Implementation of the reservoir-wave hypothesis
with CMR-derived aortic area and flow velocity
Robert D. M. Gray
CoMPLEX, University College London
Supervisors: Dr. Giovanni Biglino and Prof. Andrew Taylor
4381 words
6th March 2015
Abstract
The reservoir-wave hypothesis states that the blood pressure waveform can be
usefully divided into a reservoir pressure that is related to the global compliance
and resistance of the arterial system and an excess pressure that depends on local
conditions. The formulation of the reservoir-wave hypothesis applied to the area
waveform is shown, and the analysis is applied to area and velocity data from highresolution phase-contrast cardiovascular magnetic resonance. A feasibility study
shows the success of the principle, with the method producing largely robust and
physical parameters, and the linear relationship between flow and wave pressure seen
in the pressure formulation is retained. A small proof of concept study suggests the
potential of the technique by comparing patient and control data. Finally, the analysis is applied to the ascending and descending aorta and interestingly the invariant
property of the reservoir pressure is shown to be lost in the area formulation.
1 Background
As background to this project, I will give a brief overview of the anatomy and physiology
of the heart and arteries. Experiments in cardiovascular magnetic resonance (CMR)
provide the data for this project so an outline of this new and non-invasive method is
also instructive. Finally I will present the basic theory behind the arterial models which
underpins the reservoir-wave theory developed in the later sections.
1.1 Basic Cardiovascular Anatomy and Physiology
The heart pumps blood around the circulatory system. It consists of four chambers.
The right atrium collects blood from the systemic veins; blood flows from the right
atrium into the right ventricle, which supplies the pulmonary arteries; the left atrium
analogously collects blood from the pulmonary veins; finally the left ventricle receives
blood from the left atrium and ejects it into the systemic arteries.1
1
(b)
(a)
Figure 1: A diagram of the heart and great arteries (a) from2 and an example arterial
pressure waveform (b) from3
On exiting the left ventricle blood flows from the aortic root into the ascending
aorta and up into the aortic arch. In the arch three major arteries branch off the aorta
which supply the head and upper body. These are the brachiocephalic artery, the left
common carotid artery and the left subclavian artery. Blood not flowing into any of
these continues into the descending aorta which passes down into the body, becoming
the thoracic aorta and branching to supply the rest of the body.1 Figure 1a shows the
structure of this anatomy.
The periodic sequence of events in which the heart executes its pumping function
is the cardiac cycle. Systole is the period of contraction of the heart in which blood is
ejected from the left ventricle into the arteries. The subsequent period of relaxation as
the chambers refill with blood is known as diastole.1 During this cycle blood pressure
rises as blood is forced from the left ventricle, then decays away during diastole. The
arterial blood pressure waveform over the cardiac cycle is important for this project so
an example is show in Figure 1b.
1.2 Cardiovascular Magnetic Resonance
The method by which the data for this project is acquired is cardiovascular magnetic
resonance imaging (CMR). Magnetic resonance imaging (MRI) is a general technique
for medical imaging that works by exposing the body to a very strong magnetic field.
The physics of the technique are not wholly relevant but broadly the responses of water molecules to small perturbations in the field, which depend on the local molecular
environment, are measured using radio waves. This produces contrast between tissues
which allows images of the body to be created. CMR operates on the same principles
and is optimised for use in the cardiovascular system.4
It is further possible to extract velocity data with a technique known as phase contrast cardiovascular magnetic resonance imaging (PC-CMR). Again without any detailed
physics, the phase of the response of the water molecules contains information about
2
(a)
(b)
Figure 2: Examples of magnitude (a) and phase (b) CMR images
their velocity.4 The combination of the tissue differentiation and velocity determination
allows simultaneous measurement of the area of an artery and the velocity of the blood
flowing through it. These measurements make the entire analysis involved in this project
possible.
Figure 2 shows examples of PC-CMR images. Shown are magnitude (a) and phase
(b) images of a transverse plane of a person’s chest. In the magnitude image different
shades of grey represent different responses and thus different types of tissue. In the
phase image the velocity data are encoded into the grey scale such that grey is not
moving, white is movement out of the plane and black is into the plane. The rounded
regions in the centre and towards the bottom-right of the images are the ascending and
descending aorta respectively.
Images like these are acquired over the cardiac cycle; these frames are taken at peak
systole, when you appreciate the maximum expansion of the aorta and the motion of
blood. The data for my analysis is then produced from this raw PC-CMR data in image
processing software called OsiriX (Pixmeo, Geneva, Switzerland). In this program the
blood vessel to be measured, termed the region of interest (ROI), is manually selected
from one frame. The ROIs can then be automatically propagated through the rest of the
frames using nonrigid registration.5 The area and average velocity are then extracted
from the images to produce the data to be analysed.
1.3 Arterial Blood Flow Models
Wave intensity analysis is a method of modelling arterial blood flow that has arisen in
the last 25 years. Having its roots in gas dynamics it differs from traditional Fourier
and impedance methods in being a time domain analysis, allowing application to nonperiodic or transient flow.6
At the most basic level we are considering a 1D model of the arterial system as a
deformable tube described by a position coordinate x filled with and inviscid, incompressible fluid.
Immediately we are making several simplifying assumptions. In terms of the dimension reduction, according to Willemet and Alastruey “several comparisons against in
3
vivo, in vitro and 3D numerical models have shown the ability of the 1D formulation to
capture the main features of pressure and flow waveforms in human systemic arteries”.7
The fluid mechanical assumptions can be relaxed to investigate more complex flow, but
this is not in this case useful and I continue the analysis following Parker.8
We can write down the equation for conservation of mass in a differential element of
the tube as
∂A ∂(AU )
+
=0
(1.1)
∂t
∂x
where A(x, t) is the cross-sectional area of the tube, U (x, t) is the flow velocity averaged
over the cross-section and t is time.
We also have the conservation of momentum in the differential element as
∂U
∂U
1 ∂P
+U
+
=0
∂t
∂x
ρ ∂x
(1.2)
where ρ is the density of blood and P is the pressure in the tube averaged over the cross
section. We also write down a “tube law”
A(x, t) = A(P (x, t); x)
(1.3)
which just says that the area is some function of the local pressure and distance only.
We can write down an explicit relationship for (1.3)9 or allow A to depend directly on t
but for simplicity again following Parker I keep (1.3) as is.
At this point we use the tube law to elimate the derivatives of one variable from
(1.1) and (1.2), constructing a matrix equation in two variables which can be solved by
Riemann’s method of characteristics. These variables are typically pressure and velocity
but they should be chosen depending on the application so following Biglino et al.10 we
use area and velocity which are our measured variables.
As such we write the x-derivative of P as
∂P
1
∂A
=
(1.4)
∂x t AP ∂x t
∂A
where AP = ∂P
so that (1.1) and (1.2) become
x


U
 ∂ + 1
∂t
ρAP
A


 ∂  A = 0
U ∂x
U
0
(1.5)
These equations are hyperbolic, which means according to the method of characteristics we can solve along the “characteristic directions” which are the eigenvalues of the
matrix of coefficients of the x-derivative
1
2
A
λ± = U ±
(1.6)
ρAP
4
But we note that, as we define the distensibility, which is the fractional change in
area with a change in pressure
AP
D=
(1.7)
A
this is nothing more than the wave speed c as first defined by Bramwell and Hill11 where
1
c= √
=
ρD
A
ρAP
1
2
(1.8)
so that
λ± = U ± c
(1.9)
Therefore, as Parker states: “any perturbations introduced into an artery will propagate as a wave with the speed U + c in the forward direction and speed U − c in the
backward direction.”
This analysis can be continued as it is by Parker to derive the wave intensity, which
is a useful way of quantifying the pulse waves in arteries and preceded the reservoir-wave
hypothesis but is not necessary for its development. Really the important point is that
this model admits a rigorous but straightforward description of waves travelling at a
single speed c which can be determined from measurements of two of the variables A, U
and P . Also again note that pressure and area are related through the distensibility as
dP
1
=
d ln A
D
(1.10)
and that, once c is measured, D can be determined from (1.8). This means that any
differential relationship in terms of P can be recast in terms of ln A, which is central to
the focus of this project.
2 Reservoir Wave Hypothesis
2.1 Overview
The concept of the reservoir in haemodynamics is an old one. In 1899 Frank12 presented
his Windkessel model of arterial mechanics. The terms “Windkessel” and “reservoir” are
in this context interchangeable. The model represents the arteries as a single compliant
compartment also with a single resistance. This is a very elegant model showing the
importance of compliance in the arteries: with a periodic influx of blood a rigid arterial
system would exhibit periodic outflow. Sufficient compliance allows constant outflow,
as the compliant compartment (the reservoir) fills with blood during systole and relaxes
during diastole, releasing its contents. The model predicts exponentially falling pressure
and this is indeed seen in diastole, as in Figure 1b. However, the model does not account
for the sharp rise in pressure seen in systole.
For this reason a more popular approach of late has been one which separates the
pressure waveform into a combination of forward and backward travelling waves.13 This
5
Figure 3: The separation of velocity and pressure waveforms with and without the
reservoir. The measured waveforms are in black, separated forward and backward waves
in blue and red, and the reservoir pressure is in green. From Parker.8
is done through a continuation of the above mathematics which I will not present here.
This analysis provides a good description during systole, where the separation produces
an initial forward compression wave followed by backward reflected waves. However in
diastole this approach predicts large cancelling forward and backward waves as “this is
the only way that the wave theory can explain a falling pressure and a zero velocity”.8
The reservoir-wave hypothesis is an attempt to combine these two methods and benefit
from the good predictions of the wave theory in systole and the Windkessel, or reservoir,
theory in diastole. The reservoir-wave approach is itself rather young, beginning with
Wang14 in 2003.
Figure 3 from Parker8 demonstrates this synthesis. On the left is the separation of
pressure and velocity waveforms into forward and backward waves. On the right is the
inclusion of the reservoir which is subtracted from the measured waveform. Note that
the reservoir pressure, in green, approximates the measured pressure well in diastole but
not in systole, and that it resolves the problem of unphysical large cancelling velocity
waves in diastole.
6
Figure 4: From Wang et al.,14 presented in Tyberg et al.15 The data are from anaesthetised dogs. The top panel shows the aortic, reservoir and left ventricular pressures.The
bottom shows the wave pressure and the in- and outflows. Pwave and Qin are scaled to
their peak values to show the proportinality.
7
The “wave” or “excess” (again interchangeable) refers to the remaining part of the
pressure waveform when the reservoir is subtracted. It is found to have the interesting property of being proportional to the flow into the arterial system as shown in the
bottom part of Figure 4. According to Tyberg this is “indicating that the nearly identical, superimposed Pwave and Qin waveforms result only from the forward-traveling
compression and decompression waves generated by the LV [left ventricle]”.15
The reservoir pressure is derived from its definitive properties below, but from the
nature of the reservoir as a single compartment we also have the the property that
Pres = P (t); Pwave = P (x, t)
(2.1)
The reservoir pressure is the same at all points in space and only the wave pressure
changes, hence its identification as a wave.
2.2 Mathematical Formulation
We can now derive an analytical expression for the reservoir pressure. Following Tyberg15 we say that, as the reservoir is a single compliant compartment, the rate of change
of the reservoir pressure must be equal to the rate of change of volume multiplied by a
proportionality constant which is the inverse of the compliance C = dV
dP . Secondly, the
rate of change of volume is just the instantaneous difference between the flow of blood
in Qin (t) and out Qout (t) of the system. So
dPres
1
= (Qin − Qout )
dt
C
(2.2)
Next we make the assumption that the outflow can be described by a simple resistive
relationship so that
1
Qout = (Pres − P∞ )
(2.3)
R
where P∞ is the asymptotic pressure at which flow from the arteries ceases. Substituting
(2.3) into (2.2) we get
dPres Pres − P∞
Qin (t)
+
=
(2.4)
dt
RC
C
which we can solve to
Z
e−t/τ t
0
−t/τ
Pres = P∞ + (P0 − P∞ )e
+
Qin (t0 )et /τ dt0
(2.5)
C
0
with P0 the pressure at t = 0 and τ = RC.
To evaluate the reservoir pressure for all time we need the parameters τ , C and P∞ .
To do this we say that after a certain time T in diastole Qin = 0. This is largely a
good assumption as the aortic valve is closed after systole. Under these conditions (2.5)
becomes a simple exponential
Pres = P∞ + PT e−t/τ
8
(2.6)
By fitting this expression to the data the parameters τ and P∞ can be found. Secondly,
the resistance R can be calculated from the mean of the measured pressure and flow rate
over the cardiac cycle. C is then determined from τ and R allowing Pres to be calculated
for all time from (2.5). There are a number of methods for fitting the exponential to find
the parameters but Wang and others use a Nelder-Mead simplex method to minimise
the least squares error over the last two thirds of diastole.14
2.3 Applications and Reception
The entire validity of the reservoir-wave approach comes down to its use in clinical
applications, which have turned out to be many. Differences in the reservoir or excess
pressure can be markers of cardiovascular problems and by the separation information
can be gained about the effects of pharmacological interventions.
For instance Davies et al., trying to gain understanding on the results of large-scale
drug trials, assessed the augmentation index which is a metric of cardiovascular health.
They found that “the augmentation index is principally determined by aortic reservoir
function and other elastic arteries and only to a minor extent by reflected waves.”16
They therefore concluded that “modifying the behavior of the arterial reservoir rather
than changing wave reflection may be a useful target for future therapy.”
In another study it was found that “[excess pressure] integral was a significant predictor of cardiovascular events after adjustment for age and sex” and therefore “measurement [of it] offers a potentially new tool for selection of pharmacological therapies
on a patient-specific basis.”17
Certainly this is an active area and there are a number of ongoing attempts to apply
the approach in a clinically useful way.1819 Tyberg stated in 2014 that “Clinical studies
employing the Reservoir-Wave Approach should be undertaken to verify experimental
observations and, perhaps, to gain new diagnostic and therapeutic insights.”20 These
potential applications are in large part the motivation for this project.
The concept of the reservoir-wave is certainly interesting and has many proponents.
It also however has its detractors. Mynard and Smolich21 claim the formulation “contains
internal inconsistencies and does not provide incremental clinical value compared with
conventional wave separation.” The criticism is focussed on the fact that the reservoir
pressure can appear to have wave-like properties, thus violating its spatial invariance
(2.1). Consideration of this disagreement is outside the scope of this project, but it is
important to know that this approach is not beyond controversy and has not reached its
full development.
3 Implementation with Area Data: Results and Discussion
The main focus of the original work in this project is to implement the reservoir-wave
hypothesis using area and velocity data which can be derived from non-invasive CMR
as described in section 1.2. From CMR we essentially have the area A and velocity
U as functions of time, although sampled with a certain time resolution which was is
9
typically around 10ms, and from this also the flow as Qin = U A. I have incorporated
all the analysis into a Python script.
General wave intensity analysis and separation into forward and backward components has been applied using this sort of data10 with promising results so we expect
something similar from separation into reservoir and wave.
3.1 Formulation
Mathematically, we can recast (2.4) in terms of area using the distensibility relationship
(1.10) as
d ln Ares ln Ares − ln A∞
D
+
= Qin (t)
(3.1)
dt
RC
C
and solve analogously to
Z
D −t/τ t
0
−t/τ
ln Ares = ln A∞ + (ln A0 − ln A∞ )e
+ e
Qin (t0 )et /τ dt0
(3.2)
C
0
Again if we say that during late diastole after some time t = Tld the inflow Qin = 0
then (3.2) becomes
ln Ares = ln A∞ + (ln A0 − ln A∞ + ln AT )e−t/τ
where
D
ln AT =
C
.
Z
Tld
0
Qin (t0 )et /τ dt0
(3.3)
(3.4)
0
At this point the analysis differs from the pressure formulation. Due to the differential
relationship between A and P we do not have access to the mean pressure, and so not
to the resistance. This means the fitting is not quite as simple as in section 2.2 as if we
just fit to find τ and ln A∞ we would not be able to find C and so could not calculate
ln Ares for all time. Instead we can use the data to estimate C. After Tld , ln Ares is a
function of the three reservoir parameters.
ln Ares (ln A∞ , τ, C) = ln A∞ + (ln A0 − ln A∞ + ln AT (τ, C))e−t/τ
(3.5)
To determine the parameters I first set the cutoff time Tld as the time when the
inflow Qin first goes to 0. Then I define a sum of squares function which measures the
total squared difference between this function and the data as
X
S(ln A∞ , τ, C) =
(lnAdata
− lnAres (ln A∞ , τ, C, ti ))2
(3.6)
i
i
where the sum is over all points i for which t > Tld
S is then minimised using a Nelder-Mead simplex algorithm in Scipy to produce
the three parameters ln A∞ , τ and C. Note that this is equivalent to fitting a simple
exponential to find the two parameters ln A∞ and τ and then inverse solving (3.5) to
find C but this method allowed more flexibility when I later attempted to use bounded
optimisation.
10
3.2 Feasibility
To test the feasibility of this approach I applied the analysis to 10 sets of data taken
from healthy volunteers. This was area and velocity data from the ascending aorta as
well as distensibility calculated from the wave speed. The time resolution was 9.56ms. I
ran the exponential fitting in each case and the parameters which resulted are shown in
Table 1.
The results were largely positive: for 8 out of the 10 cases the fitting algorithm
converged onto parameters than are physically realistic and with a reasonably tight
distribution. For all cases the fitting was robust in that changing the initial parameter
values did not make a difference to the result: the optimisation algorithm appears to
find a global minimum.
Four examples are shown in Figure 5. These show the reservoir which is fitted to
the raw log-area data, and the flow and excess log-area, in analogy to the pressure
graphs in Figure 4. The good fitting of the reservoir to the data can be seen along with
the noisiness of the data which made the fitting problematic. Most importantly, there
is largely a well-approximated linear proportionality between the flow and the excess,
which shows the method works as expected.
In 2 out of the 10 cases (#s 3 and 8) the fitting algorithm settles on values which are
not physically realistic. Some of the area data takes on a shape significantly different
from the expected increase in systole and decay in diastole exhibited by the pressure
graphs in Figure 4 and the log-area graphs in Figure 5. Because the data’s resemblance
to a decaying exponential is reduced the “best fit” exponential turns out to be something
unphysical and essentially the fitting fails.
I attempted to overcome this by fitting with a different method, Sequential Least
SQuares Programming (SLSQP), which allows the parameter space to be bounded. In
this way I was able to force the convergence to a physical range, but this method turned
out to be highly sensitive to the initial parameter values that are put into the search.
It was therefore unsuitable. There exist methods such as the Levenberg–Marquardt
Volunteer #
1
2
3
4
5
6
7
8
9
10
ln A∞
0.97
1.49
-1.54×108
1.30
1.61
1.77
1.37
-3.42×104
1.32
1.58
τ
2069.01
490.65
4.18×1010
1164.17
963.58
635.16
340.35
4.79×107
599.32
413.24
C
0.51
0.24
0.56
0.92
1.72
0.55
0.24
0.42
0.44
0.63
Table 1: Fitted parameters for the 10 volunteer data sets for the feasibility study.
11
Figure 5: Graphs for 4 sets of volunteer data. In each case the first graph is the log-area
data (blue) and the reservoir fitted to the exponential part (green) plotted against time.
The second graph is the excess (black), which is the data minus the reservoir, and the
flow (red) against time. These are scaled to have their maxima both at 1.
12
algorithm which may be able to overcome this issue and be both robust and bounded,
but I was unable to investigate these within the timeframe of this project. It would be
a useful extension to the project to do so.
This highlights a problem with this method of fitting the exponential. In essence,
the parameter space is relatively flat, meaning the sum of squares function (3.6) will not
drastically increase if the parameters are changed slightly from their optimised values.
The SLSQP method hops between local minima depending on the initial parameter
values because the parameter space does not converge straightforwardly to a minimum.
This is a problem with the method but also a sign that we may need to be careful
assigning too much importance to the optimised parameters as they are affected by
noise in the data.
Another issue is that there is often non-negligible flow in diastole. This breakdown of
one the assumptions of the model will cause the fitted parameters to deviate from their
true values. It may be possible to include diastolic flow in the analysis, for instance by
fitting a general form of equation (3.2) to the data, which would take into account the
Qin integral over all time. Then all of a reasonable region of the parameter space could
be searched to find the best fit. This would not be computationally trivial but it would
be possible and would resolve the issue of non-negligible diastolic flow.
3.3 Proof of Concept
As a further proof of concept study, I applied the analysis to data from two differing
groups. These were a further 5 healthy, young volunteers (4 males, age range 25-37)
and 5 older patients with confirmed coronary artery disease referred for clinical CMR (4
males, age range 53-74). Again the data was from the ascending aortic position and the
time resolution was 9.56ms. The parameters which resulted from the exponential fitting
are shown in Table 2.
I did not have access to the distensibility in each case. This means I could not
determine the value of the compliance C but in terms of τ and ln A∞ the fitting is
unchanged, so I just present these. This time 3 out of the 10 examples did not converge
satisfactorily. Again this issue may be able to be resolved but for now I had to just
exclude these cases. The mean values of the parameters excluding those which did not
converge to physical values are also shown.
There appears to be some difference between the two groups in terms of the parameters ln A∞ and τ . This is what we would expect if the analysis is sound. The groups
would have greatly different arterial systems due to the age and pathological differences
so if the parameters have any physical meaning they should certainly differ. However
the sample size is far too small to enable any statistics.
Having discussed the feasibility of the method and considering these encouraging albeit preliminary results, the next step would be to apply this technique to larger samples
of data and assess what differences could be detected between different patients groups,
or between patients and controls. CMR imaging based wave intensity has shown some
interesting results,1022 particularly in congenital heart disease where ventriculo-arterial
coupling is compromised, and there may be similar scenarios where the complexity of the
13
Healthy volunteers
Volunteer #
ln A∞
τ
1
1.40
121.05
2
1.08
468.41
3
-4.45×106 5.77×109
4
0.60
201.99
9
5
-4.17×10
8.60×109
Mean
1.03
263.82
Older patients
Volunteer #
ln A∞
τ
1
1.33
879.33
2
1.55
1665.04
3
1.90
540.30
4
-1.21×107 2.64×1010
5
1.61
124.00
Mean
1.60
802.16
Table 2: Fitted parameters for the 5 volunteer and 5 patient proof of concept study.
physiology could be further unravelled by gathering additional insight using the reservoir
method.
Indeed, as described in section 2.3, the reservoir-wave separation of the pressure
waveform has found numerous promising clinical applications, and given the parallels
between the pressure and area formulations that I have demonstrated there is every
reason to expect similar applications could be found with the reservoir-wave separation
of the area waveform.
For instance, Tyberg et al. have related diastolic hypertension to alterations in
the asymptotic reservoir pressure P∞ , saying “Diastolic hypertension would seem to be
related most directly to alterations in reservoir pressure, particularly P∞ and reservoir
resistance.”18 A similar study could look at the affect of hypertension on the asymptotic
area A∞ . It is probably inadvisable to attach too much physical significance to A∞ but
in the model it represents the asymptotic area at which flow from the arteries ceases.
This does not seem too abstracted and it would be very interesting to investigate how
it correlates with certain conditions and pharmacological interventions, particularly as
area data can be gathered non-invasively unlike pressure.
Furthermore, the parameter τ can give insight into ventricular relaxation. Nagueh
et al. state that “τ is a widely accepted invasive measure of the rate of LV relaxation”.23
A non-invasive measurement of τ may be very useful, for example in children with diastolic dysfunction. According to Dragulescu “Assessment of DD [diastolic dysfunction] in
childhood CM [cardiomyopathy] seems inadequate using current guidelines. The large
range of normal pediatric reference values allows diagnosis of DD in only a small proportion of patients”24 when using echocardiography. Therefore getting information on
τ from CMR could be desirable in children with presumed diastolic dysfunction.
3.4 Comparing Ascending and Descending Aorta
Finally, we thought it may be interesting to compare the form of the reservoir area in
two parts of the arterial network in the same volunteer. Data was recorded for healthy
volunteers in two instances, one for the ascending aorta just above the aortic root and
a separate acquisition in the descending aorta at the level of the diaphragm. The time
resolution was again 9.56ms.
14
Figure 6: Comparison of the log of the reservoir area for the ascending (magenta) and
descending (cyan) aorta.
The motivation for this was to investigate how the property (2.1), that the reservoir
pressure is theortically invariant in space, transferred into the area formulation. Immediately it was evident that it did not, as for all cases the reservoir log-area from the two
aortic sections did not relate to one another. An example is shown in Figure 6. Fitting to
the descending data was overall quite problematic because the data was extremely noisy.
However the conclusion that the reservoir area does change over the aorta is probably
not surprising. This can be considered a statement that the distensibility varies with
x, which is completely expected as the elastic properties of arteries have been shown to
vary along the aortic tree.25
4 Conclusion
I have demonstrated the validity of the reservoir-wave hypothesis when applied to area
data rather than the tradiational formulation in terms of pressure. Application of the
analysis to volunteer cardiovascular magnetic resonance data largely produced the expected functional forms of the reservoir and wave areas, analogous to those seen in
pressure data. The optimisation method failed for 2 out of 10 cases but I have suggested
variant methods which may circumvent this problem.
As a proof of concept I applied the analysis to patient and control groups which
15
should have varying reservoir-wave parameters. The mean values of these parameters
show a difference, but the sample sizes are too small to assess significance. With reference
to uses found for the pressure reservoir-wave I have suggested that the parameters of
the area reservoir-wave may have similar application that could enhance understanding
of cardivascular function under pathological or pharmacological influence.
I investigated the variation of the form of the area reservoir in different sections of
the aorta. It was found that unlike the pressure reservoir, the area reservoir does not
retain its form across the arterial system.
This work was largely a proof of principle, and there are a number of potentially
fruitful directions to take it in, some of which I have outlined in Sections 3.2 and 3.3.
In addition, it would be desirable to integrate the analysis of the reservoir-wave into
the existing framework of image analysis and wave intensity analysis in OsiriX. This
would allow immediate evaluation of the reservoir parameters and make any large-scale
analysis greatly more straightforward.
References
1 J.
Betts, Anatomy and Physiology. 2013.
2 “http://antranik.org/blood-vessels/,”
March 2015.
3 “http://people.ece.cornell.edu/land/,”
March 2015.
4 J.
Lotz, C. Meier, A. Leppert, and M. Galanski, “Cardiovascular flow measurement
with phase-contrast mr imaging: Basic facts and implementation 1,” Radiographics,
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