$) and $int$ is the intercept of the plot of FDG $K_{1}$ with respect to $C_{\mathrm{glu}}$. Practically, $s$ can be estimated using a correlation analysis between the measured FDG $K_{1}$ and $C_{\mathrm{glu}}$ data.

To reduce the dependency of FDG $K_{1}$ on $C_{\mathrm{glu}}$, we consider $f(C_{\mathrm{glu},0})$ as an adjusted FDG $K_{1}$. Based on Eq. (6), it can be calculated using the following form

$$K_{1}^{\mathrm{adj}}\triangleq f(C_{\mathrm{glu},0})=K_{1}+s\cdot\left(C_{\mathrm{glu}}-C_{\mathrm{glu},0}\right),$$

(9)

in which the blood glucose level of a patient is normalized and used to adjust the value of FDG $K_{1}$ linearly. A similar linear correction was also used by Stout et al. [37] for reducing the effects of normal physiological concentration of plasma large neutral amino acids on the $K_{1}$ of 6-[F-18]Fluoro-L-3,4-dihydroxyphenylalanine (FDOPA). As normal blood glucose levels range between 80-120 mg/dL, we use $C_{\mathrm{glu},0}=100$ mg/dL in this work.

Note that Eq. (9) is an additive adjustment, in which $s$ is fixed and is independent of individual patients. Alternatively, a varying $s$, denoted as $s^{\prime}$, may be used on patient basis,

$$s^{\prime}=\frac{K_{1}}{C_{\mathrm{glu},0}},$$

(10)

which results in a multiplicative adjustment,

$$K_{1}^{\mathrm{adj}}=\frac{C_{\mathrm{glu}}}{C_{\mathrm{glu},0}}K_{1}.$$

(11)

This approach assumes the product of $K_{1}$ and $C_{\mathrm{glu}}$ remains constant across different glucose levels.

Similar to the analysis of dynamic FDG-PET data described above, ROIs were drawn in the LV cavity and myocardium regions to extract regional TACs from the dynamic Rb-PET images. The myocardial TAC was modeled using a one-tissue compartmental model [11] with the following expression:

$$C_{T}(t)=(1-v_{b})C_{1}(t)+v_{b}C_{wb}(t),$$

(12)

where $C_{1}(t)$ denotes the concentration of ⁸²Rb in the tissue compartment with $K_{1}$ and $k_{2}$ now representing the rate constants of ⁸²Rb transport between the plasma space and tissue space. This one-tissue model is equivalent to the two-tissue model in Eq. (2) with $k_{3}=0,k_{4}=0$. All the ⁸²Rb kinetic parameters ($v_{b}$, $K_{1}$, $k_{2}$) were estimated using nonlinear least-square curve fitting in a way similar to [31].

The estimated Rb $K_{1}$ was then converted into myocardial blood flow $\mathrm{MBF}$ using the generalized RC model following Lortie’s formula for ⁸²Rb-PET data [11]:

$$K_{1,\mathrm{RB}}=\mathrm{MBF}\cdot\left[1-a_{\mathrm{RB}}\cdot\exp\left(-\frac{b_{\mathrm{RB}}}{\mathrm{MBF}}\right)\right]$$

(13)

where $a_{\mathrm{RB}}=0.77$ and $b_{\mathrm{RB}}=0.63$ (mL/min/cm³). MBF is derived from Rb $K_{1}$ through a look-up table that is pre-generated using the model.

Similar to the conversion Eq. (13) for ⁸²Rb-chloride, we can apply the generalized RC model to fit the FDG $K_{1}$ and MBF data under the assumption that capillary recruitment is involved at higher flow[12] (see Appendix II for further explanation),

$$K_{1,\mathrm{FDG}}=\mathrm{MBF}\cdot\left[1-a_{\mathrm{FDG}}\cdot\exp\left(-\frac{b_{\mathrm{FDG}}}{\mathrm{MBF}}\right)\right],$$

(14)

where $a_{\mathrm{FDG}}$ and $b_{\mathrm{FDG}}$ are to be estimated from the paired data of FDG $K_{1}$ and Rb MBF using nonlinear least square fitting. The initial values for the two parameters $a_{\mathrm{FDG}}$ and $b_{\mathrm{FDG}}$ were both set to 0.1. Note that if $a_{\mathrm{FDG}}$ is fixed at 1.0, then the model is equivalent to the classic RC model without capillary recruitment.

The first-pass extraction fraction of FDG is defined by

$$E_{\mathrm{FDG}}\triangleq\frac{K_{1,\mathrm{FDG}}}{\mathrm{MBF}}.$$

(15)

Theoretically it relates to MBF following the generalized RC model, i.e.,

$$E_{\mathrm{FDG}}=1-a_{\mathrm{FDG}}\cdot\exp\left(-\frac{b_{\mathrm{FDG}}}{\mathrm{MBF}}\right).$$

(16)

The effect of flow-dependent FDG extraction fraction can be corrected using the inverse function of Eq. (14), leading to more quantitative FDG-derived MBF from FDG $K_{1}$. Similar to ⁸²Rb-chloride, the nonlinear conversion from FDG $K_{1}$ to MBF is performed through a look-up table using Eq. (14) with predetermined $a_{\mathrm{FDG}}$ and $b_{\mathrm{FDG}}$. The EFC was separately performed for the original FDG $K_{1}$ (i.e., without glucose normalization) and the adjusted FDG $K_{1}$ (i.e., with glucose normalization).

We used the Pearson’s correlation analysis and/or the Spearman correlation when appropriate to analyze the potential correlation between FDG $K_{1}$ and MBF and biological variables such as age (years), body mass index (BMI) (kg/m²), and blood glucose (BG) level (mg/dL). A p value $\leq$0.05 was considered as statistically significant. All the analyses were done using MATLAB (MathWorks, MA). The Bland-Altman plot [34] was used to quantitatively compare FDG MBF with Rb MBF. When appropriate, the estimated standard error (SE) of a parameter estimate $x$ was also reported in the format of $x\pm$SE.

Patient characteristics are provided in table 1. Among the fourteen patients enrolled in the study, ten were diagnosed as ischemic heart disease and four were diagnosed with or suspected of cardiac sarcoidosis prior to the scans. All of the patients completed the dynamic ⁸²Rb-PET scan. Twelve patients had a dynamic FDG-PET scan of 50-60 minutes. Two other patients had a dynamic FDG-PET scan of only 30-40 minutes due to discomfort, and hence these two subjects were not included in this study.

Other characteristics of the patients, including age, sex, diabetic status, BMI, BG level (before PET imaging), and dynamic FDG-PET scan duration are also reported in table 1. Unavailable data is marked with ‘/’ in the table.

Fig. 1 (a) and (b) show an example of myocardial TAC fitting for the FDG data and Rb data, respectively. The fits demonstrated a good match between the measured time points and predicted TAC by the model in each case. The estimated ¹⁸F-FDG kinetic parameters and ⁸²Rb-chloride MBF values of all the patients are summarized in table 2. For the FDG protocol, the results of two patients were not available due to an incomplete dynamic FDG scan. The average SE values across the twelve patients are reported in table 3 for each of the FDG kinetic parameters. The result indicates that FDG $K_{1}$ was able to be estimated with a low SE ($<14\%$).

Table 4 summarizes the Pearson’s r and p values between FDG $K_{1}$ and the biological variables including age, BMI, and BG level. The results of MBF were also included in the table for comparison. The estimated fractional blood volume $v_{b}$ of FDG was not exactly equal to the $v_{b}$ of Rb as shown in table 2 but the two parameters were correlated with each other (r=0.696, p=0.012).

FDG $K_{1}$ did not correlate with age and BMI. MBF tended to inversely correlate with BMI (r=-0.565, p=0.055). FDG $K_{1}$ tended to inversely correlate with BG (r=-0.562, p=0.091) as also shown in Fig. 2(a), while a similar trend was not observed for MBF (r=-0.076, p=0.835). The negative correlation between FDG $K_{1}$ and BG is in line with the derivation in Eq. (8).

The estimated slope $s$ between FDG $K_{1}$ and BG was 0.0057$\pm$0.0030, with which the original FDG $K_{1}$ was then adjusted for blood glucose levels using the additive adjustment Eq. (9) with a reference $C_{\mathrm{glu},0}=100$ mg/dL. The FDG $K_{1}$ after the adjustment is included in table 2. The multiplicative adjustment was also implemented and the result is included in table 2 as well. Note that the adjustments were not applied if a BG value was unavailable. Therefore the two patients (#6 and #12) were not included in the subsequent analysis. The linear dependency of FDG $K_{1}$ on blood glucose diminished with glucose normalization. Fig. 2(b-c) show that the negative correlation was no longer existing (by the additive adjustment) or reduced (by the multiplicative adjustment) between the adjusted FDG $K_{1}$ and blood glucose levels. These three sets of FDG $K_{1}$ were all used in the subsequent analysis.

Fig. 3 shows the nonlinear association of FDG $K_{1}$ with MBF before and after the adjustment of FDG $K_{1}$ for blood glucose. The Spearman correlation was 0.89 (p=0.0014) before the adjustment and became 0.96 (p$<$0.0001) after the additive or multiplicative adjustment. For the three cases, the data were fitted using the generalized RC model Eq. (14). The estimated model parameters were $a_{\mathrm{FDG}}=0.73\pm 0.15$ and $b_{\mathrm{FDG}}=0.41\pm 0.21$ for the approach without adjustment, $a_{\mathrm{FDG}}=0.80\pm 0.10$ and $b_{\mathrm{FDG}}=0.63\pm 0.11$ for the approach with the additive adjustment, and $a_{\mathrm{FDG}}=0.68\pm 0.09$ and $b_{\mathrm{FDG}}=0.39\pm 0.14$ for the approach with the multiplicative adjustment.

Using the fitted model for extraction fraction correction, FDG $K_{1}$ was converted to MBF in each case based on Eq. (14). Fig. 4 shows the results of linear Pearson correlation between FDG-derived MBF and Rb MBF. The three correlations were statistically significant (p$<$0.01). The correlation coefficient was moderate (r=0.79) without adjusting FDG $K_{1}$ for blood glucose, and became improved (r$>$0.9) after the additive or multiplicative adjustment. Fig. 5 furthers shows the Bland-Altman plots of the blood flow estimates. The differences between FDG MBF and Rb MBF were large without a glucose adjustment and became smaller with the adjustments.

Table 5 further compares the statistics (mean and standard deviation) of Rb MBF and FDG-derived MBF in the ischemic heart disease (IHD) group (9 patients) and non-IHD group (3 patients). The mean and standard deviation became closer to that of Rb MBF after the additive adjustment of FDG $K_{1}$ for blood glucose in both groups.

Fig. 6 shows the extraction fraction of FDG in the myocardium as a function of MBF using the FDG $K_{1}$ estimates with and without adjustment for blood glucose levels. Both the measured $E_{\mathrm{FDG}}$ using Eq. (15) and the calculated model Eq. (16) are shown. The theoretical extraction fraction of Rb-chloride in the myocardium [11] is also included for comparison. The results show that the relation of the extraction fraction of FDG with respect to MBF was close to that of Rb-chloride in a range of MBF between 0-2 mL/min/cm³. It seems that the additive adjustment approach led to a slightly better fit to the generalized RC model than the multiplicative adjustment did. Overall the FDG extraction fraction was about 60% relative to Rb MBF at 1.0 mL/min/cm³, while Marshall et al [38] reported an extraction fraction of 70% in rabbit heart.

As shown in table 2, some of the FDG $k_{2}$ estimates hit the upper bound (UB) 2.0. With a higher UB, the $k_{2}$ estimates can go higher, which in turn increases the $K_{1}$ value due to the coupling between $K_{1}$ and $k_{2}$. Fig. 7(a) shows the examples with two UB values (2.0 and 4.0) for fitting a myocardial TAC. Both options fitted the TAC reasonably well. UB=4.0 provided a lower fitting error (mainly in the early time), which however may be because of over-fitting of the noise. Fig. 7(b) further shows the effect of the $k_{2}$ UB on the Spearman correlation of FDG $K_{1}$ with Rb MBF in all patients. The correlation reached the peak at UB=2.0 and became lower at higher UB values. The use of the upper bound can be explained as adding a regularization in TAC fitting, which has the role of preventing over-fitting and stabilizing the kinetic estimation.

All the results reported above were obtained with $\mathrm{PBR}(t)=1$, i.e. no PBR correction was used. The effect of using the PBR correction in Eq. (5) is shown in Fig 8(a). Here the FDG $K_{1}$ was obtained with the additive adjustment. The FDG $K_{1}$ with the PBR correction was related to the PBR-uncorrected $K_{1}$ by a scaling factor $0.911\pm 0.002$. The result is consistent with the past studies that reported an average PBR of approximately 1.1 (see the references in [33]), which in turn should result in a global scaling of 0.91 in the $K_{1}$ estimates. Fig 8(b) shows the fit of the PBR-corrected FDG $K_{1}$ with respect to Rb MBF using the generalized RC model. The estimated FDG extraction fraction shown in Fig 8(c) was lower when comparing to Fig 6(b). However, the resulting change (not shown) on the final FDG-derived MBF is negligible because the scaling factor was compensated in the generalized RC model.

Note that the lower FDG extraction fraction after the PBR correction may be not reflecting the ground truth because ideally the Rb data should also be corrected for PBR. However, it is practically difficult provided that there is no published model to use.

The relation between FDG $K_{1}$ and blood flow $F$ can be described by the Renkin-Crone (RC) model without a specific consideration of capillary recruitment [23],

$$K_{1}=F\cdot\left[1-\exp\left(-\frac{PS}{F}\right)\right]\triangleq RC(F;PS),$$

(17)

where $P$ denotes the permeability of FDG and $S$ is the surface area of a given section of the capillary bed. Here, the RC model is also denoted as a function of $F$ and $PS$ for convenient use. Following the classic Michaelis-Menten model, the product $PS$ relates to the blood glucose concentration $C_{\mathrm{glu}}$ following the form (Eq. 11 of [23], Eq. 8 of [48], Eq. 2 of [49]),

$$PS=\frac{V_{\mathrm{m},\mathrm{FDG}}}{K_{\mathrm{m},\mathrm{FDG}}+\frac{K_{\mathrm{m},\mathrm{FDG}}}{K_{\mathrm{m},\mathrm{glu}}}C_{\mathrm{glu}}}\triangleq M\!M(C_{\mathrm{glu}}),$$

(18)

where $V_{\mathrm{m},\cdot}$ is the maximal rate of FDG (or glucose) transfer and $K_{\mathrm{m}}$ denotes the concentration of FDG (or glucose) at which the half-maximum transfer rate is reached.

Based on the above two models, FDG $K_{1}$ can be further expressed as a direct function of $C_{\mathrm{glu}}$, i.e.,

$$K_{1}\triangleq f(C_{\mathrm{glu}})=RC\Big{(}F;M\!M(C_{\mathrm{glu}})\Big{)}.$$

(19)

By expanding $K_{1}$ at a reference blood glucose level $C_{\mathrm{glu},0}$ using the first-order Taylor’s expansion, we will have the following linear approximation,

$$K_{1}\approx f(C_{\mathrm{glu},0})+\dot{f}(C_{\mathrm{glu},0})\cdot\left(C_{\mathrm{glu}}-C_{\mathrm{glu},0}\right),$$

(20)

where $\dot{f}(C_{\mathrm{glu},0})$ is the first-order derivative of $f$ at $C_{\mathrm{glu},0}$,

$$\dot{f}(C_{\mathrm{glu},0})=\left(\frac{\partial K_{1}}{\partial{PS}}\cdot\frac{\partial PS}{\partial{C_{\mathrm{glu}}}}\right)_{C_{\mathrm{glu}}=C_{\mathrm{glu},0}}.$$

(21)

From Eq. (17), we have

$$\frac{\partial K_{1}}{\partial{PS}}=\frac{\partial RC(F;PS)}{\partial{PS}}=\exp\left(-\frac{PS}{F}\right)>0,$$

(22)

and from Eq. (18), we have

$$\frac{\partial PS}{\partial{C_{\mathrm{glu}}}}=\frac{\partial M\!M(C_{\mathrm{glu}})}{\partial{C_{\mathrm{glu}}}}=-\frac{V_{\mathrm{m},\mathrm{FDG}}\frac{K_{\mathrm{m},\mathrm{FDG}}}{K_{\mathrm{m},\mathrm{glu}}}}{\left(K_{\mathrm{m},\mathrm{FDG}}+\frac{K_{\mathrm{m},\mathrm{FDG}}}{K_{\mathrm{m},\mathrm{glu}}}C_{\mathrm{glu}}\right)^{2}}<0.$$

(23)

Combing the above two relations together produces

$$\dot{f}(C_{\mathrm{glu},0})<0.$$

(24)

Thus, the approximate linear relation of FDG $K_{1}$ with the blood glucose level $C_{\mathrm{glu}}$ can be re-expressed as

$$K_{1}\approx-s\cdot C_{\mathrm{glu}}+int,$$

(25)

where the absolute slope $s$ and the intercept $int$ relate to the reference blood glucose level $C_{\mathrm{glu},0}$ via

$$s=-\dot{f}(C_{\mathrm{glu},0})>0,\quad int=f(C_{\mathrm{glu},0})+s\cdot C_{\mathrm{glu},0}.$$

(26)

Following the hypothesis of Yoshida et al. [12] for ⁸²Rb and ¹³N-ammonia that capillary recruitment occurs at high coronary flows, we assume the PS product of FDG, is not a constant but depends on blood flow $F$ in a multicapillary system,

$${\color[rgb]{0,0,0}PS^{\prime}=PS+\kappa\cdot F,}$$

(27)

where $PS^{\prime}$ denotes the total PS product, $PS$ is the resting PS product, and $\kappa$ is a multiplicative coefficient.

Substituting the above model into the original RC model in Eq. (17), we then have the following generalized Renkin-Crone (GRC) model for FDG,

	$\displaystyle K_{1}$	$\displaystyle=$	$\displaystyle F\cdot\left[1-\exp\left(-\frac{PS+\kappa\cdot F}{F}\right)\right]$		(28)
		$\displaystyle=$	$\displaystyle F\cdot\left[1-a_{\mathrm{FDG}}\cdot\exp\left(-\frac{PS}{F}\right)\right]{\color[rgb]{0,0,0}\triangleq GRC(F;PS)},$

where

$$a_{\mathrm{FDG}}=\exp(-\kappa){\color[rgb]{0,0,0}>0}.$$

(29)

The GRC model is equal to Eq. (14) by setting $b_{\mathrm{FDG}}=PS$. The coefficients $a_{\mathrm{FDG}}$ and $b_{\mathrm{FDG}}$ will be estimated by fitting the FDG $K_{1}$ and myocardial blood flow data.

Note that with the GRC model, we can also derive the linearized relation between FDG $K_{1}$ and $C_{\mathrm{glu}}$ (Eq. (25)) following the derivation in Appendix I for the RC model. The major change is that the term $\exp\left(-\frac{PS}{F}\right)$ in Eq. (22) should be replaced by $a_{\mathrm{FDG}}\cdot\exp\left(-\frac{PS}{F}\right)$. The resulting $\frac{\partial K_{1}}{\partial{PS}}$ remains positive because $a_{\mathrm{FDG}}>0$ as shown in Eq. (29).