Table 3.

Models from the literature that simulate the pharmacokinetics (PK) of PET tracers to predict and analyse PET scan images.

Reference	Modelling Methods	Details and Outcomes
(Kelly and Brady 2006) [25]	2-compartment F18[MISO] PK model with reversible binding, with transport via diffusion only. 2-dimensional analytical spatiotemporal model.	Michaelis-Menten techniques were used to model the conservation of O2 and cap consumption in oxic tissue (pO2 dependent equation). Randomly angled/oriented vessels, temporal dynamics modelled by changing vessel pressure and hence flow. Hypoxic tissue: gradual increase in activity then an accumulation curve. Oxic tissue: activity follows plasma levels then accumulation curve seen at later stages. Late slope of TAC curve indicated hypoxia while the beginning represented local vascular supply. Results compared to pimonidazole stained tumour sections from clinical colorectal cancer data.
(Wang et al. 2009) [26]	Iterative stochastic optimisation algorithm to delineate acute and chronic hypoxia in sequential F18[MISO] and FDG PET in 2-dimensional image maps, with comparisons to HNSCC clinical data.	Simulated images (known hypoxic regions) as well as sequential PET Data from 14 male HNSCC patients analysed assuming chronic (Gaussian histogram of number of voxels versus SUV) hypoxia remained constant while acute hypoxia (Poisson histogram) was varied. Normalisation methods forced the volume of hypoxia to be the same in both time-point scans; however the location of acute hypoxia varied. Image registration and resolution issues are discussed. Model predicted Gaussian chronic hypoxia distributions well in the generated images (r 2 = 0.93). Good fit found (13/14 cases), with acute hypoxia described well by a Poisson curve (11/14 cases) with an average of 34% (acute hypoxic volume). Suggested a third scan to increase temporal hypoxia information. 4 mm PET pixel size issue accounted for using power law distribution of chronic versus acute hypoxia within each pixel.
(Bartlett et al. 2012) [27]	Two varieties of 2-compartment, 3-rate-constant models applied to F18[MISO] PET images of human prostate tumour xenografts in rats.	One model constrained kinetic parameters k 1 and k 2 to be equal while the other did not. Intratumoural pO2 was assessed using a robotic driven probe in tumour versus plasma regions of the animal's tumour mass. Pimonidazole and perfusion Hoechst 33342 staining also analysed. Kinetic voxelised modelling (of parameter k 3) identified hypoxia with greater accuracy than tumour-to-plasma ratios. Constraining k 1 to equal k 2 during fitting was effective in controlling noise in the trapping rate constant, k 3, without introducing bias. No obvious pO2 cutoff for isolating hypoxic and nonhypoxic volumes (3.4 mm Hg applied) however noise of approx. 0.7 mm Hg in measurement technique.
(Gu et al. 2012) [28]	3-compartment F18[FLT] PK model (3 rate constants) applied to a separate GBM growth model utilising spatial MR data and considering invasion, hypoxia, necrosis, and angiogenesis.	Voxels assigned “cell density” values with hypoxic versus oxic percentages (e.g., 70 versus 30%) generated. Model simulated the dynamic clinical-scale imaging process in terms of noise and reconstruction uncertainties of PET. Clinical GBM patient data used for comparison, with patient specific virtual PET scans generated with no statistical difference to real hypoxic tumour image sets. Model could predict and distinguish hypoxic cell hyperactivity versus hyperdensity on the PET image.
(McCall et al. 2012) [29]	TACs derived from mean tissue activity concentration functions for Cu64[ATSM] (Ct) in HNSCC and muscle and compared to venous input functions (Cp).	Tracer dynamics studied in HNSCC (FaDu) xenografts in rats and analytical parameters of the model fitted to generated results matching real PET data. Influx-constants (Ki) calculated by analysis of Patlak plots of Ct/Cp ratios versus normalized time integrals of Cp. PET mean data analysed from 1 min up to 18 hours after injection. Distribution volumes (V d) calculated. High tumour to muscle uptake ratios found (4 : 1 tumour to muscle ratio at 20 min). No Cu64[ATSM] correlation to pimonidazole hypoxia staining (early or late). Cu dynamics are not only pO2 dependent, more study recommended. Early uptake of tracer in tumour at 1 min found followed by slower but steady increase, while muscle signal increased quickly then plateaued. Wash out rates in tumour and normal tissue difficult to define.
(Monnich et al. 2013 [30], M$\ddot{\text{o}}$nnich et al. 2011) [31]	O2 kinetic and F18[MISO] tracer PK model simulating 2-dimensional virtual PET maps, based on blood vessel maps from human HNSCC xenografts stained for endothelial structures	Xenografts were utilised to derive 2D vessel maps (~3% vascular fraction) and an explicit pO2-dependent binding rate, K(P). Oxygen and tracer flux across vessel walls, J T, assumed proportional to the concentration differences on the intra- and extravascular side. Tracer moved via diffusion. Irreversible binding rate modelled as dependent on pO2 only. Nonlinear Michaelis-Menten oxygen consumption versus pO2 tension. Individual time-point data did not show correlation with real data (2.5 mm Hg threshold for each voxel with median pO2 in each voxel assessed); however, ratios of 0–15 min versus 4-hour data had significant outcomes. Four-hour data did correlate but not as well as ratio data. From 2011: binding versus pO2 function described with steep initial increase (<0.5 mm Hg). Simulated local TACs share characteristics with clinical PET TACs hence it may be possible to measure perfusion from early dynamic PET. Alternative tracer dynamics (faster clearance) also simulated with earlier time point PET scans predicted optimum, although free-tracer signals limit earlier time feasibility.
(Liu et al. 2014) [32]	F18[FLT] 2- and 3-compartment PK models compared for HNSCC clinical PET images, incorporating diffusion as well as convection transport of the tracer.	A comprehensive statistical analysis of the PK model is reported. “EM-BIC” clustering methodology described, and model used to analyse raw PET images and reduce noise and hence uncertainty in the rate constant parameters derived. Model results compared to 10 × 1-hour dynamic HNSCC clinical PET data sets, with the 3-compartment (6 rate constants) “3C6K” model best fitted patient data.

[TAC: time activity curve; HNSCC: head and neck squamous cell carcinoma; GBM: glioblastoma multiforme; PK: pharmacokinetic; pO₂: partial pressure of oxygen; SUV: standard uptake value].