Supplemental Digital Content
Manuscript Title: Non-invasive iPhone measurement of left ventricular ejection
fraction using intrinsic frequency methodology
Supplement A: Pressure and vessel wall displacement in a viscoelastic vessel.
Supplement B: Carotid artery waveform capture using an unmodified iPhone camera
Supplement C: Intrinsic frequency Method
Supplement D: Cycle Selection and Signal Quality Evaluation
Supplement E: Cardiac MRI Scan and Analysis
Supplement F: Details of statistical analysis.
Supplement G: iPhone Waveform Dynamics
Supplement H: Previous attempts on approximation of LVEF using arterial pressure waveforms
Supplement I: Limitations
Supplement A: Pressure and vessel wall displacement in a viscoelastic vessel:
The radial equation of motion in a cylindrical vessel with the assumption of the incompressibility of the
vessel wall is (1):
 2ur  rr  rr   
s 2 

t
r
r
(S.1)
With the boundary condition:
 rr r R
in ( t )
Here,
  p(t )
and
 rr r  R
out ( t )
0
(S.2)
 s is the vessel wall density, ur is the radial displacement of the wall,  rr is the radial stress,  
is the circumferential stress, p (t ) is the pressure, Rin (t ) is the inner radius of the vessel wall, and Rout (t )
is the outer diameter of the vessel wall. t and r denote time and radial coordinate respectively.
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Without loss of generality, consider one sine harmonic of the pressure wave as:
p(t )  P sin t
A simple form of a viscoelastic material is (1):
  1


 E  E 2
t
t
Here, E is the modulus of elasticity,
(S.3)
 is the strain,  1 and  2 are relaxation times. Without loss of
generality, we can consider a linear isotropic viscoelastic material, so E,  1 , and  2 are all constant.
Because of the linearity of equations S.1 and S.3, we can assume that the stresses and strains all have
the same frequency of oscillation as the pressure. In the complex form one can write them as:
 
  Re ˆ e 
  Re ˆ e 
p  Re pˆ e it
i t
(S.4)
i t
Here, Re denotes the real part, and “^” denotes a complex value.
Substituting S.4 in S.3 and rearranging the terms:
ˆ  E
1  i 2
ˆ  Eˆ ˆ
1  i 1
(S.5)
Where, Ê is a complex viscoelastic modulus.
For a linear material and small strain (without loss of generality), the stress components in equation S.1
are related to radial displacement (2):
 rr  E
u r
r
 rr  E
u r
u
and    E r
r
r
Writing S.6 in the complex form and substituting it in S.1 along with S.5 and eliminating e


d 2uˆ r 1 duˆ r uˆ r
ˆ 0,

 1 
dr 2 r dr r 2
(S.6)
it
, results in:
(S.7)
2
ˆ   s r
Where, 
2 2
Eˆ
is a dimensionless number.
ˆ  1 (1). Therefore, equation S.7
Under physiological conditions, ̂ is much smaller than unity 
reduces to:
d 2uˆ r 1 duˆ r uˆ r

 0
dr 2 r dr r 2
(S.8)
ˆ
ˆ
Subject to boundary conditions of du r  0 at r  Rout and Eˆ du r  pˆ at r  Rin .
dr
dr
The real part of the solution of the equation S.8, which is the solution for a single sine harmonic of the
pressure is:
R 


2 Rout P 1   2 1 2
2 Rout P ( 1   2 )
sin t 
cos t
2
2 2
E (  1)(1    2 )
E ( 2  1)(1   2 22 )
(S.9)
Here,  is the ratio of the outer diameter to the inner diameter.
If the viscoelastic behavior of the vessel wall is negligible,  1   2  0 (1). Under this condition,
equation S9 reduces to:
R 
 2 Rout 
2 Rout P
 p(t )
sin t  
2
2
E (  1)
E
(


1
)


(S.10)
Equation S10 indicates that the vessel wall displacement is strictly in phase with the pressure. This
means that in a large blood vessel such as aorta or carotid artery where the degree of viscoelasticity is
negligible (3, 4), the vessel wall displacement has the same shape as the pressure waveform. Also, it is
obvious from equation S10 that stiff arteries or abundant soft tissue in the neck do not affect the wall
displacement waveform shape (only affecting magnitude). Therefore, the IF computed from carotid
vessel wall motion will be equal to the IF computed from carotid pressure wave ([ω1, ω2]p=[ω1, ω2]D).
This also means our EF computation from iPhone-measured waveforms is not influenced by skin color or
other physical conditions such as swelling of neck and obesity.
It is noteworthy to mention that the method introduced in this manuscript is not dependent on the nonviscoelastic wall assumption. For a highly viscoelastic vessel wall, one can assume a function that relates
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wall intrinsic frequencies to pressure intrinsic frequency. Therefore, there will be still a way to relate IFs
from displacement waves to EF that inherently contains the viscoelastic mapping between pressure and
wall motion.
Supplement B: Carotid artery waveform capture using an unmodified iPhone camera:
A custom smart phone application was written to collect noninvasive waveforms from the (unmodified)
camera of an Apple iPhone 5S (See Figure 1A in the manuscript). The application allows for control of
the camera flash as well as basic video recording. The LED (light-emitting diode) flash illuminates the
skin (over the carotid pulse) and the camera records a sequence of modulations in image intensity
across a time sequence of image arrays. To ensure that the dynamic content of the waveform is present,
the application leverages the high-speed functionality of the camera, running at an average rate of 120
frames per second (fps). By locating the phone over an artery close to the skin surface, in this case the
carotid, a displacement waveform can be measured that reveals the arterial pressure waveform (Figure
1A in the manuscript). The iPhone waveform (vessel wall displacement waveform) is a reliable proxy for
the carotid pressure waveform, and was easy to capture in a variety of body types.
Supplement C: Intrinsic Frequency Method:
Mathematically, the IF method approximates the pressure waveform by two incomplete sinusoids with
different frequencies (ω1 & ω2), which we called Intrinsic Frequencies (IFs) (5, 6). ω1 is the IF for the
coupled LV- arterial system (representing the systolic portion of the pressure wave), and ω2 is the IF for
the decoupled arterial vasculature (characterizing the diastolic portion of the pressure wave) (Figure
S1)(5).
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Figure S1. Sample pressure waveforms from iPhone (black dashed line) with superimposed reconstructed
waveforms. Using mathematical tools described above, the waveform reconstructed by ω1 is blue, and by ω2 is green.
The decoupling time (dicrotic notch) is marked in red. These waveforms are from two different subjects.
The Intrinsic Frequency algorithm is as follows:
Minimize:
f (t )   (0, T0 ) s1 (t )   (T0 , T ) s2 (t )  c 2 ,
2
(S.11)
Subject to:
a1 cos(1T0 )  b1 sin( 1T0 )  a2 cos(2T0 )  b2 sin( 2T0 ) ,
(S.12)
a1  a2 cos(2T )  b2 sin( 2T ) ,
(S.13)
s1 (t )  a1 cos(1t )  b1 sin( 1t ) ,
(S.14)
and
s2 (t )  a2 cos(2t )  b2 sin( 2t ) .
1 a  t  b
0 otherwise
where,  (a, b)  
(S.15)
(S.16)
and c is a constant.
Here a1, a2, c, b1, b2, ω1, and ω2 are unknowns. Equation S.11 and S.12 are linear constraints that guarantee
the continuity at the time T0 (dicrotic notch) and the periodicity at the time T (end of the cardiac cycle).
Note that envelopes are 𝑅1 = √𝑎12 + 𝑏12 and 𝑅2 = √𝑎22 + 𝑏22
(envelope ratio: ER=R1/R2). This
minimization problem is not convex. To find the global minimum, we use a brute-force algorithm over all
possible values of frequencies and ensure that the corresponding minimizer frequencies (ω1, ω2)m are in
fact the unique global minimizer frequencies of the original problem. In the brute-force algorithm, the
domain D was taken as
D  1 , 2  such that 0  1, 2  C.
(S.17)
Where constant C is the upper bound of the domain selected in such a way that the solution does not
include physically unaccepted values. D was discretized for pairs of (ω1, ω2). For each point (ω1, ω2) in the
discretized domain, the IF algorithm is solved and the solution is stored as P(ω1, ω2). Note that the
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minimum of the IF problem for the whole domain D corresponds to the minimum of P(ω1, ω2) over all (ω1,
ω2). After searching through all pair values, the corresponding minimum frequencies are denoted as (ω 1,
ω2)m. The brute-force algorithm looks over all possible values of frequencies and ensures that the
corresponding minimizer frequencies (ω1, ω2) m are in fact the unique global minimizer (5).
Supplement D: Cycle Selection and Signal Quality Evaluation:
Cycle selection was performed manually by a researcher blinded to patient information and MRI data.
Multiple good quality cycles were selected from each measurement (poor quality cycles identified as tilted
or distorted waveforms due to respiration or other motion). The beginning, end, and dicrotic notch of
selected cycles were identified manually. The IF method was applied to selected cycles, creating several
pairs of ω1, ω2 for each subject. The final ω1, ω2 was the averaged value of all pairs after removing outliers.
An outlier was defined as a point with ω1 (or ω2) more than 15 bpm (beats per minutes) different from
the median. As a result, a unique set of ω1, ω2 (and other IF parameters) were produced for each patient.
Five subjects’ recordings were excluded through blinded signal quality criteria. The IF method is based on
two-coupled dynamical system theory (Heart+Aorta). This assumption is violated in cases of severe mitral
or aortic valve diseases since the diseased valve acts a separate dynamical system (Heart+Valve+Aorta).
Therefore, subjects with severe Mitral or Aortic valve diseases (total of two subjects) were excluded from
analysis.
Supplement E: Cardiac MRI Scan and Analysis:
The acquisition protocol consisted of Fast Spoiled Grass (fSPGR) sagittal localizer, Fast Imaging
Employing Steady State Acquisition (FIESTA) in the sagittal, long axis, short axis and radial planes, and
phase contrast (PC) flow through the aortic valve, mitral valve, ascending aorta, and descending aorta.
All sequences were performed with breath holding at expiration to minimize diaphragmatic motion. No
contrast agents were used. DICOM (Rosslyn VA) images were transferred to an AW Workstation and
data were analyzed using ReportCARD. Volumetric analysis and EF calculation were performed using
contours drawn on the short axis FIESTA images.
Supplement F: Details of statistical analysis:
The associations between EF (MRI) and the major dimensions of the IF (ω1, ω2, T, T0, and RHDN) in this
proof-of-concept study were initially examined through calculation of correlation coefficients. As
explained in the text, we introduced new IF-related parameters called 𝜔
̅̅̅̅1 and 𝜔
̅̅̅̅2 to overcome the
nonlinear nature of the LV-arterial system in approximation of EF=f(IF). Thus, the dimensions of the IF
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(ω1, ω2, T, T0, and RHDN) and the newly formed parameters (also functions of the five major dimensions)
were used as independent variables to build the best multiple regression model and to approximate
function f(IF). Multiple regression method was performed on a training set to produce a model that
approximates f(IF) which relates intrinsic frequencies to EF (f(IF)=EF). Our training sample included
multiple carotid measurements with different devices (tonometry, optical sensor, and iPhone) from the
same subject. Subjects who participated in the study from the beginning of the enrollment (September
2014) to 15th of February 2014 were included in the training set. This includes total of 39 subjects with
iPhone measurements. These subjects’ data were used to find function f(IF) by applying linear multiple
regression analysis. There was no exclusion at this point. Of the subjects included in the training set, 38
had both iPhone and tonometry, 1 had all three measurements and five had only tonometry waveforms.
The waveforms measured by devices other than the iPhone were used only on the training set in order
to avoid overfitting (See Babyak, M. A. (2004) and references within (7)). All tests were completed in a
single session and the patients’ body positions remained unchanged during each test (see manuscript
text). A power of >80% was achieved to detect values of R2 ≥ 0.25. The regression model for calculating
EF from aforementioned IF dimensions produced R2 of 0.44 (p-value of 1.2e-07, adjusted R2=0.39, RMSE
= 9.5%) and the following equation:
EF=k0+k1 ω1+ k2 𝜔
̅̅̅̅1 + k3 𝜔
̅̅̅̅2 + k4 ER+ k5 T+ k6 RHDN+ k7 Tdys
Here, Tdys is the diastolic time (=T-T0). ̅̅̅̅
𝜔1 and 𝜔
̅̅̅̅2 are ω1 and ω2 normalized with the systolic time (T0)
and diastolic time (Tdys) respectively. The kj (j= 0 to 7) vector is [315.2, -0.65, -3.86, 0.12, -8.2, -36.0, 12.5, -12.2]. The resulting regression model was also checked to ensure that the independent variables
do not display any high multicollinearity.
Figure S2 shows the scatter graph of the iPhone waveform results for the training set and non-iPhone
waveform results used just for the training set from one single function of f(IF) with no calibration.
Finally, to further examine the precision of the results and to assess if EF-iPhone could be well calibrated
with MRI, a Bland-Altman analysis for agreement between EF-MRI and the EF-iPhone was performed,
with +/-20% as the maximal tolerated 95% limits of agreement. The limits of agreement were computed
as Mean difference ± 1.96 × SD of the differences. The statistical analysis was performed using Matlab
and SAS.
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Figure S2. Scatter graph of the results for the training set. (A) The scatter plot of the iPhone results of the training set. (B) The
scatter plot of the tonometry results of the training set (brown) and the optical sensor results of the training set (black) Y-axis is the
EF measured by MRI and X-axis is the EF computed by intrinsic frequency method using waveforms measured by an iPhone (blue)
or other devices (brown: tonometry and black: optical sensor). The red line is the ideal Y=X line.
The Regression method we used for approximating the function EF=f(IF) can only be as good as the
available training data. Therefore, this IF-iPhone method may become more accurate in computing EF
after collecting more data. This manuscript is only intended to show the proof-of-concept.
Supplement G: iPhone Waveform Dynamics
Although we did not compare iPhone App-derived waveforms with those from arterial lines, we can
compare iPhone to tonometry-derived measurements. Using an FDA approved arterial tonometry device
(Sphygmocore from AtCore Medical) we calculated IF and compared these to IF from the iPhone in the
same subjects enrolled in our pilot study. Although tonometry and iPhone waveforms were not acquired
simultaneously, they were collected in the same session with only 10-20 minutes between measurements.
The correlation between ω1 of iPhone and of tonometry was r= 0.9 (n=32). The correlation between ω2 of
iPhone and of tonometry is r= 0.95 (n=32). The high correlation between ω1 and ω2 of iPhone and
tonometry indicates that iPhone is an acceptable tool to capture the dynamics embedded in the
waveforms in comparison to an established device such as tonometry. The small differences between
them is due the fact that tonometry requires applying pressure to the skin over the pulse using the
piezoelectric sensor (deforming the absolute pressure tracing) whereas, no pressure is applied using the
iPhone. Figure S3 shows the scatter plots of ω1 and ω2 for iPhone and Tonometry from subjects whose HR
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remained within 5 bpm and left ventricle ejection time (LVET) remained within 20 milliseconds between
measurements, to ‘normalize’ cardiovascular system dynamics.
Figure S3. Comparison of iPhone and Tonometry waveform dynamics. ω1 from iPhone waveform capture is strongly correlated
with ω1 from tonometry (left plot). ω2 from iPhone is strongly correlated with ω2 from tonometry (right).
Figure S4 shows samples of iPhone waveforms taken by screenshotting of the iPhone screen. Fiducial
dynamics features of the waveform such as inflection point (associated with the reflected wave arrival
time), dicrotic notch (associated with the closure of the aortic valve), and the diastolic peak can be seen.
Figure S4. Fiducial dynamics features of the waveform. Orange arrows show inflection points.
Figure S4. Shows samples of raw tonometry waveforms (top row) and iPhone waveforms (bottom rows)
for three different subjects. Each column block belongs to the same subject.
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Figure S4. Raw Tonometry waveform (top rows) and raw iPhone waveform (bottom rows) from three different subjects. Each
column block belongs to the same subject.
Supplement H: Previous attempts on approximation of LVEF using arterial pressure waveforms
It is well accepted that arterial waveform analysis provides clinically valuable information about global
cardiovascular function (8-12) but only a small fraction of the embedded information is used clinically. For
the most part current studies have focused primarily on approximating cardiac output (liters per minute
ejected) and/or stroke volume (volume ejected with each systole). There have been few attempts to
estimate EF using waveform analysis (13, 14), and most of these require invasive arterial cannulation
and/or rely on baseline EF measurements for calibration. These studies have also not been validated for
EF measurement in humans. Additionally, most of these methodologies are based on the Windkessel
model which is limited due to inherent theoretical and modeling assumptions (15-17). One alternative
approach to Windkessel-based models is the pressure recording analytical method (PRAM) (18, 19). A
recent study showed that the PRAM method is capable of producing some correlation with EF for a specific
range of EF under specific diseases conditions. However, this study too is based on invasive measurement
of waveforms (13). To the best of our knowledge, our study is the first of these kinds of clinical approaches
to measure EF in using arterial pressure waveforms captured noninvasively.
Supplement I: Limitations
We studied only 11 HF subjects, and need to study a larger population to understand the limits and
applicability of the EF=f(IF) formula. In addition, it is important to recognize that about half of HF patients
have preserved ejection fraction(20), such that ejection fraction monitoring alone may not be useful for
them. As we explore more about the physiologic underpinnings of intrinsic frequency measures (e.g. ω1
10
and ω2), the methodology may become more applicable to this population, and we hope, more
informative in the management of critically ill patients generally. For safety reasons, MRI and iPhone
measurements cannot be performed simultaneously. However, the two measurements were completed
in a single 1.5 hour session during which heart function would have changed very little. We plan further
studies on the physiologic meaning of IF beyond calculation of ejection fraction. In particular, IF may be
capturing intrinsic features of the vasculature that deserve further exploration. We did not assess the
intra/inter observer variability and measurement variability (test-retest variation) which deserve more
attention in future studies. The iPhone measurements have been performed by research staff in cliniclike conditions, but they have not been done by any individual at home. Our future studies will include
larger populations of patients with systolic dysfunction and reduced LVEF to confirm our findings.
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