rsif.royalsocietypublishing.org

ResearchCite this article: Pahlevan NM, Tavallali P,

Rinderknecht DG, Petrasek D, Matthews RV,

Hou TY, Gharib M. 2014 Intrinsic frequency for

a systems approach to haemodynamic

waveform analysis with clinical applications.

J. R. Soc. Interface 11: 20140617.

http://dx.doi.org/10.1098/rsif.2014.0617

Received: 9 June 2014

Accepted: 13 June 2014

Subject Areas:biomechanics, bioengineering, biomathematics

Keywords:pulse wave analysis, systems science,

cardiovascular disease, ventricular/arterial

coupling, instantaneous frequency,

arterial pressure

Author for correspondence:Morteza Gharib

e-mail: mgharib@caltech.edu

Electronic supplementary material is available

at http://dx.doi.org/10.1098/rsif.2014.0617 or

via http://rsif.royalsocietypublishing.org.

& 2014 The Author(s) Published by the Royal Society. All rights reserved.

Intrinsic frequency for a systems approachto haemodynamic waveform analysiswith clinical applications

Niema M. Pahlevan1, Peyman Tavallali2, Derek G. Rinderknecht3,Danny Petrasek4, Ray V. Matthews5, Thomas Y. Hou2 and Morteza Gharib3

1Medical Engineering, Division of Engineering and Applied Sciences, California Institute of Technology,1200 East California Boulevard, MC 301-46, Pasadena, CA 91125, USA2Applied and Computational Mathematics, Division of Engineering and Applied Sciences, California Institute ofTechnology, 1200 East California Boulevard, MC 9-94, Pasadena, CA 91125, USA3Graduate Aerospace Laboratories, Division of Engineering and Applied Sciences, California Institute ofTechnology, 1200 East California Boulevard, MC 205-45, Pasadena, CA 91125, USA4Medical Engineering, Division of Engineering and Applied Sciences, California Institute of Technology,1200 East California Boulevard, MC 217-50, Pasadena, CA 91125, USA5Keck School of Medicine, University of Southern California, 1510 San Pablo Street, Suite 322, Los Angeles,CA 90033, USA

The reductionist approach has dominated the fields of biology and medicine for

nearly a century. Here, we present a systems science approach to the analysis of

physiological waveforms in the context of a specific case, cardiovascular physi-

ology. Our goal in this study is to introduce a methodology that allows for novel

insight into cardiovascular physiology and to show proof of concept for a new

index for the evaluation of the cardiovascular system through pressure wave

analysis. This methodology uses a modified version of sparse time–frequency

representation (STFR) to extract two dominant frequencies we refer to as intrin-

sic frequencies (IFs; v1 and v2). The IFs are the dominant frequencies of the

instantaneous frequency of the coupled heart þ aorta system before the closure

of the aortic valve and the decoupled aorta after valve closure. In this study, we

extract the IFs from a series of aortic pressure waves obtained from both clinical

data and a computational model. Our results demonstrate that at the heart rate at

which the left ventricular pulsatile workload is minimized the two IFs are equal

(v1 ¼ v2). Extracted IFs from clinical data indicate that at young ages the total

frequency variation (Dv ¼ v1 2 v2) is close to zero and that Dv increases with

age or disease (e.g. heart failure and hypertension). While the focus of this

paper is the cardiovascular system, this approach can easily be extended to

other physiological systems or any biological signal.

1. IntroductionThe reductionist approach has dominated the science of biology and medicine.

While this approach has provided valuable information, it has not necessarily

translated into increased understanding. From a clinical standpoint, medicine

has been driven by the perpetuation of a normal physiological system,

yet symptoms are often managed through addressing singular parameters to

be corrected and not viewed as indicative of wider systemic problems. In this

context, we will present a systems science approach to cardiovascular physi-

ology by developing the concept of intrinsic frequency (IF) as a systems

parameter. Although generally speaking this methodology is applicable to

any physiological waveform, our focus in this paper will be its application to

arterial pressure waveforms. In doing so, we will show that using such a par-

ameter as a complement to well-established pathological markers may help

us create new pathways for discovery and increase our ability to predict

clinically relevant outcomes.

Analysing arterial waveforms is clinically important because it provides

information about states of health and disease [1]. Significant efforts have been

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made in the past to elucidate the complex interaction between

the left ventricular (LV) and wave dynamics of the large cen-

tral arteries such as the aorta [1–5]. However, extracting

reliable information from haemodynamic waves about health

or disease conditions remains a significant challenge to

modern medicine. It is well accepted that the dynamics of

the left ventricle, arterial wave dynamics and the interaction

between the two determine the arterial pressure wave [1,2,6].

This means the pressure wave contains information about

these dynamic systems and their optimum coupling. This opti-

mum coupling can be impaired owing to increased arterial

stiffness, ageing, smoking or disease conditions, such as hyper-

tension and heart failure (HF) [1]. Therefore, it follows that

these waves also carry information about the diseases of the

heart, vascular disease (VD), as well as the coupling of the

heart and the arterial network [2,6–8].

There are several methods for analysing arterial pulse

waveforms [1,9]. Some of these methods are based on fre-

quency domain methodologies such as the impedance

method (Fourier method), whereas others are based on time

domain methodologies such as wave intensity analysis [10].

Both frequency and time domain methods give complemen-

tary results [9,11]. These methods however require both

pressure and flow waves to be measured simultaneously at

the same location, which is clinically difficult, if not impossible.

Our approach is based on a newly developed sparse

time–frequency representation (STFR) method [12] and an

analysis of the systemic coupling between the dynamics of

LV contraction and the dynamics of waves in the arterial net-

work. The STFR method is inspired by the empirical mode

decomposition (EMD) method [13] and provides a more

systematic way to define instantaneous frequency. Similar

to the EMD method, the STFR method is well suited to ana-

lyse nonlinear non-stationary data, is less sensitive to noise

perturbation and preserves some of the intrinsic physical

properties of the signal [12,14]. Although the application of

the EMD method to biological problems has been introduced

by Huang et al. [15,16], this paper will show the potential of

this concept to diagnose heart and VDs as well as its potential

to quantify the optimum coupling between the heart and

arterial system.

We begin with the premise that the left ventricle of the

heart and aorta construct a coupled dynamic system before

the closure of the aortic valve. The onset of aortic valve clo-

sure is marked by the dicrotic notch on the aortic input

pressure wave. This coupled dynamic system has a domi-

nant frequency that the instantaneous frequency oscillates

around (note that this frequency is not the resonant fre-

quency) which is not necessarily constant over the cardiac

cycle. This dominant frequency is influenced by the dynamics

of both the heart and the aorta. After valve closure, the heart

and the aorta are decoupled from each other. This means that

the dominant frequency is dictated only by the dynamics

of the aorta and its branches (arterial network). From this

point forward in this paper, we will refer to the heart and

arterial network as the heart þ aorta system.

By applying the STFR method, it is possible to compute the

instantaneous frequency of the coupled heart þ aorta system

and the decoupled aorta system from the aortic pressure

wave alone. The application of this technique to aortic pressure

waves led us to the observation that the instantaneous

frequency oscillates around different dominant frequencies

before and after the dicrotic notch, or closure of the aortic

valve. These instantaneous frequencies are not necessarily con-

stant in time, but do represent the dominant frequencies at any

instant time throughout the cardiac cycle beginning with the

coupled system of the heart and aorta prior to the dicrotic

notch and the decoupled system of the aorta itself afterwards.

We refer to these dominant frequencies as intrinsic frequencies

(IFs; v1 and v2). To extract the IF directly from the pressure

waveform, a modified version of STFR was developed using

a norm-2 (L2) minimization method and a brute-force algor-

ithm was applied to solve the problem. This algorithm

considers all possible values of frequencies to ensure that the

corresponding minimizer frequencies are in fact the unique

minimizers of the problem. In this regard, the piecewise con-

stant frequency before the dicrotic notch is the IF of the

heart–aorta system and the one after the dicrotic notch is the

IF of the aortic system. The main advantage of this method

in contrast to well-known and widely used impedance and

wave intensity methods is that only one arterial pressure wave-

form is required to perform the analysis [1,10].

Here, we show proof of concept of the IF as a new medical

index for the identification of the optimum left ventriculoar-

terial coupling and for diagnosis of cardiovascular disease

(CVD). The idea of the IF concept is based on our obser-

vations that when we apply the adaptive STFR method to

an aortic pressure wave to extract the instantaneous fre-

quency ( _u1(t) ¼ du1=dt) of the first intrinsic mode function

(IMF; where u1 is the phase angle of the first IMF) we see

that a dominant instantaneous frequency exists on either

side of the dicrotic notch. A computational fluid dynamics

(CFD) model was constructed to study the link between the

IFs (v1 and v2) and LV pulsatile power workload across a

range of heart rates (HRs) and aortic rigidities. Finally, our

analysis method will be applied to a small sample set of clini-

cal data from human subjects to establish proof of concept for

the IF method in the diagnosis of CVD.

2. Material and methods2.1. Adaptive method of sparse time – frequency

representationThe notion of the IMF was first introduced by Huang et al. [13].

A more mathematical definition of the IMF is given by Hou & Shi

[12], as follows.

A signal f (t) is called an IMF if there exists an envelope, a(t) .

0, and a phase function, u(t), satisfying three properties: (i) a(t)is smoother than cosu(t), (ii) u(t) is strictly increasing in

time, and (iii) the IMF has only one extremum between two

consecutive zeros,

f (t) ¼ a(t) cosu(t), t [ <: (2:1)

A real signal s(t) is called an intrinsic signal if it can be decom-

posed into a finite sum of IMFs

s(t) ¼XM

i¼1ai(t) cos ui(t): (2:2)

The essential idea behind the STFR is to find the sparsest rep-

resentation of multi-scale data within the largest possible

dictionary of IMFs. This huge dictionary consists of elements

(or bases) that are not defined a priori. The use of an infinite

dimensional highly redundant data-driven basis is what makes

the STFR truly adaptive. Based on an approximation, the STFR

method can be reduced to an L2 minimization problem [14] for

r

3

periodic signals. The description of the L2-STFR algorithm is

provided in appendix A.

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2.2. Modified sparse time – frequency representation forheart – aorta system: intrinsic frequency algorithm

In our proposed method, we assume that the instantaneous fre-

quency of the coupled heart–aorta and decoupled aorta are

piecewise constant in time. This enables us to extract the IFs

directly from the arterial pressure wave. The IF is the frequency

that carries the maximum power in equation (2.2). To extract the

IF, we propose a simple but effective norm-2 (L2) minimization

method. The envelopes of the IMF are also assumed to be piece-

wise constant in time to distinguish between the two systems.

Hence, the L2 minimization problem, for the extraction of the

trend and frequency content of the input aortic pressure wave,

is proposed as follows:

min:

f (t)� x(0, T0)s1(t)� x(T0, T)s2(t)� ck k22, (2:3)

subject to:

a1 cos (v1T0)þ b1 sin (v1T0) ¼ a2 cos (v2T0)þ b2 sin (v2T0), (2:4)

a1 ¼ a2 cos (v2T)þ b2 sin (v2T), (2:5)

s1(t) ¼ a1 cos (v1t)þ b1 sin (v1t) (2:6)

and s2(t) ¼ a2 cos (v2t)þ b2 sin (v2t): (2:7)

Here, x(a, b) ¼ 1 a � t � b0 otherwise,

�(2:8)

and c is a constant.

This problem is now reduced to solving for a1, a2, c, b1, b2, v1

and v2. Equations (2.4) and (2.5) are linear constraints that ensure

the continuity of the trend at the time T0 (dicrotic notch) and the

periodicity of the trend, respectively. This minimization states

that the aortic input pressure wave can be approximated by

two incomplete sinusoids with different frequencies (v1 and

v2), which we refer to as IFs. Where v1 is the IF for the

heart þ aorta system (before aortic valve closure ¼ before dicro-

tic notch), and v2 is the IF for the decoupled aorta (after aortic

valve closure ¼ after dicrotic notch).

The original minimization problem is not convex. Thus, we

may have several local minima. To find the global minimum,

we use a brute-force algorithm over all possible values of

frequencies to ensure that the corresponding minimizer frequen-

cies (v1, v2)m are in fact the unique global minimizer frequencies

of the original minimization problem. The details of the

brute-force algorithm are provided in appendix B.

2.3. Computational aortaA physiologically relevant computational fluid dynamics (CFD-

FSI) model of the aorta with fluid–solid interaction (FSI) was

used. The methods as well as the physical parameters of the

model were the same as those described in Pahlevan & Gharib

[7], in which full details of the computational model were pro-

vided. Simulations were performed for different levels of aortic

rigidities (compliances) labelled E1 through E3, where E1 is the

aortic rigidity of a 30-year-old healthy individual [1]. All other

Ei are multiplicative factors of E1 defined as follows: E2 ¼

1.25E1 and E3 ¼ 3E1. At each Ei, simulations were completed

for eight HRs: 70.5, 75, 89.5, 100, 120, 136.4, 150 and 187.5

beats per minute (bpm). Information about the physical model,

mathematical model, inflow boundary condition and outflow

boundary condition as well as all other model parameters such

as cardiac output (CO), terminal resistance, terminal compliance

and the shape of the inflow wave are detailed in appendix C.

2.4. Clinical dataTo examine the potential clinical relevance of the IF method, data

were first gathered from published works [1]. In addition to the

publically available data, invasive blind clinical data were

obtained from patients having clinically indicated procedures in

the cardiac catheterization laboratory at Keck Medical Center, Uni-

versity of Southern California, USA (USC). Retrospective de-

identified data were analysed for 16 consecutive blinded patient

datasets. The data were collected as part of routine medical pro-

cedures using 6F fluid-filled catheters. All clinical data were

abstracted from the Keck Medical Center cardiac catheterization

laboratory research database and approved by the University of

Southern California Institutional Review Board.

3. Results3.1. The intrinsic frequency of aortic pressure wavesA series of aortic pressure waves were examined to observe

the behaviour of the adoptive STFR method. It was observed

that the instantaneous frequency oscillates around one domi-

nant frequency range at the beginning of the cardiac cycle

and then shifts and oscillates around a second range of domi-

nant frequencies (see appendix A). This implies that there is a

different dominant frequency within each band, the first

associated with the heart–aorta system and the second

with the arterial system alone. It must be mentioned that

the IFs are the dominant instantaneous frequencies and in

this regard are fundamentally different from resonant

frequencies. To seamlessly extract these dominant frequen-

cies, we created a modified version of the STFR for the

heart–aorta system called the IF algorithm. Figure 1 shows

the application of this algorithm to a number of exemplary

aortic pressure waveforms. Figure 1a shows a typical aortic

pressure waveform as well as the location of the dicrotic

notch. Figure 1b shows the same aortic pressure waveform

with the corresponding piecewise reconstruction using only

the two IFs (v1 and v2) of the first mode IMF overlaid on

top of the original pressure waveform. For clarity, the portions

of the reconstructed waveform that correspond to v1 and v2,

namely the systolic and diastolic phases, are shown in purple

and green, respectively. To further illustrate this behaviour, over-

lays of two other types of aortic pressure waveforms and their

reconstructions are provided in figure 1c,d. In all cases shown

in figure 1, we see good agreement between the shape of the sys-

tolic and diastolic portions produced by the IFs and the original

aortic pressure waveform.

3.2. Optimum heart rate prediction from the intrinsicfrequencies

A CFD model was constructed to examine the relevance of the

IFs to the pulsatile power workload on the left ventricle. Pulsa-

tile power Ppulse was calculated using the following equation:

Ppulse ¼1

T

ðT

0

p(t)q(t) dt� pmeanqmean, (3:1)

where p(t) is the pressure, q(t) is the flow, pmean is the mean

pressure, qmean is the mean flow and T is the period of the car-

diac cycle. The computational model used to generate the aortic

pressure waveforms is described in Pahlevan & Gharib [7]. The

results of this investigation are shown in figure 2 for three levels

of aortic rigidity: E1, E2¼ 1.5E1 and E3 ¼ 3E1. When the two IF

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.070

80

90

100

110

120

130

time (s)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.850

60

70

80

90

100

110

120

130

140

150

time (s)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.880

90

100

110

120

130

time (s)

pres

sure

(m

m H

g)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.850

60

70

80

90

100

110

120

130

140

150

time (s)

pres

sure

(m

m H

g)

(a) (b)

(c) (d )

Figure 1. The IF reconstruction of aortic pressure waveforms. (a) A typical aortic pressure waveform as well as the location of the dicrotic notch (marked by the redline). (b) The piecewise reconstruction of the pressure wave given in (a) overlaid on top of the original pressure waveform (blue). The portions of the reconstructedwaveform represented by v1 and v2 are shown in purple and green, respectively. (c) An overlay of a different type of aortic pressure waveform (blue) and itsreconstruction ( purple and green). (d ) An overlay of another type of aortic pressure waveform (different characteristics from (b) and (c)) and its reconstruction. Onlythe two IFs (v1 and v2) of the first mode IMF have been used in the reconstruction of systolic (v1) and diastolic (v2) portions. A good agreement between theshape of the systolic and diastolic portions produced by the IFs and the original aortic pressure waveform can be seen.

60 80 100 120 140 160 180 20040

60

80

100

120

heart rate (bpm)

puls

atile

pow

er (

mW

)

50

100

150

200

250

300

350(a)(i) (i) (i)

(ii) (ii) (ii)

(b) (c)

IF (

bpm

)

60 80 100 120 140 160 180 20040

60

80

100

120

heart rate (bpm)

50

100

150

200

250

300

350

60 80 100 120 140 160 180 20050

100

150

200

250

300

350

400

heart rate (bpm)

50

100

150

200

Figure 2. IFs (a(i), b(i) and c(i)) and pulsatile power (a(ii), b(ii) and c(ii)) versus HR. (a) The aortic rigidity of a healthy 30 year old, E1, yields two IF curves thatintersect at an optimum HR � 110 bpm at which the pulsatile power is minimized. (b) The effect of a 50% increase in aortic rigidity (E2 ¼ 1.5E1) yields two curveswhich cross each other at an optimum HR � 140 bpm. (c) An aortic rigidity of E3 ¼ 3E1 yields two IF curves that intersect an optimum HR � 190 bpm. Note: v1

denoted by the red curve is the IF for the coupled heart þ aorta, whereas v2 denoted by the blue curve is the IF for the decoupled aorta.

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0 10 20 30 40 50 60 70 80 90−20

−10

0

10

20

30

40

50

60

70

80

90

100

110

120

age (years)

Dw (

bpm

)

+

+

+

+

+

+

+

++ +

+

++

++

healthy (published data)linear fit (healthy published data)VD (blind clinical data)HF (blind clinical data)HF (published data)

Figure 3. TFV (Dv ¼ v1 2 v2) versus age for healthy and CVD conditions. The IFs, v1 and v2, are close to one another at young ages therefore the differenceis very near zero. A linear fit shows that the difference, Dv, increases with age in healthy subjects. Furthermore, it can be observed that CVD increases Dv as thedisease shifts the ventricular – arterial system out of optimum coupling. In this study, all CVD patients displayed a Dv . 60 bpm. All published waveform dataused to produce this plot were taken from Nichols et al. [1]. Detailed explanations about the ages for the data marked with a plus sign (þ) are given in theelectronic supplementary material. HF refers specifically to HF with LV systolic dysfunction. (Online version in colour.)

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curves, v1 and v2, are graphed as a function of HR, remarkably,

the two curves always intersect at the optimum computed HR

at which the LV pulsatile workload is minimized (figure 2). In

other words, the LV pulsatile workload reaches its minimum

when the two IFs become equal.

The plots of IF and pulsatile power versus HR in figure 2

also clearly show that at increased levels of aortic rigidity the

optimum HR shifts to the right. For example, changing aortic

stiffness threefold increases the optimum HR from 110 bpm

to approximately 185 bpm (E1 versus E3). Additionally,

from figure 2 it can be noted that high aortic rigidities have

a greater effect on pulsatile workload in the range of physio-

logical resting HRs. For example, given a resting HR of

80 bpm, an aortic stiffness of E1, E2 and E3 results in pulsatile

power workloads of 50, 95 and 325 mW, respectively.

3.3. Total frequency variation (Dv): an indexfor cardiovascular health and disease

When a similar analysis examining Dv ¼ v1 2 v2 is applied

to a survey of published clinical data taken from healthy sub-

jects of increasing age as shown in figure 3, we see a clear

physiological pattern. This suggests that Dv is near zero at

young ages when the heart–arterial system is operating

close to the optimum state and that Dv increases with age.

The survey was further extended to include published clinical

data from subjects with HF with LV systolic dysfunction in

addition to clinical data from subjects with vascular disease

(VD) and with HF with LV systolic dysfunction gathered

through collaboration with the catheterization laboratory at

USC. The analysis of these aortic pressure waveforms,

shown in figure 3, demonstrates that, in addition to ageing,

CVD also increases Dv owing to the ventricular–arterial

system shifting from its optimum coupling.

3.4. First intrinsic frequency (v1): a medical indexfor heart disease

After observing the behaviour of Dv in response to ageing

and CVD, we were motivated to investigate the physiologi-

cal information contained in the individual IFs. Since the

dynamics of the heart–arterial system are dominated by

the dynamics of the heart before aortic valve closure, we

anticipated that v1 would be affected by pathophysiological

conditions that impair the pumping dynamics of the heart

such as HF with LV systolic dysfunction. As shown in

figure 4, by examining the v1 for the subset of subjects includ-

ing the published healthy and HF data from figure 3, we

observe that v1 becomes elevated in HF with LV systolic dys-

function and otherwise remains relatively constant under

healthy conditions as age advances. For example, all subjects

with HF in our data population exhibited a v1 above

120 bpm. By contrast, normal healthy subjects displayed a

v1 below 112 bpm.

3.5. Second intrinsic frequency (v2): a medical indexfor vascular disease

The aorta and arterial networks dominate the dynamics of

the heart–arterial system after aortic valve closure. Hence,

v2 is likely to be affected by VDs such as arterial stiffening

and hypertension. As seen in figure 5, if we examine v2 for

the subset of subjects including the published healthy and

VD data displayed in figure 3, we observe that among

healthy individuals v2 decreases with age, which can be

indicative of increasing arterial rigidity [8]. Figure 5 also

shows that v2 drops significantly with certain VDs such as

hypertension and peripheral VDs, in most cases dropping

below 36 bpm (figure 5).

10 20 30 40 50 60 70 8080

90

100

110

120

130

140

150

160

age (years)

w1

(bpm

)

+

+

+

+

+

+

++

++

+

+

+

+

+

healthy (published data)HF (blind clinical data)HF (published data)

Figure 4. v1 at different ages and under healthy and HF conditions. All subjects with HF (LV systolic dysfunction) show v1 . 120 bpm (above the top dashedblack line). Normal subjects show v1 , 112 bpm (below the bottom dashed red line). Data points referred to as published were computed from waveforms takenfrom Nichols et al. [1]. Detailed explanations about the ages for the data marked with a plus sign (þ) are given in the electronic supplementary material. (Onlineversion in colour.)

20 30 40 50 60 70 80 900

20

40

60

80

100

120

age (years)

w2

(bpm

)

++

+ ++

+

+++

+

healthy (published data)

linear fit (healthy published data)

VD (blind clinical data)

Figure 5. v2 at different ages under healthy and VD conditions. All VD patients show v2 , 36 bpm (below the horizontal dashed red line). A linear fit shows thatv2 decreases with age by approximately 10 bpm per decade. Data points referred to as published were computed from waveforms taken from Nichols et al. [1].Detailed explanations about the ages for the data marked with a plus sign (þ) are given in the electronic supplementary material. (Online version in colour.)

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4. DiscussionIn this study, we formulated a modified version of the STFR

method allowing the direct extraction of the dominant instan-

taneous frequency ( _u1(t) ¼ du1=dt) of the first IMF from an

aortic pressure wave. In addition to directly outputting the

two IFs (v1 and v2), the main advantage of this method is

that only one arterial waveform, namely the pressure wave,

is required to perform the analysis in contrast to well-

known and widely used impedance and wave intensity

methods where both pressure and flow waves are required

[1,10]. Additionally, as only the shape of the waveform is

required to calculate the IFs, a wide range of both invasive

and non-invasive arterial pressure waveform measurement

techniques can be used.

4.1. Total frequency variation and optimum heart rateTo examine the physiological significance of the IFs, a

computational model was constructed to explore the relation-

ship between the IFs and the pulsatile workload on the heart.

Additionally, to isolate the pulsatile power contributions of

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the aorta, parameters related to the left ventricle were kept

constant (see Material and methods and appendix C) [7].

Although this model is not entirely physiological, since

both HR and stroke volume increase in response to an

increase in required CO, it provides a framework with

which to explore the IF method and illustrates the importance

of aortic wave dynamics on the workload of the cardiovascu-

lar system. As shown in figure 2, based on the conditions of

the aorta, changing aortic stiffness by a factor of 3 from that of

a healthy 30 year old can increase the pulsatile workload on

the heart from 50 to 325 mW at a resting HR of 80 bpm. The

potential significance of this additional load on the heart is

clear when one considers that this is nearly a sevenfold

increase in pulsatile power at an HR of 80 bpm and that the

average hydraulic power of the heart is only about approxi-

mately 1 W [7,17]. Figure 2 also shows that regardless of

aortic stiffness the two IF curves intersect at an HR at

which the pulsatile power on the heart is minimized. In

other words at this optimum HR, the IFs of the heart–aorta

system before and after decoupling are equal (Dv ¼ 0).

These results reiterate those of our previous work, which

suggested that there is an optimum HR at which LV pulsatile

power is minimized, and this optimum HR shifts to a higher

value as the aortic rigidity increases [7]. Additionally, these

findings are generally in agreement with those observed pre-

viously by researchers examining ventriculoarterial matching

and optimal power output by the left ventricle [18–20].

4.2. The intrinsic frequencies as indicesfor cardiovascular disease

In the light of the relationship between the IFs and minimal

pulsatile workload on the heart, it follows that the total fre-

quency variation (TFV) should be very close to zero at young

ages when there is an optimum balance between heart pump-

ing dynamics and the dynamics of the aorta and its branches.

From the clinical data presented in figure 3, we observed that

TFV (Dv ¼ v1 2 v2) increases naturally with age as optimum

coupling is disrupted and that in cases of HF or VDs we see

TFV more rapidly deviate from the ageing line. This means

that subjects with very different ages can have the same TFV

given the severity of their CVD. In this regard, the results of

the clinical data suggest that the TFV can be considered as a

possible marker of left ventricle–arterial coupling as well as

being strongly correlated to CVD.

Taken individually, the IFs also contain information

related to the respective systems which are engaged during

the cardiac cycle. Namely, v1 reflects the dynamics of the

heart and v2 the dynamics of the aorta and arterial network.

For example, in figure 4 it was shown that v1 increases above

120 bpm in patients with HF with LV systolic dysfunction. By

contrast, in healthy individuals, v1 remains below 112 bpm.

Future work will aim at confirming the above observations

using more diverse clinical datasets at various stages of HF.

Likewise, changes in the dynamics of the aorta and arterial

network due to ageing or VD will be reflected in the value of

v2. As seen in figure 5, v2 decreases linearly with age. This be-

haviour is similar to the observations of other researchers

monitoring arterial stiffness through techniques such as

pulse wave velocity [8]. On the contrary, however, it is crucial

to point out that v2 is indirectly proportional to arterial rigid-

ity or in other words v2 decreases with increasing arterial

stiffness. In this regard, it is important to note that these IFs

should not be confused with the resonance frequency from

classical dynamical systems (e.g. mass–spring system), which

increases with rigidity. Additionally, as illustrated in figure 5

under VD conditions v2 prematurely drops below 36 bpm

independent of age. Although a more rigorous population

study is needed, these results suggest that v2 has potential as

a marker of vascular ageing as well as for diagnosis of VD

and the quantification of their severity (e.g. hypertension).

Clinical studies commonly challenge medical science to

interpret confounding or paradoxical results. A relevant

example is the observation noted in the Framingham Heart

Study that the risk of sudden death was increased by three-

fold in treated hypertensive subjects compared with

untreated [21]. The subjects were treated with a thiazide

diuretic and the explanation was relegated to a potential

electrolyte imbalance. In the light of our new method of eval-

uating the dynamic vascular physiology, a new explanation

may be forthcoming. When the aetiology of the hypertension

is a result of the vasculature alone, the effect of diuretic

therapy (reducing preload) may affect the v1 in an unfavour-

able way and shift the Dv to a level that is unsustainable and

cause sudden death. Clearly, more data are needed to evalu-

ate these possibilities, but the methodology presented here

may prove to be a very useful clinical tool.

4.3. Critique of methodsIn this study, we have proposed a new method for analysing

cardiovascular physiology using aortic pressure waveforms.

The clinical study data used for the analysis were collected

from previously published work or from retrospective

blinded patient datasets (see Material and methods). The

number of aortic pressure waveforms we could attain were

limited and not from a designed study. In this regard, a

focused clinical study would be required before extracting

any true statistical correlations. Our analysis however

shows a general trend that fulfils our intention of demonstrat-

ing a proof of concept. With regards to the CFD model, the

following assumptions have been made: (1) the blood was

assumed to be an incompressible Newtonian fluid; (2) the

aortic wall was assumed to be elastic and isotropic; (3) the

aortic arch and bifurcations were excluded; (4) the truncated

vasculature was modelled with an extension tube boundary

model [22]; and (5) the left ventricle was assumed to be a

flow source [2,23,24]. The effect of these modelling assump-

tions has been thoroughly explained in Pahlevan & Gharib

[7]. Nevertheless, the results from both clinical data and

CFD data are complementary.

4.4. ConclusionWe have shown the proof of concept for a new medical index,

the IF, and introduced a quantitative method based on instan-

taneous frequency theory. Using only one pressure waveform,

the IF concept can be used to quantify the impaired balance

between the heart and aorta under various disease conditions.

One important advantage of this method is that only the

shape of the pressure waveform, not the magnitude, is required

to extract the IFs. In this study, the IFs of the cardiovascular

system were extracted from clinical data under resting con-

ditions. From these data, we observed that the two IFs, v1 and

v2, representing the coupled heart þ aorta system and

decoupled aortic system, respectively, are close at young ages

and gradually deviate through the progression of age or disease.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

20

40

60

80

100

120

140

time (s)

inst

anta

neou

s fr

eque

ncy

(bpm

)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.850

60

70

80

90

100

110

120

130

140

150(a) (b)

time (s)

pres

sure

(m

m H

g)

Figure 6. Instantaneous frequency of the pressure wave’s first IMF. The range of instantaneous frequency oscillation (marked by the grey band) changes after the dicroticnotch (marked by the vertical red line). (a) The aortic input pressure. (b) The instantaneous frequency of the aortic input pressure wave. (Online version in colour.)

rsif.royalsocietypublishing.orgJ.R.Soc.Interface

11:20140617

8

Additionally, we established a link between the closeness of

these frequencies and minimal pulsatile workload on the left

ventricle. Examined individually, the IFs contain information

relating to LV systolic dysfunction and VD. Further investi-

gations are needed to analyse the IF indices under non-resting

conditions such as exercise. Future studies are planned to

verify the predictive value of this concept in the detection of

CVD states. While this paper was focused on the heart–arterial

system, these principles may be extended to the full vascular

system including venous return or generalized to other systems:

for example, the gastrointestinal system, where there are natural

rhythm and waves amenable to a similar analysis.

Funding statement. The authors (N.M.P., D.G.R. and M.G.) acknowledgethe support from Caltech innovative initiative grant (CII).

Appendix A. Sparse time – frequencyrepresentationA.1. Sparse time – frequency representation algorithmThe adaptive STFR method consists of two major steps. The

first step is to construct a highly redundant dictionary of all

IMFs, D. The second step is to find the sparsest decomposition

by solving a nonlinear optimization problem

min: Msubject to: s(t) ¼

PMi¼1 ai(t) cos ui(t), ai(t) cos ui(t) [ D,

(i ¼ 1, . . . , M):

9>=>;

(A 1)

This problem is an L0 minimization problem. Solving this

problem is extremely difficult. It is a nonlinear and non-

convex optimization problem [12,14]. To overcome this diffi-

culty, a nonlinear matching pursuit method is proposed to

approximate the original L0 minimization problem. Based

on an approximation, the STFR method can be reduced to

an L2 minimization problem [14]. A brief description of this

algorithm is as follows:

min: f(t)� a(t) cos u(t)k k22, (A 2)

subject to: a(t) cos u(t) [ D: (A 3)

In this formulation, the dictionary D is defined as

D ¼ a(t) cos u(t):du

dt¼ _u(t) � 0, a(t), _u(t) [ V(u)

� �,

(A 4)

where V(u) is a linear space consisting of functions smoother

than cos u(t):

V(u) ¼ span 1, cosku

2Lu

� �, sin

ku2Lu

� �, k ¼ 1, . . . , Lu

� �:

(A 5)

More detail about the dictionary, D, can be found in Hou &

Shi [14].

At each step of the algorithm, an IMF is extracted. The

residual is treated as a new signal and the L2 minimization

is again applied to the residual. By this nonlinear matching

pursuit method, one can extract the different scales of a

multi-scale, non-stationary and nonlinear signal [14].

A.2. Instantaneous frequency of aortic pressure wavesA demonstration of the STFR method applied to an exemp-

lary aortic pulse pressure waveform and the corresponding

instantaneous frequency curve of the first IMF are shown in

figure 6. In both plots, the location of the dicrotic notch is

denoted by a vertical line. As shown in figure 6, on either

side of the dicrotic notch there is a distinct band of

frequencies around which the instantaneous frequency

oscillates, marked by the grey band.

Appendix B. Brute-force algorithmIn order to solve the modified STFR problem, a brute-force

algorithm was used. First, the domain D was taken as

D ¼ {(v1, v2) such that 0 , v1, v2 � C}: (B 1)

In domain D, the frequencies v1 and v2 are bounded above

by some constant C. This is a valid assumption since the

aortic pressure wave signal has a certain level of smoothness,

and the signal is not rough; therefore, certain frequencies

rsif.royalsocietypublishing.orgJ.R.Soc.

9

cannot be accepted physically and mathematically as the sol-

ution of the problem.

Next, we discretize D for pairs of (v1 and v2). For each

point (v1 and v2) in the discretized domain, the modified

STFR problem is solved and the solution is stored as P(v1,

v2). Note that the minimum of the modified STFR problem

for the whole domain D corresponds to the minimum of

P(v1, v2) over (v1, v2); this is a simple search problem. The

corresponding minimum frequencies are denoted as (v1,

v2)m. The original minimization problem is not convex.

Thus, we may have several local minima. However, the

brute-force algorithm looks over all possible values of fre-

quencies and ensures that the corresponding minimizer

frequencies (v1, v2)m are in fact the unique global minimizer.

Interface11:20140617

Appendix C. Computational fluid dynamics/fluid – solid interaction model of aortaC.1. Physical modelThe methods, the physical parameters of the model as well as

the relevance and accuracy of the model assumptions were

described previously [7]. The geometrical data such as length,

diameter and wall thickness were all within the average physio-

logical range [25]. The change in rigidity along the wall of the

aorta and tapering of the aorta were considered in the model;

however, the aortic arch and bifurcations were excluded. The

blood was assumed to be an incompressible Newtonian fluid.

The aortic wall was assumed to be elastic and isotropic. The

material properties of the wall were taken from Nichols et al. [1].

C.2. Mathematical and computational modelAn arbitrary Lagrangian–Eulerian (ALE) formulation was

applied to solve the FSI problem. In an ALE formulation,

the Navier–Stokes equations (for an incompressible fluid)

take the following form [22,26]:

r � V ¼ 0

rf

@V

@tþ (V �W)rV

� �þrp ¼ mfr2V þ Fb,

8<: (C 1)

where W is the mesh velocity, V is the flow velocity, p is the

static pressure, mf is the dynamic viscosity of the fluid and Fb

is the body force.

A no-slip boundary condition was assumed at the wall.

The coupling equations, applied to the solid–fluid interface,

were displacement compatibility and traction equilibrium at

the wall.

Large deformation–small strain theory was considered

for the solid domain (wall of the aortic model). The solid

mechanics equations, constitutive relation (equation (C 2))

and balance of momentum (equation (C 3)), for a linear elastic

isotropic material in Lagrangian form, were used to calculate

the dynamic motion of the elastic wall [27],

sij ¼ l1kkdij þ 2ml1ij (C 2)

and

sij,j þ Fi ¼ rs€ui: (C 3)

In these equations, sij is the wall stress tensor, F is the external

force, u is the displacement vector, rs is the wall density and

l, ml are Lame constants.

The finite-element method with the direct two-way coup-

ling method of FSI was used. The time integration scheme

was the implicit Euler method. The commercial package

ADINA, v. 8.6 (ADINA R&D, Inc., MA, USA) was used to

run the simulations. Full details of the formulation of the

FSI model and numerical method can be found in our

recent publication [7].

C.3. Inflow and outflow boundary conditionA physiological flow wave with a flat velocity profile, the same

as Pahlevan & Gharib [7], was imposed at the inlet. It was

scaled to give a CO of 4.6 l min21 for any desired HR. The

choice of an outflow boundary condition is important since

aortic waves can be greatly affected by changes in the radial

arteries. These arteries can affect wave dynamics in the aorta

by altering the wave arrival time at the inlet of the aorta as

well as by changing the terminal volume compliance and

resistance. We used the extension tube boundary model for

the outflow boundary condition [22]. This outflow boundary

model involves extending the computational domain by an

elastic tube connected to a rigid contraction tube. This outflow

boundary model takes into account the effects of the truncated

vasculature (resistance, compliance and wave reflection). The

geometrical and material properties of the outflow boundary

model are the same as in Pahlevan & Gharib [7].

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