Exchange tissues

The two-site exchange tissue, or exchange tissue for short, is the most common tissue type used in applications such as brain, cancer, prostate, muscle, and more. It models tissues that are defined by a vascular bed which receives blood flow and exchanges indicator with an extravascular space.

Data measured on exchange tissues can be analysed most conveniently using the user interface in dcmri.Tissue. Developers can access the core functionality through the functions listed in Exchange tissues.

Definitions and notations

Tissue parameters

Short name	Full name	Definition	Units
Fb	Blood flow	Flow of blood into the vascular space of a unit tissue	mL/sec/cm3
Ktrans	Volume transfer constant	Volume of arterial plasma cleared of indicator per unit time and per unit tissue	mL/sec/cm3
E	Extraction fraction	The fraction of entering particles that will pass through the interstitum at least once	None
vb	Blood volume fraction	Volume fraction of the blood space	mL/cm3
H	Hematocrit	Volume fraction of the red blood cells in whole blood	None
vi	Interstitial volume	Volume fraction of the interstitial space	mL/cm3
ve	Extracellular volume	Combined volume fraction of plasma and interstitium	mL/cm3
vc	Cellular volume	Volume fraction of the tissue cells	mL/cm3
PSe	Endothelial water permeability	Flow of water across the endothelium per unit tissue volume	mL/sec/cm3
PSc	Cytolemmal water permeability	Flow of water across the cell wall per unit tissue volume	mL/sec/cm3

Kinetic models

Short name	Full name	Parameters	Assumptions
2CX	Two-compartment exchange	H, vb, vi, Fb, PS	\(PS_i = PS\)
2CU	Two-compartment uptake	H, vb, Fb, PS	\(PS_i = 0\)
HF	High-flow, AKA extended Tofts model, extended Patlak model, general kinetic model.	H, vb, vi, PS	\(F_b = \infty\)
HFU	High flow uptake, AKA Patlak model	H, vb, PS	\(F_b = \infty\), \(PSi = 0\)
FX	Fast indicator exchange	H, ve, Fb	\(PS = \infty\)
NX	No indicator exchange	vb, Fb	\(PS = 0\)
U	Uptake	Fb	\(F_v = 0\)
WV	Weakly vascularized, AKA Tofts model.	H, vi, Ktrans	\(v_b = 0\)

Configurations

The configuration of an exchange tissue is fully defined by: a pharmacokinetic model (see section Indicator kinetics), which describes the transport of indicator through the tissue; and a water-exchange model (see section Water exchange), which describes the transport of water and magnetization. Any pharmacokinetic model can be combined with any water exchange model to build a complete tissue model.

The parameters that characterise an exchange tissue depend on the configuration. Table Tissue parameters lists all relevant parameters, and a list of parameters for each configuration can be found through the function dcmri.params_tissue or in its documentation. The most commonly used tissue types have fast water exchange across all barriers, in which case the parameters are those listed in table Kinetic models.

Indicator kinetics

The most general pharmacokinetic model implemented in dcmri.Tissue is the 2-compartment exchange model (2CX). It assumes that the indicator distributes over two compartments: the plasma p and the interstitium i, with volume fractions vp and vi, respectively. The plasma volume is related to the blood volume vb by the hematocrit H; plasma and interstitium combined form the extracellular space e with volume fraction ve:

\[v_p = (1-H)v_b \qquad \textrm{and} \qquad v_e = v_p + v_i\]

Fp is the flow of plasma into p via the arterial inlet, which equals the flow of plasma out of the venous outlet. It is related to the blood flow Fb via the hematocrit H:

\[F_p = (1-H)F_b\]

The transport of indicator across the endothelium (the barrier separating p and i) is quantified by the indicator’s permeability-surface area product PS. The most general 2CX model assumes that the transport of indicator across the endothelium is the same in each direction, i.e. the PS from interstitium to plasma (PSi) is the same as the PS from plasma to interstitium. Any leakage from the interstitium via lymphatic flow or otherwise is ignored.

The indicator extraction fraction E and the volume transfer constant Ktrans measure the uptake of indicator into the interstitium. E is the fraction of the indicator that enters the interstitium at least once in a transit through the tissue. Ktrans is the rate at which indicator is delivered to the interstitium. E and Ktrans are related to the other parameters:

\[E=\frac{PS}{PS+F_p} \qquad \textrm{and} \qquad K^{\mathrm{trans}}=EF_p\]

The other kinetic models available through dcmri.Tissue are all special cases of 2CX. An overview can be found in table Kinetic models:

• The two-compartment uptake model (2CU) applies when the acquisition time is short so that the return of indicator to the vasculature is not detectable.

• The high-flow model (HF) applies when the temporal resolution of the measurement is too low to see any dispersion in the vasculature, in which case the blood flow is above the detection limit.

• The high-flow uptake model (HFU) combines the assumptions of the above.

• The fast-exchange model (FX) assumes that the transport across the endothelium is so rapid that plasma and interstitium are effectively well-mixed.

• The no-exchange model (NX) applies when the opposite is the case, when no measureable amounts of indicator leak out of the vasculature.

• The uptake model (U) applies when data are truncated to the first seconds after indicator arrival when any effect of venous outflow (Fv) is not detectable.

• The weakly vascularised model (WV) applies when the indicator in the vasculature is below the detection limit.

Water exchange

For two-compartment exchange tissues, the three tissue compartments exchanging water are the blood, interstitium and tissue cells. Water exchange refers to the transport of water across the barriers between them: transendothelial water exchange between blood and interstitium, and transcytolemmal water exchange between interstitium and tissue cells. The water exchange in the blood compartment between plasma and red blood cells is assumed to be in the fast exchange limit throughout.

Since water occupies intracellular spaces, water exchange models introduce a dependence on the intracellular volumes. The volume fraction of the tissue cells is measured by the parameter vc, and it is assumed that cells, blood and interstitium compse the entire tissue:

\[v_b + v_i + v_c = 1\]

The rate of water exchange across a barrier is quantified by the permeability-surface area (PS) of water, a quantity in units of mL/sec/cm3. PSe is the transendothelial water exchange rate and PSc is the transcytolemmal water rate.

Water exchange across either of these two barriers can be in the fast-exchange limit (F), restricted (R), or there may be no water exchange at all (N). Since there are two barriers involved this leads to 3x3=9 possible water exchange regimes. dcmri.Tissue denotes these 9 regimes by a combination of the letters F, R and N: the first letter refers to the water exchange across the endothelium, and the second to the water exchange across the cell wall. Examples of possible water exchange regimes are:

• RF: Restricted water exchange across the endothelium (\(0\lt PS_e\lt\infty\)) and fast water exchange across the tissue cell wall (\(PS_c=\infty\)).

• NF: No water exchange across the endothelium (\(PS_e=0\)) and fast water exchange across the tissue cell wall (\(PS_c=\infty\)).

• FR: Fast water exchange across the endothelium (\(PS_e=\infty\)) and restricted water exchange across the tissue cell wall (\(0\lt PS_c\lt\infty\)).