Perfusion is the process of body delivering blood to tissue. MRI provides several techniques to study perfusion. In this post, I will summarize the model behind a non-invasive technique called arterial spin labelling (ASL).

## perfusion

The tissue perfusion can be described by the following mass conservation equation \begin{align} \dot m = f (m_a - m_v) \end{align} where $$m(t)$$ is the amount of the tracer per gram tissue, $$f$$ is the cerebral blood flow (CBF) in the unit of $$ml\cdot g^{-1}\cdot s^{-1}$$, and $$m_a (m_v)$$ is the amount of tracer per $$ml$$ arterial (venous) blood. This equation is also known as Fick’s principle.

There are four pools of water protons in the voxel: extravascular tissue, arterial blood, capillary, and venous blood. Under the assumption that water protons in capillary, tissue and venous blood freely exchange, these three components can be considered as one pool. In fact, $$m$$ in the first equation should be understood as both tissue and capillary. The rapid exchange assumption then leads to $$m_v = m/\lambda$$ , where $$\lambda$$ is the brain-blood partition coefficient for water defined as (quantity of water $$/g$$ of brain)/(quantity of water $$/ml$$ of blood). The accepted value is $$\lambda = 0.9 ml/g$$ for the whole brain. It also varies with hematocrit. It is also known that grey matter has higher $$\lambda$$ than white matter on average and there are spatial variations even in one type of tissue. But it is hard to measure thus in practice, only one number for the whole brain is used.

Thus we arrive at the model \begin{align} \dot m = f m_a - \frac{f}{\lambda} m \end{align} Effectively, the venous clearance introduces a temporal decay with decay constant $$f/\lambda$$.

If steady state is achieved, we also have \begin{align} m_a^0 = m^0 / \lambda \end{align} where the superscript denotes steady state.

## ASL perfusion

For ASL, the tracer is the tagged water protons, an extra decay mechanism is thus introduced \begin{align} \dot m = \frac{m^0-m}{T_1} + f m_a - \frac{f}{\lambda} m, \end{align} where $$T_1$$ is the decay constant for tissue (tissue and capillary). Note an implicit assumption is made here: water is completely extracted from the vascular space immediately after arrival in the voxel.

The solution given initial condition $$m(0) = m^0$$ is \begin{align} m(t) = m^0e^{-R_1’t} + \int_0^t e^{-R_1’(t-\tau)}\left[\frac{m^0}{T_1} + fm_a(\tau)\right] d\tau, \end{align} where the apparent decay rate is given by \begin{align} R_1’ = \frac{1}{T_1} + \frac{f}{\lambda}. \end{align}

Note tagging affects $$m_a(t)$$ and typically one reverses $$m_a$$. Thus the measured signal difference between tagged and control is \begin{align} \Delta m(t) = 2\alpha f \int_0^t e^{-R_1’(t-\tau)} m_a(\tau)d\tau, \end{align} where $$\alpha$$ is the tagging efficiency.

At normal perfusion rates of $$40-100 ml/100 g/min$$, the signal change is in the order $$0.5-1.5\%$$ of the full signal. Thus $$30$$ to $$40$$ pairs of control and tagged images need to be repeated for desired SNR.

In an early paper of ASL perfusion, it is assumed that steady states are obtained for the arterial blood and tissue. Furthermore, the arterial blood is assumed not to decay. Thus the control signal is $$m = m^0$$ and $$m_a(t) = m^0/\lambda$$. The signal difference is $$\Delta m = \frac{2\alpha f m^0}{\lambda R_1'}$$. The CBF can be computed as \begin{align} f = \frac{\lambda R_1’}{2\alpha} \frac{\Delta m}{m^0}. \end{align}

### the standard model

In the standard model, the two former assumptions of rapid exchange and immediate water extraction are assumed. One further assumption is made about the arterial input $$m_a(t)$$: it is assumed to be uniform plug flow:

\begin{align} m_a(t) = \begin{cases} 0, \quad 0 < t < t_a \\ m_a^0 e^{-t/T_{1b}} \text{(pulsed)}, \quad t_a< t< \tau + \Delta t \\ m_a^0 e^{-t_a/T_{1b}} \text{(continuous)} \\ 0, \quad t> t_a + \tau \end{cases} \end{align}

Here $$T_{1b}$$ is the decay constant for blood, $$t_a$$ is the arterial transit time (ATT), defined as the time difference from tagging to the arrival of tagged blood. For healthy gray matter, the ATT is approximately $$0.5$$ to $$1.5s$$. For patients and deep white matter, the ATT can be greater than $$2s$$.

Thus, the ASL difference signals are given by

\begin{align} \Delta m(t) = \begin{cases} 0, \quad t< t_a \\ 2\alpha f m_a^0 T_1' e^{-\frac{t_a}{T_{1b}}}q_{ss}(t), t_a < t < t_a+\tau \\ 2\alpha fm_a^0T_1'e^{-\frac{t_a}{T_{1b}}}e^{-R_1'(t-t_a-\tau)}q_{ss}(t), t> t_a+\tau \end{cases} \end{align}

for CASL, where

\begin{align} q_{ss}(t) = \begin{cases} 1-e^{-R_1'(t-t_a)}, \quad t_a < t < t_a+\tau \\ 1-e^{-R_1'\tau}, \quad t> t_a+\tau \end{cases} \end{align}

and

\begin{align} \Delta m(t) = \begin{cases} 0, \quad t< t_a \\ 2\alpha fm_a^0(t-t_a)e^{-\frac{t}{T_{1b}}}q_p(t), t_a < t < t_a+\tau \\ 2\alpha fm_a^0\tau e^{-\frac{t}{T_{1b}}}q_p(t), t>t_a+\tau \end{cases} \end{align}

for PASL, where

\begin{align} q_p(t) = \begin{cases} \frac{1-e^{-k(t-t_a)}}{k(t-t_a)}, t_a< t< t_a+\tau \\ \frac{e^{-k(t-t_a-\tau}\left(1-e^{-k\tau}\right)}{k\tau}, t>t_a+\tau \end{cases} \end{align}

In practice, it is commonly assumed that $$T_{1b}\simeq T_1'$$, i.e., $$k\simeq0$$, thus $$q_p(t)\simeq 1$$. Similarly, it is commonly assumed that $$t>t_a+\tau$$, or $$R_1'\tau\gg1$$ such that steady state is reached, i.e., $$q_{ss}(t)\simeq 1$$. The late acquisition has the advantage of eliminating the signal dependence on $$t_a$$, which is hard to measure and varies voxel by voxel.

Under these assumptions, the CBF can be calculated from the CASL result as \begin{align} f \simeq \frac{\lambda e^{\frac{t-\tau}{T_{1b}}}}{2\alpha T_{1b}\left(1-e^{-\frac{\tau}{T_{1b}}} \right)}\frac{\Delta m(t)}{m^0} \end{align} Here we have used $$m_a^0 = m^0/\lambda$$.