European Journal of Applied Physiology

, Volume 118, Issue 11, pp 2443–2454 | Cite as

A multiscale model for the simulation of cerebral and extracerebral blood flows and pressures in humans

  • Giacomo Gadda
  • Marcin Majka
  • Piotr Zieliński
  • Mauro Gambaccini
  • Angelo TaibiEmail author
Original Article



Brain hemodynamics is fundamental for the functioning of the human being. Many biophysical factors affect brain circulation, so that a satisfactory understanding of its behavior is challenging. We developed a mathematical model to simulate cerebral and extracerebral flows and pressures in humans.


The model is composed of an anatomically informed 1-D arterial network, and two 0-D networks of the cerebral circulation and brain drainage, respectively. It takes into account the pulse-wave transmission properties of the 55 main arteries and the main hydraulic and autoregulation mechanisms ensuring blood supply and drainage to the brain. Proper pressure outputs from the arterial 1-D model are used as input to the 0-D models, together with the contribution to venous pressure due to breathing that simulates the drainage effect of the thoracic pump.


The model we developed is able to link the arterial tree with the venous pathways devoted to the brain drainage, and to simulate important factors affecting cerebral circulation both for physiological and pathological conditions, such as breathing and hypo/hypercapnia. Finally, the average value of simulated flows and pressures is in agreement with the available experimental data.


The model has the potential to predict important clinical parameters before and after physiological and/or pathological changes.


Mathematical modeling Circulation Cerebral and extracerebral parameters Breathing \(\mathrm{CO}_{2}\) pressure change 

List of symbols


Parent vessel


Cross-sectional area


Area of the vessel when the transmural pressure is zero


Parameter related to \(\mathrm{CO}_{2}\) autoregulation


Coriolis coefficient


Constant parameter related to \(\mathrm{CO}_{2}\) autoregulation


Coefficient related to C


Wave speed


Capacitance per unit length of vessel


Capacitance in the upper segment of the right IJV


Capacitance of pial arterioles


Central value of the capacitance of pial arterioles


Cerebrospinal fluid


First daughter vessel


Second daughter vessel

\(\Delta C_{\mathrm{pa}}\)

Amplitude of the sigmoidal autoregulation curve for the capacitance of pial arterioles


Young modulus


Frictional force per unit length


Autoregulation gain


Conductance of the upper right anastomotic connection


\(\mathrm{CO}_{2}\) reactivity gain


Conductance of the middle segment of the right IJV


Conductance of the upper segment of the right IJV


Wall thickness


Mock CSF possibly injected into or subtracted from the cranial cavity


Internal jugular vein


Parameter related to the collapsibility of the IJV segments


Constant parameter related to \(\mathrm{CO}_{2}\) autoregulation


Constant parameter related to capacitance of pial arterioles


Basal conductance of the upper segment of the right IJV


Constant parameter related to resistance of pial arterioles


Inductance per unit length of vessel


Blood viscosity


Pressure inside the vessel


Arterial pressure


Arterial \(\mathrm{CO}_{2}\) pressure


Set point of the arterial \(\mathrm{CO}_{2}\) pressure


Pressure in the upper segment of the collateral network


Central venous pressure


External pressure


Intracranial pressure


Pressure outside the upper segment of the right IJV


Pressure in the middle segment of the right IJV


Pressure in the upper segment of the right IJV


Venous sinuses pressure


Volumetric flow


Cerebral blood flow


CSF fluid outflow rate


Flow in the distal azygos


Flow in the proximal azygos


Flow in the lower segment of the collateral network


Flow in the middle segment of the collateral network


Flow in the upper segment of the collateral network


Flow in the lower anastomotic connection (left side)


Flow in the upper anastomotic connection (left side)


Flow in the lower anastomotic connection (right side)


Flow in the upper anastomotic connection (right side)


Flow in the external carotid arteries (flow to face and neck)


CSF formation rate


Flow in the lower segment of the left IJV


Flow in the middle segment of the left IJV


Flow in the upper segment of the left IJV


Flow in the lower segment of the right IJV


Flow in the middle segment of the right IJV


Flow in the upper segment of the right IJV


Flow in the lumbar vein


Set point of Q


Flow in the renal vein


Flow in the upper segment of the superior vena cava (IJV confluence)


Flow in the lower segment of the superior vena cava


Flow in the vertebral veins


Flow in the left vertebral vein


Flow in the right vertebral vein


Resistance per unit length of vessel


Resistance of small vessels and terminal branches


Reflection coefficient


Reflection coefficient for wave propagating in the parent vessel a


Blood density


Resistance of pial arterioles


Reflection coefficient of the aortic valve


Differential cross-sectional area



\(\tau _{\mathrm{aut}}\)

Time constant of the autoregulation mechanism

\(\tau _{\mathrm{CO}_{2}}\)

Time constant of the \(\mathrm{CO}_{2}\) reactivity effect


Blood velocity


Average axial blood velocity


Blood volume in the pial arterioles


Longitudinal direction


State variable that accounts for the effect of autoregulation


State variable that accounts for the effect of \(\mathrm{CO}_{2}\) reactivity


Coefficient related to R


Characteristic impedance


Characteristic impedance of the peripheral segment



No funding was received.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

Research involving human participants and/or animals

This article does not contain any studies with human participants or animals performed by any of the authors.

Data availability

All data generated or analyzed during this study are included in this published article.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Giacomo Gadda
    • 1
  • Marcin Majka
    • 2
  • Piotr Zieliński
    • 2
  • Mauro Gambaccini
    • 1
  • Angelo Taibi
    • 1
    Email author
  1. 1.Department of Physics and Earth SciencesUniversity of FerraraFerraraItaly
  2. 2.Institute of Nuclear PhysicsPolish Academy of SciencesKrakówPoland

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