## Abstract

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 physiology. 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 intrinsic frequencies (IFs; *ω*_{1} and *ω*_{2}). The IFs are the dominant frequencies of the instantaneous frequency of the coupled heart + 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 (*ω*_{1} = *ω*_{2}). Extracted IFs from clinical data indicate that at young ages the total frequency variation (Δ*ω* = *ω*_{1} − *ω*_{2}) is close to zero and that Δ*ω* 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. Introduction

The 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 physiology 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 parameter 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 made in the past to elucidate the complex interaction between the left ventricular (LV) and wave dynamics of the large central 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 optimum coupling can be impaired owing to increased arterial stiffness, ageing, smoking or disease conditions, such as hypertension 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 frequency 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 complementary 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 network. 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 analyse 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 closure is marked by the dicrotic notch on the aortic input pressure wave. This coupled dynamic system has a dominant frequency that the instantaneous frequency oscillates around (note that this frequency is not the resonant frequency) 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 + aorta system.

By applying the STFR method, it is possible to compute the instantaneous frequency of the coupled heart + 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 constant 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; *ω*_{1} and *ω*_{2}). To extract the IF directly from the pressure waveform, a modified version of STFR was developed using a norm-2 (*L*_{2}) minimization method and a brute-force algorithm 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 constant 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 waveform 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 ventriculoarterial coupling and for diagnosis of cardiovascular disease (CVD). The idea of the IF concept is based on our observations that when we apply the adaptive STFR method to an aortic pressure wave to extract the instantaneous frequency () of the first intrinsic mode function (IMF; where *θ*_{1} 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 (*ω*_{1} and *ω*_{2}) 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 clinical data from human subjects to establish proof of concept for the IF method in the diagnosis of CVD.

## 2. Material and methods

### 2.1. Adaptive method of sparse time–frequency representation

The 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*, *θ*(*t*)*,* satisfying three properties: (i) *a*(*t*) is smoother than cos*θ*(*t*), (ii) *θ*(*t*) is strictly increasing in time, and (iii) the IMF has only one extremum between two consecutive zeros,
2.1

A real signal *s*(*t*) is called an *intrinsic signal* if it can be decomposed into a finite sum of IMFs
2.2

The essential idea behind the STFR is to find the sparsest representation 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 *L*_{2} minimization problem [14] for periodic signals. The description of the *L*_{2}-STFR algorithm is provided in appendix A.

### 2.2. Modified sparse time–frequency representation for heart–aorta system: intrinsic frequency algorithm

In our proposed method, we assume that the instantaneous frequency 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 (*L*_{2}) minimization method. The envelopes of the IMF are also assumed to be piecewise constant in time to distinguish between the two systems. Hence, the *L*_{2} minimization problem, for the extraction of the trend and frequency content of the input aortic pressure wave, is proposed as follows:

min:
2.3subject to:
2.4
2.5
2.6
2.7
2.8and *c* is a constant.

This problem is now reduced to solving for *a*_{1}, *a*_{2}, *c, b*_{1}, *b*_{2}, *ω*_{1} and *ω*_{2}. Equations (2.4) and (2.5) are linear constraints that ensure the continuity of the trend at the time *T*_{0} (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 (*ω*_{1} and *ω*_{2}), which we refer to as IFs. Where *ω*_{1} is the IF for the heart + aorta system (before aortic valve closure = before dicrotic notch), and *ω*_{2} is the IF for the decoupled aorta (after aortic valve closure = 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 frequencies (*ω*_{1}, *ω*_{2})_{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 aorta

A 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 provided. Simulations were performed for different levels of aortic rigidities (compliances) labelled *E*_{1} through *E*_{3}, where *E*_{1} is the aortic rigidity of a 30-year-old healthy individual [1]. All other *E _{i}* are multiplicative factors of

*E*

_{1}defined as follows:

*E*

_{2}= 1

*.*5

*E*

_{1}and

*E*

_{3}= 3

*E*

_{1}. At each

*E*, 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.

_{i}### 2.4. Clinical data

To 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, University 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 procedures 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. Results

### 3.1. The intrinsic frequency of aortic pressure waves

A 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 dominant frequency range at the beginning of the cardiac cycle and then shifts and oscillates around a second range of dominant 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 frequencies, 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 1*a* shows a typical aortic pressure waveform as well as the location of the dicrotic notch. Figure 1*b* shows the same aortic pressure waveform with the corresponding piecewise reconstruction using only the two IFs (*ω*_{1} and *ω*_{2}) of the first mode IMF overlaid on top of the original pressure waveform. For clarity, the portions of the reconstructed waveform that correspond to *ω*_{1} and *ω*_{2}, namely the systolic and diastolic phases, are shown in purple and green, respectively. To further illustrate this behaviour, overlays of two other types of aortic pressure waveforms and their reconstructions are provided in figure 1*c,d*. In all cases shown in figure 1, we see good agreement between the shape of the systolic and diastolic portions produced by the IFs and the original aortic pressure waveform.

### 3.2. Optimum heart rate prediction from the intrinsic frequencies

A CFD model was constructed to examine the relevance of the IFs to the pulsatile power workload on the left ventricle. Pulsatile power *P*_{pulse} was calculated using the following equation:
3.1where *p*(*t*) is the pressure, *q*(*t*) is the flow, *p*_{mean} is the mean pressure, *q*_{mean} is the mean flow and *T* is the period of the cardiac 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: *E*_{1}, *E*_{2} = 1*.*5*E*_{1} and *E*_{3} = 3*E*_{1}. When the two IF curves, *ω*_{1} and *ω*_{2}, 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 (*E*_{1} versus *E*_{3}). Additionally, from figure 2 it can be noted that high aortic rigidities have a greater effect on pulsatile workload in the range of physiological resting HRs. For example, given a resting HR of 80 bpm, an aortic stiffness of *E*_{1}, *E*_{2} and *E*_{3} results in pulsatile power workloads of 50, 95 and 325 mW, respectively.

### 3.3. Total frequency variation (Δ*ω*): an index for cardiovascular health and disease

When a similar analysis examining Δ*ω* = *ω*_{1} − *ω*_{2} is applied to a survey of published clinical data taken from healthy subjects of increasing age as shown in figure 3, we see a clear physiological pattern. This suggests that Δ*ω* is near zero at young ages when the heart–arterial system is operating close to the optimum state and that Δ*ω* 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 Δ*ω* owing to the ventricular–arterial system shifting from its optimum coupling.

### 3.4. First intrinsic frequency (*ω*_{1}): a medical index for heart disease

After observing the behaviour of Δ*ω* in response to ageing and CVD, we were motivated to investigate the physiological 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 *ω*_{1} 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 *ω*_{1} for the subset of subjects including the published healthy and HF data from figure 3, we observe that *ω*_{1} becomes elevated in HF with LV systolic dysfunction and otherwise remains relatively constant under healthy conditions as age advances. For example, all subjects with HF in our data population exhibited a *ω*_{1} above 120 bpm. By contrast, normal healthy subjects displayed a *ω*_{1} below 112 bpm.

### 3.5. Second intrinsic frequency (*ω*_{2}): a medical index for vascular disease

The aorta and arterial networks dominate the dynamics of the heart–arterial system after aortic valve closure. Hence, *ω*_{2} is likely to be affected by VDs such as arterial stiffening and hypertension. As seen in figure 5, if we examine *ω*_{2} for the subset of subjects including the published healthy and VD data displayed in figure 3, we observe that among healthy individuals *ω*_{2} decreases with age, which can be indicative of increasing arterial rigidity [8]. Figure 5 also shows that *ω*_{2} drops significantly with certain VDs such as hypertension and peripheral VDs, in most cases dropping below 36 bpm (figure 5).

## 4. Discussion

In this study, we formulated a modified version of the STFR method allowing the direct extraction of the dominant instantaneous frequency () of the first IMF from an aortic pressure wave. In addition to directly outputting the two IFs (*ω*_{1} and *ω*_{2}), 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 rate

To examine the physiological significance of the IFs, a computational model was constructed to explore the relationship between the IFs and the pulsatile workload on the heart. Additionally, to isolate the pulsatile power contributions of 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 cardiovascular 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 approximately 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 (Δ*ω* = 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 previously by researchers examining ventriculoarterial matching and optimal power output by the left ventricle [18–20].

### 4.2. The intrinsic frequencies as indices for cardiovascular disease

In the light of the relationship between the IFs and minimal pulsatile workload on the heart, it follows that the total frequency variation (TFV) should be very close to zero at young ages when there is an optimum balance between heart pumping dynamics and the dynamics of the aorta and its branches. From the clinical data presented in figure 3, we observed that TFV (Δ*ω* = *ω*_{1} − *ω*_{2}) 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, *ω*_{1} reflects the dynamics of the heart and *ω*_{2} the dynamics of the aorta and arterial network. For example, in figure 4 it was shown that *ω*_{1} increases above 120 bpm in patients with HF with LV systolic dysfunction. By contrast, in healthy individuals, *ω*_{1} 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 *ω*_{2}. As seen in figure 5, *ω*_{2} decreases linearly with age. This behaviour 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 *ω*_{2} is indirectly proportional to arterial rigidity or in other words *ω*_{2} 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 *ω*_{2} prematurely drops below 36 bpm independent of age. Although a more rigorous population study is needed, these results suggest that *ω*_{2} 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 threefold 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 evaluating 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 *ω*_{1} in an unfavourable way and shift the Δ*ω* to a level that is unsustainable and cause sudden death. Clearly, more data are needed to evaluate these possibilities, but the methodology presented here may prove to be a very useful clinical tool.

### 4.3. Critique of methods

In 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 demonstrating 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 assumptions has been thoroughly explained in Pahlevan & Gharib [7]. Nevertheless, the results from both clinical data and CFD data are complementary.

### 4.4. Conclusion

We have shown the proof of concept for a new medical index, the IF, and introduced a quantitative method based on instantaneous 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 conditions. From these data, we observed that the two IFs, *ω*_{1} and *ω*_{2}, representing the coupled heart + aorta system and decoupled aortic system, respectively, are close at young ages and gradually deviate through the progression of age or disease. 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 investigations 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.) acknowledge the support from Caltech innovative initiative grant (CII).

## Appendix A. Sparse time–frequency representation

### A.1. Sparse time–frequency representation algorithm

The 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
A1

This problem is an *L*_{0} minimization problem. Solving this problem is extremely difficult. It is a nonlinear and non-convex optimization problem [12,14]. To overcome this difficulty, a nonlinear matching pursuit method is proposed to approximate the original *L*_{0} minimization problem. Based on an approximation, the STFR method can be reduced to an *L*_{2} minimization problem [14]. A brief description of this algorithm is as follows:
A2
A3

In this formulation, the dictionary *D* is defined as
A4where *V*(*θ*) is a linear space consisting of functions smoother than cos *θ*(*t*):
A5More 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 *L*_{2} 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 waves

A demonstration of the STFR method applied to an exemplary 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 algorithm

In order to solve the modified STFR problem, a brute-force algorithm was used. First, the domain *D* was taken as
B1

In domain *D*, the frequencies *ω*_{1} and *ω*_{2} 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 cannot be accepted physically and mathematically as the solution of the problem.

Next, we discretize *D* for pairs of (*ω*_{1} and *ω*_{2}). For each point (*ω*_{1} and *ω*_{2}) in the discretized domain, the modified STFR problem is solved and the solution is stored as *P*(*ω*_{1}, *ω*_{2}). Note that the minimum of the modified STFR problem for the whole domain *D* corresponds to the minimum of *P*(*ω*_{1}, *ω*_{2}) over (*ω*_{1}, *ω*_{2}); this is a simple search problem. The corresponding minimum frequencies are denoted as (*ω*_{1}, *ω*_{2})_{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 frequencies and ensures that the corresponding minimizer frequencies (*ω*_{1}, *ω*_{2})_{m} are in fact the unique global minimizer.

## Appendix C. Computational fluid dynamics/fluid–solid interaction model of aorta

### C.1. Physical model

The 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 physiological 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 model

An 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]:
C1where ** W** is the mesh velocity,

**is the flow velocity,**

*V**p*is the static pressure,

*μ*

_{f}is the dynamic viscosity of the fluid and

*F*_{b}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],
C2and
C3In these equations, *σ*_{ij} is the wall stress tensor, *F* is the external force, *u* is the displacement vector, *ρ*_{s} is the wall density and *λ*, *μ*_{l} are Lamé constants.

The finite-element method with the direct two-way coupling 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 condition

A 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 min^{−1} 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].

- Received June 9, 2014.
- Accepted June 13, 2014.

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