Brain Vasculature and Intracranial Dynamics
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Brain Vasculature and Intracranial Dynamics. February 22 nd , 2007 LPPD lab meeting. Michalis A. Xenos and Andreas A. Linninger Laboratory for Product and Process Design , Departments of Chemical and Bio-Engineering, University of Illinois, Chicago, IL, 60607, U.S.A.

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Brain Vasculature and Intracranial Dynamics

February 22nd, 2007

LPPD lab meeting

Michalis A. Xenos and Andreas A. Linninger

Laboratory for Product and Process Design,

Departments of Chemical and Bio-Engineering, University of Illinois, Chicago, IL, 60607, U.S.A.


Motivation for studying intracranial dynamics
Motivation for studying Intracranial Dynamics

  • Many biological phenomena in the brain need better interpretation

    • Large deformation of biological tissues under hydrocephalic conditions.

    • Interaction of vasculature with soft brain tissue.

  • Fundamental transport processes in soft tissues are poorly understood.

  • Interpretation of experiments is inadequate.

Hydrocephalic Brain MRI

Normal Brain MRI


Equations of motion for a vessel bifurcations and unions

Continuity

Momentum

Distensibility of vasculature

are equal for i = 1, …,n

Equations of motion for a vessel, Bifurcations and Unions

Straight tube

parent

children

l : tube length

F: Poiseuille coef.

E : Young modulus

A0 : Initial cross sectional area

p* : External pressure

Unknowns: fin , p , A

Bifurcation

child

parents

Union


Bifurcated tubes vs lumped tubes

T’2

T2

T4

T4

T1

T1

T3

Bifurcated tubes vs. lumped tubes



Blood signals and fitting procedure
Blood signals and fitting procedure

Pulsatile blood pressure from MRI


Spectral analysis for blood and ventricular signals from mri data

Arterial spectrum

Ventricular spectrum

Phase lag

Spectral analysis for blood and ventricular signals from MRI data

Applying FFT

the spectrum for

both pulses :



Model generator and obtained networks
Model Generator and obtained networks

Model Generator

Blood network

same to Zagzoule’s


A simple blood flow model zagzoule network
A simple blood flow model – Zagzoule network

Hademenos, Massoud, The physics of Cerebrovascular Diseases, 1998



A holistic brain compartmental model

QA

QFfr

Subarachnoid space (SAS) (r)

QFfb

Arteries (a)

QFcf

Ventricles (f)

Qac

QFpf

Qap

Choroid Plexus (p)

Capillaries (c)

Qpc

Qpv

QFcb

Qcv

QFfv

Venous

Sinus (n)

Qvn

QFbr

Veins (v)

QFbv

Brain (b)

QFri

QFrn

QA

QFir

Spinal Cord (s)

A holistic brain compartmental model

Blood flow

CSF flow





Open vs closed network systems

Continuity

Momentum

Continuity

Distensibility of vasculature

Momentum

Total stress

p*

d

solid

p

p

fluid

Open vs. Closed network systems

Open system

Closed system


Egnor s mathematical model
Egnor’s mathematical model

Newton’s Law

or

we differentiate both sides of the above equation

Forcing function

the particular solution of the above equation is

Damping force

Elastic resistance

Pulsatile resistance

Elastic force

Resonance : In an oscillating system with one degree of freedom, resonance is the state of minimal impedance.

Acceleration


Rlc vs 2 nd order oscillators
RLC vs. 2nd order oscillators

  • Series RLC Circuit notations:

    • V - the voltage of the power source (V)

    • I - the current in the circuit (A)

    • R - the resistance of the resistor (Ohms = V/A);

    • L - the inductance of the inductor (H = V·s/A)

    • C - the capacitance of the capacitor (F = C/V = A·s/V)

  • Given the parameters v, R, L, and C, the solution for the current i using Kirchhoff's voltage law is:

  • For a time-changing voltage V(t), this becomes

  • Rearranging the equation (dividing by L and differentiating with respect to t gives the following second order differential equation:

Damping factor

  • We now define two key parameters:

Resonance frequency


Our model vs rlc

Continuity

Momentum

Distensibility of vasculature

Straight tube

R

C

Electrical analogy (RC)

Our model vs. RLC

I


Assumptions of egnor s model
Assumptions of Egnor’s model

1. Intracranial CSF pulsations are normally synchronous with

arterial pulsations.

2. Normal intracranial CSF pulsations are of similar amplitude

and morphology in the ventricles and the subarachnoid space.

3. Arterial pulsations are normally filtered from the cerebelar

circulation, so the capillary blood flow is nearly smooth (the

windkessel mechanism).

Oscillations of CSF with a single degree of freedom are

described by the same differential equation that describes the

oscillations of electrons in an alternating current electrical circuit

Windkessel Effect

The mechanism by which arterial pulsations are progressively dissipate to render the capillary circulation almost pulseless is called the windkessel effect.

Egnor et al, Pediatric Neurosurgery, 2001; 2002


The role of resonance in egnor s model and the reality
The role of resonance in Egnor’s model and the reality

  • With the help of resonance Egnor assumes that the Intracranial CSF pulsations are synchronous with arterial pulsations. This is not true:

Phase lag:

aqueduct: – 52.5 ± 16.5o

prepontine cistern: – 22.1 ± 8.2o

C-2: +5.1 ± 10.5o (consistent with flow

synchronous with the arterial pulse)

Wagshul et al, J. Neurosurg, 2006

Zhu et al, J. MRI, 2006


Conclusions
Conclusions

  • Current systems describe the brain until a point

  • Electrical analogies of the models

  • Egnor’s model and his assumptions were presented

  • The flow between blood and CSF has a phase lag and the assumption of synchronous pulsations is not valid

  • Advancement of these models with the inclusion of porosity!

  • Future steps – distributed systems



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