Spectral analysis
In SA, the tissue activity at time t, C
tiss(t), is modelled as a convolution of the delivery input function, C
in(t), with the sum of M + 1 distinct exponential terms as in Equation 1.
(1)
α
j
and β
j
are assumed to be real non-negative values. The upper limit, M, represents the maximum number of terms to be included in the model. The values of the rates β
j
are predetermined and fixed in order to cover a range of all possible kinetic components measurable from the data. The values of the amplitudes α
j
are estimated from the input and tissue time-activity curves by non-negative least square algorithm, and normally, only a few components with α
j
> 0 are detected.
Spectral analysis was implemented as previously reported [15], with C
tiss(t) referring to the activity of the tracer measured in the whole heart and C
in(t) associated to the measured delivery input function. The grid of values of the rates β
j
(β
1 < β
2 < … < β
M
) (Equation 1) was defined as a logarithmic distribution with lower limit β
1 = 1/(3T
end), where T
end was the end time of the experiment, and upper limit β
M
= 3/T
in in agreement with previous studies [16]. T
in was the duration of the first time frame of the experiment. The number of points M between β
1 and β
M
was chosen equal to 100. A component for β = 0 was included in the model, corresponding to a fully trapped component. The M + 1 unknown values of α
j
were estimated through a non-negative least square estimator. Weights were inversely proportional to the variance of decay-corrected measured activity. The SA algorithm was implemented in Matlab (the MathWorks, Natik, USA).
Non-linear spectral analysis
For the implementation of NLSA, it is not necessary to specify a grid of values of β
j
. The M + 1 unknown values of β
j
and α
j
were estimated through non-linear fitting with the initial conditions chosen within a physiological range (and in agreement with [13]) maintained constant for all datasets analysed. The number of exponentials necessary to give a good fit of the data M (Equation 1) was fixed between 1 and up to a maximum of 4, in agreement with previous studies [13]. The Akaike information criterion (AIC) [17] was used to choose the model that best fit the data. Weights were inversely proportional to the variance of decay-corrected measured activity. The NLSA algorithm was implemented in Matlab (the MathWorks, Natik, USA).
Graphical method: Patlak plot
Due to the demonstrated irreversible tissue retention of the tested tracers [13, 18], the Patlak plot (given by the expression below) was chosen among the available graphical methods [14]:
(2)
where K represents the net uptake rate of the tracer and V the distribution of the tracer in the compartment that is in rapid equilibrium with the plasma. The unknown constants K and V were obtained by linear regression from a graph of C
tiss(t)/C
in(t) against computed for t > 7.5 min. Because of the large number of data points (five samples per second), both quantities were interpolated on a 6-s sub-sampled grid. K was the ultimate parameter of interest considered, while V was excluded from the tracer kinetic analysis.
Normalised activity
We defined the NA as a surrogate of the semi-quantitative index standard uptake value (SUV) used for the characterisation of tracer trapping in vivo. SUV is given by the ratio of the tissue radioactivity and the injected radioactivity divided by the body weight [19]. In ex vivo experiments, the input function is delivered as an impulse with no recirculation, and therefore, the plasma radioactivity rapidly decays to zero. Additionally, due to the experimental set-up, the measured activity is not directly proportional to the radioactivity concentration. For these reasons, the semi-quantitative index SUV could not be computed in this study as it is defined for in vivo experiments. We define a new index, NA, in which the maximum measured activity in the target tissue substituted the normalised injected dose of the tracer normally used in SUV. NA was calculated as a ratio of the mean tissue activity (C
tiss(t)) measured in counts per second (CPS) over a small interval at the end of the experiment [end - 0.075 min, t
end] and the maximum value of C
tiss(t) (CPS).
(3)
Simulation studies
Datasets were simulated, reproducing the characteristics of ex vivo time-activity curves from isolated hearts in normoxia and at different levels of hypoxia. Bi- and tri-exponential ex vivo time-activity curves with known kinetics were simulated using Equation 1. The values of β
j
were fixed (for tri-exponentials, β
1 = 0.5 min-1, β
2 = 5 min-1 and β
3 = 15 min-1; and for bi-exponentials, β
1 = 0.5 min-1and β
2 = 15 min-1), whereas α
j
were randomly generated within a chosen interval (for tri-exponentials, 0.011 min-1 < α
1 < 0.11 min-1, 0.12 min-1 < α
2 < 0.2 min-1, and 5 min-1 < α
3 < 8 min-1; and for bi-exponentials, 0.011 min-1 < α
1 < 0.11 min-1 and 5 min-1 < α
2 < 8 min-1). Bi- and tri-exponential curves were simulated with and without a trapping component α
0 (for β
0 = 0). Four different values of irreversible trapping were simulated (α
0,1 = 0.006 min-1, α
0,2 = 0.06 min-1, α
0,3 = 0.2 min-1 and α
0,4 = 0.6 min-1) (see Figure 2). The values used in the simulations were chosen within a physiological range and in agreement with previous studies [13]. To obviate further sources of variability between one dataset and another, the same C
in(t) was taken from an experimental dataset, representative of these experiments, and used for all simulations. Each dataset was simulated for a low and a high SNR (SNR = 60 and SNR = 120), consistent with the experimental data, and N = 100 and a time resolution Δt = 0.0033 min were chosen. SNR was computed as the ratio of the maximum value of the time-activity curve and the standard deviation of the signal in the last minute of each experiment [20].
The percent bias (%BIAS) of quantitative and semi-quantitative indices (α
0, Patlak and NA) was calculated as a performance index (Equation 4).
(4)
where p
j
and p
TRUE are the estimated and true value of the indices p.
Experimental protocol
All procedures were performed in accordance with the United Kingdom Home Office Guide on the Operation of the Animals (Scientific Procedures) Act 1986 and KCL's Ethical Review Process Committee.
[18 F]-FDG was provided by the clinical PET Centre, St. Thomas' Hospital, whereas [18 F]-FMISO was prepared following a previously reported method [21].
Mature male Wistar rats (250 to 300 g) were fed ad libitum with regular animal feed. Hearts (n = 4 for [18 F]-FDG and n = 3 for [18 F]-FMISO) were harvested under terminal anaesthesia (sodium pentobarbitone (100 mg/kg) intraperitoneal injection with heparin (200 IU)) and plunged into ice-cold Krebs Henseleit buffer (KHB), with the following composition: NaCl 118 mM, NaHCO3 25 mM, MgSO4 1.2 mM, KCl 5.9 mM, Na2EDTA 0.6 mM, glucose 11.1 mM and CaCl2 2.5 mM, pH 7.4. Hearts were cannulated via the aorta and retrogradely perfused at constant flow (14 ml/min) with KHB at 37°C. Cardiac contractile function was monitored with a water-filled balloon inserted into the left ventricular lumen inflated to record an end diastolic pressure of 6 to 8 mmHg, which was connected to a pressure transducer and recording apparatus. Perfusion pressure was measured by a further pressure transducer inserted into the arterial line. A bolus of each radiotracer (1 MBq in 100 μl KHB) was administered via an in-line injection port, and its transit through the perfusion apparatus was monitored by NaI detectors (1) in the arterial line above the heart cannula and (2) directly opposite the heart (to quantify the tracer accumulation in the heart) connected to a GinaSTAR TM ITLC unit (see Figure 1). All datasets were acquired with a time resolution Δt = 0.0033 min.
After a stabilisation period, where KHB perfusate was gassed with 95% O2-5% CO2, the first bolus of either [18 F]-FDG or [18 F]-FMISO was injected into the perfusion line, and their passage through the system under 'normoxic’ conditions was recorded. After 5 min, hypoxia was induced by switching to a second reservoir (KHB gassed with 95% N2-5% CO2). Further boli of each tracer were injected into the system after 5 min and 15 min of hypoxic buffer perfusion, respectively, and the kinetics of cardiac retention monitored. A scheme of the experimental protocol used to acquire [18 F]-FDG and [18 F]-FMISO plasma and tissue time-activity curves is shown in Figure 3A, whereas a representative experimental dataset is shown in Figure 3B.
Data correction
All datasets were corrected for radioactive decay. Additionally, to adjust the experimental data for residual activity due to prior injections in the same heart, a model prediction correction was used. The normoxic input/tissue time-activity curve was fitted with the algorithm assessed (SA or NLSA), and the result was extrapolated in the time interval relative to the second injection. The contribution of the activity from the first injection to the second was therefore calculated and subtracted. After the background correction, the curve relative to the second injection was fitted and its contribution subtracted to the third injection.
Statistical analysis
Statistical analysis was performed using GraphPad Prism® (GraphPad Software Inc, USA). All values are expressed as the mean ± SD. Data were analysed using a one-way ANOVA with Dunnett's test to compare each group of datasets acquired in hypoxia with the corresponding control group measured in normoxia [22].