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Avestia Publishing
Journal of Biomedical Engineering and Biosciences
Volume 1, Year 2014
Journal ISSN: TBD
DOI: TBD
Pulsatile Aortic Pressure-Flow Analysis using
Fractional Calculus for Minimally-invasive
Applications
Glen Atlas 1,2, Sunil Dhar 3, John K-J. Li4,5
1Rutgers
New Jersey Medical School, Dept. of Anesthesiology
Newark, New Jersey, USA, 07039
atlasgm@njms.rutgers.edu
2
Stevens Institute of Technology, Dept. of Chemistry, Chemical Biology, and Biomedical Engineering
Hoboken, New Jersey, USA, 07030
3
New Jersey Institute of Technology, Dept. of Mathematical Sciences
Newark, New Jersey, USA, 07102
4
Rutgers University, Department of Biomedical Engineering
Piscataway, New Jersey 08854, USA
5
College of Biomedical Engineering and Instrument Science
Zhejiang University
Hangzhou, China
Abstract- Time-dependent pulsatile pressure and flow
waveforms in the aorta carry with them considerable
information regarding the underlying dynamic behavior of the
cardiovascular system. The aortic pressure-flow relationship
has traditionally been described using integer calculus. As such,
periodicity and linear system assumptions are necessarily
imposed to extract hemodynamic information. We introduce the
use of fractional calculus (FC) to relate minimally-invasive
measurements, of the velocity of aortic blood flow with an
esophageal Doppler monitor, to the derived aortic pressure. The
basis for this research is a Taylor series model of the velocity of
aortic blood flow with subsequent term-by-term fractional
integration as well as fractional differentiation. These results
demonstrate that this FC approach could potentially generate
the aortic pressure waveform throughout systole. Further
studies of its first derivative, or the time rate of pressure change,


, may also allow its maximal value,  ( ), to be computed

terms
(http://creativecommons.org/licenses/by/3.0).
Unrestricted use, distribution, and reproduction in any medium
are permitted, provided the original work is properly cited.
Disclaimer: Presented in part at the International Conference on
Biomedical Engineering and Systems, Prague, Czech Republic,
August 14-15, 2014 and on the NJIT Center for Applied
Mathematics
and
Statistics
website:
http://m.njit.edu/CAMS/Technical_Reports/CAMS13_14/repor
t1314-18.pdf
1. Introduction
Hemodynamic diagnosis and clinical management
often focus on blood pressure and flow measurements
and subsequent analysis of their temporal relationship
[1]. Routine catheterization can readily provide left
ventricular pressure and aortic pressure, as well as
cardiac output, from thermodilution. However, beat-tobeat flow measurements are less common. On the other
hand, ultrasound Doppler echocardiography can provide
images of cardiac and large vessel structures as well as
blood flow velocity. Nonetheless, the aortic blood
pressure waveform cannot be obtained.
Noninvasive or minimally-invasive approaches are
preferred methods for routine clinical diagnosis and
follow-up. Tonometer-based measurements, of blood

for use as an index of left ventricle contractility when
noninvasive ultrasound Doppler flow velocity is available in the
clinical setting.
Keywords: Fractional
Esophageal
Doppler
Differintegration.
calculus, Aortic blood flow,
monitor,
Differintegral,
© Copyright 2014 Authors - This is an Open Access article
published under the Creative Commons Attribution License
Date Received: 2014-08-21
Date Accepted: 2014-12-12
Date Published: 2014-12-15
1
pressure overlying the carotid artery, have been
common [2], but are much less frequently used in
conjunction with Doppler ultrasound in the clinical
setting. It is possible to derive central aortic pressure
when the Doppler ultrasound aortic flow velocity is
known [3], [4]. These latter traditional hemodynamic
models are typically based upon a linear second-order
system utilizing the acceleration, velocity, and
displacement of blood flow [5]. By convention,
acceleration is defined as the first derivative of velocity,
with respect to time, whereas displacement is its
indefinite integral. In contradistinction, fractional
calculus (FC) is based upon both integer and non-integer
differentiation as well as integer and non-integer
integration [6], [7].
 ′ () =  (−1)
.
(2)
 ′ ′() = ( − 1) (−2)
.
(3)
 ′′′ () = ( − 1)( − 2) (−3) .
(4)
The nth repetitive integer differentiation process can
therefore be summarized as:

 
=
(!) (−)
(−)!
ℎ  ≥  .
(5)
In a likewise manner, the nth repetitive integer
integration process can also be examined for a power
function of time:
() =   .
2. Related Work
(6)
 (+1)
+1
FC, although three centuries-old, has recently
found applications in the analysis of biological systems.
For instance, Djordjevic et al. [8] developed a rheological
model of airway smooth muscle cells using a method
incorporating FC and a least-squares data fitting
technique. They showed that FC could be effectively
utilized to account for a weak power law frequency
dependence of cell rheological behavior. This effect could
not be explained with traditional viscoelastic theory.
Recently, an FC dynamic model has been applied to
generate electrocardiogram (ECG) signals based upon
oscillations and a global optimization scheme. This
technique subsequently generates a realistic time series,
of the ECG signal, and may find potential applications in
modeling abnormal and irregular patterns of cardiac
conduction [9].
We have considerable experience with the use of
esophageal Doppler monitor (EDM) for minimallyinvasive measurement of the aortic flow velocity
waveform [5]. This paper provides the first such
application of FC to minimally-invasive hemodynamic
studies and demonstrates how FC-based modelling
could be effectively utilized in understanding the aortic
pressure-flow relationship during systole.
∫ () =
3. Methods
Note that the gamma function is not defined for
values of x equal to either zero or negative integer values.
Furthermore, when x is a positive integer, the gamma
function has the following property:
3.1. Fractional Calculus
To cognize this application of FC, traditional
integer differentiation is first examined for a power
function of time:
() = 
.
(7)
 (+2)
∫ ∫ () = (+1)(+2) .
(8)
 (+3)
∫ ∫ ∫ () = (+1)(+2)(+3) .
∫ … ∫ () …  =
⏟
 − 
 −
=
(!) (+)
(+)!
(9)
.
(10)

Note that a constant of integration can be utilized
after the completion of the repetitive integration process.
Thus, using either (5) or (10), repetitive differentiation
or repetitive integration can be similarly accomplished
using either positive or negative values for n
respectively.
The gamma function Γ() can be defined as [10]:

1
2

( ) (+2) … (+)
→∞  +1
Γ() = lim
1

→∞

 −1

=    ∏=1 (1 + )
Γ() = ( − 1)!

.
(1)
2
.
.
(11)
(12)
Additionally, Γ() “smoothly connects” the integer
values of the factorial function. It is therefore suitable for
defining non-integer factorial values. The gamma
function is illustrated in Figure 1.
3.2. Fractional Calculus and the Taylor Series of an
Exponential Function
The Taylor series for an exponential function is:
(x)
Gamma Function
15
10
5
0
-5
-10
-15


-2
-1
0
1
2
3
Figure 1. The gamma function is useful in determining noninteger values of the factorial function. It is not defined for
zero and negative integer values
Equations (5) and (10) can then be modified to
utilize the gamma function:
=
(−)
Γ(+1)
Γ(−+1)
.
(13)
Equation (13) can be used as the definition of the
differintegral [11]. Where q can have a positive value;
either integer or non-integer. 1, 2 Note that q can also take
on integer or non-integer negative values:3
∫
⏟… ∫ () …  =
 − 
 −
=
Γ(+1) (+)
Γ(++1)
.
(14)
The term q is referred to as the order of
differintegration [12]. Additionally, () =
(− ∙ )
!
(−∙)0
0!
+
(−∙)1
1!
+
(−∙)2
2!
…+
(−∙)
!
.
(15)
Thus, for a sufficiently large N, an exponential
function can be accurately approximated as a summation
of power functions. Using the above methodology, the
Taylor series for an exponential can therefore be termby-term fractionally differentiated or fractionally
integrated:

(−) Γ( + 1) (−)
 ( −∙ )
= ∑{
∙[
]} .
 
!
Γ( −  + 1)
(16)
=0
For mathematical purposes, t cannot equal zero
and be raised to a negative power. However, t can take
on positive near-zero values. Negative values of t can also
yield complex results. To further reiterate, care must be
used when selecting integer values of q to prevent
undefined values of the gamma function from occurring.
4. Examining the Velocity of Aortic Blood Flow

 
|
  =0
. As
previously stated, (0) is not defined. The gamma
function is also not defined for negative integer values.
Thus, specific fractional derivatives, or fractional
integrals, may be unattainable.
Owing to either the positive or negative value of q
in (13), the differintegral can therefore be utilized for the
fractional differentiation, or fractional integration, of
q cannot take on integer values equal to such
quantities as: (m + 1) or (m + 2) or (m + 3), etc.
1However,
=∑
=
4
x
 
−∙
=0
-3
 
power functions. Furthermore, using FC, differentiation
and integration may possibly be represented as a
continuous process rather than discrete processes.
The esophageal Doppler monitor (EDM) is
frequently utilized to assess the velocity of aortic blood
flow during systole. The EDM allows clinicians to
accurately assess patients’ cardiac output and stroke
volume during anesthesia and critical care conditions
[13]. Figure 2 illustrates this waveform.
This velocity, v(t), can be modelled as [5]:

() =  − (1 − ) 
0 <  < 
.
(17)
2Imaginary and complex values of q can also be utilized.
However, these will not be addressed in this introductory
paper.
3Note that an alternative terminology could be that of
fractional derivatives and fractional antiderivatives.
3
Where  represents an acceleration term and  is
a dimensionless gain. The time spent in systole is
referred to as flow time, FT. It should be noted that can
be determined [5]:

)

2–(
0 <  < 
( – )
.
(18)
< 
(20)
Figure
3
demonstrates
the
continuous
differintegral (20) over the range: −1 ≤  ≤ 1.
magnitude →
magnitude →
=
(+1−)

 
 (−) ∙Γ(+2)∙()
=

−
[∑
=0


!∙Γ(+2−)
(−) ∙Γ(+3)∙()(+2−)
1
∑
0<
]
=0

!∙Γ(+3−)
Figure 2. The velocity of aortic blood flow as measured by an
EDM. Note that PV represents peak velocity whereas FT
signifies the time spent in systole. The time at which PV
occurs is referred to as FTp [5]
Note that FTp represents the time at which peak
velocity (PV) occurs. This is illustrated in Figure 2. Using
a Taylor series, v(t) can subsequently be approximated
as a time-based power function:
() = (∑
=0
(−) ∙()(+1)
(−) ∙()(+2)
1
∑
)
=0

!
!
associated with 0 < q < 1. Furthermore, () =
(19)
By means of the aforementioned technique,
fractional derivatives and fractional integrals of v(t) can
then be determined:
|
  =0
Using MATHCAD (PTC Corp., Needham, MA, USA)
v(t) can be calculated utilizing the numerical values from
Table 1. Subsequently, its differintegrals of order −0.7
and 0.1 can both be determined. These functions are
illustrated in Figure 4.
Note that the dimension associated with v(t) is m/s
 (−0.7) 
is m/s(−0.7).4
 (−0.7)
 (0.1) 
is m/s0.1.
 (0.1)
whereas that of
dimension of
4
 
5. Numerical Assessment
−
0 <  < .
Figure 3. Equation (20), the velocity of aortic blood flow
during systole, v(t), represented as a continuous
differintegral. Note that fractional integration is associated
with −1 < q < 0 whereas fractional differentiation is
This dimension is equivalent to m·s0.7.
4
Furthermore, the
Table 1. Numerical values used for initial computational
purposes
dimensionless
6. The Systolic Pressure-Flow Relationship in
the Aorta
Using (20), a straightforward model of aortic blood
pressure, P(t), as a function of the velocity of aortic blood
flow during systole is:
() =
0 
 0
 
 
=  2 (   +    ) +  .
(21)
Where r represents the radius of the aorta, and a
and b are both velocity-based differintegrals of order
−0.7 and 0.1 respectively. The term Za is “reactance-like”
and would be analogous to a combination of elastance
and resistance. Whereas Zb would be analogous to a
combination of inertia and resistance. Furthermore, C is
a constant of integration and k converts units of Pascals
to mmHg. In addition:

|
  =0
=
 
|
  =0
=0
.
(22)
0.2
differintegr
al of order
0.1
0
-0.2
0.36
0.1
differintegr
al of order
−0.7
0.4
0.30
b
0.6
0.24
s
s
dimensionless
velocity
0.8
0.18
0.36
0.1
−0.7
1
0.12
FT
FTp
a
Notes
acceleration
gain
exponential
decay
flow time
time to peak flow
order of
fractional
differintegration
order of
fractional
differintegration
0.06
Units
m/s2
dimensionless
s-1
0.00



Value
7.25
3.00
6.154
magnitude
Term
1.2
time (s)
Figure 4. Velocity as a function of time, v(t), and both its
associated differintegrals of order −0.7 and 0.1 are displayed
So that C also functions as an initial condition.
Moreover, for the purposes of this preliminary
assessment, a “trial and error” technique was employed
to determine numerical values for a, b and Za and Zb.
These are displayed in Table 2. Note that Za and Zb have
magnitudes which are “ballpark approximate” to those of
traditionally-derived resistance, elastance, and inertia.

The above model can also be utilized to assess 
during systole:


 (+1) 
 (+1) 
=  2 (  (+1) +   (+1) ) .
(23)

Both P(t) and  are illustrated in Figure 5. Note
that a positive near-zero initial value for t, instead of
zero, has to be used in (23) to prevent a “division by
zero” singularity error from occurring.
Table 2. Numerical values used for final computational
purposes
5
Term
C
Value
80
Units
mmHg
k
0.0075
mmHg/Pascal
r
Za
0.011
3.157·107
m
N·sa/m5
Zb
7.015·106
N·sb/m5
Notes
constant of
integration
unit
conversion
aortic radius
“reactancelike” term
“reactancelike” term
Figure 5. Using fractional calculus, P(t) is modelled using
differintegrals which are based upon the velocity of aortic

blood flow during systole. Note that is also displayed

Straightforward linear algebraic techniques could
also be applied. This would allow “real-time” or beat-tobeat assessment of the “reactance-like” terms, Za and Zb.
The following matrix relationship is derived using (21):

|
  =
 
|
  =

|
  =
 
|
  =
 2 ∙ [


] ∙ [ ] = [  ] .


(24)
Thus:
[

|
  =
 
|
  =
|
 
 
|
  =

1
] = 2 [ 

 
{
=

−1
]
demonstrated as a useful minimally-invasive clinical tool
[15]. However, central aortic pressure cannot be
obtained noninvasively.
Applanation tonometry, applied to peripheral
arteries such as the carotid and radial, has been reported
for noninvasive monitoring of both pulsatile pressure
waveforms and arterial compliance [16]. Subsequently,
these recorded peripheral waveforms have also been
used to derive central aortic pressure waveforms via a
transfer function. Clinically, these have been applied to
obtain the augmentation index [17] and to assess
vascular stiffness. But the temporal aspects, of the aortic
pressure-flow relationship through minimally-invasive
means, has yet to be demonstrated.
We have introduced a technique, based on FC, for
potentially deriving the temporal relationship of aortic
pressure and flow throughout systole. The minimallyinvasive measurement of the velocity of aortic blood flow
is accomplished with an EDM; although this can also be
obtained
with
a
transthoracic
Doppler
echocardiographic monitor; which is commonly utilized
in many clinics.
Our approach is novel in that the Taylor series
expansion of an exponential function was applied to
allow term-by-term fractional differentiation and
fractional integration. A closely related work using FC
was performed by Craiem and Armentaro [18] who
examined the power-law stress-strain relationship
within
sheep
aorta;
through
simultaneous
measurements of pulsatile aortic pressure and diameter.
They were able to account for a dynamic frequencydependent elastic modulus [19].

The rate of rise of aortic pressure  and flow

∙[ ]

.
(25)
}
6. Discussion and Conclusion
Noninvasive assessment, of the hemodynamics
within large arteries such as the aorta, has been limited,
although invasive means have been commonplace in the
clinical setting through catheterization [1], [14].
Advanced imaging modalities can provide geometric and
detailed structural changes, but the dynamic properties
of the pressure-flow relationship is typically not
obtainable.
Esophageal
Doppler
ultrasound
measurements, of aortic blood flow velocity, have been
velocity
both have a close relationship with

myocardial performance; particularly their peak values,


 ( )and  ( ) [20]. Thus, the minimally

invasive assessment of left ventricle contractility is
potentially possible using an FC model.
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