Haemodynamics

The following equations represent mass balance equations for blood volume and deoxyhaemoglobin into the cortical column as the venous compartment expands (like a balloon). These equations describe the physiological aspect of large amounts of blood inflow increasing locally to support small increases in oxygen metabolism rates and an associated change in oxygen extraction fraction. ¹

\[\dot{V}=\frac{f_{in}(t) − f_{out}(t)}{\tau} \]

\[\dot{q} =\frac{ f_{in}(t)\frac{E_f}{E_0} − f_{out}(t)\frac{q(t)}{V(t)} }{τ} \]

where:

\[ \dot{E_{f}} = 1 − \frac{1 − E_0}{1 − f_{in}(t)} \]

\[ \dot{f_{out}} = V (t) \frac{1}{\alpha} + \tau_{vs} \]

• \(V \) = blood volume (normalized with respect to resting state value)
• \(Q \) = deoxyhemoglobin levels (normalized with respect to resting state value)
• \(f_{out}\) = outflow of blood from the model
• \(\tau \) = time constant
• \(E_f\) = oxygen extraction fraction
• \(E_0\) = baseline oxygen extraction fraction
• \(τ_{vs}\) = viscoelastic time constant

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1: Martin Havlicek et al. “Physiologically informed dynamic causal modeling of fMRI data”. In: NeuroImage 122 (Nov. 2015), pp. 355–372. issn: 1053-8119. doi: 10 . 1016 / J . NEUROIMAGE.2015.07.078.