Intracranial Dynamics
BioE 310 - Biological Systems Analysis
Intracranial Dynamics and Harmonic Analysis
Richard Hickey
[email protected]
As part of the coursework for BioE 310
Report produced under supervision of Professor Andreas Linninger
Dept. of Bioengineering, University of Illinois at Chicago
Abstract
A quantitative model of blood flow in intracranial vasculature, in tandem with cerebrospinal fluid
(CSF) flow in the intrathecal space, is created to better understand and predict how pressure
changes in one system will affect the other. A compartmental approach is used to analyze flow and
pressure changes between major areas of the intracranial space. Finally a harmonic analysis
provides insight into the role of arterial pulsation in whole-brain fluid dynamics at ultimate
periodic response. We find that the amplitude and phase angle of the pulse differ between each
compartment, illustrating how the various compartments all react uniquely to each heartbeat.
1. Introduction
2. Methods
The dynamic nature of fluid flow throughout the
cranium is of significant interest to many disciplines
of medicine. The minute pressure changes within the
intracranial space that occur during each cardiac
cycle (heartbeat) have profound effects on the
efficacy of intrathecal drug delivery [2]. The
relationship between arterial, capillary, and venous
pressures within the body are well studied. However
there are many compartments within the cranium that
are difficult to assess directly. The activity within
these areas is of paramount importance when
studying pathological changes in intracranial pressure
(ICP) such as in hydrocephalus or traumatic brain
injury [7]. Despite a demand for detailed information
of this domain, the invasive measures currently
required
to
obtain
such
parameters
are
contraindicated in all but the most extreme cases
[5,7].
This model works with the 6 major compartments of
intracranial space: Cerebral arteries, capillaries, veins
and venous sinus, as well as brain and CSF [5,6].
Limited numeric data regarding intracranial flow is
available. Some of the more comprehensive work,
done by Sorek et al, provides constants for resistance
and compliance of the various intracranial
compartments [5]. These values were used in a
matrix of coefficients that describe a mass
conservation balance equation. Boundary conditions
were obtained using the Monro-Kellie hypothesis of
the fixed-volume cranial space and standard
assumptions of mean arterial pressure [1,4]. Pressures
and flows of the steady state system were solved
using MATLAB matrix operations.
Given that the system can be measured at the inlet
(carotid arteries) and outlet (jugular veins), it is
reasonable to attempt to construct a model of what
activity takes place within the intracranial system.
The arterial pulse can be treated as a waveform and
as a sort of “forcing function” that propagates activity
all through the brain and surrounding compartments.
The study of how this one waveform interacts to
stimulate other similar, but not identical, waveforms
is called harmonic analysis. It is through an
understanding of these harmonics that we can model
the activity in areas not accessible for direct
instrumentation.
The harmonic analysis was done with single-variable
forcing equation (1), and a frequency of 80bpm was
chosen to roughly approximate an arterial waveform
[8].
𝐼0 + sin(𝜔0 𝑡)
Finally, pressures from the earlier computation were
introduced into the harmonic system to create a more
comprehensive simulation of intracranial dynamics.
A 5-variable, single-frequency, first order system was
input into MATLAB in the form of a 5x5 matrix Z.
𝑍𝑛,𝑛 = 𝑖 ∗ 𝜔0 ∗ 𝐶𝑛 ∗
[
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(1)
𝑟𝑒𝑎𝑙(𝑍)
𝑖𝑚𝑎𝑔(𝑍)
1
𝑅𝑛
−𝑖𝑚𝑎𝑔(𝑍) 𝑋𝑟𝑒𝑎𝑙
] (𝑋
) = 𝑓0
𝑟𝑒𝑎𝑙(𝑍)
𝑖𝑚𝑎𝑔
(2)
(3)
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Intracranial Dynamics
When f0 is a column vector consisting of values for I0
(from the forcing function) and 0, a solution X exists
with complex entries that describe the harmonic
waves.
BioE 310 - Biological Systems Analysis
(Bottom) Equivalent MATLAB generated plot, with flows
(Q) and pressures (P) labeled
3.1 Pressure Model
3. Results
A MATLAB script was used to generate a map of the
intracranial space. This flow diagram indicates which
compartments communicate with one another.
Solutions for mean pressure and flow from the steady
state equation are plotted on the diagram (Fig. 1).
The inflow/outflow calculated are consistent with
literature values at approximately 750ml/min [3,6].
Using the same set of equations plus a periodic
function for arterial waveform, a simple graph of
pressures/time can be visualized (Fig. 2). This is
useful for observing trends in pressure and is
consistent with the work of Zhou, Xenos et al [8].
Each of the 6 compartments can be seen oscillating at
different levels of pressure. However, this model
(like [8]) is incomplete because it does not account
for any damping of the waveform. As it travels
through a significant amount of vasculature and
tissue around the brain, and as the mean pressure
drops, it would be expected that the amplitude of the
lower pressure waves would be lower than the initial
arterial one. Indeed that is the case, which is
something we can observe using the more thorough
mathematical modeling that is harmonic analysis.
Figure 2: The arterial waveform reproduced at steady state
pressures for all compartments
3.2 Harmonic Analysis
Figure 1: (Top) Conceptual flow network showing
communication
between
compartments
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Harmonic analysis was applied to the sinusoidal
function representing arterial flow using 5x5 matrix Z
as in equation (2). Because of the sine wave
representing arterial flow (1), only the imaginary part
of X is preserved. The results of one complete
oscillation, omitting the baseline pressures, are
shown in (Fig. 3) and demonstrate the phase shift
between compartments. These harmonic waves are
all shown at the ultimate periodic response and have
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Intracranial Dynamics
BioE 310 - Biological Systems Analysis
the same period as the arterial pulse. They can also be
visualized individually (Fig. 4).
Figure 3: Over a single cycle of arterial flow (black), each
compartment can be seen with a unique amplitude and
phase shift.
These diagrams are made using the assumption that
the heartbeat is a continuous signal (we hope!) and
so does not include the introductory period in which
the responding waves are still adapting to the forcing
(arterial) function. Rather they are shown at their
ultimate periodic response, as time t becomes
arbitrarily large.
Although the waves still appear to vary greatly, they
all share the same frequency as the initial (arterial)
wave. Where the waves differ is in two values: their
amplitude and their phase shift. The baseline for each
measurement comes from the arterial wave, as that is
the forcing function. All other waves are shown as
percentage of arterial pressure and delay vs. arterial
frequency. There is a predictable stepwise drop in
pressure as the fluid moves from artery to capillary to
vein. Interestingly, the brain compartment
experiences a markedly different phase from the
other compartments. A working theory is that
because it is being fed by more than one inlet (see
flow diagram in Fig. 1) its waveform has a pattern
unique from the others.
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Figure 4: (Top) Harmonic waves of individual
compartments influenced by the arterial wave at 80bpm
(Bottom) Amplitude and phase shifts relative to the arterial
pulse.
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Intracranial Dynamics
BioE 310 - Biological Systems Analysis
5. Discussion
Figure 5: Pressure values are combined with the harmonic
waves to create a comprehensive plot of intracranial
activity
With this better understanding of the waveforms as
they exist in a real system with resistance, we can
revisit the earlier plot of pressure waveforms vs.
time. This new analysis represents our most complete
model, which incorporates mean pressures as well as
harmonics (Fig. 5).
4. Validation
The results seen in this study accurately predict those
seen in a selection of other works [6,8]. The shift in
amplitude observed in Fig. 2 reflect reasonable
physiologic
differences
between
intracranial
compartments.
These preliminary results suggest a significantly
pulsatile nature of pressure within the brain tissue
itself, which if accurate has interesting implications
warranting further study. Such a finding is consistent
with some previous literature including the work of
Wagshul et al and leads to a more thorough
understanding of how changes in blood pressure
affect the pulsation of CSF [7].
A shortcoming of this model is the simplistic
waveform chosen to represent the cerebral artery
pulse. A single sine wave was used because its
harmonic influence on a multivariable system is well
understood. In future studies it would be of interest to
perform a Fourier transform of a measured
physiological signal in order to obtain a more
accurate representation of the pulse waveform. A
similar model could then be created using this
multiple-frequency forcing function. This would
make for a more complete model but no significant
change in results would be anticipated.
One major assumption made in this study is that flow
into the intracranial space is always equal to flow out.
Aspects of this Monro-Kellie theory have been
upheld for over 2 centuries [4]. Principally that due to
the rigid bony structure of the skull there must be a
fixed volume inside of it at all times. The balance of
this volume between brain, CSF, blood, and other
matter may vary but the total volume must not. A
caveat that has not been properly examined by
subsequent “compartmental” type approaches is that
not all of the compartments reside fully within the
intracranial space. The major culprit, of course, is the
CSF. With each cardiac cycle a significant amount of
CSF (2ml) pulsates down into the spinal cord. The
spine does not follow the same set of rules regarding
compressibility – namely, the vertebrae do not
provide the same level of confinement as does the
skull. A modified flow network that ventures outside
of the intracranial space and into the spine is outlined
below (Fig. 7). In future studies it would be
beneficial to examine what sort of impact the detour
of CSF down the spinal column has on the rest of the
intracranial network.
Figure 6: Reproduced from Zhou, Zenos, et al [8].
Generally similar pressures were observed in both systems.
However this image, like Fig. 2, lacks a complete harmonic
adjustment of wave amplitude.
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Intracranial Dynamics
BioE 310 - Biological Systems Analysis
Dr. Linninger’s lab are subject to IRB review
procedures and intellectual property procedures.
Therefore, the use of these data and procedures are
limited to the coursework only. Publications need to
be approved and require joint authorship with staff of
Dr. Linninger’s lab.
8. References
Figure 7. Is there another compartment not adequately
represented? The spinal cord is not within the intracranial
space but communicates with it through the CSF.
1.
Dixon, W., and W. Halliburton. The cerebrospinal fluid. II. Cerebro-spinal pressure. J.
Physiol. , 1914.at
<http://jp.physoc.org/content/48/23/128.full.pdf>
2.
Hsu, Y., H. D. M. Hettiarachchi, D. C. Zhu,
and A. Linninger. The frequency and
magnitude of cerebrospinal fluid pulsations
influence intrathecal drug distribution: key
factors for interpatient variability. Anesth.
Analg. 115:386–94, 2012.
3.
Lakin, W., and S. Stevens. A whole-body
mathematical model for intracranial pressure
dynamics. J. Math. Biol. 383:347–383, 2003.
4.
Mokri, B. The Monro-Kellie hypothesis:
Applications in CSF volume depletion.
Neurology 56:1746–1748, 2001.
5.
Sorek, S., J. Bear, and Z. Karni. A NonSteady Compartmental Flow Model of the
Cerebrovascular System. Biomechanics
21:695–704, 1988.
6.
Stevens, S. A. Mean Pressures and Flows in
the Human Intracranial System, Determined
by Mathematical Simulations of a SteadyState Infusion Test. Neurol. Res. 809–814,
2000.
7.
Wagshul, M. E., P. K. Eide, and J. R.
Madsen. The pulsating brain: A review of
experimental and clinical studies of
intracranial pulsatility. Fluids Barriers CNS
8:5, 2011.
8.
Zhou, X., M. Xenos, and S. Kontapalli.
Response on Harmonic Excitation Analysis.
Laboratory for Product and Process Design,
University of Illinois at Chicago: 2005.
6. Conclusion
It is possible to create a compartmental model of the
intracranial system to mimic what occurs inside that
difficult-to-access part of the human body. The
combination in this model of pressures and
harmonics appears to represent a novel and more
complete simulation of intracranial flow. Much
previous work in the field had not accounted for
signal damping in the resistive tissue in the
intracranial system [8]. Based on the results from the
combined pressure & harmonics model (Fig. 5) we
can see that the waveform within intracranial
structures varies greatly.
This model demonstrates the power of harmonic
analysis to describe the inner workings of what is
otherwise an inaccessible system. These findings may
be used to better design systems for intrathecal drug
delivery or to better understand pathological changes
that occur with rising ICP.
7. Acknowledgments
Report produced under the supervision of
Bioengineering Professor Andreas Linninger, with
the generous aid of Chih-Yang Hsu and Sebastian
Pernal.
Intellectual Property: Biological and physiological
data and some modeling procedures provided to from
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Intracranial Dynamics
BioE 310 - Biological Systems Analysis
Appendix A: IntracranialPressure.m
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clear all; close all; clc;
%% Create vectors
% Positions to graph flow network
ptCoordMx = [5 10; 5 7; 2.4 6.3; 7.5 6; 5 4; 5 3; 5 2;];
%Constitutive relationships
pointMx = [-1 0 0 0;
1 -2 -5 -3;
2 -4 -7 0;
3 4 -6 0;
5 6 -8 0;
7 8 -9 0;
9 0 0 0];
faceMx = [1 2; 2 3; 2 4; 3 4; 2 5; 4 5; 3 6; 5 6; 6 7];
%% Given values
Rac = 0.106667; %mmHg s / ml
Rcv = 0.020;
Rbv = 0.6;
Rvs = 0.0027;
Rfs = 2.80;
Rcb = 1.4;
Rfb = 10.00;
Rcf = 1;
alpha = [Rac Rcf Rcb Rfb Rcv Rbv Rfs Rvs 0];
[Plength,Pwidth]=size(pointMx);
[Flength, Fwidth] = size(faceMx);
Findex = 1:Flength;
Pindex = (Flength+1):(Flength+Plength);
Matindex = [Findex Pindex];
Matlength = length(Matindex);
b = zeros(Matlength,1);
k = 1;
Pin=1;
Pout=7;
b(Pin)=102;
b(Pout)=5;
t=0:.01:5;
I0 = 6; % amplitude
HR = 80; % in BPM
w0 = 2*pi*(HR/60);
Pa = 102 + I0*sin(w0*t); % Arterial waveform
Ps = 5 + I0*sin(w0*t); % Outflow waveform
for T =1:length(t) % BEGIN TIME LOOP
k=1;
C = zeros(Matlength,3); %C is column vectors Row, Col, Value
b(Pin) = Pa(T);
b(Pout) = Ps(T);
%% Conservation Eqs: F1 - F2 = 0
for i=1:(Plength)
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BioE 310 - Biological Systems Analysis
%Exception: first
if(i==Pin)
C(k,1) = i;
C(k,2) = Pindex(i);
C(k,3) = 1;
k=k+1;
%Exception: last
elseif(i==Pout)
C(k,1) = i;
C(k,2) = Pindex(i);
C(k,3) = 1;
k=k+1;
else %NOT first or last
for j=1:Pwidth
if pointMx(i,j) < 0
C(k,1) = i;
C(k,2) = abs(pointMx(i,j));
C(k,3) = -1;
k=k+1;
end
if pointMx(i,j) > 0
C(k,1) = i;
C(k,2) = abs(pointMx(i,j));
C(k,3) = 1;
k=k+1;
end
end
end
end
%% Constitutive Eqs: a1F1 - P1 + P2 = 0
for i=1:Flength
for j=1:Fwidth
if j==1
C(k,1) = i+Plength;
C(k,2) = abs(faceMx(i,j)+Flength);
C(k,3) = -1;
k=k+1;
end
if j==2
C(k,1) = i+Plength;
C(k,2) = abs(faceMx(i,j))+Flength;
C(k,3) = 1;
k=k+1;
end
end
C(k,1) = i+Plength;
C(k,2) = i;
C(k,3) = alpha(i);
k=k+1;
end
% Populate SPARSE Matrix
B = sparse(C(:,1),C(:,2),C(:,3));
% Determinant and solution
d = det(B);
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BioE 310 - Biological Systems Analysis
x = B\b;
Yplot(T,:)=x(Pindex);
end %END TIME LOOP
%% Plot Pressures
plot(t,Yplot(:,1:end-1));
axis([0 5 0 112])
legend('Pa','Pc','Pf','Pb','Pv','Ps','Location','East');
%% Create Network Diagram
figure
hold on
for i=1:length(ptCoordMx)
scatter(ptCoordMx(i,1),ptCoordMx(i,2));
pointS = sprintf(' P%i=%.1f',i,x(Pindex(i)));
text(ptCoordMx(i,1),ptCoordMx(i,2),pointS);
end
for j=1:Flength
plot([ptCoordMx(faceMx(j,1),1) ptCoordMx(faceMx(j,2),1)] , ...
[ptCoordMx(faceMx(j,1),2) ptCoordMx(faceMx(j,2),2)]);
xspot = (ptCoordMx(faceMx(j,1),1)+ptCoordMx(faceMx(j,2),1))/2;
yspot = (ptCoordMx(faceMx(j,1),2)+ptCoordMx(faceMx(j,2),2))/2;
pointS = sprintf(' Q%i=%.1f',j,x(Findex(j)));
text(xspot,yspot,pointS);
end
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BioE 310 - Biological Systems Analysis
Appendix B: Harmonics.m
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clear all; close all; clc;
% P0 values from IntracranialPressures.m
P0 =[102; %a
21.8122607460216; %c
16.9962366130515; %f
11.6796974728199; %b
7.01817800731953; %v
5]; %s
% Constants, from SOREK et al
Cab = 0.0155;
Ccf = 0.0364;
Cbv = 0.3180;
Cfs = 0.0334;
Cfb = 0.1830;
Rac
Rcv
Rbv
Rvs
Rfs
Rcb
Rfb
=
=
=
=
=
=
=
0.106667; %mmHg s / ml
0.020;
0.6;
0.0027;
2.80;
1.4;
10.00;
I0 = 6; % amplitude
HR = 80; % rate in BPM
w0 = 2*pi*(HR/60); % angular frequency (rad)
% Time to plot (s)
% t=0:.01:(60/HR); % Single cycle
t = 0:.01:5; % 5 second plot
% (a) c f b v s
Z = [ ...
1i*w0*Ccf+1/Rac
0
0
0
0;
0
1i*w0*Cfb+1/Rcv
0
0
0;
0
0
1i*w0*(Cbv-Cab)+1/Rbv
0
0;
0
0
0
1i*w0*(-Cbv)+1/Rvs
0;
0
0
0
0
1i*w0*(-Cfs)+1/Rvs];
A = [real(Z) -imag(Z);
imag(Z) real(Z)];
y = [I0 I0 I0 I0 I0
0 0 0 0 0]';
X = A\y;
Xc = [ ...
X(1) +
X(2) +
X(3) +
X(4) +
X(5) +
1i*X(6);
1i*X(7);
1i*X(8);
1i*X(9);
1i*X(10)];
for n=1:length(t)
V(n,:) = imag(Xc.*exp(1i*w0*t(n))); % take only imaginary part
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end
Y = I0 * sin(w0*t); % original signal
plot(t,Y+P0(1),'k',t,V(:,1)+P0(2),':r',t,V(:,2)+P0(3),'-.m',t,V(:,3)+P0(4),'-b',t,V(:,4)+P0(5),'-g',t,V(:,5)+P0(6),'--y')
legend('1: Arteries','2: Capillaries','3: CSF','4: Brain','5: Veins','6:
Venous Sinus','Location','East')
xlabel('Time (s)'); ylabel('Pressure (mmHg)');
title('All compartments, with initial values');
grid on;
figure;
subplot(3,2,1);
plot(t,Y+P0(1),'k');
title('1: Arteries');
subplot(3,2,2);
plot(t,V(:,1)+P0(2),':r');
title('2: Capillaries');
subplot(3,2,3);
plot(t,V(:,2)+P0(3),'-.m');
title('3: CSF');
subplot(3,2,4);
plot(t,V(:,3)+P0(4),'--b');
title('4: Brain');
subplot(3,2,5);
plot(t,V(:,4)+P0(5),'--g');
title('5: Veins');
subplot(3,2,6);
plot(t,V(:,5)+P0(6),'--y');
title('6: Venous Sinus');
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