balloon_RL

balloon_RL(
    phi=1.0,
    kappa=1 / 1.54,
    gamma=1 / 2.46,
    E_0=0.34,
    tau=0.98,
    alpha=0.33,
    V_0=0.02,
    v_0=40.3,
    TE=40 / 1000.0,
    epsilon=1.43,
    r_0=25,
)

A balloon model with revised coefficients and linear BOLD equation derived from Stephan et al. (2007).

Equivalent code:

balloon_RL = BoldModel(
    parameters = dict(
        second    = 1000.0,
        phi       = 1.0,
        kappa     = 1/1.54,
        gamma     = 1/2.46,
        E_0       = 0.34,
        tau       = 0.98,
        alpha     = 0.33,
        V_0       = 0.02,
        v_0       = 40.3,
        TE        = 40/1000.,
        epsilon   = 1.43,
        r_0       = 25.,
    ),
    equations = [
        # Single input
        ann.Variable('I_CBF = sum(I_CBF)', init=0.0),
        ann.Variable('ds/dt = (phi * I_CBF - kappa * s - gamma * (f_in - 1))/second', init=0.0),
        ann.Variable('df_in/dt = s / second', init=1.0, min=0.01),

        ann.Variable('E = 1 - (1 - E_0)**(1 / f_in)', init=0.3424),
        ann.Variable('dq/dt = (f_in * E / E_0 - (q / v) * f_out)/(tau*second)', init=1.0, min=0.01),
        ann.Variable('dv/dt = (f_in - f_out)/(tau*second), init=1.0, min=0.01),
        ann.Variable('f_out = v**(1 / alpha)', init=1, min=0.01),

        # Revised coefficients
        ann.Variable('k_1 = 4.3 * v_0 * E_0 * TE'),
        ann.Variable('k_2 = epsilon * r_0 * E_0 * TE'),
        ann.Variable('k_3 = 1 - epsilon'),

        # Linear equation
        ann.Variable('BOLD = V_0 * ((k_1 + k_2) * (1 - q) + (k_3 - k_2) * (1 - v))'),
    ],
    inputs="I_CBF",
)

Parameters

Name Type Description Default
phi input coefficient 1.0
kappa signal decay 1 / 1.54
gamma feedback regulation 1 / 2.46
E_0 oxygen extraction fraction at rest 0.34
tau time constant (in s!) 0.98
alpha vessel stiffness 0.33
V_0 resting venous blood volume fraction 0.02
v_0 frequency offset at the outer surface of the magnetized vessel for fully deoxygenated blood at 1.5 T 40.3
TE echo time 40 / 1000.0
epsilon ratio of intra- and extravascular signal 1.43
r_0 slope of the relation between the intravascular relaxation rate and oxygen saturation 25