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Computers and Chemical Engineering 35 (2011) 1304–1316

Contents lists available at ScienceDirect

Computers and Chemical Engineering

journa l homepage: www.e lsev ier .com/ locate /compchemeng

n impedance model for blood flow in the human arterial system. Part I: Modelevelopment and MATLAB implementation

avid A. Johnsona,b, Justin R. Spaetha,1, William C. Rosec, Ulhas P. Naika,b,d,e, Antony N. Berisa,∗

Department of Chemical Engineering, University of Delaware, Newark, DE 19716, USADelaware Biotechnology Institute, University of Delaware, Newark, DE 19716, USADepartment of Health, Nutrition, and Exercise Sciences, University of Delaware, Newark, DE 19716, USADepartment of Biological Science, University of Delaware, Newark, DE 19716, USADepartment of Chemistry and Biochemistry, University of Delaware, Newark, DE 19716, USA

r t i c l e i n f o

rticle history:eceived 8 April 2010eceived in revised form 7 September 2010ccepted 17 September 2010vailable online 24 September 2010

a b s t r a c t

An impedance model capable of predicting the time-dependent blood pressure and flow profiles in all ofthe vessels in the human arterial network has been developed. The model is based on a Womersley-typeone-dimensional in space approximation of the pulsatile flow of a viscous fluid within elastic vessels.Nominal values from the literature are used to provide the input aortic pressure wave, the geometricdimensions of large arteries, various blood properties, vessel elasticity, etc. The necessary information tocharacterize the smaller arteries, arterioles and capillaries is taken from a physical scaling model [West,

eywords:ATLAB

ecursive functionlood flow modelulsatile flowrterial network

G. (1999). The origin of scaling laws in biology. Physica Acta, 263, 104–113]. The parameters, input setup,and the subsequent solution to the model equations have been efficiently implemented within MATLAB,which also allows for a variety of output information displays. The MATLAB implementation also allowsfor a comprehensive sensitivity analysis of the results to various input parameter values to be effortlesslyobtained.

ensitivity analysis

. Introduction

The human arterial network characterizes a complex system ofillions of vessels, ranging over four orders of magnitude of dimen-

ions from the largest one (the aorta, of a diameter of the order ofm) to the smallest capillaries of a diameters of the order of a fewicrons (Nichols & O’Rourke, 1998). In many regards this is not dis-

imilar from other networks encountered in chemical engineeringractice, such as heat exchanger networks, pipe distribution andtorage tanks, separation units networks, typically encounteredithin chemical plants (Turton, Baille, Whiting, & Shaeiwitz, 2009).s such, the system analysis approach that has proven so success-

ul with the design of chemical processes can also be applicablen the analysis of the arterial flow. In addition to the multitude ofessels, which are eventually arranged in a fractal topology (West,999) in the terminal regions of the arterial tree, one has also to

ake into account: the pulsatile character of the flow input fromhe heart (Remington & Wood, 1956), the elastic and sometimesiscoelastic nature of the vessels wall (Pedley, 1980), the non-

∗ Corresponding author. Tel.: +1 302 831 8018; fax: +1 302 831 1048.E-mail address: beris@udel.edu (A.N. Beris).

1 Present address: Department of Chemical Engineering, Princeton University,rinceton, NJ 08544, USA.

098-1354/$ – see front matter © 2010 Elsevier Ltd. All rights reserved.oi:10.1016/j.compchemeng.2010.09.006

© 2010 Elsevier Ltd. All rights reserved.

Newtonian characteristics of the blood rheology (Owens, 2006)etc. This explains why the development of a model for the flowof blood in the human circulatory system is a difficult task (Taylor& Draney, 2004). However, it is one with many applications, rang-ing from pharmacokinetics and pharmacodynamics (Macheras &Illiadis, 2006) to cardiovascular disease monitoring (Nguyen, Clark,Chancellor, & Papavassiliou, 2008), and physiological control theory(Batzel, Kappel, Schneditz, & Tran, 2007), etc.

The difficulties associated with resolving all the flow details aresuch that it makes any effort of addressing them in the full flowproblem prohibitive in the whole human arterial network. Rathertwo opposite avenues have emerged. One is focusing on detailed3D and time-dependent analyses of the flow in a selected compo-nent of the full arterial tree (Milner, Moore, Rutt, & Steinman, 1998;Xu, Long, Collins, Bourne, & Griffith, 1999; Torii et al., 2009) andeven there the level of the physical detail widely varies. The other,exploiting some drastic approximation of the flow (typically origi-nating from a linear superposition of the solution to the 1D pulsatileflow for small vessel disturbances) is focusing on developing anefficient solution of the flow through the whole network (Olufsen,1999, 2004; Raymond, Merenda, Perren, Rufenacht, & Stergiopulos,

2009). However, note, that due to the interconnectivity of all thevessels, even when one is interested on 3D simulation in a subnet-work it is important to have some model for the rest of the networkif in vivo conditions are to be simulated (Grinberg & Karniadakis,

D.A. Johnson et al. / Computers and Chemical Engineering 35 (2011) 1304–1316 1305

Nomenclature

APm, AQ

m coefficients for forward pressure and flow waves,[Pa] and [m3/s], respectively

BPm, BQ

m coefficients for backward pressure and flow waves,[Pa] and [m3/s], respectively

c complex wave velocity [m/s]c0 Moens–Korteweg wave velocity [m/s]D diameter of a vessel (D = 2r) [m]d reflection coefficient for flow and pressure wavesE elasticity of blood vessel [Pa]f,f0 heart rate [1/s]Hct blood hematocritHctc critical blood hematocrit (0.04)HctT local hematocrit of smaller vesselsh wall thickness of blood vessel [m]i imaginary numberJn nth order Bessel Function of the first kindK exponential factor used in lubrication approxima-

tion for tapering vesselsk′ ratio of Moens–Korteweg velocity to complex wave

velocityL length of vessel [m]l length of vessel segment [m]m harmonic number for Fourier decomposition of

wavesnd number of daughter vessels at bifurcationP pressure [Pa]�P pressure drop [Pa]�P steady state pressure drop [Pa]Q volumetric flow [m3/s]Q̄ steady state flow [m3/s]qPoiseuille steady state flow associated with the

Hagen–Poiseuille formula [m3/s]R hydraulic resistance [Pa s/m3]RHP hydraulic resistance of the Hagen–Poiseuille for-

mula [Pa s/m3]Re dimensionless Reynolds Numberr radius of vessel [m]rbottom distal radius of vessel segment [m]rtop proximal radius of vessel segment [m]S constant for complex wave velocity (=1 in West,

=1/(1 − �2) in Womersley)T period of pulse waveform [s]t time [s]u axial velocity [m/s]u* dimensionless axial velocityv̄ cross-section averaged velocity [m/s]w, w0 angular frequency for wave [1/s]Z characteristic impedance to flow [Pa s/m3]z length along axial direction [m]˛ dimensionless Womersley number˛c complex Womersley number, defined as i3/2˛�̇ shear rate [1/s]�0 dimensionless number based on Moens–Korteweg

velocity�rel relative viscosity of blood for smaller vessels� dimensionless viscous time constant entering in

Casson’s equation� viscosity of blood [Pa s]�N viscosity of blood at high shear rates [Pa s]�P viscosity of blood plasma [Pa s]� density of blood [kg/m3]�P density of plasma [kg/m3]

�RBC density of red blood cells [kg/m3]�w density of the vessel wall [kg/m3]� Poisson ratio0 yield stress [Pa]w wall shear stress [Pa]

2008; Formaggia, Quarteroni, & Veneziani, 2009). This explains arenewed interest in obtaining good 1D models and their efficientsolution—see (Raymond et al., 2009) and references therein.

A novel 1D arterial flow model is introduced in this work basedon a systems approach. It is important to note that one area wheremost of the previous 1D models lag behind is in lumping the distalvasculature using a three-element Windkessel model, which trun-cates the final sub-networks of vessels of the arterial circulationdown to the capillary level. This is where parameter lumping andregression is entered into the system. In contrast, in the approachfollowed in the present work, by modeling all the way down to thelast capillary, the full wave effects in the network are retained, andthere is no necessity to fit any parameters for optimizing the sys-tem. Another modeling component that can enter the system (as isseen, for example, in Raymond et al., 2009) is that of a heart modelthat is used to replace the input of pressure or flow data. Howeverthat can introduce some artificial development of the waveformcontours. By keeping in our approach the input as obtained fromexperimental or literature data, the evolution of the waveforms canbe specifically retained for the individual, which has better validityfor medical applications.

The objective of the present work is to show how one can use theMATLAB environment to efficiently implement the solution to the1D arterial network model (using the recursive function feature). Inaddition this implementation allows taking advantage of the inter-active environment within MATLAB for enabling a user-friendlyinterface and an easy input of the relevant parameters, which isalso of critical need to assist in parametric sensitivity studies. Thefacilitation within MATLAB of the use of various output forms andplots represents another advantage. Finally note that although theanalysis here is restricted to the arterial network there is no concep-tual difficulty to transfer these ideas to other network flow analysesof physiological importance, such as the venous system, the respi-ratory system, the lymph system etc.

In the next section of this paper we thoroughly describe themodel equations. In Section 3 we describe the network construc-tion of the arterial circulation starting with the large arteries forwhich detailed information is available and ending with the smallervessels and capillaries for which scaling laws are utilized. Section 4describes the MATLAB implementation of the 1D blood flow model.Then in Section 5 we present and discuss some characteristic resultsobtained with our model involving parameter sensitivity analysisand comparisons with literature. Section 6 contains our conclu-sions.

2. Model development

For viscous flow within thin-walled elastic tubes the one dimen-sional linearization approximation of the time-dependent flowequations leads to a hyperbolic system of partial differential equa-tions with respect to time and space (Womersley, 1955). As a result,for oscillatory flow the solution is expected to be in the form of a

superposition of forward and backward moving waves in each ves-sel. More specifically, using the linear superposition assumption,

1 hemic

t

Q

atflwcfl

�

t(mo

Z

d

R

p

itrmb

2

iog&

Q

R

nv2rru

R

pe

306 D.A. Johnson et al. / Computers and C

he total flux Q can be written as (Womersley, 1955):

= Q̄ + Real

( ∞∑m=1

AQmeimw0(t−z/c) +

∞∑m=1

BQmeimw0(t+z/c)

). (1)

For future use, let us also define wm ≡ 2fm = 2mf0 for thengular frequency corresponding to the m-th harmonic. It is impor-ant to note that in Eq. (1), from the individual frequency oscillatoryow solutions, AQ

m and BQm, representing the coefficients of the for-

ard and backward moving waves, respectively, are, in general,omplex. The pressure waves are of the exact same form as theow waves, albeit with different coefficients:

P = �P + Real

( ∞∑m=1

APmeimw0(t−z/c) +

∞∑m=1

BPmeimw0(t+z/c)

). (2)

Backward waves are created in the system due to wave reflec-ions at bifurcation junctions (Fung, 1996). Based on Eqs. (1) and2), the local, or characteristic, impedance can be defined for the-th harmonic as the complex quantity Zm obtained from the ratio

f the pressure to the flux complex amplitudes:

m ≡ APm

AQm

= −BPm

BQm

. (3)

Similarly, corresponding to the steady-state solution, we canefine a resistance (or real, steady-state, impedance) R as

≡∣∣�P

∣∣Q̄

. (4)

The steady-state and oscillatory flow impedances are the basicredictions of any 1D model.

The separate analysis of the steady and oscillatory flow effectss a key part of our model. Furthermore, in Section 3, we describehe arterial network as a tree of bifurcating, or beyond a certainadius, trifurcating (West, 1999), vessels. In the following we esti-ate the pressure-flow response of each vessel, and then consider

ifurcation effects.

.1. Steady-state solution

The steady-state solution for the flow of a viscous fluid of viscos-ty � through a cylindrical vessel of uniform circular cross sectionf radius r and of length L upon the action of a pressure drop �P isiven by the well-known Hagen–Poiseuille formula (Bird, Stewart,Lightfoot, 1987)

¯ = |�P| r4

8�L≡ qPoiseuille, (5)

The associated resistance is therefore:

≡ |�P|Q̄

= 8�L

r4≡ RHP. (6)

Large vessels in the arterial tree are tapered, resulting in aon-constant pressure drop per unit length over the length of theessel. We account for this using Olufsen’s method (Olufsen, 1999,004). This assumes that the vessel’s radius varies exponentially(z) = rtopeKz along the axial direction. To determine the true flowesistance of the vessel, the lubrication approximation approach issed (Deen, 1998; Spaeth, 2006).

tapered(z) = 8�4

[e−4Kz − 1

−4K

]≡ RHP

L

[e−4Kz − 1

−4Kz

]. (7)

rtop

Blood is a rheologically complex fluid consisting mainly of a sus-ension of blood cells and proteins in plasma. Due to this mixture oflements in the blood, the blood flow behavior is non-Newtonian.

al Engineering 35 (2011) 1304–1316

While more complicated analyses are available (Owens, 2006) (andare certainly warranted when more complex multidimensionalflow analyses are carried through), in the spirit of our simple one-dimensional flow approximation used in the present work, weaccount for non-Newtonian effects using the simplified analysisdeveloped by Truskey, Yuan, and Katz (2004). They propose theCasson equation as a constitutive relationship for wall shear stress:

w1/2 = 0

1/2 + (�N�̇�)1/2. (10)

An important parameter in this analysis is the concentrationof the red blood cells which is given (on a volume fraction basis)as the hematocrit ratio, Hct (Berne & Levy, 2001; Turitto, 1982).The dependence of yield stress and viscosity on the hematocrit wasgiven by Truskey et al. (2004) (in Pa) as

0∼10(Hct − Hctc)3, (11)

�N = �P(1 + 2.5Hct + 7.35Hct2). (12)

Truskey then arrives at the following expression for the steadystate flux in a cylindrical geometry

Q̄ = r4�P

8�NL

[1 + 11

21

(0

w

)4− 16

7

(0

w

)1/2+ 8

3

(0

w

)]

≡ �P

RHP

[1 + 11

21

(0

w

)4− 16

7

(0

w

)1/2+ 8

3

(0

w

)]. (13)

The steady Poiseuille resistance given by Eq. (6) was thereforepatched in this work using the lubrication approximation, Eq. (7),and the Casson relationship, Eq. (13), corrections to obtain bettervalues for the local steady resistance. Since in this process somevalues for the wall shear rate are needed, this is typically an iterativeprocess, albeit one that converges very fast, within 2–3 iterations.

2.2. Pulsatile flow of a viscous fluid within a single elastic tube

Womersley (1955) arrived at closed-form solutions for pulsatileincompressible viscous flow within elastic cylindrical vessels bymaking simplifications to the governing equations. Since Womer-sley’s original work several refinements appeared. Here we useone of them as outlined in the following paragraphs. Note thatwe emphasize their predictions for the complex impedance Z = Zm

corresponding to an arbitrary (m-th) harmonic.Further analysis of Womersley’s model equations and elab-

oration of the solution for periodic flow through a thin-walledelastic tube by West (1999) predict that the complex character-istic impedance Z (the ratio of the local pressure gradient to localflow), is approximated by

Z = c20�

r2c, (14)

where c0 is the Moens–Korteweg wave velocity

c0 ≡√

Eh

2�r, (15)

the wave velocity in pulsatile inertial flow through an elastic tube,ignoring viscous effects. In Eq. (14) c is the complex wave veloc-ity introduced in order to take into account viscous effects. In Eq.(15), E is the modulus of elasticity of the vessel wall and h is thewall thickness. Note that in some of Womersley’s and later work, afactor 1/1 − �2 where � is the Poisson ratio, has also been used toscale the nominal Moens–Korteweg velocity, and subsequently the

impedance (Olufsen, 1999, 2004; Pedley, 1980; Womersley, 1957):

c0 ≡√

11 − �2

Eh

2�r. (16)

D.A. Johnson et al. / Computers and Chemical Engineering 35 (2011) 1304–1316 1307

F ). Thed sub-n

t(vi

c

c

waW

˛

wd1t

2

ttPd

P

ig. 1. Model vascular network. The network is the same as that of Olufsen (2004ownstream radii. The network also includes 23 branching terminal arterial trees (

However, both this factor and an imaginary component addedo the modulus of elasticity to account for wall viscoelastic effectsPedley, 1980) can be taken into account by suitably adjusting thealue of E within the original expression provided by Eq. (15) whichs also the approach used in the present work.

The relationship between the Moens–Korteweg velocity and theomplex velocity, c, is given by Womersley (1955) as

= c0

√−S

J2(˛c)J0(˛c)

= c0

√S(

1 − 2J1(˛c)˛cJ0(˛c)

), (17)

here Jn is the nth order Bessel function, ˛c ≡ i3/2˛ (Pedley, 1980),nd ˛ is the dimensionless Womersley number (Pedley, 1980;omersley, 1955)

≡ r

√2f�

�, (18)

here f is the frequency of the pulsatile flow, and � and � are theensity and viscosity of the fluid, respectively. In Eq. (17), S equals/(1 − �2) in Milnor (1982), but West (1999) omits it (i.e. he sets ito unity).

.3. Pulsatile flow in a bifurcating vessel tree

As a result of the pressure and flux continuity at every bifurca-

ion, we can formulate the following recursive formulae to describehe relationship between the parent, Pf,0, and daughters, Pf,1 andf,2, forward pressure coefficients (for, say, a one parent to twoaughters bifurcation)

f,0e(−twL0/c0)(1 + d0) = Pf,1(1 + d1e(−2iwL1/c1))

= Pf,2(1 + d2e(−2iwL2/c2)), (19)

network includes 45 arteries, each characterized by its length and upstream andetworks) denoted by “B”s. See text for details.

Pf,0

Z0e(−twL0/c0)(1 − d0)

= Pf,1

Z1(1 − d1e(−2iwL1/c1)) + Pf,2

Z2(1 − d2e(−2iwL2/c2)), (20)

where w is the wave frequency under consideration and Li, ci, diand Zi are the length, (complex) wave velocity, reflection coeffi-cient (defined as the ratio of the backward to the forward amplitudewave) and impedance, for i = 0, 1, 2 (parent, daughter 1 and daugh-ter 2) vessels, respectively. These particular expressions remainvalid across the arterial network until reaching the capillary level,where the reflection ratio reaches a value of zero and only forwardpressure waves are exhibited (shown here for two identical endcapillary daughter vessels):

Pf,0e(−twL0/c0)(1 + d0) = Pcap, (21)

Pf,0

Z0e(−twL0/c0)(1 − d0) = 2Pcap

Zcap. (22)

At this level, the equations are sufficient to determine the,in general complex, Pcap/Pf,0, di coefficients. Starting with Eqs.(21)–(22), Eqs. (19) and (20) can be used recursively up the net-work to the aorta, for each one bifurcation at a time, leading to theevaluation of all the reflection ratios and all the ratios for the daugh-ter to parent forward pressures. At the end, simple imposition ofthe forward pressure of the top artery (aorta) allows then for theevaluation of all pressure coefficients through a simple forward-substitution recursive use of the previously established ratios.

3. Arterial tree model construction and vessel-dependentphysical properties

The blood flow model considers the full time-dependent behav-ior of the pressure and flow waves throughout the entire arterialnetwork based on the analysis outlined in the previous sections.

1 hemical Engineering 35 (2011) 1304–1316

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Table 1Input parameters to 1D model.

Property Typical value incommon units

Typical value in SIunits

DischargeHematocrit (HctD)

0.45(dimensionless)

0.45

Plasma viscosity(�p)

1.35 cP 1.35 × 10−3 Pa s

Plasma density (�p) 1.03 g/cc 1030 kg/m3

Red blood celldensity (�RBC)

1.09 g/cc 1090 kg/m3

Terminal arteryradius (rterm)

3 �m 3 × 10−6 m

308 D.A. Johnson et al. / Computers and C

ere we describe the modeling of the geometry and the physicalroperties.

.1. Large arteries

We have chosen to follow the layout for the arterial networksed by Olufsen (2004), with some small modifications. The largerrteries in the network, for which length and diameter data haveeen provided (Olufsen, 2004), are assigned numbers and are con-ected as shown in Fig. 1. This information is entered in an inputle in a format that allows for easy modifications as needed. Theserteries shall henceforth be referred to as “named arteries”. Ithould be noted that the coronary arteries and the intercostals areodeled as a single artery with a resistance equivalent to that of

ach true sub-network.

.2. Arterial trees

In Fig. 1, below each of the terminal large arteries (those markedith a “B”) exists what shall be referred to as an “arterial tree”. Theimensions of the vessels in each arterial tree are dependent uponhe dimensions of the “parent vessel” from which their tree stems.caling laws developed by West (1999) are used to size the ves-els within each arterial tree. West utilizes an energy minimizationrgument to show that the most efficient scaling of consecutiveessel radii in a fractal network obeys the following recursive law

k+1 = rkn−1/ndd . (23)

We follow here West’s recommendations (West, 1999) accord-ng to which the best modeling structure of the smaller arteries isbtained when

d ={

2 rdaughter ≥ 900 �m3 rdaughter < 900 �m

. (24)

Additionally, West explains that the volume serviced by a fractaletwork is proportional to the sum of the cubes of the lengths ofll vessels on a given level, provided the vessels are sufficientlymall (West, 1999). This leads to the following recursive law forhe scaling of consecutive vessel lengths

k+1 = lkn−1/3d

. (25)

Ultimately, it is the size of red blood cells that determines howmall capillaries can be guiding us to the selection of the terminaladius. Hence, we set the length-to-diameter ratio of the vesselsn the last level of each tree to a constant and implemented theecursive scaling law for vessel lengths in reverse, working fromhe end of each arterial tree back towards the top.

.3. Vessel compliance

The compliance of each vessel in the network determines theoens–Korteweg wave velocity, given by Eq. (15). Olufsen (2004)

resents experimental data for the term Eh/r versus vessel radiusver for vessels ranging from capillaries to the aorta in size. Theata is neatly fit by

Eh

r= 2 × 107e(−22.53r) + 8.65 × 105. (26)

The modulus of elasticity, representing only elasticity effects is aeal number. However, in the body, the blood vessels are not purelylastic. There is always some viscoelastic effect associated with the

essels and surrounding tissues (Pedley, 1980; Womersley, 1957).n the linear analysis of oscillatory flows this viscoelastic effect cane represented by an imaginary component of the elasticity. Thisesults to additional pressure losses. Given the lack of more specific

Maximum l/D ratio 50 (dimensionless) 50Residual capillarypressure (Pres)

15 mmHg 2000 Pa

information, we follow here Pedley (1980) in taking the imaginarypart to be 15% of the real.

3.4. Variations in blood viscosity, density and hematocrit

Due to the Fahraeus and Fahraeus–Lindqvist effects (Truskey etal., 2004), the effective physical and rheological properties of bloodchange throughout the arterial network. As the vessels becomesmaller, the volume exclusion associated with the red blood cellsand plasma change, such that the local hematocrit, density, andviscosity are affected. To account for this change, the model imple-ments a relationship by Pries, Secomb, Gaehtgens, and Gross (1990)between vessel diameter (in �m) and discharge hematocrit to pre-dict the local hematocrit:

Hct

HctD= HctD + (1 − HctD)(1 + 1.7e−0.415D − 0.6e−0.011D), (27)

One key feature to note with this expression is that the hema-tocrit is hardly affected until vessels near the capillary level arereached.

The effects of local hematocrit changes on the density are thenaccounted for using a model presented by Eloot, de Wachter, VanTricht, and Verdonck (2002) linearly varying the density betweenthat of blood plasma, �p, and red blood cells, �RBC:

�blood = �p × (1 − Hct) + �RBC × Hct. (28)

Oscillatory viscosity changes are accounted for in associationwith the vessel diameters (in �m), in a model also presented byPries et al. (1990):

�

�p= 1 + eHct·a−1

e0.45·a−1(110e−1.1424D + 3 − 3.45e−0.035D), (29)

where �p is the plasma viscosity and the term a is given as

a = 41 + e−0.593(D−6.74)

. (30)

4. Implementation of 1D impedance model in MATLAB

4.1. Input information and model parameter representation

Developing the 1D impedance model in MATLAB has the bene-fit of separating the input parameters from the output results andthe calculations. The input parameters first involve characteristicproperties for the 45 named arteries shown in Fig. 1 (such as theirgeometrical dimensions and their inter-connectivity within thearterial network). Second, they involve general parameters (such as

those shown in Table 1) in terms of which the vessel-specific prop-erties are calculated based on the relations described in Sections 3.3and 3.4, respectively. Note that for easy reference to the literaturethe parameter values in Table 1 are offered in their customarily used

hemica

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titgsaba

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nt(aitgt

D.A. Johnson et al. / Computers and C

nits, albeit consistently within the MATLAB code SI units are used.he input parameters information is stored in a .mat file. The char-cteristic properties of the 45 main arteries are built into an array,here the first index in the array represents a specific artery (from

he named arteries) in the network (e.g. 1 = aorta, 13 = left carotidrtery, 21 = left radial artery, 41 = left femoral artery, etc.) followinghe numbering scheme described in Fig. 1. The .mat file is uploadednto the core of the model code before running. In addition, geo-

etric network connectivity information is also contained withinhe .mat parameter file in two arrays with information about thearent and daughter vessels. These arrays contain links to informhe program which vessel is the parent and which the two daugh-ers of the reference vessel. In the case of a terminal sub-networks the daughter vessel, zeros are used to notify the program toegin the use of the fractal subnetwork (arterial tree) calculationss described in Section 3.

For running the program, a pressure profile over a cycle at one ofhe major (named) arteries is also needed. For flexibility, this is builtnto a separate specific parameter file to also allow for the modelo have patient specific applications in future use. To allow for areater variability of applications, the input pressure profile can bepecified in any named artery of the network (the only requirementt this time is to represent the artery with its particular index num-er). The input pressure profile needs to be what MATLAB considerss a text tensor of two columns (time and pressure).

In the simulations performed in the present work, the inputressure data used were at the top of the arterial tree, taken fromlufsen (2004). The image was scanned into the computer andata points obtained via using the graphical digitizing programN-SCAN IT©. The program uses the input profile given as a sin-le period for a pressure waveform, but it can be adjusted to allowor multiple periods to be analyzed if so desired. The allowance ofpecifying the parameter file, a specific pressure profile, and theocation for the measured profile permit for the model to be eas-ly adjusted to specific predictions for any particular patient (if soesired).

.2. Input parameters processing and solution of the governingquations

The core code is used for both further processing of the inputnformation and its reorganization in a form suitable for the solu-ion of the network equations as well as for the systematic solutionf those equations. A key component therefore of it (and anotherajor advantage of implementing it within the MATLAB environ-ent) is the natural array structure of the components of the

roblem. Thus, in the core code, right after the input informations loaded, the apparent physical properties (like density, viscosity,lasticity, etc) are calculated pertaining to each one of the mainessels following the models described in Sections 2.1, 3.3 and 3.4.hose are stored in one-dimensional arrays following, again, theumbering scheme described in Fig. 1. Additional derivative prop-rties pertaining to the solution (such as the steady state resistance,alculated following Section 2.1 and the Fourier components of thempedances, calculated following Section 2.2) are also evaluatednd stored in similar 1D (2D for the Fourier components, with theomponent index as the second index) arrays.

Further input parameter processing involves evaluating perti-ent property information for the rest of the vessels of the networkhat are not explicitly accounted for among the 45 named vesselsarterioles, secondary arterioles, all the way to capillaries). This isgain where MATLAB array notation becomes handy, with the array

ndices n, k used to represent a vessel belonging to the arterialree that originates at the end of the n-th named vessel in the k-theneration of bifurcating vessels—see Fig. 1. All vessels belongingo the kth-generation are considered equivalent and represented

l Engineering 35 (2011) 1304–1316 1309

together, with the network topology that we have considered hereconsisting of successive bifurcations or trifurcations, following thespecification described in Section 3.2. Note that this approach hasbeen followed consistently for all arterial trees, independent of thecorresponding point of origin (or, equivalently, the organ/tissueserviced by that arterial tree in the body). However, it is straight-forward to adjust this evaluation procedure, as needed, to take intoaccount organ-specific information.

The processing of the information for the secondary vesselsproceeds therefore as follows. First the pertinent geometric infor-mation is generated in a recursive fashion using the scalingformulae supplied in Section 3.2. For each arterial tree (indicatedby B in Fig. 1) we start with the first generation of daughter ves-sels the geometrical dimensions of which are specified based onthe scaling laws and the dimensions of the parent named vessel. Aseach generation progresses, we evaluate successively the daugh-ter geometric information based on the parent one, following thescaling relations presented Section 3. If the radius of the currentgeneration is found to be less than the terminal artery radius (asprescribed in Table 1), the current generation becomes the lastgeneration (and is assumed to be at the capillary level). After thegeometric information is constructed, the apparent values of thephysical properties are then generated, as for the main (named)vessels, using the same models described in Sections 2.1, 3.3 and3.4. Following that, derivative properties pertaining to the solution(such as the steady state resistance and the Fourier components ofthe impedances) are also evaluated and stored in a similar fashionas for the named arrays. This is a very powerful and quite generalscheme that allows handling of rather complex network topologies.

The next step in the core code involves the systematic construc-tion of the solution to the problem. First, the input pressure data isconverted in its corresponding Fourier series. Only the steady state(zeroth component) and the first ten harmonics are used in thiswork as their superposition gives a nearly identical reproduction ofthe data. In the following we describe the process followed whenthe input pressure is considered to represent the pressure profileat the top of the arterial tree. If the input profile corresponds to anyother of the named arteries location, a rescaling of the solution fol-lows the process outlined below. The rescaling factor is establishedfor the steady state and each harmonic component of the solutionseparately by requiring the equality of the calculated data to theinput ones at the desired location. This establishes the consistencyof the solution to the supplied data.

Starting at the capillaries and working up to the aorta using anelectric circuit analogy, the total network steady state resistanceup to that point is calculated using the individual vessel segmentssteady resistance and the network topology information. Of partic-ular interest is the total steady state network resistance Rtot. Then,the overall flux, Qtot, is evaluated based on the inlet steady statevalue for the pressure, Pss and the residual pressure at the capillariesPres, as

Qtot = Pss − Pres

Rtot. (31)

Using that flux information, and the first vessel resistance, wethen evaluate the pressure at the end of the first vessel, i.e. whereit bifurcates at its two daughter vessels. From there on, we proceedthen, using the vessel connectivity information, and their steadystate resistances, to evaluate the steady state fluxes and pressuresat all vessels.

After this is accomplished, the transient portion of the solutionis calculated separately for each harmonic using the recursive for-

mulae described by Eqs. (19) and (20) in Section 2.3, implementedwithin MATLAB. More specifically we evaluate all the daughter-to-parent pressure harmonic coefficient ratios as well as theircorresponding reflection coefficients, using a recursive function

1 hemical Engineering 35 (2011) 1304–1316

ccTaWrc

rttcfaitw(dtsaaptkptavd

4

cs.trtseorsdtvn

5

5

twacpiific

the heart pumps the blood into the aorta during systole. There issome backflow into the heart at the onset of diastole which endswhen the aortic valve closes to stop regurgitation into the heart.Time-averaging the flow of blood for the aorta in this case showed

310 D.A. Johnson et al. / Computers and C

all, starting from the final capillaries (where the reflection coeffi-ients are taken to be zero) and working backwards up to the aorta.here the aortic parameters (pressure and flux) can be adjustedccording to the Fourier coefficients of the input pressure profile.e then proceed and adjust all remaining solution parameters, also

ecursively, but moving forward this time from the aorta to the finalapillaries.

The pressure and flow data are saved into separate tensors cor-esponding to the named arteries and the various generations ofhe arterial trees. This information is further processed (followinghe analytical solution offered by Eqs. (1) and (2)) to generate spe-ific spatio-temporal profiles at discrete time and axial locationsor a fixed �t = T/(10N) (where N is the number of harmonics used)nd �l = l/10 (where l is the specific vessel’s length). For example,n order to describe spatio-temporal variations of vessel proper-ies (such as pressure and flow) for each one of the named arteries,e used the following form: p(j, k, l) and q(j, k, l), where the first

jth) index locates the main artery number, the second (kth) indexescribes the time point, and the last (lth) index refers to the spa-ial axial position down that particular vessel. Similarly, to describepatio-temporal variations of vessel properties (such as pressurend flow) for each one of the secondary arteries belonging to therterial trees indicated with B in Fig. 1, we used the following form:gen(j, k, l, n) and qgen(j, k, l, n), where, as before, the jth index locateshe main artery number where the sub-network comes from, theth index describes the time point, the lth index refers to an axialosition, and the new nth index is the generation number withinhe sub-network. Additional information is also stored in a similarrray fashion (such as the total resistances/impedances up to thatessel), which allows for further and future use of the data (if soesired).

.3. Output

The pressure/flow information that is obtained running the coreode at each artery, time point, and axial point along the ves-el segment as explained above is stored in a tensor format in amat file. These results can therefore be used subsequently manyimes in various post-processing applications without the need toerun the application. Alternatively, the .mat file can be loaded andhen individual 3D plots be generated to present graphically thepatio-temporal variations of the results for selected vessels (anxample is Fig. 7 in Section 5). A separate file has also been devel-ped to look at the pressure spatial profiles as they evolve withespect to time, and plots this information in an .avi movie formato that one can visually track the movement of the systolic andicrotic peaks in the network. With the data readily available inhe .mat files, one can analyze readily the solution at any vessel,isualize it, as well as compare the solutions between vessels, aseeded.

. Results and discussion

.1. The base case

Using the input pressure profile given in Olufsen (2004), andhe parameter values described in Table 1, a “base case scenario”as built with pressure and flow contours developed for the entire

rterial network. The flow profiles were decomposed using a ten-oefficient (harmonics) Fourier analysis to capture the shape of therofile accurately with respect to time. The pressure profile shown

n Fig. 2 is the pressure profile that was digitized for decompos-ng with a Fourier analysis. After obtaining the harmonics for therst ten Fourier coefficients, the Fourier model was executed foromparison between the data points digitized and our model rep-

Fig. 2. The input pressure profile used for the 1D network model from Olufsen(2004), with its corresponding Fourier reconstructed profile.

resenting the data. As seen in Fig. 2, the model representing thepressure profile (given by the solid line) matches fairly closely withthe digitized data (given be the dotted line). The input pressure pro-file could be defined at any of the forty-five vessels in the arterialtree. Usually in this study the input pressure was considered tobe the pressure at the aortic root. Note that although SI units areused everywhere else, we report the pressure results in mmHg, forconsistency with the medical literature (1 mmHg = 133.2 Pa).

In some simulations, a physiologically realistic flow input wascreated by Fourier-synthesis. In other simulations, pressure atone or another vessel in the network was used as an input, andthe pressure and flow at the aortic tree was then determined byback-calculation. By using this methodology, experimental pres-sure measurements at a particular vessel in a test subject, can beapplied to the pertinent vessel can be used to estimate the flowwaveform exiting the left ventricle into the aorta. Fig. 3 shows theexpected flow profile entering the aorta. Fig. 3 shows the calcu-lated flow profile entering the aorta for the aortic pressure profileused in this study. The flow pulse has a large positive and negativeflux into the aorta, which matches well with the flow of blood as

Fig. 3. The corresponding flow profile for the pressure waveform given in Fig. 2 andapplying to the arterial network model.

D.A. Johnson et al. / Computers and Chemica

Fig. 4. (A) Input aortic pressure from Olufsen (2004); reconstruction of samepressure using first ten harmonics; forward and backward pressure waves. Meanpwa

aa

ttlcwtppfataa

ppitprp

ressure (98.2 mmHg) has been added to all traces. (B) Total, forward, and back-ard components of aortic flow. The mean flow has been also added to the forward

nd backward components to facilitate plotting on the same scale as total flow.

bout 103 ml/s (=1.03 × 10−4 m3/s) of blood flowing into the aortand the vascular network.

The forward and backward waves for blood pressure and flowhrough the arterial tree was simulated by taking each of the firsten harmonics of the Fourier series and solving for the respectiveocal impedances in each vessel segment from the aorta down to theapillary level. At the capillary level, the recursive formula for for-ard and backward coefficients was solved, and recursively applied

o the vessels working back up to the aorta level to predict the totalressure and flow waveforms in the entire network. The sum ofressure forward and backward waves and the difference of floworward and backward waves are used to develop the total pressurend flow profiles, as shown in Fig. 4. Our software also provideshe option to use as input the pressure profile at any major artery,lthough in the results presented here we only used aortic pressures input.

In Fig. 4a we also show the decomposition of the transient com-onent of the input pressure profile into its forward and backwardropagating components as calculated from our model. As shown

n Fig. 4a, the early part of the total input pressure pulse is due

o a large rise in the forward propagating pressure wave. This isartially reflected at the first and subsequent bifurcations, givingise to a backward propagating wave. The time delay between theeaks of forward and backward pressure waves is due to the finite

l Engineering 35 (2011) 1304–1316 1311

propagation time along the aorta. The peak for reflected pressureis significantly smaller than the peak for forward pressure. Thesefeatures follow closely with experimental data—see page 113 inLi (2000, 2004). The transient part of the pressure has a nega-tive region which occurs from diastole when the pressure built upwithin the vessel becomes larger than the pressure at the heart asit is refilling with blood.

In Fig. 4b we show the flow which results from the given inputpressure profile. The pressure pulse results first in a large positiveflux, which matches well with the flow of blood as the heart pumpsthe blood into the aorta during systole, followed by a backflow.In actuality this is typically suppressed by the aortic valve. How-ever, since we do not explicitly model the valves, we cannot fullyavoid this backflow. This is indeed the main difference when wecompare the flow results to experimental data—see page 113 in Li(2000, 2004). The mean aortic flow in this case is about 103 ml/s,i.e. 6.1 l/min.

In Fig. 4b we also show the decomposition of the transientpart of the flux into forward and backward waves as calculatedby our model. We see that during systole, as the forward pres-sure is at its peak, the forward wave of blood flow is also at itsmaximum. In contrast, as the heart enters diastole, the forwardpressure wave falls and the reflected pressure wave rises. The neg-ative peak in the reflected flow wave is aligned with the positivepeak in the reflected pressure. It is worth noting that all these fea-tures, the shape and size of the forward and backward flow waves,are also seen in the experimental data reported by Li (2000, 2004) inpage 113.

5.2. Predictions of pressure and flux in selected arteries

If the pressure or flow time-periodic profile is specified at anypoint of the main arterial network, the pressure and flow profilescan be generated at all other points. In particular, we have evalu-ated the blood pressure and flow through the entire arterial treepresented in Fig. 1 using as input an aortic pressure profile givencase discussed by Olufsen (2004), as also discussed in the previ-ous section. This is indicated in Fig. 2 and by the continuous blackline in Fig. 5a. The predicted flow profile at the same input locationis presented by the continuous black line in Fig. 5b. For compar-ison purposes, and as an illustration of the plotting capabilitieswithin our MATLAB code, pressure and flow profiles effortlesslyobtained at several other arteries are also shown in Fig. 5a and b,respectively.

In the pressure results shown in Fig. 5a we see that at moredistant arteries from the heart, the main systolic pressure peakincreases. We also see that the subsequent peak (also termed thedicrotic peak) that appears after the main systolic one begins toshift away from the main systolic peak and forms its own sepa-rate shape in the time-dependent pressure profiles at more distantarteries from the heart. Those predictions are in good qualitativeagreement with available experimental observations, as judged bycomparison to some literature data by (Remington & Wood, 1956)also reproduced here (for convenience) as Fig. 6.

Remington and Wood (1956) performed experimental measure-ments of cardiac pressure pulses as they evolved down the vessels,primarily in the brachial and radial arteries, but a few measure-ments were also performed in the aorta. As shown in Fig. 6 (Fig. 8from the Remington & Wood, 1956, paper), they measured pulsesof pressure along the main aortic trunk, and observed pressurewave amplification (i.e. increase in pulse pressure) in distal vessels(e.g. the abdominal aorta and femoral arteries). This pressure wave

amplification is similar to that presented in Nichols and O’Rourke(1998), Olufsen (2004), Li (2000, 2004), and Nichols, Conti, andWalker (1977). In experimental measurements, Nichols, Avolio,and Kelly (1993) and Nichols and O’Rourke (1998) show that there

1312 D.A. Johnson et al. / Computers and Chemical Engineering 35 (2011) 1304–1316

Fot

ide

eptbp

FdPS

ig. 5. Model predictions for several arteries in the full network. Note: the continu-us black line in (A) represents the input pressure profile. In (B) the right scale referso flow in the aorta, and the left scale to all the rest.

s a peaking effect along the vessel away from the heart, but thisisappears with age, which can be due to changes in vessel walllasticity.

From the model predictions of the pressure profiles in sev-ral of the major arteries shown in Fig. 5a, there appears to be a

eaking effect in the systolic pressure peak in arteries most dis-ant from the heart. This peaking phenomenon has been reportedefore by Duan and Zamir (1992,1995), whose work looked at theressure profiles in canine arteries and who developed methods to

ig. 6. Experimentally measured pressure data of main aortic trunk and points with-rawn along the trunk. [Fig. 8 reprinted from Remington and Wood (1956), J. Appl.hysiol., 9, pp. 433–442, used with permission from the American Physiologicalociety].

Fig. 7. Contour of blood pressure along the main aortic trunk with respect to bothtime and distance from the heart.

form wave propagation representations for one-dimensional mod-els. Similarly, work by Olufsen (2004) also showed the developmentof pressure peaking in distal arteries. Olufsen (2004) refers to themaximum pressure (i.e. systolic peak) increasing further from theheart due to the effects of vessel tapering.

5.3. Spatio-temporal evolution of the pressure along the aortictrunk

As another example of the plotting capabilities within MAT-LAB, the local pressure spatio-temporal variations are presentedin Fig. 7 as a 2D surface in 3D space. Note that the pulse pres-sure increase from the aorta root to 40 cm away is about 10 mmHg,similar to the pressure peaking effects shown in Fig. 6 in the exper-iments of Remington and Wood (1956). Similarly, the time delayin the systolic pressure wave transduction from the aorta to thefemoral arteries is approximately 0.1 s for both the experimentaldata of Remington and Wood (1956) and the model predictions.The good comparison between the experimental data and themodel predictions for the pressure peaking and the time delayfor the systolic peak show that lower frequency elastic and wavetransduction/reflection effects are captured well with the linearapproximations of the model.

Furthermore note that even though there is an increase inthe systolic pressure with distance, the mean pressure, otherwiseknown as the zeroth frequency (Poiseuille associated) pressure,shows a decrease in vessels further from the heart due to the vis-cous losses on the vessel walls. This decrease is expected accordingto Olufsen (2004) and Noordergraaf (1978), due to the effects ofimpedance from the vascular beds and this is also observed in themodel predictions. Although this decrease is not easy to evaluatefrom the raw data shown in Figs. 5 and 6, it is easy to evalu-ate in our model. Some indicative results in the aortic trunk are:98.2133 mmHg at the beginning of the aorta, 98.1835 mmHg in thethoracic aorta, and 98.1065 mmHg in the abdominal aorta, con-tinuing down into the femoral artery, the mean pressure drops to97.7590 mmHg.

The occurrence of a subsequent pressure peak (i.e. the dicroticpeak) in the model-predicted pressure profiles is observed in arter-ies further from heart, and the more distant arteries yield greaterprominence of the secondary peak—see Fig. 7. The dicrotic peak

typically is merged or contained with the systolic peak closer tothe heart, as can be seen in the aortic profile of Figs. 2, 4a and 5a,but as the pressure wave moves further away (as seen in Fig. 7) thedicrotic peak separates from the systolic peak and moves later in the

hemica

ptdsis

F0pr(mpf

D.A. Johnson et al. / Computers and C

rofile. It would appear that from the nature of the wave transduc-ion and reflection within the arterial network, constructive and

estructive interferences of forward and backward moving pres-ure waves lead to the synergy of large pressure peaks and dropsn the pressure profile between the systolic and secondary pres-ure peak formation, but with the work of Olufsen (2004) this is

ig. 8. Flow and pressure profiles with respect to time and distance within the aorta usi.0049 (kg/m s), a terminal capillary pressure of 0 Pa (gauge), a maximum length over drofile (D) flow profile for our viscosity function, a terminal capillary pressure of 2000 Pa (adius of 400 �m. (E) Pressure profile (F) flow profile for constant viscosity 0.0049 (kg/m s)l/D) ratio of 25, and a minimum arteriole radius of 400 �m. (G) Pressure profile (H) flow

aximum length over diameter (l/D) ratio of 25, and a minimum arteriole radius of 3 �mressure of 0 Pa (gauge), a maximum length over diameter (l/D) ratio of 50, and a minimuunction, a terminal capillary pressure of 2000 Pa (gauge), a maximum length over diame

l Engineering 35 (2011) 1304–1316 1313

also due to a pressure dependence on wave propagation, wherethe higher pressure waves move faster than the lower pressure

waves, hence the lower pressure regions begin to drop further andthe peaks get steeper further away from the heart. Similar findingshave been presented in work by Nichols and O’Rourke (1998), Li(2000, 2004), and O’Rourke (1967), where the dicrotic peak shifts

ng different paramerers. (A) Pressure profile (B) flow profile for constant viscosityiameter (l/D) ratio of 25, and a minimum arteriole radius of 400 �m. (C) Pressuregauge), a maximum length over diameter (l/D) ratio of 25, and a minimum arteriole, a terminal capillary pressure of 2000 Pa (gauge), a maximum length over diameter

profile for our viscosity function, a terminal capillary pressure of 2000 Pa (gauge), a. (I) Pressure profile (J) flow profile for our viscosity function, a terminal capillarym arteriole radius of 400 �m. (K) Pressure profile (L) flow profile for our viscosity

ter (l/D) ratio of 50, and a minimum arteriole radius of 3 �m.

1314 D.A. Johnson et al. / Computers and Chemical Engineering 35 (2011) 1304–1316

(Conti

ab(msbpfmwmw

Fig. 8.

way from the systolic peak, while also the lower pressure regionsecome lower, making the peaks steeper. Remington and Wood1956) show fairly little development in the dicrotic wave in their

easurements, though in work by O’Rourke (1967), these peakshow slightly more development in measurements made in a wom-at. The predictions as seen in the model, show that peaks are morerominent due to Olufsen’s (2004) idea of wave propagation beingaster for higher pressure, though in the contour measurements

ade in man they appear to not have formed a separate dicroticave as early in the vessels. The variation for the dicrotic peaksay arise from inertial and/or non-linear mode interaction effects,hich are not incorporated in the model.

nued ).

5.4. Sensitivity analysis of model parameters

A special advantage of a user-friendly MATLAB implementa-tion of the model is the easiness to vary parameter values andperform sensitivity analysis of the results. One such example is dis-cussed here. Several parameters that were of interest for varyingwithin the model were that of oscillatory viscosity (i.e. using thePries model for viscosity as a function of vessel diameter given

by Eq. (29), or a constant value of 0.0049 [Pa s] as was imple-mented in work by Olufsen, 2004), residual capillary resistance (i.e.a final pressure of 15 mmHg–2000 Pa (gauge) or zero as used byOlufsen, 2004), maximum length over diameter ratio for terminal

hemica

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6

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D.A. Johnson et al. / Computers and C

rtery trees (the base value 50 or 25 as used by Olufsen, 2004),nd minimum arterial radius for terminal networks (i.e. 3 �m or00 �m as used by Olufsen, 2004). The implementation of varia-ion of these parameters showed variations of model results in theay the pressure would peak in subsequent vessels, the way flow

nd backflow would peak, and the time delay for pressure waveso be felt down the network. Results from these studies are shownn Fig. 8.

For sensitivity of parameters tested in Fig. 8, the most promi-ent effects upon the model appeared when changing restrictionspon the minimum radius allowed in the terminal artery treesnd the maximum allowed vessel length to diameter (l/D) ratio.y increasing the (l/D) ratio, decreasing the minimum radius forerminal branches, or doing both simultaneously, the pressuref the distant arteries peak to higher values, while the backflowf blood would peak as well. Whether using a constant viscos-ty or the model regressed from previous literature findings, theffects are minimal compared with the (l/D) ratio or the mini-um arterial radius, but it does affect the peak pressure a little.

he terminal arterial pressure also affects the pressure peak, but itlso has a small effect on the peak flow through the network. Byllowing the minimum radius to decrease, a large decrease occurso the peak flow into the network. An additional phenomenoneen is that the (l/D) ratio appears to affect the lag time for pres-ure peaking in the arteries by increasing the lag as the rationcreases.

. Conclusions

A one-dimensional arterial network model has been presentedased on a separate analysis of the steady-state and transientehavior. Detailed geometric information was used for the mainrteries while refined scaling relations were employed for the rest.he steady state analysis took into account non-Newtonian vis-oelastic behavior of the blood rheology, tapering effects for thearger vessels, and the Fahraeus and Fahraeus–Lindqvist effectsor the smaller ones. The transient analysis uses the linearizationpproximation of the equations for pulsatile flow in an elastic tube.his network model has been implemented within the MATLABnvironment thereby allowing one to take advantage of several keyharacteristics of that environment: a user-friendly interface, anfficient mapping of the relevant model information into array datatructures, a natural implementation of recursive solution proce-ures and an effortless interface of output information to differentlots.

The model we present has been developed using only physi-al parameters of the arterial network available in the literature.o parameters were optimized or fitted for best agreement withny specific data. The results for pressure showed good agree-ent with experimental results for pressure pulse evolution along

he vascular network. Furthermore, taking advantage of the MAT-AB environment, a parameter sensitivity study was performedhat showed the terminal capillary radius and aspect ratio �/Do be the most critical to the model predictions parameters. Aurther application of the model is in connection with three dimen-ional computational fluid dynamic (3D CFD) simulators, in order toevelop more physically relevant input/output conditions allowinghe more detailed simulation of the local fluid dynamics of partic-lar sections of interest in the arterial tree (Johnson et al., 2010).he MATLAB implementation was particularly handy for this appli-ation as it allowed an of-line evaluation of the proper boundary

onditions independent to the full 3D simulations. This is a majortep of refining ways to perform multi-scale hybrid simulations toave computational time and energy, while still staying true to theystem being modeled.

l Engineering 35 (2011) 1304–1316 1315

Acknowledgements

DAJ acknowledges the generous support of the National Sci-ence Foundation’s Integrative Graduate Education and ResearchTraineeship (NSF-IGERT) Program at the University of Delawareand the National Aerospace Administration Delaware Space GrantConsortium at the University of Delaware. JPS acknowledges thesupport received by a grant from the Department of the Health,Nutrition and Exercise Sciences of the University of Delaware.

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