Introduction

Positron emission tomography (PET) is increasingly used in animal models of various human diseases. In particular, the 2-deoxy-2-[18F]fluoro-d-glucose ([18F]FDG) PET technique that has been used for the in vivo measurement of local glucose utilization in man is playing a pivotal role in rodent models and transgenic mice to study human diseases and tissue responses due to various drug treatments and therapies. Absolute quantification of glucose utilization rate in a local tissue requires either an operational equation calculated along with a predefined set of rate constants when only a single static scan is performed at a specific time postinjection [1] or compartmental/graphical analysis on the time course of tissue kinetics acquired with dynamic imaging [2, 3]. Inherent in these methods is an invasive and tedious blood sampling procedure to measure the [18F]FDG activity concentration in arterial plasma at timed intervals during the uptake/imaging period to form the input function (IF) for kinetic analysis [4]. While direct blood sampling is considered the reference standard for determination of the IF, the procedure is technically challenging in rodents because of the small caliber of blood vessels and the limited total blood volume.

Several alternative methods for the derivation of IF have been proposed for [18F]FDG PET studies in humans and small animals. These methods include arteriovenous shunting [5, 6], β-microprobe measurement [7, 8], automatic micro-blood-sampling system [9], population-averaged measurement [10, 11], image-derived measurement from a large vascular structure [12–17], model-based estimation [18–20], and a combination of the last two approaches [21–23]. Although the use of arteriovenous shunts [5, 6] or automatic micro-blood-sampling system [9] may allow minimization of blood loss and standardization of blood sampling protocol, they require a time-consuming and invasive surgical procedure to implant the catheters, making these techniques not possible for use in repetitive study on the same animal. Likewise, the β-microprobe technique [7, 8] has similar issues as an invasive microsurgery is needed to expose the blood vessel so that the β-microprobe can be positioned inside or in direct contact with the blood vessel. The size of the β-microprobe, which needs to be small when comparing to the blood vessel, has also limited the technique to be applicable to rat studies so far [7]. Population-based method [10, 11] calibrates a standardized IF obtained from a large population by one to two blood samples for an individual by assuming that the shape of the IF is identical among subjects and groups. However, the shape discrepancies in the IF resulting from different injection rates or study conditions can cause erroneous estimation of physiological parameters. Repeated measurements in a group of subjects are required for constructing a new IF template whenever a new tracer or different injection schedule is used or when a different physiological condition of the animals is studied. Derivation of IF from region-of-interest (ROI) defined manually over a vascular structure on the PET images [12–14] or by means of multivariate statistical approaches [15–17] has also been investigated. Due to the limited spatial resolution of PET and the small size of the vascular ROI, the image-derived IF is prone to partial-volume and spillover effects, and its noise level cannot be assumed to be negligible for kinetic modeling. In this context, a model-based approach such as simultaneous estimation (SIME) method [18, 19] has been proposed to simultaneously estimate, with the aid of one or two late-time blood samples for scaling purpose, the kinetic parameters in multiple tissue ROIs, and the underlying plasma IF, which can be parameterized by a mathematical model. This approach has recently been extended to small-animal [18F]FDG PET studies [21–23] with reduced number of parameters to be estimated and improved identifiability of the estimated IF using an image-derived time-activity curve (TAC) from the left ventricle (LV).

Nevertheless, it has been shown in all those studies that at least one blood sample is required to scale the estimated IF and determine the plasma glucose level. Therefore, taking a blood sample appears to be unavoidable for [18F]FDG PET. However, we speculate that pharmacokinetic parameters equivalent to data from full blood sampling can be obtained if one can utilize this single blood sample measurement and take other physiological information into account. We have recently shown in mice that [18F]FDG clearance rate in plasma is strongly correlated with [18F]FDG accumulation rate in urinary bladder [24]. In this study, we propose a hybrid modeling approach that based partially on the renal physiology to estimate the plasma curve at late times by use of the total accumulated radioactivity in the urinary bladder and the blood sample collected at the end of the PET experiment, and we validate it with dynamic mouse [18F]FDG PET data. The results of IF estimation are expected to be improved when comparing to those using only one blood sample drawn at the end of the PET study as extra physiological information is incorporated into the estimation process. The use of liver TAC as an alternative IF is also investigated, and the results are compared with those obtained with the proposed hybrid approach and the gold standard method of arterial blood sampling.

Materials and Methods

Theory

Unlike glucose, [18F]FDG is largely excreted by the kidneys without being resorbed by the renal proximal tubules and is continuously accumulated in the urinary bladder [25]. Let C W(t) be the radioactivity concentration of [18F]FDG in whole-blood, k F be the glomerular filtration rate of kidney, f u(t) be the time-varying urine flow, and Q B(t) be the total accumulated activity of [18F]FDG at time t in the urinary bladder of volume V B(t). Fig. 1 illustrates a simplified compartmental model configuration that describes the tracer kinetics in the renal system. Here, the renal tubules system of the kidney is modeled as a single-exponential dispersion kernel, D(t), given by:

Fig. 1
figure 1

A simplified compartmental model for [18 F]FDG excretion in the renal system.

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$$ D(t)=\frac{1}{\tau }{e^{{-(t-\Delta )/\tau }}} $$
(1)

where Δ and τ represent, respectively, the time shift and the dispersion for the blood to transit across the kidney. According to the model configuration of Fig. 1, the radioactivity concentration of [18F]FDG in urine at the outflow of kidney, C u, is given by:

$$ {C_{\mathrm{u}}}(t)={k_{\mathrm{F}}}D(t)\otimes {C_{\mathrm{W}}}(t) $$
(2)

where \( \otimes \) denotes convolution integration. To simplify the renal modeling procedure and to improve parameter identifiability, it is assumed that the urine flow (f u) associated with the accumulation of [18F]FDG in the urinary bladder is proportional to the rate of change of bladder volume (dV B/dt) and can be empirically modeled by a one-exponential function, i.e., \( {V_{\mathrm{B}}}(t)=A\left( {1-{e^{{\text{-}\upomega \mathrm{t}}}}} \right) \) and \( {{{d{V_{\mathrm{B}}}}} \left/ {dt } \right.}={f_{\mathrm{u}}}(t)=A\omega {e^{{\text{-}\upomega \mathrm{t}}}} \), where A and ω are determined by fitting V B to the bladder volume measurements. The following inhomogeneous linear differential equation can be derived for Q B(t):

$$ \frac{d}{dt }{Q_{\mathrm{B}}}(t)={f_{\mathrm{u}}}(t){C_{\mathrm{u}}}(t) $$
(3)

With an intravenous tracer administration, the tracer kinetics observed in blood can generally be described by sum-of-exponentials models [26], and the [18F]FDG clearance in whole-blood can be assumed to follow a one-exponential function at late times:

$$ {C_{\mathrm{W}}}(t)={C_{\mathrm{W}}}\left( {{t_{\mathrm{m}}}} \right){e^{{\lambda \left( {{t_{\mathrm{m}}}-t} \right)}}} $$
(4)

where λ (per minute) represents the clearance rate of [18F]FDG in whole-blood at late times and \( {C_{\mathrm{W}}}\left( {{t_{\mathrm{m}}}} \right) \) represents the whole-blood radioactivity concentration measured at the end of the experiment (at time t m). Inserting Eqs. 2 and 4 into Eq. 3 and integrating from \( {t_i} \) to \( {t_{i+1 }} \) yields:

$$ {Q_{\mathrm{B}}}\left( {{t_{i+1 }}} \right)-{Q_{\mathrm{B}}}\left( {{t_i}} \right)={k_{\mathrm{F}}}{C_{\mathrm{W}}}\left( {{t_{\mathrm{m}}}} \right)\int_{{{t_i}}}^{{{t_{i+1 }}}} {{f_{\mathrm{u}}}(t)D(t)\otimes {e^{{\lambda \left( {{t_{\mathrm{m}}}-t} \right)}}}dt} $$
(5)

Since multiple PET frames are available at late times (t > 35 min), improved parameter estimation is expected by minimizing the following cost function with \( {Q_{\mathrm{B}}}(t) \) measured at multiple late-time intervals \( \left[ {{t_i},{t_{i+1 }}} \right] \):

$$ \min \left\{ {\sum\limits_i {{{{\left\| {{Q_{\mathrm{B}}}\left( {{t_{i+1 }}} \right)-{Q_{\mathrm{B}}}\left( {{t_i}} \right)-{k_{\mathrm{F}}}{C_{\mathrm{W}}}\left( {{t_{\mathrm{m}}}} \right)\int_{{{t_i}}}^{{{t_{i+1 }}}} {{f_{\mathrm{u}}}(t)D(t)\otimes {e^{{\lambda ({t_{\mathrm{m}}}-t)}}}dt} } \right\|}}^2}} } \right\} $$
(6)

and the whole-blood sample measurements at late times (e.g., t i  > 35 min) can be estimated by Eq. 4 once λ is determined.

Animal Preparations and Experimental Procedures

All animal experiments were performed in accordance with institutional guidelines and a protocol approved by the Animal Research Committee of the University of California, Los Angeles. Nine male C57BL/6 mice (six nonfasting and three fasting; 26.6 ± 4.9 g) were studied. Details of the experimental procedures have been reported in previous studies [21, 24, 27]. In brief, the animals were anesthetized with inhalant anesthesia (1.5 %–2 % isoflurane in 100 % oxygen), and the body temperature of the mice was maintained at 37 °C by a thermostatically controlled heating pad. The left femoral artery was cannulated with a surgically inserted polyethylene catheter (PE10; Intramedic, Clay-Adams). Small-animal PET was performed in list-mode on a microPET Focus 220 scanner (Concorde Microsystems, LLC); immediately after a bolus of [18F]FDG (20.9 ± 5.7 MBq; 50–60 μL) was administered. Serial blood samples were collected manually at the femoral catheter and were subsequently pipetted to preweighed test tubes, weighed, and then counted in a gamma counter (WIZARD 3"; PerkinElmer Life Sciences). After the PET scan was completed, a 10-min computed tomography (CT) scan was acquired with a small-animal CT scanner (MicroCAT II; ImTek Inc.). The list-mode PET data were binned to a framing protocol of 1 × 3, 10 × 0.5, 1 × 2, 1 × 4, 6 × 5, 1 × 10, 2 × 30, 2 × 120, 1 × 180, 2 × 600, and 2 × 900 s (for 60-min scan) or 3 × 900 s (for 90-min scan). After correction for radioactive decay, random coincidences, and dead-time losses, the PET data were reconstructed using Fourier rebinning and a two-dimensional filtered backprojection algorithm with a ramp filter cut off at the Nyquist frequency and a zoom factor of 4.74, resulting in a voxel size of 0.4 × 0.4 × 0.8 mm3. The small-animal CT images were coregistered with PET data for attenuation correction [28].

[18F]FDG Kinetic Model

The standard three-compartment [18F]FDG model [2] was used to fit the liver and muscle TACs. The state equations for the kinetics of [18F]FDG in tissue are given by:

$$ \begin{array}{*{20}c} \frac{d}{dt }{C_{\mathrm{E}}}(t)={K_1}{C_{\mathrm{P}}}(t)-\left( {{k_2}+{k_3}} \right){C_{\mathrm{E}}}(t)+{k_4}{C_{\mathrm{M}}}(t) \\ \frac{d}{dt }{C_{\mathrm{M}}}(t)={k_3}{C_{\mathrm{E}}}(t)-{k_4}{C_{\mathrm{M}}}(t) \\\end{array} $$
(7)

where \( {C_{\mathrm{E}}}(t) \) and \( {C_{\mathrm{M}}}(t) \) represent free and phosphorylated forms of [18F]FDG in extravascular space of tissue, respectively; \( {C_{\mathrm{P}}}(t) \) represents arterial plasma IF; \( {K_1}\ \left( {\mathrm{milliliters}\ \mathrm{per}\ \mathrm{minute}\ \mathrm{per}\ \mathrm{gram}} \right) \) and \( {k_2}\ \left( {\mathrm{per}\ \mathrm{minute}} \right) \) are the forward and reverse transport constants of [18F]FDG between tissue and plasma; and \( {k_3}\ \left( {\mathrm{per}\ \mathrm{minute}} \right) \) and \( {k_4}\ \left( {\mathrm{per}\ \mathrm{minute}} \right) \) are the phosphorylation and dephosphorylation rate constants in tissue. A time-dependent plasma-to-whole-blood [18F]FDG equilibrium ratio, \( {R_{\mathrm{PB}}}(t), \) previously calculated at our laboratory [9] was used to account for the differences between [18F]FDG concentrations in plasma and whole-blood:

$$ {R_{\mathrm{PB}}}(t)=0.386{e^{-0.191t }}+1.165. $$
(8)

The model-predicted radioactivity concentration in tissue, \( {C_{\mathrm{T}}}(t) \), is given by:

$$ {C_{\mathrm{T}}}(t)={C_{\mathrm{E}}}(t)+{C_{\mathrm{M}}}(t)+{v_{\mathrm{b}}}\left( {\frac{{{C_{\mathrm{P}}}(t)}}{{{R_{\mathrm{P}\mathrm{B}}}(t)}}} \right) $$
(9)

where \( {v_{\mathrm{b}}}\ \left( {\mathrm{milliliters}\ \mathrm{per}\ \mathrm{gram}} \right) \) represents the blood volume in tissue. The [18F]FDG uptake constant \( \left( {{K_{\mathrm{i}}}} \right) \) can be calculated from the kinetic constant estimates [1, 2]:

$$ {K_{\mathrm{i}}}=\frac{{{K_1}{k_3}}}{{{k_2}+{k_3}}} $$
(10)

which can also be determined by the Patlak graphical analysis [3].

Plasma Input Function

Standard kinetic modeling in PET assumes that the complete time course of plasma IF is available so that the kinetic constant parameters can be estimated by fitting the compartment model to the measured tissue TACs [4]. In contrast, the SIME approach [18, 19] assumes that (1) the plasma IF is unknown (or partially known) and can be parameterized by a mathematical model/equation and (2) multiple tissue TACs of distinct kinetics are driven by the same IF. In this study, the parametric IF model was described by a sum of four-exponential functions [29]:

$$ {C_{\mathrm{P}}}(t)={A_1}{e^{{-{\beta_1}t}}}+{A_2}{e^{{-{\beta_2}t}}}+{A_3}{e^{{-{\beta_3}t}}}-\left( {{A_1}+{A_2}+{A_3}} \right){e^{{-{\beta_4}t}}} $$
(11)

where \( {A_1}-{A_3}\ \left( {\mathrm{Becquerel}\ \mathrm{per}\ \mathrm{milliliter}} \right) \) and \( {\beta_1}-{\beta_4}\ \left( {\mathrm{per}\ \mathrm{minute}} \right) \) can be determined by nonlinear least-squares fitting. Thus, the input function estimated by the proposed hybrid approach (IFHYBRID) consists of two steps: The late-time blood samples are first derived by the renal modeling technique (Eqs. 4 and 6); the kinetic constants in the tissue TACs and the parameters that described the estimated plasma IF are determined by simultaneously fitting the [18F]FDG model (Eqs. 7 and 9) to the liver and muscle TACs, and the parametric IF model (Eq. 11) to the plasma curve constructed by joining the early-time LV data \( \left( {t\leq 1\min } \right) \) corrected for partial-volume, delay and dispersion [24], and erythrocyte uptake (Eq. 8) and the late-time blood data \( \left( {t>35\min } \right) \) derived in the first step, while blood data between 1 and 35 min postinjection that were supposed to be unavailable were interpolated by the parametric IF model (Eq. 11). The “gold standard” input function (IFGS) for each study, with which the estimated IF was compared, was constructed in the same way but with all serial blood samples taken after 1 min postinjection.

Data Analysis

Results are presented as mean ± standard deviation (SD) unless otherwise stated. Statistical analyses were performed using Prism 4.03 (GraphPad Software, Inc.). A P value of less than 0.05 was considered statistically significant.

Both PET image data and blood data were converted to absolute radioactivity concentration (Becquerel per milliliter) with the cross-calibration factor derived from routine cylinder phantom experiments. Images were analyzed using AMIDE (http://amide.sourceforge.net/). Three-dimensional ellipsoid volumes of interest (VOIs) were manually drawn over the brain, myocardium, skeletal muscle of forelegs, liver, and kidney on the coregistered PET/CT images, which provided anatomic references for VOI placement. The VOIs were applied to each time frame of the PET images, and the average radioactivity concentration was computed to obtain the corresponding TACs. Because uptake in the tissues was relatively homogeneous within the VOIs and the background radioactivity was low, background subtraction and partial-volume correction were not performed.

Derivation of the complete time course of the plasma IF can be improved with the TAC obtained from the LV. Spherical VOI of 1.5 mm in diameter was drawn on the PET image in the region that corresponded to the LV, which could be visualized during the first 10 s postinjection. Due to continuous excretion of [18F]FDG by the kidneys, the volume of [18F]FDG in the urinary bladder increases as a function of time. The PET data were scaled with an empirical threshold value of about 65–75 % of the global maximum, which was found to provide good boundary delineation of bladder VOIs on the fused PET/CT image [30], and VOIs that covered the entire bladder were drawn frame-by-frame, and the total accumulated activity curve of the urinary bladder was then calculated.

The best estimates of [18F]FDG clearance rate in plasma and delay and dispersion of blood across the kidney and accumulation of 18F-activity in urinary bladder were determined by minimizing the cost function given by Eq. 6. Simultaneous fitting of [18F]FDG model (Eqs. 7 and 9) and parametric IF (Eq. 11) to the experimental data was performed using SAAM II [31]. The accuracy of the input function obtained without using the renal modeling technique was also evaluated using the last blood sample collected at the end of PET imaging, i.e., only the second step of the proposed method was applied to estimate the IF \( \left( {\mathrm{I}{{\mathrm{F}}_{1\text{-}\mathrm{BS}}}} \right) \). Under the assumption that [18F]FDG is rapidly cleared from the liver [25], the image-derived liver TAC (IFLIVER) has been suggested as an alternative plasma IF [14, 32] and was also investigated in this study.

Areas under the curve (AUCs) were calculated for IFHYBRID (AUCHYBRID), IF1-BS (AUC1-BS), and IFLIVER (AUCLIVER). Their correspondences with those calculated using IFGS (AUCGS) were assessed by means of linear regression. The [18F]FDG uptake constant (K i) values in the brain, myocardium, skeletal muscle, and liver were estimated by the Patlak graphical analysis [3] using various IFs and tissue data from 15–60 min [24]. Limits of agreement of AUC and Patlak K i estimates obtained by different methods and those by IFGS were assessed by the Bland–Altman analysis [33], where the differences between measurements on the same animal by two methods were plotted against the average, and the estimated bias and its 95 % confidence interval (CI) were computed. The Friedman’s test followed by Dunn’s multiple comparison test [34] was used to assess whether the AUCs and Patlak K i obtained by various approaches were statistically different from those obtained by IFGS.

Results

Radioactivity and Urine Volume in Urinary Bladder

Fig. 2 shows typical total accumulated radioactivity and the volume in urinary bladder as a function of time obtained in three mice. It can be seen that the increase in volume was not linear and that the accumulation of [18F]FDG in the urinary bladder varied differently among different animals. Table 1 summarizes estimates of various parameters, including the clearance rate of [18F]FDG in whole-blood at late-time \( \left( \lambda \right) \), dispersion \( \left( \tau \right) \), and delay \( \left( \varDelta \right) \) time constants for accumulating 18F-activity to the urinary bladder in the form of urine. The average dispersion and delay time constants were 1.89 ± 0.98 and 2.13 ± 0.85 min, respectively, and the parameter estimates were reasonably consistent among different animals. In spite of the nonlinear increase in bladder volume and the difference in accumulation rate of [18F]FDG in the urinary bladder, the proposed renal modeling technique was able to provide the blood clearance estimate comparable to that estimated using multiple late-time blood samples (0.0212 ± 0.0106 min−1 versus 0.0177 ± 0.0063 min−1, Wilcoxon signed rank test, P = 0.425).

Fig. 2
figure 2

Representative total accumulated radioactivity (top row) and the corresponding volume (bottom row) of 18F-activity in urinary bladder measured in three mice.

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Table 1 Estimated [18F]FDG clearance rate in blood (λ), dispersion (τ), and delay (Δ) time constants for filtrating blood in the kidney and accumulating 18F-activity to the urinary bladder
Full size table

Estimation of Input Function

Fig. 3 shows plasma IFs and their estimation by various approaches in all studies. The partial-volume-, delay-, dispersion-, and R PB-adjusted early-time LV TAC was used to represent the plasma data within 1 min postinjection, and the late-time plasma data were corrected for the uptake by erythrocytes. The estimated IFHYBRID closely resembled \( \mathrm{I}{{\mathrm{F}}_{\mathrm{GS}}} \) that required arterial cannulation and continuous blood sampling. The mean absolute differences between \( \mathrm{I}{{\mathrm{F}}_{\mathrm{HYBRID}}} \) and \( \mathrm{I}{{\mathrm{F}}_{\mathrm{GS}}} \) were 8.3 % ± 7.0 % and 4.5 % ± 2.7 % at 30 and 60 min, respectively. However, the image-derived liver TAC (IFLIVER) differed substantially from IFGS.

Fig. 3
figure 3

Comparison of input functions obtained in all animals using different methods.

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The AUC of the estimated \( \mathrm{I}{{\mathrm{F}}_{\mathrm{HYBRID}}} \), \( \mathrm{I}{{\mathrm{F}}_{1\text{-}\mathrm{BS}}} \), and \( \mathrm{I}{{\mathrm{F}}_{\mathrm{LIVER}}} \) was compared with that of the \( \mathrm{I}{{\mathrm{F}}_{\mathrm{GS}}} \), which was calculated by fitting the [18F]FDG model with all available blood sample data. The correspondences and the limits of agreement between \( \mathrm{AU}{{\mathrm{C}}_{\mathrm{GS}}} \) and \( \mathrm{AU}{{\mathrm{C}}_{\mathrm{HYBRID}}} \), \( \mathrm{AU}{{\mathrm{C}}_{1\text{-}\mathrm{BS}}} \), and \( \mathrm{AU}{{\mathrm{C}}_{\mathrm{LIVER}}} \) are shown in Fig. 4. Although none of the regression lines was significantly different from the line of identity (extra sum-of-squares F-test, P > 0.05), the slope of the regression line between \( \mathrm{AU}{{\mathrm{C}}_{\mathrm{LIVER}}} \) and \( \mathrm{AU}{{\mathrm{C}}_{\mathrm{GS}}} \) tended to deviate from one (P = 0.066). There were no significant differences between the AUC values of plasma IFs obtained by different methods when compared with \( \mathrm{AU}{{\mathrm{C}}_{\mathrm{GS}}} \) (Dunn’s multiple comparison test, P > 0.05). The mean biases from \( \mathrm{AU}{{\mathrm{C}}_{\mathrm{GS}}} \) were similar for \( \mathrm{AU}{{\mathrm{C}}_{\mathrm{HYBRID}}} \) and \( \mathrm{AU}{{\mathrm{C}}_{1\text{-}\mathrm{BS}}} \) (1.82 versus 2.31 MBq/ml × min), but the 95 % limits of agreement (mean ± 1.96 SD) with \( \mathrm{AU}{{\mathrm{C}}_{\mathrm{GS}}} \) were better for \( \mathrm{AU}{{\mathrm{C}}_{\mathrm{HYBRID}}} \) (95 % CI, −4.37–8.01 MBq/ml × min) than for \( \mathrm{AU}{{\mathrm{C}}_{1\text{-}\mathrm{BS}}} \) (95 % CI, −9.80–14.42 MBq/ml × min). In contrast, the mean bias (−4.70 MBq/ml × min) and the 95 % CI (−22.08–12.67 MBq/ml × min) for the difference between \( \mathrm{AU}{{\mathrm{C}}_{\mathrm{LIVER}}} \) and \( \mathrm{AU}{{\mathrm{C}}_{\mathrm{GS}}} \) were found to be much larger.

Fig. 4
figure 4

Graphs show the linear regression between AUCGS and a AUCHYBRID, c AUC1-BS, and e AUCLIVER. The limits of agreement between AUCGS and AUCHYBRID, AUC1-BS, and AUCLIVER are shown in subpanels b, d, and f, in which the dotted line represents the mean difference in AUC and the dashed lines represent the 95 % confidence interval for the difference (mean ± 1.96 SD).

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[18F]FDG Uptake Constant

Fig. 5 shows the comparison of K i calculated for the brain, myocardium, skeletal muscle, and liver using Patlak analysis with \( \mathrm{I}{{\mathrm{F}}_{\mathrm{HYBRID}}} \) and \( \mathrm{I}{{\mathrm{F}}_{\mathrm{GS}}} \). The slopes of the regression lines were close to 1, and the intercepts were nearly equal to 0, with the Spearman correlation coefficients of 0.983, 0.983, 1.0, and 0.933 for the brain, myocardium, skeletal muscle, and liver, respectively. The limits of agreement between Patlak K i estimates were further examined by the Bland–Altman analysis and are shown in Fig. 6. The Bland–Altman plots show a slight bias in K i estimated with \( \mathrm{I}{{\mathrm{F}}_{\mathrm{HYBRID}}} \) (−0.00094, −0.0044, −0.00029, and −0.0001 ml/min/g in brain, myocardium, muscle, and liver, respectively) and \( \mathrm{I}{{\mathrm{F}}_{1\text{-}\mathrm{BS}}} \) (−0.00095, −0.0024, −0.00022, and −0.00027 ml/min/g in brain, myocardium, muscle, and liver, respectively) in comparison with \( {K_{\mathrm{i}}} \) estimated using \( \mathrm{I}{{\mathrm{F}}_{\mathrm{GS}}} \). A little wider spread of K i values for \( \mathrm{I}{{\mathrm{F}}_{1\text{-}\mathrm{BS}}} \) compared with \( \mathrm{I}{{\mathrm{F}}_{\mathrm{HYBRID}}} \) was observed, and the differences and discrepancies were much more prominent for \( \mathrm{I}{{\mathrm{F}}_{\mathrm{LIVER}}} \). Table 2 shows the mean ratios of Patlak K i values in various tissues determined with \( \mathrm{I}{{\mathrm{F}}_{\mathrm{HYBRID}}} \), \( \mathrm{I}{{\mathrm{F}}_{1\text{-}\mathrm{BS}}} \) and \( \mathrm{I}{{\mathrm{F}}_{\mathrm{LIVER}}} \) to those obtained with \( \mathrm{I}{{\mathrm{F}}_{\mathrm{GS}}} \). The K i ratios for \( \mathrm{I}{{\mathrm{F}}_{\mathrm{HYBRID}}} \) and \( \mathrm{I}{{\mathrm{F}}_{1\text{-}\mathrm{BS}}} \) were close to 1. However, for \( \mathrm{I}{{\mathrm{F}}_{\mathrm{LIVER}}} \), the K i ratios in brain and skeletal muscle ranged from −0.28 to 1.02 and from −0.21 to 1.04, respectively, and differed significantly from 1 (P < 0.05). The negative K i ratios were caused by two different cases in which the use of liver as IF produced negative K i estimates for the brain and skeletal muscle, respectively.

Fig. 5
figure 5

Comparison of Patlak K i in a brain, b myocardium, c skeletal muscle, and d liver, determined with IFHYBRID and IFGS.

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Fig. 6
figure 6

Limits of agreement between Patlak K i obtained for a brain, b myocardium, c skeletal muscle, and d liver, using IFGS (K i,GS), IFHYBRID (K i,HYBRID, left column), IF1-BS (K i,1-BS, middle column), and IFLIVER (K i,LIVER , right column). Dotted line represents the mean difference in K i; dashed lines represent the 95 % confidence interval for the difference (mean ± 1.96 SD). Note the different ranges for the ordinate axis when liver TAC was used as the input function (right column).

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Table 2 Mean ratios of Patlak K i estimate in various tissues determined with IFHYBRID, IF1-BS, IFLIVER, and the gold standard IF (IFGS)
Full size table

Discussion

Accurate quantification of dynamic PET data is directly related to the precise definition of the IF, which is usually obtained via arterial catheterization and continuous blood sampling. The invasiveness of aseptic microsurgery, the small size of rodents, and the limitation in total blood volume pose considerable technical complexities and methodological challenges to acquiring IF measurements without causing much distress and disturbance to the animals. Our results show that the proposed hybrid method is able to estimate the IF reliably and to provide good AUC and \( {K_{\mathrm{i}}} \) estimates when compared with those obtained with the gold standard IF, which requires serial blood sampling.

The ideal situation would be if blood sampling could be eliminated completely. However, a number of studies have demonstrated that at least one blood sample is needed for scaling the estimated IF, and significantly more accurate IF estimation is achieved with increasing number of blood sample measurements [18–20, 22, 23]. Consistent with the findings of those studies, the use of one blood sample in the estimation process provided much better estimated IF than without using any blood sample (data not shown). On the contrary, the main advance of this study is that the late-time blood measurements are estimated by making use of the relationship between plasma clearance and total accumulated radioactivity in bladder [24] and the excretion pathway of [18F]FDG [25] instead of collecting multiple blood samples to improve the accuracy of the estimated IF as in many previous studies.

The proposed renal model is certainly a simplified physio-anatomical model for the complex and heterogeneous structure of kidney. Nonetheless, the estimated plasma clearance rate at late-time using the renal model is in line with the published data [21–23], whereas the mean tracer appearance time in the urinary bladder was found to be about 2–3 min and is in good agreement with previously reported values [24], thus indicating that useful information can be extracted from the proposed model. Further studies are clearly required to formulate a more representative compartmental model configuration that is able to resolve and describe different functional compartments of kidney under the limited spatial resolution of small-animal PET and to deal with renal failure condition (e.g., diabetes mellitus) when the kidneys could not excrete [18F]FDG into the urine completely.

A more accurate approach to determine the total radioactivity and the capacity of the bladder would be to collect urine samples at timed intervals through urethral catheterization. However, the procedure is impractical in mice due to various anatomical, physiological, and technical considerations, including the gender of the animal, the size and material of the catheter, the disease condition, and the potential contamination of the urine specimens. Therefore, a completely noninvasive image-based approach was sought in this study. The total radioactivity and the volume of urinary bladder were calculated from the small-animal PET images with the aid of the threshold value determined using the fused PET/CT image. Potential errors in their calculations are mainly attributed to the spillover of the radioactivity from the surrounding tissues which is comparable to the radioactivity in the bladder at early times. However, the effect of spillover of radioactivity from the surrounding tissues can be assumed to be small and negligible at late-time because the size of bladder VOI is usually much bigger than the spatial resolution, which is 1.7 mm full-width at half-maximum with the selected reconstruction settings, and the measured radioactivity is dominated by that in the urinary bladder. The bladder volume (∼0.1–0.3 ml) determined at late-time is well within the published maximum observed in generic laboratory mice (0.5 ml; [35]), the value of which depends also on a number of factors, including the strain, gender, and size of the mice, the study condition, the time of measurement, and the measurement technique being used. Therefore, the bladder capacity determined from the fused PET/CT image at late-time is expected to be close to the “actual” value, which was, however, not measured for each mouse because the urinary bladder was not harvested at the end of the experiment as the animal would micturate soon after euthanasia, resulting in erroneous urine volume determination and total 18F-activity measurement.

Different from many other organs or tissues, which are supplied solely by arterial blood, the liver has dual blood supply from the hepatic artery and the portal vein. The use of dual blood supply in fitting [18F]FDG kinetics of liver has been shown to provide more physiological reasonable estimates than the use of single blood supply [36, 37]. However, surgical cannulation of both hepatic artery and portal vein for blood sampling in rodents is not a trivial task and will only further complicate the small-animal imaging procedure. Therefore, we used the arterial input to approximate the dual input to the liver, and this is not expected to hamper the estimation capability of the proposed method.

Assuming the liver has relatively large blood volume and rapid [18F]FDG clearance [25], the image-derived liver TAC has been suggested as a possible input function for small-animal [18F]FDG PET studies [14, 32]. To date, however, no results have been published regarding the errors and the bounds of applicability of using this “reference tissue” as the input function for quantitative analysis of [18F]FDG PET data. It should be noted that [18F]FDG uptake in liver is dependent upon the physiological status and metabolic conditions (e.g., hyperinsulinemia or fasting) of the animal. Our results show that Patlak \( {K_{\mathrm{i}}} \) estimates derived using liver TAC as the input function had large differences when compared with those obtained with the gold standard input function (Fig. 6 and Table 2), despite the good correlations in AUCs between liver and plasma TACs (Fig. 4e). The discrepancies are likely caused by the differences in the shape of liver and plasma TACs (Fig. 3).

As [18F]FDG does not cross murine erythrocyte membrane freely [38], the plasma-to-whole-blood activity concentration ratio will not reach relatively constant (i.e., equilibrium) until after ∼25 min postinjection [9, 24]. Therefore, when a late-time whole-blood sample is available, the “equivalent” plasma radioactivity concentration can be obtained by adjusting the whole-blood radioactivity concentration with the equilibrium ratio. Because the arterial and venous blood concentrations are usually very similar at late-time and only one blood sample taken at the end of the experiment is needed for the proposed method, it is possible to replace the arterial blood sample by arterialized-venous or venous blood sample, which is easy to collect at the lateral caudal vein. This could further simplify the proposed procedure and make it useful in repeated/longitudinal dynamic [18F]FDG PET studies on the same animal, where repetitive invasive blood sampling during multiple imaging sessions is infeasible.

It is anticipated that the accuracy and robustness of the proposed method can be further improved. For example, the threshold value for bladder boundary detection on the PET image can be determined using a calibrated phantom similar in shape and size to the bladder. Also, the procedure of defining bladder VOI on multiple image frames which is performed manually can be streamlined by some automatic VOI delineation algorithms so that the VOI defined will have less human intervention and higher reproducibility. Since the photon attenuation was changed as the bladder got larger, the attenuation correction was only an approximation. Conceptually, this attenuation change can be accounted for, but the procedure is not expected to be simple and easy. As a result, the determination of total 18F-activity and bladder volume at early times may be biased due to over-correction for photon attenuation using the attenuation map derived from the CT image acquired at late-time. Nonetheless, the renal modeling procedure depends only on the late-time measurements of total 18F-activity and bladder volume, both of which are expected to be more accurate than their early-time measurements because of the more uniform distribution of 18F-activity in the bladder volume at late times and the close approximation of photon attenuation resulting from a better alignment between CT and late-time PET images. As for modeling the time-varying urine flow, the one-exponential function is certainly an empirical choice. Other analytical functions (e.g., polynomials) may also be used for this purpose. To our knowledge, however, no data have yet been published to calculate the urine flow rate in murine over a short time interval, probably because of the technical difficulties associated with frequent urine collection without increasing levels of pain and discomfort caused to the animal, and the urine flow (as well as the urine production rate) of 1 ml per day has been reported [39, 40]. Since the frequency of voiding and the void volume may change from time to time (e.g., the mouse may urinate more frequently during awake time when water intake is higher), the urine production rate is likely varying throughout the day. The nonlinear increase in the bladder volume curves indicates that the urine production rate is not constant over time and decreases as the bladder gets filled up. As for the computation and implementation aspects, the two-step estimation of the proposed hybrid approach can be reformulated into a one-step estimation procedure by unifying Eqs. 4 and 11. Fitting an analytical function form for the input function is a common strategy to reduce statistical uncertainties in the measured blood data [41, 42]. In this study, a sum of four-exponentials was used to describe the plasma IF characterized by the abrupt changes in radioactivity concentration immediately after tracer administration and to account for differences in injection schedule. An alternative approach is to use a physiological-based model to describe the tracer kinetics in blood/plasma. One such model has been proposed for human [18F]FDG PET [29], but an optimal functional form has yet to be determined for rodents. We are currently exploring various approaches to address the issues as mentioned above.

Conclusions

In this study, we proposed and validated a hybrid approach that uses the total accumulated radioactivity in the urinary bladder and a blood sample collected at the end of the study as well as time-activity data from multiple tissues to estimate the plasma input function for dynamic mouse [18F]FDG PET studies. By combining the proposed renal modeling technique with the simultaneous estimation, the estimated blood samples agreed well with those derived from the standard reference but invasive blood sampling approach. Importantly, this hybrid approach obviates the need of arterial catheterization as it requires only one late-time blood sample, which can be collected from the tail vein, thus making it possible to perform quantitative dynamic [18F]FDG PET studies longitudinally on the same animal. Conversely, liver time-activity curve is unsuitable as an input function for absolute quantification of [18F]FDG PET data, as it provides significantly biased estimates of uptake constant in various tissues.