# Dynamic simulation of aortic valve stenosis using a lumped parameter
cardiovascular system model with flow regime dependent valve pressure loss
characteristics

Ryno Laubscher  Institute for Biomedical Engineering, Department of Mechanical
and Mechatronic Engineering, Stellenbosch University, Stellenbosch, South
Africa  Jacques Liebenberg and Philip Herbst  Division of Cardiology, Faculty
of Medicine and Health Sciences, Stellenbosch University, Cape Town, South
Africa

###### Abstract

Valvular heart diseases are growing concern in impoverished parts of the
world, such as Southern-Africa, claiming more than 31 % of total deaths
related to cardiovascular diseases. The ability to model the effects of
regurgitant and obstructive lesions on the valve body can assist clinicians in
preparing personalised treatments. In the present work, a multi-compartment
lumped parameter model of the human cardiovascular system is developed, with a
newly proposed valve modelling approach which accounts for geometry and flow
regime dependent pressure drops along with the valve cusp motion. The model is
applied to study various degrees of aortic stenosis using typical human
cardiovascular parameters. The results generated with the proposed model, are
compared to predictions using previously published valve modelling approaches
and both sets of results are compared to typical local and global
physiological parameters found in literature such left-ventricular systolic
pressures, peak and mean aortic valve pressure drops and vena contracta
velocities. The results show that the previously published valve models under
predicts expected severely stenosed peak and mean transvalvular pressure drops
by approximately 47% and 30% respectively, whereas the newly proposed model
under predicts the peak pressure drop by 20% and over predicts mean pressure
drop by 7%.

###### keywords:

cardiovascular system, aortic stenosis, lumped parameter modelling, valve
dynamics, valve pressure losses

††journal: Medical Engineering and Physics

##  1 Introduction

Cardiovascular diseases (CVDs) are a growing social and healthcare burden on
the world. Recent studies show that globally an estimated 19 million people
died [1] from CVDs in 2020 compared to 17.9 million in 2017 [2]. The leading
cause of CVD related deaths in high-income countries such as in Europe and
Northern America is ischaemic heart disease (IHD), which accounts for
approximately 50% of deaths from CVDs. In impoverished countries, such as
those located in Africa, IHD accounts for less than 10% of deaths from CVDs.
In Sub-Saharan Africa, valvular heart diseases such as rheumatic heart disease
(RHD), bicuspid aortic valve disease and degenerative aortic valve disease,
accounts for approximately 31% of CVD related deaths [3],[4]. Recent
statistical projections also show that valvular diseases are set to double in
the United States and Europe before 2050 [1].

Valvular heart disease is broadly categorised as obstructive and regurgitant
lesions of the valve body. Valve malfunction, whether due to regurgitation or
obstruction is associated with abnormal intracardiac haemodynamic behaviour.
Left untreated this can lead to reduced cardiac output or high ventricular
pressures [5]. The ability to quantify the influence of valvular diseases on
the local and global haemodynamics of the cardiovascular system would assist
clinicians in evaluating patient specific cardiovascular performance and
preparing personalised treatments [6]. One approach to non-invasively predict
and study valve disease effects on local and global haemodynamic functions is
to use electrical-hydraulic analogue lumped-parameter mathematical models
(LPMs) of the cardiovascular system [7]. LPMs can be used to quickly simulate
variations in blood pressures, flow rates and volumes of various compartments
that make up the circulatory system as a function of time.

Recently several authors have utilised LPMs to study various CVDs [8], [9],
[10], [11], [12], [13]. Garber et al. [6] stated that the heart valve
modelling approaches predominantly used in literature, are merely
simplifications of the actual valve mechanics and does not take into account
the local haemodynamics such as mechanical forces and valve motion. Some
researchers have developed more advanced valve models for LPMs, which included
valve motion and area-dependent pressure losses. Korakianitis and Shi [14],
[15] studied the effect of aortic regurgitation and mitral stenosis using a
LPM and an advanced valve model which accounted for valve motion and variable
area effects on pressure losses. Similarly, Mynard et al. [16] proposed an
advanced valve model which incorporates the valve motion as a function of the
pressure forces. The proposed model used tuned coefficients to accurately
capture the valve motion and pressure losses through the valves.

In the present work, a 0D valve model is proposed which accounts for valve
motion and Reynolds number dependent pressure losses. The model, similarly to
[15], solves the valve cusp dynamics as a set of differential algebraic
equations (DAEs) and uses the valve position to calculate the pressure losses.
To calculate the pressure drop through the valve, the domain is approximated
as a gradual contraction (nozzle) rather than a typical orifice and includes
the entrance, exit, frictional and local loss effects. The proposed model is
applied to investigate the influence of increasing degrees of aortic stenosis
on a cardiovascular system using a multi-compartment LPM. To investigate if
the proposed level of valve modelling detail is necessary, the model outputs
are compared to results generated using the approaches presented by [15].
According to the best knowledge of the authors the present work is the first
instance of using a dynamic valve model with Reynolds number dependent
pressure losses to study the local and global effects of aortic stenosis on a
LPM of the circulatory system.

The computer models of the cardiovascular system, were developed using the
_Julia_ v.1.7.0 [17] programming language and the _DifferentialEquations.jl_
[18] and _NLSolve.jl_ open-source libraries.

##  2 Material and methods

In the current section, the mathematical models for the different components
in the cardiovascular system seen in figure 1 will be discussed. The LPM
developed in the present work consists of the four heart chambers and their
respective downstream valves, the systemic circulation network and the
pulmonary circulation network.

Figure 1: Lumped parameter network model of cardiovascular system. aortic -
AO, mitral - MI, pulmonary - PO, tricuspid - TI, aortic sinus - AS, systemic
arteries - SAT, systemic arterioles - SAR, systemic capillaries - SCP,
systemic vein - SVN, pulmonary sinus - PS, pulmonary arteries - PAT, pulmonary
arteriole - PAR, pulmonary capillaries - PCP, pulmonary veins - PVN

###  2.1 Modelling of heart chambers

The heart chambers act as non-linear pressure sources in the hydraulic
network, which pressurises the low-pressure blood coming from the upstream
veins or chambers. To simulate the pressure-volume-time relationships of the
left ventricle (LV), left atrium (LA), right ventricle (RV) and right atrium
(RA) the time-varying elastance model of Suga et al. [19] is used. The
instantaneous static pressure inside the $i^{th}$ chamber (ventricles or
atriums) is calculated using the relation shown in equation 1. The
instantaneous volume $V_{i}(t)$ $\left[mL\right]$ of the $i^{th}$ chamber is
calculated using a simple mass balance over the respective heart chamber, as
shown in equation 2.

$P_{i}(t)=P_{i,0}+e_{i}(t)\left(V_{i}(t)-V_{i,0}\right)$ (1)

$\frac{dV_{i}}{dt}=Q_{i,in}(t)-Q_{i,ex}(t)$ (2)

In equations 1, 2: $P_{i,0}$ $\left[mmHg\right]$ is the unstressed chamber
internal pressure, $V_{i,0}$ $\left[mL\right]$ is the unstressed chamber
volume, $Q_{i,in}(t)$ $\left[mL/s\right]$ is the inlet flow rate into the
chamber and $Q_{i,ex}(t)$ the exit flow rate out of the chamber. $e_{i}(t)$ is
the time-varying elastance of the $i^{th}$ chamber and is estimated
differently for the atriums and ventricles. For the ventricles the time-
varying elastances are modelled using equation 3 [15] and the ventricular
activation functions with equation 4 [10].

$e_{i}(t)=E_{i,d}+\frac{E_{i,s}-E_{i,d}}{2}\cdot f_{i}(t)$ (3)

$f_{i}(t)=\begin{cases}1-\cos\left[\left(\frac{t}{T_{1}}\pi\right)\right],&\text{if
}0\leq t<T_{1}\\\ 1+\cos\left[\frac{t-T_{1}}{T_{2}-T_{1}}\pi\right],&\text{if
}T_{1}\leq t<T_{2}\\\ 0,&\text{if }T_{2}\leq t<T\par\end{cases}$ (4)

In equation 4, $T$ $\left[s\right]$ is the heart beat period, $T_{1}$ is the
time at end of systole and $T_{2}$ is the time at end of ventricular
relaxation. The time-varying elastances for the atriums are calculated using
equation 5 and the atrial activation functions are estimated using equation 6.

$e_{i}(t)=E_{i,min}+\frac{E_{i,max}-E_{i,min}}{2}\cdot f_{i}(t-D)$ (5)

$f_{i}(t)=\begin{cases}0,&\text{if }0\leq t<T_{a}\\\
1-\cos\left[2\pi\frac{t-T_{a}}{T-T_{a}}\right],&\text{if }T_{a}\leq
t<T\end{cases}$ (6)

In equations 5, 6: $E_{i,max}$, $E_{i,min}$ are the maximum and minimum
pressure-volume relations inside the atriums, $T_{a}$ is the time at onset of
atrial contraction and $D$ is the time of atrial relaxation. As seen above, in
the time-varying elastance model equations, various empirical values are
required, such as $T_{1}$, $T_{2}$ and $E_{i,s}$. Table 1 contains the
empirical values used for the ventricles and atriums. In the current work the
heart beat period is fixed at $T=1$ $\left[s\right]$.

Table 1: Time-varying elastance model parameters for atriums. Ref. [10], [14] Parameters | Left heart | Right heart
---|---|---
| _Atriums_ |
$E_{max}$ $\left[\frac{mmHg}{mL}\right]$ | $0.25$ | $0.15$
$E_{min}$ $\left[\frac{mmHg}{mL}\right]$ | $0.25$ | $0.15$
$T_{a}$ | $0.8T$ | $0.8T$
$D$ $\left[s\right]$ | $0.04$ | $0.04$
$V_{0}$ $\left[mL\right]$ | $4$ | $4$
$P_{0}$ $\left[mmHg\right]$ | $1$ | $1$
| _Ventricles_ |
$E_{d}$ $\left[\frac{mmHg}{mL}\right]$ | $0.1$ | $0.1$
$E_{s}$ $\left[\frac{mmHg}{mL}\right]$ | $2.5$ | $1.15$
$T_{1}$ | $0.3T$ | $0.3T$
$T_{2}$ | $0.45T$ | $0.45T$
$V_{0}$ $\left[mL\right]$ | $5$ | $10$
$P_{0}$ $\left[mmHg\right]$ | $1$ | $1$

###  2.2 Systemic and pulmonary networks

The systemic and pulmonary vasculature networks are modelled using five
components each, as seen in figure 1. These components are the sinuses,
arteries, arterioles, capillaries and veins [14]. Depending on the type of
vasculature, the local haemodynamics is simulated using a combination of
hydraulic resistance ($R$), blood inertia (inductance, $L$) and vessel
compliance (capacitance, $C$) to capture the time-dependent pressure-volume
behaviour of the specific vessel.

The flow rate through the aortic sinus is simulated using the ordinary
differential equation (ODE) shown in equation 7. The time-dependent blood
pressure inside the aortic sinus can be evaluated by solving the flow rate
differential equation and equation 8 simultaneously [20], [14].

$L_{AS}\frac{dQ_{AS}}{dt}=\left(P_{AS}-P_{SAT}\right)-R_{AS}Q_{AS}$ (7)

$C_{AS}\frac{dP_{AS}}{dt}=Q_{AO}-Q_{AS}$ (8)

In the equations above, $L_{AS}$ $\left[mmHg\cdot s^{2}/mL\right]$ is the
blood inertia at the aortic sinus, $Q_{AS}$ $\left[mL/s\right]$ is the flow
rate of blood out of the sinus, $P_{AS}$ $\left[mmHg\right]$ is the inlet
aortic sinus pressure, $P_{SAT}$ $\left[mmHg\right]$ is the systemic arterial
inlet pressure, $R_{AS}$ $\left[mmHg\cdot s/mL\right]$ is the aortic sinus
hydraulic resistance, $C_{AS}$ $\left[mL/mmHg\right]$ is the aortic sinus
compliance and $Q_{AO}$ is the flow rate of blood leaving the aortic valve.

For the pulmonary sinus the flow rate and blood pressure can be simulated by
solving equations 9, 10. The descriptions and units of the variables are
similar to that of the aortic sinus above.

$L_{PS}\frac{dQ_{PS}}{dt}=\left(P_{PS}-P_{PAT}\right)-R_{PS}Q_{PS}$ (9)

$C_{PS}\frac{dP_{PS}}{dt}=Q_{PO}-Q_{PS}$ (10)

The systemic arteries are modelled similarly to the aortic sinus. The
governing differential equations for flow rate and inlet blood pressure for
the systemic arteries is given in equations 11, 12.

$L_{SAT}\frac{dQ_{SAT}}{dt}=\left(P_{SAT}-P_{SAR}\right)-R_{SAT}Q_{SAT}$ (11)

$C_{SAT}\frac{dP_{SAT}}{dt}=Q_{AS}-Q_{SAT}$ (12)

The variations in pulmonary arteries outlet flow rate and inlet blood pressure
are simulated by solving equations 13, 14.

$L_{PAT}\frac{dQ_{PAT}}{dt}=\left(P_{PAT}-P_{PAR}\right)-R_{PAT}Q_{PAT}$ (13)

$C_{PAT}\frac{dP_{PAT}}{dt}=Q_{PS}-Q_{PAT}$ (14)

The systemic arteriole and capillaries are modelled using only hydraulic
resistance components due to the rigidity and small diameter of the vessel
walls [13]. The flow though these vasculature sections are assumed to be
steady, thus $Q_{SAT}=Q_{SAR}=Q_{SCP}$. The pressure drop through these
sections are calculated using equations 15, 16.

$R_{SAR}Q_{SAR}=P_{SAR}-P_{SCP}\equiv R_{SAR}Q_{SAT}=P_{SAR}-P_{SCP}$ (15)

$R_{SCP}Q_{SCP}=P_{SCP}-P_{SVN}\equiv R_{SCP}Q_{SAT}=P_{SCP}-P_{SVN}$ (16)

Similarly, for the pulmonary network arteriole and capillaries the pressure
drops are calculated using equations 17, 18.

$R_{PAR}Q_{PAR}=P_{PAR}-P_{PCP}\equiv R_{PAR}Q_{PAT}=P_{PAR}-P_{PCP}$ (17)

$R_{PCP}Q_{PCP}=P_{PCP}-P_{PVN}\equiv R_{PCP}Q_{PAT}=P_{PCP}-P_{PVN}$ (18)

The systemic and pulmonary veins are simulated as compliance-resistance
components with negligible blood inertia. The governing differential equations
for the systemic (equation 19) and pulmonary (equation 20) vasculature inlet
pressures are shown below.

$C_{SVN}\frac{dP_{SVN}}{dt}=Q_{SCP}-Q_{SVN}$ (19)

$C_{PVN}\frac{dP_{PVN}}{dt}=Q_{PCP}-Q_{PVN}$ (20)

To determine the flow rate through the venous system the hydraulic resistance
expressions are used. The driving pressure for the systemic vein flow rate is
the pressure difference between the vein inlet and the right atrium chamber
pressure [20]. For the pulmonary network, the driving pressure is between the
inlet to the veins and the left atrium chamber pressure.

$Q_{SVN}R_{SVN}=P_{SVN}-P_{RA}$ (21)

$Q_{PVN}R_{PVN}=P_{PVN}-P_{LA}$ (22)

All parameters used in the vasculature model are shown in table 2. In the next
section the modelling of the different valve modelling approaches will be
discussed.

Table 2: Systemic and pulmonary vasculature network parameters Parameters |  $R$ $\left[\frac{mmHg\cdot s}{mL}\right]$ |  $L$ $\left[\frac{mmHg\cdot s^{2}}{mL}\right]$ |  $C$ $\left[\frac{mL}{mmHg}\right]$ | Ref.
---|---|---|---|---
_Systemic network_ |  |  |  |
Aortic sinus | 0.003 | $6.2\cdot 10^{-5}$ | 0.08 | [14]
Systemic arteries | 0.05 | 0.0017 | 1.6 |  [14], [21]
Systemic arterioles | 0.5 | - | - |  [14], [15]
Systemic capillaries | 0.52 | - | - |  [14], [15]
Systemic veins | 0.075 | - | 22.0 |  [14], [15]
_Pulmonary network_ |  |  |  |
Pulmonary sinus | 0.002 | $5.2\cdot 10^{-5}$ | 0.18 |  [14], [15]
Pulmonary arteries | 0.01 | 0.0017 | 5.0 |  [14],[10]
Pulmonary arterioles | 0.05 | - | - |  [14], [15]
Pulmonary capillaries | 0.05 | - | - | -
Pulmonary veins | 0.006 | - | 30.0 |  [10], [14]

###  2.3 Heart valve models

The purpose of the four heart valves is to prevent reverse flow or
regurgitation of blood back into the heart chambers, similar to the operation
of check valves used in the process industry. The opening and closing
processes of the valve cusps are governed by a combination of pressure, vortex
and frictional forces [14] acting on the surfaces of the valves. In typical
lumped parameter modelling of cardiovascular systems, heart valves are
approximated as resistance components with a single diode component [22].

In the present work, three heart valve models are used along with the
cardiovascular LPM (figure 1) discussed above, to study the effects of
increasing degrees of aortic stenosis.

####  2.3.1 Valve model 1

The simplified heart valve model presented by [14], estimates the flow rate of
blood through the $i^{th}$ valve using equation 23, which is based on a
typical orifice pressure drop relation [23]. In equation 23, $A_{base}$ is the
base area of the inlet conduit to the valve, $\rho_{l}$ is the blood density,
$K$ is the flow coefficient, $A_{r}$ is the valve area opening ratio defined
as $\frac{A_{valve,open}}{A_{base}}$ and $\Delta P(t)$ is the absolute
pressure gradient across the valve which is calculated using equation 24.

The authors of this diode-based orifice model, designated the variable $CQ$ as
the valve flow coefficient. These coefficients are set to constant values that
differ for the semilunar and atrioventricular (AV) valves. For the semilunar
valves, $CQ_{AO}=CQ_{PO}=350\left[\frac{mL}{s\cdot mmHg^{0.5}}\right]$ and for
the AV valves, $CQ_{MI}=CQ_{TI}=400\left[\frac{mL}{s\cdot mmHg^{0.5}}\right]$.
These values were tuned manually, to produce near physiological cardiovascular
behaviour over a range of mitral stenosis and aortic regurgitation cases
analysed in [14].

$Q_{i}(t)=\sqrt{\frac{2A_{base}^{2}}{\rho K}}A_{r}(t)\sqrt{\Delta
P(t)}=CQ\cdot A_{r}(t)\sqrt{\Delta P(t)}$ (23)

$\Delta P(t)=\begin{cases}P_{in}(t)-P_{ex}(t),\text{if }&P_{in}\geq P_{ex}\\\
P_{ex}(t)-P_{in}(t),\text{if }&P_{in}<P_{ex}\end{cases}$ (24)

The valve area opening ratio $A_{r}$ shown in equation 23, is estimated using
simple diode behaviour, where $A_{r}$ either takes a value of 0 or 1 depending
on the pressure gradients across the specific heart valve, as shown in
equation 25. In equation 25, $P_{in}(t)$ $\left[mmHg\right]$ is the
instantaneous valve upstream static blood pressure and $P_{ex}(t)$
$\left[mmHg\right]$ is the instantaneous valve downstream static blood
pressure.

$A_{r}(t)=\begin{cases}1,\text{if }&P_{in}(t)\geq P_{ex}(t)\\\ 0,\text{if
}&P_{in}(t)<P_{ex}(t)\end{cases}$ (25)

####  2.3.2 Valve model 2

The advanced heart valve modelling methodology presented by [14], also uses
equation 23 and accompaning model constants to calculate the pressure losses
through the four heart valves. The difference between valve model 1 and 2, is
in the estimation of the valvular area opening ratio $A_{r}$. For valve model
2, the valve opening ratio is calculated as a function of the valve cusp
opening angle $\theta_{v}$, as shown in equation 26, where $\theta_{v,max}$ is
the maximum opening angle of the valve cusps.

$A_{r}(t)=\frac{\left(1-\cos\theta_{v}(t)\right)^{2}}{\left(1-\cos\theta_{v,max}\right)^{2}}$
(26)

To find the time-dependent $\theta_{v}(t)$
$\left[\text{rad}/\text{degrees}\right]$ for each heart valve, four
differential equations governing the valve cusp motion are solved. These
dynamic equations, takes into account the pressure forces acting on the valve
cusps, frictional forces due to neighbouring tissue resistance and the fluid-
structure interactions of the vortexes forming downstream of the valves.
Therefore, to simulate the rotational cusp motion of the $i^{th}$ heart valve,
equation 27 is numerically solved.

$\frac{d^{2}\theta_{v,i}}{dt^{2}}=\frac{F_{p,i}(t)-F_{f,i}(t)-F_{v,i}(t)}{I_{o,i}}$
(27)

In the equation above, $F_{p,i}(t)$ is the pressure forces, $F_{f,i}(t)$ the
tissue frictional forces and $F_{v,i}(t)$ the vortex forces acting on the
$i^{th}$ heart valve at time step $t$. $I_{o,i}$ is the mass moment of inertia
of the heart valve cusps.

As discussed in the works of Korakianitis and Shi [14], [15] and Mynard et al.
[16], the pressure force is the dominant factor which dictate valvular motion.
Therefore, in the current work only the pressure forces are considered and
frictional and vortex forces ignored ($F_{f}=F_{v}=0$). To estimate the
pressure forces on the cusps of the $i^{th}$ heart valve the constitutive
relationship shown in equation 28 is utilised.

$F_{p,i}(t)=\left(P_{in,i}(t)-P_{ex,i}(t)\right)\cdot
K_{p,i}\cos\left(\theta_{v,i}(t)\right)$ (28)

In the equation above, the valvular force coefficient $K_{p}$
$\left[\frac{\text{rad}}{s^{2}\cdot mmHg}\right]$ is assumed to have a
constant value of 5500 for all four heart valves. This force coefficient value
was determined through manual model tuning to produce valvular motion similar
to that found in literature. From the above discussion it is apparent that the
proposed model, does not account for all the mechanical forces acting on the
AV valves such as cord tensions. Nonetheless it has been shown that the valve
model 2 dynamics adequately captures the motion of the AV valves.

####  2.3.3 Valve model 3

The challenge associated with implementing the valve modelling methodologies
discussed in sections 2.3.1 and 2.3.2 in patient-specific LPMs, is that the
flow ($CQ$) and valvular force ($K_{p}$) coefficients are manually tuned and,
therefore, not applicable to a wide range of valve geometric and haemodynamic
parameters. In the present section, valve pressure loss and motion models will
be presented that are based on basic valve parameters such as valve cusp
thickness, cusp height, valve opening angle and instantaneous valve flow rate.
Furthermore, the pressure loss calculations will distinguish between AV and
semilunar valves due to the different configurations relative to the adjacent
heart chambers. The proposed model does not consider the individual valvular
disease morphologies such as bicuspid aortic disease and rheumatic heart
disease. Rather all diseases are assumed to only reduce the maximum valve
opening angle.

Figure 2, shows a schematic of a realistic semilunar heart valve geometry
(left) and the approximated valve geometry (right) for the current modelling
methodology. For both the AV and semilunar valves the cusps are approximated
as thin straight rectangular plates rotating around hinge points located at
the base of the valve.

Figure 2: Schematics of semilunar valve geometries

To simulate the valve cusp motion a similar approach is used to the one
discussed in section 2.3.2. The governing dynamic differential model equations
for the $i^{th}$ heart valve are shown in equations 29 and 30, where
$\omega_{i}$ $\left[\frac{rad}{s}\right]$ is the valve cusp angular velocity,
$\bar{L}_{i}$ $\left[m\right]$ is the distance from the base of the cusp to
its cusp centre of gravity and $A_{base}=\pi d_{base}^{2}/4$ is the closed
valve flow area. The factor of $133.\dot{3}$ is used to convert from SI units
to $\left[\frac{\text{rad}}{s^{2}\cdot mmHg}\right]$.

$\frac{d\theta_{v,i}}{dt}=\omega_{i}(t)$ (29)

$I_{o,i}\frac{d\omega_{i}}{dt}=\left(\left[P_{in,i}(t)-P_{ex,i}(t)\right]\cdot
A_{base,i}\cdot\bar{L}_{i}\cdot\cos\theta_{v,i}\right)\cdot 133.\dot{3}$ (30)

The mass moment of inertias for the cusps are calculated using equation 31
which is based on the standard expression for mass moment of inertia of a
rectangular plate rotating about its edge [24]. The mass of the $i^{th}$ valve
is calculated as $m_{valve,i}=\pi d_{base}H_{cusp,i}t_{cusp,i}\rho_{cusp}$,
where $H_{cusp,i}$ $\left[m\right]$ is the height of the valve cusps when
fully open (see figure 2), $\rho_{cusp}$ $\left[\frac{kg}{m^{3}}\right]$ is
the density of the cusp material and $t_{cusp,i}$ $\left[m\right]$ is the
thickness of the valve cusps. For stenosed valves it was assumed that the cusp
thicknesses remain unchanged.

$I_{o,i}=\frac{m_{valve,i}}{12}\cdot\left(H_{cusp,i}^{2}-t_{cusp,i}^{2}\right)+m_{valve,i}\cdot\bar{L}_{i}^{2}$
(31)

If one studies the geometry of the heart valve shown in figure 2 from a pure
fluid dynamics point-of-view, the valve geometry and accompanying flow losses
are more akin to a gradual contraction or a nozzle with variable wall angles,
rather than an variable area orifice plate. Therefore, the proposed model uses
the calculated valve cusp angles, cusp heights and valvular flow rates to
determine the time-dependent flow coefficients of each valve based on a
simplified nozzle/contraction geometry. Below the calculation procedures for
the AV and semilunar valves will be discussed.

Blood is forced into the semilunar valves by the corresponding ventricular
contraction and through the valve body and into the downstream sinus. For the
semilunar valves the total pressure loss comprises of a local inlet loss due
to the blood entering the valve body, a contraction loss due to frictional and
geometrical effects as the blood travels through the nozzle-shaped geometry
and a sudden expansion loss due to blood exiting the valve body and entering
the larger flow area of the sinus. The location of these losses can be seen in
figure 3. Seeing as the nozzle formed by the valve cusps protrudes into the
sinus, the difference in flow area experienced by the blood as it exits the
valve, is not $d_{vena,contracta}\rightarrow d_{wall,max}$ but rather
$d_{vena,contracta}\rightarrow d_{wall,vc}$, where
$d_{vena,contracta}=d_{base}\cdot\sqrt{A_{r}}$. The typical ratio of base
diameter $d_{base}$ to maximum sinus diameter $d_{wall,max}$ for semilunar
valves are 1.46 [25], [26], in the present work the sinus diameter at the exit
of the nozzle is assumed to be $1.23\cdot d_{base}$ as seen in figure 2.

Figure 3: AV and semilunar valve losses

To calculate the flow rate through the semilunar valves using the proposed
valve model, equation 32 is used, where the $133.\dot{3}$ and $10^{6}$ factors
are again used to convert from SI to medical units $\left(mL/s\right)$.

$Q_{i}(t)=\left(\sqrt{\frac{2A_{base}^{2}A_{r}^{2}}{\rho_{l}K_{t}(t)}\Delta
P(t)\cdot 133.\dot{3}}\right)\cdot 10^{6}$ (32)

In the equation above $K_{t}$ is the total flow coefficient for the semilunar
valves and consists of $K_{c,SL}$ the local contraction loss coefficient,
$K_{f,SL}$ the semilunar valve friction loss coefficient and $K_{se,SL}$ the
semilunar valve sudden expansion coefficient. The contraction loss coefficient
is calculated as shown in equation 33 [27], [28].

$K_{c,SL}(t)=0.0696\sin(\alpha_{i}(t))\cdot(1-\beta_{i}(t)^{5})(\lambda_{i}(t)^{2})+(\lambda_{i}(t)-1)^{2}$
(33)

In equation 33, $\alpha_{i}(t)=\frac{\pi}{2}-\theta_{v,i}(t)$ and
$\beta_{i}(t)=\sqrt{A_{r}(t)}$ for the $i_{th}$ valve. The contraction ratio
is calculated as
$\lambda_{i}(t)=1+0.622\cdot\left(\frac{\alpha_{i}(t)}{180}\right)^{4/5}\cdot\left(1-0.215\beta_{i}(t)^{2}-0.785\beta_{i}(t)^{5}\right)$.
The frictional loss coefficient is calculated using equation 34.

$K_{f,SL}(t)=\frac{f_{i}\left(1-\beta_{i}(t)^{4}\right)}{8\sin\left(\frac{\alpha_{i}(t)}{2}\right)}+\beta_{i}(t)^{4}\frac{f_{i}(t)H_{cusp,i}}{0.5\left(d_{base,i}(t)+d_{vena,contracta}(t)\right)}$
(34)

The friction factor $f_{i}$ per semilunar valve is calculated using the well-
known smooth surface Colebrook-White equation [23] shown equation 35. The
Reynolds number used in the calculation of the friction factor is determined
using equation 36, and is based on the velocity at the vena contracta of the
semilunar valve.

$\frac{1}{\sqrt{f_{i}}}=-2\log_{10}\left(\frac{2.51}{Re_{i}\sqrt{f_{i}}}\right)$
(35)

$Re_{i}=\frac{\rho_{l}v_{i}(t)d_{vena,contracta}(t)}{\mu_{l}},\ \text{where}\
v_{i}(t)=\frac{Q_{i}(t)}{A_{base}A_{r}(t)}$ (36)

The sudden expansion loss coefficient for the semilunar valves is calculated
using equation 37 [28], where
$\beta_{2,i}(t)=\frac{d_{vena,contracta}(t)}{d_{wall,vc}}$.

$K_{se,SL}(t)=\left(1-\beta_{2,i}(t)^{2}\right)^{2}$ (37)

For the AV valves, blood is ejected from the atriums through the valves into
the ventricles. Seeing as the ventricles act as blood reservoirs and have
larger volumes than the upstream atriums, the AV valve geometries are
approximated as smooth nozzles discharging into large static fluid volumes, as
seen in figure 3. Similar to the semilunar valves, the flow rates through the
AV valves are calculated using equation 32. The total loss coefficient for the
AV valves are calculated using equation 38, where the AV valve frictional loss
coefficient $K_{f,AV}$ is calculated using the same approach as for the
semilunar valves. It should be noted that AV valves typically have
$\theta_{v,max}$ values larger than $75^{0}$, but for the purpose of the
present study the values proposed by Korakianitis and Shi will be used.

$K_{t}(t)=K_{f,AV}(t)+1$ (38)

The physiological parameters used in valve model 3 such as cup thicknesses,
maximum valve opening angles and valve base diameters were taken from
literature for an average human male, and the values can be seen in table 3.

Table 3: Valve model 3 physiological parameters Parameter | Value | Units | Reference
---|---|---|---
_Aortic valve_ |  |  |
base diameter, $d_{base,AO}$ | 24.7 | $\left[mm\right]$ | [29]
cusp height, $H_{cusp,AO}$ | 17.5 | $\left[mm\right]$ |  [25], [26]
cusp thickness, $t_{cusp,AO}$ | 0.61 | $\left[mm\right]$ | [30]
sinus diameter, $d_{wall,vc,AO}$ | 30.4 | $\left[mm\right]$ |  [25], [26]
maximum opening angle, $\theta_{v,max,AO}$ | 75 | $\left[\text{degrees}\right]$ | [14]
minimum opening angle, $\theta_{v,min,AO}$ | 5 | $\left[\text{degrees}\right]$ | [14]
_Pulmonary valve_ |  |  |
base diameter, $d_{base,PO}$ | 25 | $\left[mm\right]$ | [29]
cusp height, $H_{cusp,PO}$ | 17.6 | $\left[mm\right]$ | $1.42\frac{d_{base,PO}}{2}$
cusp thickness, $t_{cusp,PO}$ | 0.4 | $\left[mm\right]$ | [30]
sinus diameter, $d_{wall,vc,PO}$ | 30.6 | $\left[mm\right]$ | $1.23d_{base,PO}$
maximum opening angle, $\theta_{v,max,PO}$ | 75 | $\left[\text{degrees}\right]$ | [14]
minimum opening angle, $\theta_{v,min,PO}$ | 5 | $\left[\text{degrees}\right]$ | [14]
_Mitral valve_ |  |  |
base diameter, $d_{base,MI}$ | 27.0 | $\left[mm\right]$ | [31]
cusp height, $H_{cusp,MI}$ | 19.17 | $\left[mm\right]$ | $1.42\frac{d_{base,MI}}{2}$
cusp thickness, $t_{cusp,MI}$ | 1.3 | $\left[mm\right]$ | [30]
maximum opening angle, $\theta_{v,max,MI}$ | 75 | $\left[\text{degrees}\right]$ | [14]
minimum opening angle, $\theta_{v,min,MI}$ | 5 | $\left[\text{degrees}\right]$ | [14]
_Tricuspid valve_ |  |  |
base diameter, $d_{base,TI}$ | 28.0 | $\left[mm\right]$ | [31]
cusp height, $H_{cusp,TI}$ | 19.9 | $\left[mm\right]$ | $1.42\frac{d_{base,TI}}{2}$
cusp thickness, $t_{cusp,TI}$ | 0.9 | $\left[mm\right]$ | [30]
maximum opening angle, $\theta_{v,max,TI}$ | 75 | $\left[\text{degrees}\right]$ | [14]
minimum opening angle, $\theta_{v,min,TI}$ | 5 | $\left[\text{degrees}\right]$ | [14]

For valve model 3, the blood viscosity, blood density and cusp material
density used are $\mu_{l}=0.0035$ $\left[Pa\cdot s\right]$ [32],
$\rho_{l}=995$ $\left[\frac{kg}{m^{3}}\right]$ [33], $\rho_{cusp}=1060$
$\left[\frac{kg}{m^{3}}\right]$ [32].

###  2.4 Case studies

In the current work, the valve models presented in sections 2.3.1, 2.3.2,
2.3.3 are applied to simulate the local and global haemodynamics of a typical
human male cardiovascular system. The results generated using the three valve
models are compared for different aortic stenosis case studies. Each
successive case study corresponds to a reduction in maximum opening flow area
of the valve. For valve model 1, the area opening ratio $A_{r}$ is simply
limited to a specified value and for valve models 2 and 3 the maximum opening
angles are limited $\theta_{v,max,AO}$. Table 4, contains the limits applied
to the aortic valve for the 5 case studies.

Table 4: Aortic stenosis case studies Case # | Description | Valve area $\left[cm^{2}\right]$ | $A_{r,max,AO}$ |  $\theta_{v,max,AO}$ $\left[\text{degrees}\right]$
---|---|---|---|---
1 | None | 4.81 | 1 | 74.9
2 | Moderate | 1.63 | 0.333 | 55.1
3 | Severe | 1.06 | 0.222 | 49.4
4 | Very severe | 0.64 | 0.1333 | 43.2
5 | Critical | 0.43 | 0.0899 | 38.8

###  2.5 Numerics

To simulate the dynamics of the cardiovascular system, the differential-
algebraic equations (DAEs) for the time-varying elastance model (section 2.1),
vasculature dynamics (section 2.2) and the heart valve dynamics (section 2.3)
are solved simultaneously using an ODE system of equations integrator. For the
current work the Tsitouras 5/4 Runge-Kutta method [34] is used with the
relative and absolute tolerances set to $1\cdot 10^{-4}$ and $1\cdot 10^{-6}$
respectively. The maximum integrator iteration count of $10^{6}$ was also
specified.

To numerically integrate and solve the mentioned DAEs, certain constraints and
conditions had to be incorporated such as valve motion limits, implicit
equation solving and initial conditions. To incorporate the discontinuities
arising from the valve motion limits the following condition is included in
the simulation procedure of each valve

$\theta_{v}=\begin{cases}\theta_{v}=\theta_{v,max},\ \omega=0.0&\text{ if
}\theta_{v}\geq\theta_{v,max}\\\ \theta_{v}=\theta_{v,min},\ \omega=0.0&\text{
if }\theta_{v}\leq\theta_{v,min}\\\ \theta_{v}&\text{ if
}\theta_{v,min}<\theta_{v}<\theta_{v,max}\end{cases}$ (39)

To find the friction factor for each valve per time step requires the solution
of the implicit Colebrook-White equation (see equation 35). In the DAE
integrator solver loop of the cardiovascular system, a nested Newton-Raphson
solver is used to find the friction factors for a given valve, flow rate, and
valve opening fraction.

To solve the DAEs, the initial conditions are required. For the current model,
initially the semilunar valves were set to fully open and the AV valves to
fully closed. Furthermore, the initial volume of the left and right ventricles
are set to 800 and 500 $mL$ respectively and the initial volumes of both
atriums to 2.5 $mL$. For the remaining pressure and flow rate dependent
variables the initial conditions are set to 1 $mmHg$ and 0 $\frac{mL}{s}$
respectively. The initial valve cusp angular velocities are set to $0.0$
$\frac{\text{rad}}{s}$.

##  3 Results and discussions

To demonstrate that the proposed LPM, along with the three heart valve
modelling extensions, can reproduce typical physiological haemodynamic
parameters the results for case study 1 was compared to data from literature
as seen in table 5. Note that M1, M2 and M3 corresponds to the LPM implemented
with valve models 1, 2 and 3 respectively.

Table 5: Comparison between LPMs and typical physiological haemodynamic parameters Parameters | M1 | M2 | M3 | Physiology
---|---|---|---|---
LV Stroke volume, $mL$ | 82.7 | 73.9 | 73.1 | 51-111 [8]
Cardiac output, $L/min$ | 4.9 | 4.4 | 4.4 | 4.0-8.0 [35]
_Left ventricle_ |  |  |  |
Systolic pressure, $mmHg$ | 123.9 | 115.8 | 115.6 | 90-140 [36]
Diastolic pressure, $mmHg$ | 7.5 | 7.3 | 7.0 | 4-12 [36]
Systolic volume, $mL$ | 53.5 | 50.4 | 50.4 | 37-57 [8]
Diastolic volume, $mL$ | 137.2 | 140 | 140.8 | 121-163 [8]
_Right ventricle_ |  |  |  |
Systolic pressure, $mmHg$ | 29.7 | 28.9 | 28.5 | 15-30 [8]
Diastolic pressure, $mmHg$ | 6.4 | 6.7 | 6.3 | 2-8 [8]
Systolic volume, $mL$ | 37 | 36.7 | 36.7 | 36-84 [8]
Diastolic volume, $mL$ | 122 | 126.3 | 124.9 | 121-167 [8]
_Systemic arterial pressures_ |  |  |  |
Systolic, $mmHg$ | 123.6 | 115.6 | 115.4 | 90-140 [36]
Diastolic, $mmHg$ | 84.2 | 76.4 | 75.6 | 60-90 [36]
_Pulmonary arterial pressures_ |  |  |  |
Systolic, $mmHg$ | 28.9 | 28.3 | 28.3 | 15-30 [8]
Diastolic, $mmHg$ | 15.9 | 16.6 | 16.7 | 5-16 [36]

The results in table 5 shows that the three models can capture typical
physiological parameters of the human cardiovascular system. It is seen that
M2 and M3 produce very similar results for case study 1, whereas M1 generates
results with a higher cardiac output and higher blood pressures across the
network. These differences between models are due to the different valvular
pressure drops calculated by each approach and are reflected in table 6.

Table 6: Peak pressure drops, $\text{max}\left(\Delta P\right)$, over valves for case study 1 Valve | M1 | M2 | M3
---|---|---|---
AO | 9 | 11.8 | 9.5
PO | 4.7 | 5.2 | 2.9
MI | 4.2 | 6 | 6.9
TI | 2.2 | 4.2 | 4.6

Figure 4 below shows the mitral and aortic valve area opening ratios ($A_{r}$)
as a function time for the three valve models and selected case studies
(1,2,4). The diode-like behaviour of the M1 valves can be seen in the results
as the valves abruptly opens and closes, with no inertia. For case studies 1
and 2, M2 and M3 produces very similar results. It is observed that the
inclusion of the valve motion effects results in the valves opening later and
closing more slowly compared to the M1 results. In general, it can be seen
that the developed fundamental valve motion model (valve model 3) which uses
geometric and fluid property parameters rather than empirically tuned values
can recreate expected valve motion [15].

Figure 4: Time-dependent aortic and mitral valve area opening ratios (a) Case
study 1, (b) Case study 2, (c) Case study 4

To investigate the effect that the different valve modelling methodologies
have on global haemodynamic parameters for increasing degrees of aortic
stenosis, the left ventricle pressure-volume (PV) loops for the case studies
are plotted in figure 5. Comparing M1 and M2 results, it is observed that the
left ventricular systolic pressures (LVSPs) predicted by M2 are lower than the
values generated using M1. This is in agreement with the observations by
Korakianitis and Shi [14]. The percentage differences between M1 and M2 LVSPs
over the range of stenosis cases analysed stay relatively constant at 7.6%. M1
predicts on-average 9% higher stroke volumes compared to the values predicted
by M2. Therefore, M2 predicts lower cardiac output, which highlights the
importance of including valve motion effects seeing as cardiac output is a
critical clinical parameter [8]. A larger range of LVSP differences are
observed when comparing the PV loops of M1 and M3. For case study 1, the M3
predicted LVSP is approximately 8.8% lower than the M1 predicted LVSP and for
case study 5, the M3 predicted LVSP is 10.4% higher than the value predicted
by M1. This shows a changing relationship in LVSP differences driven by the
valvular pressure drop calculation methodology. Similarly, a larger variation
in stroke volume differences between M1 and M3 are observed. For case study 1,
the stroke volume predicted by M3 is 11.6% lower than the value predicted by
M1 and for case study 5, the M3 predicted stroke volume is 18.2% lower than
the M1 value. The LVSP and stroke volume trends observed in figure 5,
indicates that the M3 modelling approach predicts higher pressure drops
(ventricular afterload) in the stenosed valve compared to the M1 and M2
models.

Figure 5: Left ventricle pressure-volume loops for increasing degrees of
aortic stenosis calculated using valve models 1,2,3

Figure 6 shows the aortic valvular pressure drops calculated using the three
different modelling approaches over the range of analysed aortic stenosis
cases. The results show for the different cases, that M1 and M2 valve
modelling approaches calculates similar pressure drops, with an average peak
pressure drop difference between the modelling approaches of approximately 9%.
The M3 valve modelling approach, on the other hand, calculates significantly
higher pressure drops for the stenosed cases. The peak pressure drops
calculated by M2 and M3 for case study 1 are 12 and 9.5 $mmHg$ respectively.
For case study 4, 79.1 $mmHg$ (M2) and 113.1 $mmHg$ (M3) and for case study 5,
117 $mmHg$ (M2) and 170.1 $mmHg$ (M3). Handke et al. [37] performed in-vivo
transesophageal 3D echocardiography to investigate aortic valve dynamics. The
authors reported for mild ($A_{r}=0.5$) to severe ($A_{r}=0.15$) aortic
stenosis the range of peak valvular pressure drops are 40-130 $mmHg$. Using
the data generated in the present work, the peak pressure drops for M2 and M3
were interpolated to the $A_{r}$ values mentioned by Handke et al. For
$A_{r}=0.5$, the M2 and M3 values are 26.68 $mmHg$ and 38.5 $mmHg$
respectively. For $A_{r}=0.15$, the M2 and M3 values are 73.1 $mmHg$ and 104
$mmHg$ respectively. Messika-Zeitoun and Lloyd [38], reported that for
stenosed aortic valve flow areas of 1.5 $cm^{2}$ and 1 $cm^{2}$ the mean
transvalvular gradients are 20 $mmHg$ and 40 $mmHg$ respectively. For similar
flow areas the M3 approach generates aortic valve gradients of 21.8 $mmHg$ and
42.5 $mmHg$, where the M2 approach generates gradients of 16.2 $mmHg$ (M1 -
17.7 $mmHg$) and 30.0 $mmHg$ (M1 - 33 $mmHg$). These results show that the M3
modelling approach can capture the experimentally measured range of typical
aortic valve pressure drops more accurately than the M1 and M2 approaches.

Figure 6: Pressure drops over aortic valve for different degrees of stenosis

To understand why the M3 modelling approach is predicting significantly higher
pressure drops over the range of aortic stenosis cases, the time-dependent
valve opening diameters, vena contracta velocities and Reynolds numbers are
plotted in figure 7. The opening diameters of the aortic valve are calculated
as $d_{vena,contracta}=2\sqrt{\left(A_{r}\cdot A_{base}\right)/\pi}$. In
figure 7 (a,b), it is seen as the maximum opening diameter of the vena
contracta reduces due to stenosis, the peak and duration of the high flow
velocities in the valve body increases. For the case with no stenosis the
initial peak vena contracta velocity is approximately 2.5 $\frac{m}{s}$, but
this velocity corresponds to the time step at which the valve starts to open.
The peak vena contracta velocity for case study 1 (no stenosis) when
transvalvular flow is established is approximately 1.25 $\frac{m}{s}$ as seen
in figures 7 (a) and (b). For the severe and very severe cases the peak
velocities are above 4.0 $\frac{m}{s}$. These values agree well with
physiological values found in literature. Garcia et al. [39], stated that for
healthy aortic valves the peak vena contracta flow velocity is between 1-1.5
$\frac{m}{s}$ for males between 21 to 59 years of age. Messika-Zeitoun and
Lloyd [38], stated velocities of $>4$ $\frac{m}{s}$ for cases where the valve
areas are below $1$ $cm^{2}$.

The increase in duration of high velocities in the valve, results in sustained
high flow Reynolds numbers in the valve as seen in figure 7 (c). From the
results, it is seen that the blood flow Reynolds numbers in the valve changes
significantly over the ejection period as well as over the various degrees of
stenosis. Therefore, based on equation 34 the flow coefficient will not remain
constant as proposed by M1 and M2. Studying the Reynolds number results
further, it is seen that the blood flow is turbulent for most of the ejection
time duration, therefore, the assumption of using the turbulent smooth tube
friction factor model of Colebrook-White is vindicated.

Figure 7: Time-dependent (a) valve opening diameters, (b) vena contracta
velocities and (c) Reynolds numbers for valve model 3

To show the effect of taking into account the variation in aortic valve
Reynolds number for the range of analysed cases, the time-dependent flow
coefficients for models M1, M2 and M3 are plotted in figure 8. The figures
show that the flow coefficient values for the semilunar and AV valves remain
constant for M1 and M2, as discussed in section 2.3.1. The flow coefficient
values calculated by the M3 approach for the mitral valve is seen to be lower
than the constant value proposed by [14], which results in M3 calculating
higher pressure drops through the mitral valve when compared to the values of
M2 (table 6). Studying the aortic valve data, it is observed for case study 1,
that the calculated flow coefficients by M3 are significantly larger than the
constant value used by M1 and M2. This explains why in table 6, the M3 model
predicts a lower aortic valvular pressure drop compared to M2. In figure 8
(b), for the moderate stenosis case we seen that the flow coefficients
calculated by the M3 approach, are lower compared to the constant value used
in M1 and M2. The lower flow coefficient values result in larger pressure
drops over the valve during valvular ejection. For the very severe aortic
stenosis case it is observed that the flow coefficients are lower than in the
moderate stenosis case. This reduction in flow coefficients as the degree of
stenosis is increased, is due to the higher sustained blood flow Reynolds
numbers in the valve which leads to larger frictional and geometrical
contraction losses.

The approximate maximum and minimum flow coefficients calculated by the M3
modelling approach are 650 and 150 respectively, over the range of case
studies. This gives an average flow coefficient value of 400 which is close to
the tuned value proposed by Korakianitis and Shi [14] (350 and 400). The
advantage of using the M2 approach, is that it solves significantly faster
than the M3 approach. The M1 and M2 modelling approaches, simulates 10 s, in
approximately 15 s, whereas M3 takes 10-15 minutes.

Figure 8: Time-dependent flow coefficients for models M1, M2 and M3. (a) No-
stenosis, (b) Moderate stenosis and (c) Very severe stenosis.

##  4 Conclusions

The results in the present work shows that the newly proposed valve modelling
approach, predicts higher pressure drops at more severe degrees of aortic
stenosis when compared to the results of the models from literature. This is
due to the fact that the models from literature are insensitive to changes in
blood flow Reynolds number through the valve. Comparing the calculated ranges
of pressure drops over stenosed valves with experimental data from literature,
shows that the proposed model captures the expected pressure drops more
accurately compared to the models from literature.

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# Goniometers are a Powerful Acoustic Feature for Music Information Retrieval
Tasks

Tim Ziemer1
1 _University of Hamburg, 20354 Hamburg, Germany, E-mail: tim.ziemer@uni-
hamburg.de_

## Introduction

Music Information Retrieval (MIR) task reach from blind source separation over
automatic music transcription to music recommendation. Musicologists often use
acoustic features and machine learning methods as explorative tool to analyze
ethnographical recordings [4, 6], popular music [2, 5] and large music
collections [2, 3]. One of the most-used acoustic features for music analysis
are Mel-Frequency Cepstral coefficients (MFCCs), even though they neither
correlate with any aspect of music perception nor music production and mixing
[8]. To be explanatory and intuitive, a feature should relate to the way music
is produced, or the way music is perceived. The goniometer is a potential
candidate because music producers and mixing engineers consult it to monitor
and tune spatial aspects of their music mix. The present work is an
explorative study to evaluate the intuitiveness and usefulness of the
goniometer as a feature for music information retrieval of mixed stereo music.

## The Goniometer

Music producers and audio engineers know several recording and mixing
techniques to control spatial aspects of a music production [9]. Many of these
spatial aspects are monitored using a goniometer. A goniometer is illustrated
in Fig. 1. It consists of two meters, one point cloud that looks like
Lissajous figures or a phase space diagram, and one bar diagram.

Figure 1: Typical goniometer. The time signal of the left channel (L) is
plotted over the time signal of the right channel (R) yielding a point cloud
(purple). The normalized correlation of both signals is indicated by a bar on
a bar diagram (right). Often, a high correlation is represented in green,
while a high negative correlation is indicated in red.

In the phase space diagram, the time series of the left channel is plotted
over the time series of the right channel for all samples of a short-term
buffer that may contain $4096$ samples. This buffer is also referred to as a
time frame. Each sample pair is represented as a point, so all samples of a
time frame produce a point cloud. Here, the time is represented as the
opacity, i.e., the first samples from the buffer are transparent, while the
last samples are opaque. The plot tends to be tilted by $-45^{\circ}$ for a
simple reason: When a signal is routed to the first stereo channel, the signal
will only be played by the left loudspeaker and the corresponding plot will
look like a dotted line distributed along the L-axis. A mono signal equally
routed to the left and the right channel produces a vertical dotted line. Due
to the tilt, the plot gives a rough indication of signal panning in the stereo
triangle. The Length of the line indicates the volume of the sound. All points
in the upper square and in the lower square indicate that the signal is in
phase, i.e., both loudspeakers are either positively or negatively elongated.
Points in the left and the right square indicate that the signal is out of
phase, i.e., that one loudspeaker is elongated positively while the other one
is elongated negatively. A perfectly horizontal line through the axis origin
is produced when the signal of the left and the right channel are perfectly
out of phase. This exception is typically avoided, because it will lead to
cancellation of bass frequencies in the typical listening setup, i.e., in a
stereo triangle constellation. Furthermore, a mono mixdown of the signal will
completely cancel the out-of-phase signal. Mono mixdowns are still quite
common in at least two typical listening situations. Firstly, in nightclubs,
when the loudspeakers are very far apart. Here, many mising engineers decide
to mix both stereo channels at least partially. This way they avoid that a
hard-panned instrument is only audible in a single corner of the dancefloor.
Secondly, when music is played back over the internal loudspeakers of a mobile
phone. In the good old days grandmother’s portable radio was completely mono,
while the younger generation sometimes listens to music through the mono
tweeter of their mobile phone, or through portable bluetooth speakers whose
stereo channels are installed in one tiny cabinet. The extreme opposite of a
single line in the phase space is when the complete area is occupied by dots.
This is only produced when each channel is fed with an individual white noise.
Only when both white noises are normalized to the maximum sound pressure
level, the complete space is filled with points. In all other cases, some of
the regions in the field are unoccupied. Music producers and mixing engineers
know several recording techniques and digital audio effects that widen the
distribution of points to make instruments and ensembles sound wider and to
make the listening experience more immersive, enveloping, spatial and vivid
[7]. At the same time, techniques exist to make wide distributions narrower,
for example to increase the presence and the bass of single instruments, and
to make them clearer and easier to localize. In summary, the width of the
point cloud gives an indication of how spatial the overall mix is. The size of
the point represents the volume. The overall direction of the distribution
indicates the panning. In the plotted example, the sound from the left
loudspeaker is louder, because a loud instrument panned to the left dominates
the overall mix. Music producers and mixing engineers have their own
philosophy concerning the appearance of the point cloud, e.g., how wide, how
large, how fluctuating, and how tilted it should be, depending on the music
material that they are mixing.

The second plot of the goniometer is a simple bar diagram. It indicates the
normalized correlation of both buffered stereo channels. Consequently, the
channel correlation can take values from $-1$ to $1$. Some goniometers also
plot a short-term history, i.e., the highest and the lowest channel
correlation during the last couple of seconds, or the minimum and maximum
channel correlation that has been measured overall. Often, colors indicate
whether a stereo mix is perfectly mono compatible (green), fairly mono
compatible (yellow) or not mono compatible (red). Again, music producers and
mixing engineers have their own philosophy concerning the region in which the
bar should lie, the lowest position they will accept, and the dynamics of the
bar position that they want, depending on the type of music that they mix.

### The Goniometer Feature

In practical use, music producers and mixing engineers consult the goniometer
to observe the spatial aspects of the mix and to monitor how modifications
through recording and mixing steps affect it. The goniometer is a music
metering and monitoring tool. This procedure is rather qualitative than
quantitative, i.e., practitioners evaluate the appearance of the two
goniometer plots. To serve as an acoustic feature, the output of the
goniometer has to be quantified.

To quantify the appearance of the point cloud, a simple box counting method is
applied. The square that represents all possible coordinate pairs is divided
into $20\times 20$ sub-squares, as indicated in Fig. 1. This yields $400$ sub-
squares. Then, it is simply counted how many sub-squares are occupied by one
or more points during one time frame. When we divide this number by $400$, we
get a percentage. This percentage is stored for each analyzed time frame in a
one-dimensional feature vector. In python, this can be computed efficiently by
the following code for each time frame fs:

    import numpy as np
    bc = []#empty box counting feature vector
    for i in range(fs):
       box[i]=[int(round(l[i]*10)),int(round(r[i]*10))]
    bc.append(len(np.unique(box))/400)

I call this feature _Phase Scope_ or _Box Counting_. The variables $l$ and $r$
are arrays with a length of fs that contain the time series of the left and
the right channel.

To quantify the bar graph, its magnitude for each time frame is stored in a
one-dimensional feature vector. In python this can easily be done, e.g., by:

    import numpy as np
    corr=[]#empty array for corelation feature vector
    corr.append(np.min(np.corrcoef(l,r)))

I call this feature _Channel Correlation_.

The code for extracting the feature and writing feature vectors into a CSV-
file is publically available on my GitHub repository [1].

In [2] we could already show that the two goniometer features can — in
combination with other recording studio features, like crest factor and root-
mean-square energy — predict which Disk Jockey would play which song. In this
study, I evaluate the goniometer alone in an explorative way that reveals how
the goniometer can serve as a feature for linear and nonlinear music
classification and clustering tasks.

## Method

To evaluate the usefulness of the goniometer features, I chose an explorative
approach. I analyzed some hundred songs. First, I imported stereo mp3 files
and downsampled them to $22050$ Hz. Then, I took $500$ frames from the middle
of the music piece. Each frame was $2048$ samples long and had no overlap, so
I analyzed around $46$ seconds per file. Naturally, only stereo files were
analyzed, as goniometers represent spatial aspects of stereophonic sound. To
see how intuitive and explanatory the goniometer features were, I compared
somewhat contrasting types of music. My first comparison was between $16$
songs from a meditation album by David Miles Huber, an American Composer and
$15$ songs by Jimmy-J & Cru-L-T, a UK duo that produced drum and bass music in
the 1990s, also referred to as breakbeat hardcore. My second comparison was
between $90$ classical music pieces and $93$ hip hop pieces, both in a broad
sense. I visually inspected scatter plots of the two goniometer features for
linear observations. When a linear separation between the two types of music
was not possible, I trained a self-organizing map using the goniometer to see
how explanatory the goniometer is when used with a nonlinear clustering
method. The results were analyzed regarding causal relationships between the
intention or sound ideal of the music producers/mixing engineers and the
goniometer values.

## Results & Discussion

The analysis results of the first comparison are plotted in Fig. 2. This
scatter plot represents the feature space. Here, the drum & bass and
meditation music are very far apart and can be separated linearly. Every
clustering algorithm, like k-means or self-organizing maps should be able to
cluster them, and even linear classifiers, like single layer perceptrons,
should be able to classify them with perfect accuracy. Note that in this case
the phase scope is already sufficient to distinguish the two, while the
channel correlation is not. This clear picture reveals a lot about the
aesthetic ideal and the intention of the music producers. Jimmy-J & Cru-L-T’s
drum and bass music is loud and compressed and has a high channel correlation
to keep bass frequencies in phase for phat, percussive drums. David Miles
Huber on the other hand produced much more dynamic music, with a wider, less
correlated distribution. Of course, this is obvious to music experts. The fact
that the goniometer features reflect this intention is evidence that the
features may have a high explanatory value, as they help to find causal
relationships between music and the producers’ intention.

Figure 2: Mean channel correlation over mean phase scope for drum and bass
(orange) and mediation music (purple). These two goniometer features perfectly
separate the two.

The scatter plot of the classical and hip hop music can be seen in Fig. 3. One
can see similar trends as before, i.e., hip-hop music is found in the upper-
right corner and classical music in the lower-left. But there are much larger
overlaps along both dimensions. A linear separation between the two types of
music is not possible with the goniometer features.

Figure 3: Simply plotting the distribution of goniometer mean values shows
obvious trends, classical music and hip-hop genres are not linearly separable.

To evaluate the usefulness of the goniometer features, they have been used to
train a Self-Organizing Map (SOM) [6]. SOMs are neural networks for
unsupervised learning, i.e., clustering. A grid consisting of $23\times
23=529$ neurons was fitted to the training data in $500$ iteration steps, with
a learning rate of $\eta=0.025$ and an initial neighborhood size of $\nu=25$,
which is allowed to decrease linearly to $1$. The result is plotted in terms
of a unified distance matrix in Fig. 4. Here, dark blue fields represent
clusters, yellow fields represent boarders. Each labeled red point represents
one music piece. It can be seen that the SOM, trained on the goniometer
features, clearly separates hip hop from classical music (red border) and even
manages to cluster a single album by the Knabenchor St. Petersburg (orange
oval mark) that is not visible in the linear feature space, Fig. 3. The SOM
reveals that the goniometer features enable a nonlinear separation between the
two analyzed types of music, and it even identified the Knabenchor St.
Petersburg as a cluster, probably because the whole album was recorded in one
session, with a fixed stereo microphone setup, yielding a homogeneous phase
scope and channel correlation.

Figure 4: Trained with the mean values of goniometer outputs, a self-
organizing map perfectly separates hip-hop from classical music (red line) and
even identifies a single album (orange oval).

The two component matrices are illustrated in Fig. 5. They show how much each
feature contributes to each field on the SOM. It can be seen that the weight
of the phase scope is aligned from top to bottom, whereas the weight of the
channel correlation is aligned from right-to-left. This orthogonality
indicates that the two are orthogonal and therefore perfectly compliment each
other. Orthogonal features are very important for machine learning, as
redundancy reduces the accuracy of algorithms and redundancy reduction methods
may be computationally costly.

Figure 5: Component matrices of the phase scope (left) and channel correlation
(right) reveal that the two aspects of the goniometer are linearly
independent, i.e., one is aligned vertically and the other one, horizontally.

## Conclusion

In this work, I suggest the goniometer as an explanatory acoustic feature for
music information retrieval tasks. I define a way to quantify the qualitative
view that music producers and mixing engineers have on the goniometer.
Analyzing different types of music revealed that the goniometer yields
intuitive, plausible results that serve to cluster different music styles
explained by spatial and dynamic aspects of the music recording and production
process. The phase scope and the channel correlation are orthogonal features
that perfectly complement each other. In contrast to MFCCs, extracting the
goniometer features needs very little computational power and the features are
causally linked to the intention of the music producers. Additional analyses
will reveal whether the goniometer features are also suitable to explore the
other stereo sound files, such as podcasts, audio drama and the sound from
movies, or recordings of the sound radiation characteristics of musical
instruments, binaural recordings of architectural acoustics and soundscape
recordings.

## References

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