Relaxation

Contrast agents and their concentration are visible in MRI because the agents are designed to modify the relaxation rates of tissues. This section deals with the relationship between contrast agent concentration and magnetic relaxation rates.

The detailed interaction between contrast agent molecules and magnetic tissue properties can be complex, but fortunately the relationship between concentrations and relaxation rates can be modelled relatively easily with simple approximations. See the table with definitions for a summary of relevant terms and notations.

Definitions and notations

Models of magnetic relaxation are determined by the following parameters:

Relaxation model parameters

Short name	Full name	Definition	Units
\(R_1\)	Longitudinal relaxation rate	Reciprocal of longitudinal relaxation time	Hz
\(M_z\)	Longitudinal tissue magnetization	Component of the tissue magnetization parallel to the magnetic field	A/cm/cm3
\(M_{ze}\)	Equilibrium longitudinal tissue magnetization	Longitudinal magnetization at rest	A/cm/cm3
\(m_z\)	Longitudinal magnetization	Longitudinal magnetization per unit water volume	A/cm/mL
\(m_{ze}\)	Equilibrium longitudinal magnetization	Longitudinal magnetization per unit water volume at rest	A/cm/mL
\(m_{zi}\)	Inlet longitudinal magnetization	Magnetization of the water flowing into the tissue	A/cm/mL
\(J\)	Tissue magnetization flux	Magnetization flux per unit of tissue volume	A/cm/sec/cm3
\(N_z\)	Normalized longitudinal tissue magnetization	\(M_{z}/m_{ze}\)	mL/cm3
\(n_z\)	Relative longitudinal tissue magnetization	\(N_{z}/v\)	dimensionless
\(j\)	Normalized tissue magnetization flux	\(J/m_{ze}\)	mL/sec/cm3
\(R_{10}\)	Precontrast longitudinal relaxation rate in tissue	Native longitudinal relaxation rate in the absence of contrast agent	Hz
\(r_1\)	Longitudinal relaxivity	Increase in longitudinal relaxation rate \(R_1\) per unit concentration	Hz/M
\(r^*_2\)	Transverse relaxivity	Increase in transverse relaxation rate \(R^*_2\) per unit concentration	Hz/M
\(v\)	Water volume fraction	Volume fraction of the space occupied by water	mL/cm3
\(f_i\)	Water inflow	Volume of water flowing in per unit of time and per unit of tissue	mL/sec/cm3
\(f_o\)	Water outflow	Volume of water flowing out per unit of time and per unit of tissue	mL/sec/cm3
\(PS_{kl}\)	Magnetization permeability-surface area from l to k	Magnetization transfer rate from compartment l to compartment k	mL/sec/cm3

Longitudinal relaxation

Fast water exchange

We consider a tissue with uniform magnetization. Magnetization is carried in by inflow of magnetized water and carried out by water flow and relaxation. The longitudinal magnetization is governed by the Bloch equation:

(1)\[ v\frac{dm_z}{dt} = f_i m_{zi} - f_o m_z + R_1 v (m_{ze} - m_z)\]

After regrouping terms and writing this in terms of the total magnetization \(M_z=vm_z\):

(2)\[ \frac{dM_z}{dt} = J - KM_z\]

where we define influx and rate constants:

(3)\[\begin{split} J &= R_1 v\, m_{ze} + f_i m_{zi} \\ K &= R_1 + \frac{f_o}{v}\end{split}\]

Note this is in fact just another one-compartment model (see section Compartment), with the magnetization \(M_z\) playing the role of tracer. If \(K\) is a constant, or we are considering sufficiently short time scales so that it can be assumed to be constant, the solution is:

(4)\[ M_z = e^{-tK}M_z(0) + e^{-tK}*J(t)\]

If additionally the influx \(J\) can be assumed constant, we can compute the convolution:

(5)\[ M_z = e^{-tK}M_z(0) + \left(1-e^{-tK}\right)K^{-1} J\]

If the flow terms are negligible compared to the relaxation rates, then we have:

\[J = KM_{ze}\]

This is also true whenever the inflowing magnetization is in equilibrium - as can be seen from applying Eq. (1) to the equilibrium state. In either of these scenarios we have \(J/K=M_{ze}\), which produces the familiar solution for free longitudinal relaxation:

(6)\[ M_z = e^{-tR_1}M_z(0) + \left(1-e^{-tR_1}\right)M_{ze}\]

Restricted water exchange

The above solution assumes the tissue magnetization is uniform, i.e. the water moves so quickly between tissue compartments that any differences in magnetization are immediately levelled out. If that is not the case, the exchange of magnetization between the tissue compartments must be explicitly incorporated.

We consider this for the example of two interacting water compartments \(1,2\). The generalization to N compartments is then straightforward. We can write a Bloch equation for each and now explicitly include the exchange of magnetization between them. As there is no confusion possible we drop the z-indices for this section to avoid overloading the notations:

\[\begin{split}v_1\frac{dm_1}{dt} &= f_{i,1}m_{i,1} - f_{o,1}m_1 + R_{1,1}v_1(m_{e,1}-m_1) + PS_{12}m_2 - PS_{21}m_1 \\ v_2\frac{dm_2}{dt} &= f_{i,2}m_{i,2} - f_{o,2}m_2 + R_{1,2}v_2(m_{e,2}-m_2) + PS_{21}m_1 - PS_{12}m_2\end{split}\]

The magnetization transfer \(PS_{lk}m_k\) will be mediated by physical water flow, but other mechanisms of magnetization transfer between compartments may also be at play. The basic assumption is that the transfer is proportional to the water magnetization - as long as this is true the equation is valid and the precise mechanism of transfer only affects the physical interpretion of \(PS\).

Gathering terms and expressing the result in terms of the total magnetization \(M=vm\), this takes the familiar form of a two-compartment model (see section N-compartment system):

\[\begin{split}\frac{dM_1}{dt} &= J_1 - \Lambda_1M_1 + \Lambda_{12}M_2 \\ \frac{dM_2}{dt} &= J_2 - \Lambda_2M_2 + \Lambda_{21}M_1\end{split}\]

Here we define rate constants:

\[\begin{split}\Lambda_1 &= R_{1,1} + \frac{f_{o,1} + PS_{21}}{v_1} \qquad \Lambda_{12}=\frac{PS_{12}}{v_2} \\ \Lambda_2 &= R_{1,2} + \frac{f_{o,2} + PS_{12}}{v_2} \qquad \Lambda_{21}=\frac{PS_{21}}{v_1}\end{split}\]

and an influx of magnetization:

\[\begin{split}J_1 &= R_{1,1}v_1 m_{e,1} + f_{i,1}m_{i,1} \\ J_2 &= R_{1,2}v_2 m_{e,2} + f_{i,2}m_{i,2}\end{split}\]

In matrix form the Bloch equations are exactly the same as the n-compartment kinetic equations:

(7)\[ \frac{d\mathbf{M}}{dt} = \mathbf{J} - \mathbf{K} \mathbf{M}\]

Here \(\mathbf{K}\) is a square matrix which has off-diagonal elements \(-\Lambda_{ij}\) and diagonal elements \(\Lambda_i\).

The equations, and therefore their solutions, are formally identical to the fast-exchange situation (Eq. (2)). If the relaxation rates \(R_1\) are constant in time, or changing slowly on the time scale we are interested in, the solution is a direct generalization of the fast exchange case (see Eq. (4)):

\[\mathbf{M}(t) = e^{-t\mathbf{K}}\mathbf{M}(0) + e^{-t\mathbf{K}}*\mathbf{J}\]

If additionally the influx \(\mathbf{J}\) is constant, the result is formally the same as Eq. (5):

(8)\[\mathbf{M}(t) = e^{-t\mathbf{K}}\mathbf{M}(0) + \left(1-e^{-t\mathbf{K}}\right) \mathbf{K}^{-1}\mathbf{J}\]

As for the one-compartment case, if the flow terms are negligible, or when the inflowing magnetization is in equilibrium, we have:

\[\mathbf{J} = \mathbf{K}\mathbf{M}_{e}\]

And the solution simplifies:

(9)\[\mathbf{M}(t) = e^{-t\mathbf{K}}\mathbf{M}(0) + \left(1-e^{-t\mathbf{K}}\right) \mathbf{M}_e\]

The effect of contrast agents

With standard doses of contrast agents used in in-vivo MRI acquisitions, the contrast agent increases the longitudinal relaxation rate of tissue in proportion to its concentration:

(10)\[R_1(c) = R_{10} + r_1 c\]

The relaxivity \(r_1\) is a constant which depends on the contrast agent. It generally has at most a weak dependence on tissue type, except for contrast agents which exihibit stronger levels of protein binding. This linear relationship is a very good approximation under most conditions.

In the absence of contrast agent, tissues with different \(R_1\) values nevertheless show mono-exponential longitudinal relaxation because of the fast water exchange between them. The magnetization in this fast water-exchange limit relaxes with a single \(R_1\) which is a weighted average of the \(R_1\) values of the different compartments:

\[R_1 = \sum_i v_i R_{1,i}\]

The result can be proven by considering the limit \(PS>>R_1\) in a multi-compartment model.

If each tissue component has a different concentration \(c_i\), but each compartment has the same relaxivity \(r_1\), the relaxation rate shows a linear dependence on the total tissue concentration \(C\):

\[R_1 = R_{10} + r_1 C \quad\textrm{with}\quad R_{10} = \sum_i v_i R_{10,i} \quad\textrm{and}\quad C = \sum_i v_i c_i\]

In this regime the longitudinal relaxation is not affected by how the indicator is distributed over the compartments exactly. This is no longer the case if the tissue compartments have different relaxivities. In that case the result must be generalized:

\[R_1 = R_{10} + \sum_i r_{1,i} v_ic_i\]

In this case, the change in \(R_1\) is explicitly dependent on the exact distribution of the indicator over the tissue compartments. In other words, two states with the same total tissue concentration \(C\) can nevertheless have different \(R_1\) values. In such a scenario, the concentrations cannot be derived directly from the relaxation rates. A relevant example is the use of the hepatobiliary agent gadoxetate, which at most field strengths shows a 2-fold increase in relaxivity as soon as it enters the hepatocytes.

If the tissue is not in the fast water exchange limit, it is no longer characterised by a single \(R_1\) value, and the effect of concentration must be determined by applying Eq. (10) to the relaxation rates of each compartment individually.

Transverse relaxation

Like longitudinal relaxation, transverse magnetization is often approximated by a linear relationship:

\[R^*_2(C) = R^*_{10} + r^*_2 C\]

However, unlike the longitudinal relaxivity \(r_1\), the transverse relaxivity \(r^*_2\) is strongly dependend on tissue type. Hence using literature values is not usually realistic.

[… coming soon …] The effect of contrast agent leakage.