Pharmacokinetics

An indicator is a foreign substance added to a body fluid which is separately detectable. A tracer is a special kind of indicator which behaves in exactly the same way as a component of the body fluid. Labelled water such as that used in arterial spin labelling is a tracer. MRI contrast agents are an example of an indicator that is not a tracer.

Pharmacokinetics creates the critical link between tissue parameters and concentration-time profiles of an indicator. In DC-MRI the indicator is usually an MR contrast agent. It is initially added to venous blood, and then carried through the circulation and distributed through the rest of the body.

dcmri provides a library of simple pharmacokinetic models, which can be assembled to build more complex pharmacokinetic models (see section PK blocks). For each system, functions are included which return the tissue concentration C(t) and outflux J(t) of the system.

The sections below define the models and their solutions in more detail. See the table with definitions for a summary of relevant terms and notations. This introduction aims to provide sufficient detail to unambiguously define all models included in dcmri, but is not intended to provide a full introduction to pharmacokinetics. For that we refer to classic textbooks or review papers.

Definitions and notations

Pharmacokinetic models: definitions and notations

Short name	Full name	Definition	Units
J	Indicator flux	The amount of indicator molecules entering or leaving a system in a unit of time.	mmol/sec
c	Indicator concentration	The amount of indicator molecules relative to the volume of distribution.	mmol/mL
C	Indicator tissue concentration	The amount of indicator molecules relative to the volume of tissue.	mmol/cm3
v	Indicator volume of distribution	The fraction of the tissue accessible to the indicator.	mL/cm3
T	Indicator mean transit time	The average time an indicator molecule needs to pass through the tissue.	sec
R	Residue function	The fraction of indicator left at time t of an injection at time t=0.	dimensionless
h	Propagator	The transit time distribution	1/sec
K	Compartment rate constant	The ratio between outflux and tissue concentration.	1/sec

Indicator transit times

Explicit definitions of pharmacokinetic models are built on the conservation of indicator mass:

\[\frac{dC}{dt}(t) = J_\mathrm{in}(t) - J_\mathrm{out}(t)\]

For linear and stationary systems, solutions can always be written in terms of the residue function R(t) and the propagator h(t):

\[J_\mathrm{out}(t) = h(t)*J_\mathrm{in}(t) \qquad\textrm{and}\qquad C(t) = R(t)*J_\mathrm{in}(t)\]

Here the symbol \(*\) denotes the convolution product of two functions, which is defined in this context as:

\[(f*g)(t) = \int_0^t d\tau f(\tau)g(t-\tau)\]

The propagator h(t) can be interpreted as the probability distribution of transit times through the tissue. The expectation value of h(t) is therefore the mean transit time T of the system. Substituting the definitions of R(t) and h(t) in the equation of mass conservation we find a relationship between them:

\[R(t) = 1 - \int_0^t h(\tau) d\tau\]

This implies that the mean transit time is also the area under R(t).

dcmri includes discrete approximations of R(t) and h(t) for most systems. In some cases these are trivial, but implementations are nevertheless provided for completeness. We are assuming throughout this section and all implementations that t=0 at the start of the acquisition and the system contains no indicator concentrations at that time.

Trap

The simplest pharmacokinetic system is the trap, which captures all the indicator that enters it (see section Trap for implementations):

\[C(t) = \int_0^t J_\mathrm{in}(\tau)d\tau\]

In practice a system behaves like a trap when the shortest transit times are longer than the acquisition time. In that case any loss of indicator falls beyond the measurement window and the concentrations are accurately predicted by modelling the system as a trap.

Pass

A pass is a space where the concentration is proportional to the input (see section Pass for implementations). For a pass with mean transit time T and volume fraction v, the tissue concentration is proportional to the influx or inlet concentration:

\[C(t) = T J_\mathrm{in}(t) = vc_\mathrm{in}(t)\]

In practice it is used to model tissues where the transit times are shorter than the temporal sampling interval. Under these conditions any bolus broadening is not detectable.

Compartment

A compartment is a space where the outflux is proportional to the concentration (see section Compartment for implementations). This is particularly true in systems that are well-mixed, i.e. have a uniform concentration throughout.

Expressing conservation of indicator mass provides the mathematical definition of a compartment:

\[\frac{dC}{dt}(t) = J_\mathrm{in}(t) - KC(t)\]

Here K is the rate constant of the compartment. The solution is:

\[C(t) = e^{-Kt}*J_\mathrm{in}(t)\]

This shows that the residue function of a compartment is a mono-exponential, and its mean transit time is therefore the reciprocal 1/K. The propagator of a compartment is a normalized exponential:

\[h(t) = \frac{e^{-t/T}}{T}\]

Plug flow

A plug-flow system is a space where all indicator particles have a constant velocity u (see section Plug flow for implementations). Indicator motion through a plug-flow system can be modelled as a one-dimensional system with mass conservation at each point:

\[\frac{\partial C}{\partial t}(x,t) = -u\frac{\partial C}{\partial x}(x,t)\]

A plug flow system is in many ways the opposite of a compartment as it does not allow for any mixing at all. Indicator concentrations at the outlet are shifted in time but are not otherwise distorted:

\[J_\mathrm{out} (t) = J_\mathrm{in}(t-T)\]

The mean transit time T equals u/L, where L is the distance between in- and outlet. The concentration inside a plug flow system is found by integrating the mass conservation:

\[C(t) = \int_0^t d\tau \left(J_\mathrm{in}(\tau) - J_\mathrm{in}(\tau-T)\right)\]

Chain

A chain is a serial arrangement of n identical compartments, each with a transit time T/n (see section Chain for implementations). The mean transit time of a chain is T and the propagator is a convolution of n exponentials (see also dcmri.nexpconv). This takes the form of a normalized gamma-variate function which is known to provide a good model for concentration-time curves after rapid indicator injection:

\[h(t) = \frac{1}{\Gamma(n)}\left(\frac{t}{T/n}\right)^{n-1} \frac{e^{-t/T/n}}{T/n}\]

With \(n\to\infty\) a chain becomes a plug flow system, and with \(n=1\) a chain is a single compartment. If we introduce a dispersion parameter D = 1/n with values in the range of [0,1], then a chain is fully characterized by two numbers (T,D) which has a compartment (D=1) and a plug flow system (D=0) as special cases. Moreover while the physical definition involves a discrete system of n compartments, the solution allows for D to take any value in the range [0,1].

In practice a chain can therefore be used to model tissues that cause an unknown level of indicator dispersion in between the extremes of no dispersion (D=0) and maximal dispersion (D=1).

Plug-flow compartment

A plug-flow compartment is a serial arrangement of a compartment with mean transit time DT and and plug-flow system with mean transit time (1-D)T (see section Plug-flow compartment for implementations). The total mean transit time of a plug-flow compartment is T. The dimensionless parameter D can take any values in the interval [0,1] and the system has a plug-flow system (D=0) and a compartment (D=1) as special cases.

The propagator of a plug-flow compartment is a delayed exponential function:

\[\begin{split}h(t<(1-D)T) &= 0 \\ h(t>(1-D)T) &= \frac{e^{-(t-(1-D)T)/DT}}{DT}\end{split}\]

A plug-flow compartment is similar to a chain in that it can be used to model tissues with an unknown level of dispersion by varying the dispersion parameter D. Its internal structure is coarser but it is computationally more efficient than a chain.

Step

A step is a system where the transit time distribution is a step function with a constant non-zero value between the time points (1-D)T and (1+D)T (see section Step for implementations):

\[h( (1-D)T < t < (1+D)T ) = 1/(2DT)\]

And h(t)=0 otherwise. The mean transit time of a step is T and just like the chain and the plug-flow compartment, the dispersion parameter D takes values in the range [0,1] where D=1 represents maximum dispersion, and D=0 is a plug-flow system without dispersion.

Free

A free system is one where the transit time distribution can take any required form. The transit time distribution is parametrized as a histogram with any number of bins. The model parameters are the n+1 boundaries of the n bins, and the n frequencies of each bin (see section Free for the available functions). For model fitting the boundaries are usually treated as fixed parameters, and the frequencies are treated as unknowns.

N-compartment system

An n-compartment system is a collection of n interacting compartments (see section N-compartment models for the available functions). Each compartment in the system can exchange with any other compartment, and with the external environment. The system is therefore characterized by n equations of the following form:

\[\frac{dC_i}{dt}(t) = J_i(t) - \sum_{j=1}^n K_{ji}C_i(t) + \sum_{j\neq i}K_{ij}C_j(t)\]

Here \(J_i(t)\) is the influx into compartment i fom the environment. The system is fully determined by the \(n^2\) rate constants \(K_{ji}\) which represent the rate constants for the outflux from i to j if \(i\neq j\), and the rate constant for the outflux from i to the environment if \(i= j\). Arranging the n concentrations and influxes in arrays \(\mathbf{C}\) and \(\mathbf{J}\) we can write this in a form very similar to the one-compartment case:

\[\frac{d\mathbf{C}}{dt} = \mathbf{J} - \mathbf{\Lambda} \mathbf{C}\]

Here \(\Lambda\) is a square matrix which has off-diagonal elements \(-K_{ij}\) and diagonal elements \(K_i = \sum_j K_{ji}\). The general solution also has the same form as the one-compartment case, except that it now involves a matrix exponential:

\[\mathbf{C}(t) = e^{-\mathbf{\Lambda}t} * \mathbf{J}(t)\]

The mean transit time of each compartment is \(T_i=1/K_i\) and the extraction fraction \(E_{ji}\) from i to j is the ratio \(K_{ji}/K_i\). An alternative way of characterizing the system is therefore in terms of the n mean transit times \(T_i\) and the n(n-1) extraction fractions \(E_{ji}\).

2-compartment exchange

A two-compartment exchange model is a 2-compartment model with a central compartment that exchanges with the environment, and a second compartment that only exchanges with the central compartment (see section 2-compartment exchange). It is characterised by 3 parameters: the mean transit times of both compartments, and the extraction fraction from the central compartment into the second compartment. Since this is an example of an n-compartment model, the solution can be obtained from these more general functions, but a dedicated solution for the 2-compartment exchange model is more convenient to use.

Non-stationary compartment

A non-stationary compartment is a compartment with a rate constant that is a function of time (see section Non-stationary compartment for implementations):

\[\frac{dC}{dt}(t) = J_\mathrm{in}(t) - K(t) C(t)\]

In this case the solution can no longer be expressed as a convolution. Instead the equation must be solved numerically, for instance by forward propagation over small time steps dt:

\[C(t+dt) = dt J_\mathrm{in}(t) + (1 - dt K(t)) C(t)\]

The solution is stable if the time steps are small enough, i.e. \(dt K(t) < 1\) for any time t. This also states that the time step dt must be smaller than the shortest mean transit time T(t) of the compartment. When fitting data with unknown transit time, suitable lower-bounds must be placed on the values of T(t) to avoid very small values of dt and correspondingly large computation times.

A non-stationary compartment would be used in situations where the tissue properties themselves change in the course of the measurement - for instance because the acquisition is very long, or because rapid physiological changes are taking place. In practice the number of free parameters can be reduced by interpolating between values at particular times. For instance, parameters \(K(t)\) can be determined for each \(t\) by interpolating between two values \((K_i, K_f)\) at the initial and final time points, respectively. When the model is used to explain measured data, those two values would then be treated as free parameters.

Michaelis-Menten compartment

The Michaelis-Menten compartment is a compartment where the rate constant depends on the concentration (see section Michaelis-Menten compartment for implementations):

\[\frac{dC}{dt} = -K(C) C \qquad\textrm{with}\qquad K(C) = \frac{V_\max}{K_m+C}\]

For small enough concentrations \(C << K_m\) this reduces to a standard linear compartment with \(K=V_\max/K_m\). The Michaelis-Menten compartment would therefore mainly be used in situations where higher doses of contrast agent are injected. It is a classic example of a non-linear system and an analytical solution is available through the work of Schnell and Mendoza.