Do tracer-kinetic models represent the real tracer kinetics in tissue closely?

What is the impulse residue function and why is there a convolution relationship?

Can there be an all-purpose model?

Can we do without the problematic arterial input function?

What are some of the implementation issues to be aware of?

Researchers and clinicians who are interested in understanding the various existing tracer-kinetic models, or who wish to develop new models specific to their needs.This lecture will explain the fundamentals behind the various tracer-kinetic (TK) models used to analyze dynamic medical imaging, such as DCE, DSC, ASL MRI, and other dynamic imaging modalities. We will focus on the model based analysis of the tracer concentration curves, C(t), assuming that it has been measured accurately. With a clearer understanding of these TK models, learners will have a better idea of the limitations, challenges and implementation issues when applying these models.

Dynamic medical imaging data can be analyzed qualitatively by visual inspection of the signal time curves, or semi-quantitatively by extracting values, such as peak value and area under curve. However, these observations are dependent on injection protocol, acquisition sequence and physiological state of the subject (Koh et al., 2012). Using a TK model to analyze C(t) measured from dynamic medical imaging data permits the extraction of quantitative parameter values from the measured tracer concentrations over time. These describe some physiological aspects of the tissue region of interest. Although many TK models have been around longer than MRI, the much higher spatial and temporal resolutions achieved by modern dynamic imaging such as MRI, CT, and US are pushing the limitations of some early TK models (Cheong et al., 2004; Donaldson et al., 2010).

In a typical TK experiment using a medical imaging modality, an imaging contrast material, acting like a tracer, is injected intravenously shortly after repeated imaging has started and imaging will continue for a duration to capture the concentration changes of the tracer in each image voxel. A TK model describes how this tracer concentration changes over time after an ideal impulse of tracer enters a tissue region of interest. This description is expressed as the impulse residue function, IRF(t) which is equal to the fraction of this tracer impulse remaining in the tissue region at time t. Solving for the IRF requires a deconvolution process, usually performed by curve fitting, of the convolution relationship between the IRF and tracer concentrations:

$$C(t)=\int_{-\infty}^{t} IF(t)\cdot IRF(t-\tau)d\tau$$

where IF is the input function. This convolution relationship is based on the assumptions of linearity and stationarity about the tracer and the tissue region (Zierler, 1962). Note that IF refers to the actual or local input function (LIF), which is technically different for each tissue region and is not the arterial input function (AIF). AIF will be discussed subsequently. In order to describe the IRF, or to describe how the tracer will behave in the tissue region, we consider how the tissue region appears in the real world (Pries and Secomb, 2014) and identify the relevant parameters that affect the way the injected tracer would move in the tissue region. With these parameters as a basis, some possible TK models, from those that make the least assumptions or restrictions to the most, will be discussed.

Realistically, tracers enter a tissue region, such as an image voxel, via multiple arterioles and capillaries (multiple input functions), and travel through the voxel via multiple passage ways (distributed vascular transit times). There would be different tracer concentrations along these blood vessels (tissue homogeneity and distributed parameter models). Some tracer would influx into the interstitial space and diffuse about within it (e.g. multiple interstitial spaces) until eventually efflux back into the vasculature.

All existing TK analysis consider only one or two (in the case of the liver) input functions (Koh et al., 2008), but the number of input functions is not relevant to the IRF, whose input is an impulse. Early models are mostly compartmental that assume infinitely fast tracer diffusion in both the vascular and interstitial spaces because the temporal resolutions of the data were quite low. Thus, the concept of a vascular transit time and multiple passage ways only exists for non-compartmental models such as tissue homogeneity models (Johnson and Wilson, 1966; St. Lawrence and Lee, 1998a,b) and distributed parameter models. Among the latter, both catenary (Larson et al., 1987) and mammillary (Koh et al., 2003) multiple interstitial spaces were proposed but have not found useful applications. Multiple interstitial spaces compartmental models are common in dynamic PET but irrelevant when non-metabolic and non-binding imagining contrast agents are used as tracers. For example, if one assumes or approximates a tissue region as a single input, single vascular transit time, different tracer concentrations in vascular space, and infinite tracer diffusion coefficient in interstitial space, this would be the tissue homogeneity model proposed by Johnson and Wilson (1966), which was further approximated with an adiabatic approximation by St. Lawrence and Lee (1998a,b).

As mentioned previously, the AIF is different from LIF. In early studies when the whole organ was being studied, the AIF was rightfully the LIF. However, this is not the case if we are looking at a small tissue region. Yet, virtually no imaging system is able to measure the local input function. Furthermore, in MRI, an additional concern is the derivation of the tracer concentrations from MRI signal, affected by the rapid blood flow and high tracer concentrations in major arteries (Sourbron, 2010). How suitable, or what are the assumptions necessary in order to estimate the LIF from the AIF (Calamante, 2013)? Perhaps, we could just treat this input function as an unknown too (Schabel et al., 2010a,b). Could we avoid the input function altogether by using TK field theory proposed by Sourbron (2014)?

Lastly, we briefly discuss some technicalities in implementation. The different bolus arrival times between an AIF and the C(t) or LIF is an old problem. Predetermining this value may not be always feasible (Cheong et al., 2003). If we include this time difference as a parameter to be determined via curve fitting, then we need to deal with the discontinuity issue at the initial time when the IRF jumps from 0 to 1 (Koh et al., 2011). This is also an issue in single vascular transit time models at the end of the vascular transit time (Bartos et al., 2010). Curve fitting may not always be successful and may produce extreme parameter values when it fails. How can we filter out these “bad” voxels (Balvay et al., 2005)? Related to this is the issue of model selection, where model complexity and data quality should match (Cheong et al., 2004; Donaldson et al., 2010).

In this lecture, we learned about the assumptions behind different TK models, the limitations of the AIF, and some technical issues in the curve fitting process. Ideally, the most realistic TK models should be preferred. However, the required high data quality might not be achievable, where less realistic TK models are sufficient. The limitations of AIF have led some researchers to explore new TK analysis approaches such as simultaneous estimation of AIF and TK parameters, and the TK field theory. Curve fitting, or optimization in general, is a tricky process itself. It is important to ensure proper implementation and verify the fitted results to avoid contamination from wrongly fitted results.