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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 Taibi
Original Article
  • 98 Downloads

Abstract

Purpose

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.

Methods

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.

Results

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.

Conclusions

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

Keywords

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

List of symbols

\(a\)

Parent vessel

A

Cross-sectional area

\(A_{0}\)

Area of the vessel when the transmural pressure is zero

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

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

\(\alpha\)

Coriolis coefficient

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

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

\(\beta\)

Coefficient related to C

c

Wave speed

C

Capacitance per unit length of vessel

\(C_{\mathrm{jr}3}\)

Capacitance in the upper segment of the right IJV

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

Capacitance of pial arterioles

\(C_{\mathrm{pan}}\)

Central value of the capacitance of pial arterioles

\(\text{CSF}\)

Cerebrospinal fluid

\(d_{1}\)

First daughter vessel

\(d_{2}\)

Second daughter vessel

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

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

E

Young modulus

f

Frictional force per unit length

\(G_{\mathrm{aut}}\)

Autoregulation gain

\(G_{\mathrm{cjr}3}\)

Conductance of the upper right anastomotic connection

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

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

\(G_{\mathrm{jr}2}\)

Conductance of the middle segment of the right IJV

\(G_{\mathrm{jr}3}\)

Conductance of the upper segment of the right IJV

h

Wall thickness

\(\text{hbf}\)

Mock CSF possibly injected into or subtracted from the cranial cavity

\(\text{IJV}\)

Internal jugular vein

k

Parameter related to the collapsibility of the IJV segments

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

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

\(k_{\mathrm{C}_{pa}}\)

Constant parameter related to capacitance of pial arterioles

\(k_{\mathrm{jr}3}\)

Basal conductance of the upper segment of the right IJV

\(k_{\mathrm{R}}\)

Constant parameter related to resistance of pial arterioles

L

Inductance per unit length of vessel

\(\mu\)

Blood viscosity

P

Pressure inside the vessel

\(P_{\mathrm{a}}\)

Arterial pressure

\({\text{Pa}}_{\mathrm{CO}_{2}}\)

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

\(\mathrm{Pa}_{\mathrm{CO}_{2}n}\)

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

\(P_{\mathrm{c}3}\)

Pressure in the upper segment of the collateral network

\(P_{\mathrm{cv}}\)

Central venous pressure

\(P_{\mathrm{ex}}\)

External pressure

\(P_{\mathrm{ic}}\)

Intracranial pressure

\(P_{\mathrm{j}3ext}\)

Pressure outside the upper segment of the right IJV

\(P_{\mathrm{jr}2}\)

Pressure in the middle segment of the right IJV

\(P_{\mathrm{jr}3}\)

Pressure in the upper segment of the right IJV

\(P_{\mathrm{vs}}\)

Venous sinuses pressure

q

Volumetric flow

Q

Cerebral blood flow

\(Q_{0}\)

CSF fluid outflow rate

\(Q_{\mathrm{azy}1}\)

Flow in the distal azygos

\(Q_{\mathrm{azy}2}\)

Flow in the proximal azygos

\(Q_{\mathrm{c}1}\)

Flow in the lower segment of the collateral network

\(Q_{\mathrm{c}2}\)

Flow in the middle segment of the collateral network

\(Q_{\mathrm{c}3}\)

Flow in the upper segment of the collateral network

\(Q_{\mathrm{cjl}2}\)

Flow in the lower anastomotic connection (left side)

\(Q_{\mathrm{cjl}3}\)

Flow in the upper anastomotic connection (left side)

\(Q_{\mathrm{cjr}2}\)

Flow in the lower anastomotic connection (right side)

\(Q_{\mathrm{cjr}3}\)

Flow in the upper anastomotic connection (right side)

\(Q_{\mathrm{ex}}\)

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

\(Q_{\mathrm{f}}\)

CSF formation rate

\(Q_{\mathrm{jl}1}\)

Flow in the lower segment of the left IJV

\(Q_{\mathrm{jl}2}\)

Flow in the middle segment of the left IJV

\(Q_{\mathrm{jl}3}\)

Flow in the upper segment of the left IJV

\(Q_{\mathrm{jr}1}\)

Flow in the lower segment of the right IJV

\(Q_{\mathrm{jr}2}\)

Flow in the middle segment of the right IJV

\(Q_{\mathrm{jr}3}\)

Flow in the upper segment of the right IJV

\(Q_{\mathrm{lv}}\)

Flow in the lumbar vein

\(Q_{\mathrm{n}}\)

Set point of Q

\(Q_{\mathrm{rv}}\)

Flow in the renal vein

\(Q_{\mathrm{svc}1}\)

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

\(Q_{\mathrm{svc}2}\)

Flow in the lower segment of the superior vena cava

\(Q_{\mathrm{vv}}\)

Flow in the vertebral veins

\(Q_{\mathrm{vvl}}\)

Flow in the left vertebral vein

\(Q_{\mathrm{vvr}}\)

Flow in the right vertebral vein

R

Resistance per unit length of vessel

\(R_{1}\)

Resistance of small vessels and terminal branches

\(R_{\mathrm{f}}\)

Reflection coefficient

\(R_{\mathrm{f}}^{\mathrm{a}}\)

Reflection coefficient for wave propagating in the parent vessel a

\(\rho\)

Blood density

\(R_{\mathrm{pa}}\)

Resistance of pial arterioles

\(R_{\mathrm{v}}\)

Reflection coefficient of the aortic valve

\(\sigma\)

Differential cross-sectional area

t

Time

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

Time constant of the autoregulation mechanism

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

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

u

Blood velocity

U

Average axial blood velocity

\(V_{\mathrm{pa}}\)

Blood volume in the pial arterioles

x

Longitudinal direction

\(x_{\mathrm{aut}}\)

State variable that accounts for the effect of autoregulation

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

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

\(\xi\)

Coefficient related to R

Z

Characteristic impedance

\(Z_{0}\)

Characteristic impedance of the peripheral segment

Notes

Funding

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
  1. 1.Department of Physics and Earth SciencesUniversity of FerraraFerraraItaly
  2. 2.Institute of Nuclear PhysicsPolish Academy of SciencesKrakówPoland

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