final paper aims

7/25/2019 Final Paper Aims

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Mathematical modeling and sensitivity analysis of

the cardiovascular arm arteries with anastomosis

R. Gul and S. Bernhard

Abstract

Cardiovascular disease is one of the major problems in todays medicine

and the number of patients increase worldwide. To find the correct treatment,prior knowledge about function and dysfunction of the cardiovascular systemis required and method need to be developed that identify the disease in anearly stage.

Mathematical modeling is a powerful tool for prediction and investigationof the cardiovascular system. It has been shown, that the Windkessel model,drawing an analogy between electrical circuits and fluid flow, is a simple buteffective method to model the human cardiovascular system.

In this paper we have applied parametric local sensitivity analysis to a lin-ear elastic model of the arm arteries including anastomosis to find and ranksensitive parameters that may be helpful in clinical diagnosis. A computa-tional model for anastomosis (anterior ulnar recurrent) is carried out to studythe effects of some clinically relevant haemodynamic parameters like bloodresistance and terminal resistance on pressure and flow at different locations

of the arm artery. In this context we also discuss the spatio-temporal depen-dency of local sensitivities. The percentage effect on the measurable statevariables pressure and flow, with respect to percentage change in cardiovas-cular input parameters, is quantified using norms. This novel methodologyallows us to verify the qualitative results obtained by sensitivity analysis.

The sensitivities with respect to flow resistance (R), arterial compliance(C) and flow inertia (L), reveal that the flow resistance and diameter of thevessels are most sensitive parameters. Those parameters play a key role indiagnoses of severe stenosis and aneurysms. In contrast, wall thickness andelastic modulus are found to be less sensitive. Results from time dependentsensitivities show that R is sensitive at diastole and systole of the flow andpressure wave respectively. While C and L are most sensitive at systole ofboth flow and pressure waves respectively.Sensitivities w.r.t. Rand Zare obtained from inferior ulnar collateral anasto-

mosis with anterior ulnar recurrent (IUC-AUR), which show that by increasingRor equavalently by decreasing diameter of the anastomosis, the flow withinanastomosis decreases and the pressure difference yields back flow near end-to-side anastomosis.Keywords: computational cardiovascular model, cardiovascular parameters,

sensitivity analysis, anastomosis, Windkessel model.

Fachbereich Mathematik, Freie Universitat Berlin, Germany. [email protected]. of Electrical Engineering and Information Technology, Pforzheim University of Applied

Sciences, Germany. [email protected]

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1 Introduction

With growing interest in the prediction and diagnosis of cardiovascular diseases,different mathematical models have been developed and applied. Windkessel models(electrical analogy to fluid flow) have shown to be an effective approach in modelingthe human cardiovascular system [18]. Westerhof, N. et al. [3] studied the design,construction and evaluation of an electrical analog model. Quarteroni, A. et al. [1]introduced a multiscale approach, where local and systemic models are coupled ata mathematical and numerical level. He also introduced the Windkessel models fordifferent inlet and outlet conditions.

Within the Windkessel model the hemodynamic state variables (pressure (p)and flow (q)), are interrelated to the model parameters like elastic modulus (E),vessel length (l) and its diameter (d), wall thickness (h), the density of blood ()and the network structure. Provided that the model parameters applied correspond

to a certain cardiovascular disease, the Windkessel model is a good way to studyvariations of parameters, which are difficult to modify directly.The basis for robust parameter estimation is on the one hand an optimal ex-

perimental measurement setup and on the other hand the development of modelsthat describe the hemodynamic state variables in a set of relevant parameters thatcan be estimated with high accuracy. The design consists of several logical steps,dealing with questions like:

Which vascular system parameters are most influential on the hemodynamicstate variables pressure and flow?

Which vascular system parameters are insignificant and may be fixed or elim-inated?

Which region in flow and pressure waves are sensitive w.r.t. cardiovascularparameters?

What are the parameters, variables and experimental measurement locationswithin a clinical setting?

Sensitivity analysis is a powerful approach to find sensitive and therefore importantcardiovascular system parameters. The sensitive parameters can be further used todesign a measurement setup and to interpret measurements. in [9] for example, Sato,T. et al. studied the effects of compliance, volume and resistance on cardiac outputusing sensitivity analysis. Yih-Choung Yu et al. [10] used parameter sensitivity toconstruct a simple cardiovascular model. Leguy, C.A.D et al. [15] applied globalsensitivity analysis on the arm arteries and showed that the elastic modulus is mostsensitive parameter, while arterial length is a less sensitive parameter.

In this paper we are interested in the effect of anastomosis on local sensitivities.Anastomoses are the interconnection between vessels.

Beside providing a collateral circulation anastomoses also act as a second routof blood flow when main vessels are blocked by plaque, atherosclerosis or steno-sis, which minimized the demage at tissue level. Another important concept whilestudying anastomosis is valveless flow. William Harvey published a report explain-ing impedance defined flow , which explained a mechanism for valveless flow [11].Later on, Weber [12] stated that heart is not able to pump blood alone but thereare other forces which help in circulation. There are several parameters in cardio-vascular system which control the blood flow or simply create valveless flow like,

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viscous and inertial effects and also elastic properties of two vessels [13]. Details of

the valveless flow mechanism are given in [14]. In this paper, we present a computa-tional model of anastomosis around elbow joint (interior ulnar collateral anastomosiswith anterior ulnar recurrent, IUC-AUR). As luminal diameter or equavalently theblood resistance play an impotant role in blood flow through anastomosis, so, herewe will study the effects of blood resistance and terminal impedance on pressureand flow through anastomosis.The methods developed in this paper, are seen as the first step towards cardiovas-cular system identification from cardiovascular measurements. Within this work wederive a computational cardiovascular model for the arm arteries (with and withoutanastomosis) by using the lumped parameter approach. In a first instance we applylocal sensitivity analysis (without anastomosis) to study the effects of cardiovascularparameters on the hemodynamic state variables. We study time dependent sensi-tivities w.r.t. R, C and L to find sensitive regions in the pressure and flow waves.

Due to the network structure of the cardiovascular system it is helpful to determineand discuss the location dependent sensitivities i.e which locations in arm arteriesare sensitive w.r.t. R, C and L. Finally to quantify and compare our results, weapply the concept of norms.

2 Derivation of the model equations

Under the assumption that the arterial tree is decomposed into short arterial seg-ments of lengthl with a constant circular cross-section and linear elastic wall behav-ior, the following one dimensional flow equations can be derived from the linearizedNavier-Stokes equation, the equation of continuity and the shell-equation for thinwalled, linear elastic tubes [1, 16]

p

x=Rq+ L

q

t, (1)

q

x=

p

Z + C

p

t. (2)

Within these equations the state variables are the flowq and the relative pressure p.The viscosity and inertial forces of the blood flow are described by the viscous flowresistance R and blood inertia L per unit length respectively. The elastic propertiesof the wall are modeled by a compliance C per unit length, while Z is terminalresistance [16]. Integration of the two partial differential eqns. (1) and (2) alongflow axis leads to a system of equations (3) commonly used to describe electricalcircuits (see Figure??). In this type of model each segment of the arterial system isdescribed by a set of two equations that are known as Windkessel equations. Here(pin, qout) and (pin, pout) are the boundary conditions for non-terminal and terminalnodes respectively. To model a mean venous pressure with a value of 15mmHg forterminal nodes the equation system is setup by including an additional terminalresistance Z. The matrix form of the Windkessel eqns. with boundary conditionsis

dX

dt =AX+ B (3)

Where,X = {p1q1,...,pNsqNs}is a state vector, A is a state matrix and Bcontainthe boundary conditions. A complete description of state-space representaion is

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Figure 1: Linear elastic model for fluid flow in nonterminal vessel segments (left)and for terminal vessel segments (right).

given in section 2.3For non-terminal segments (Figure1:left)X = (qin, pout)T

A=

RL

1

L1

C 0

, B(pin, qout) =

pinL

qoutC

For terminal segments (Figure1:right)X= (qin, qout)T

A=

RL

1

L1

C 1

ZC

, B(pin, pout) =

pinL

poutZC

The electrical parameters for ith segment in the arterial tree are related to thephysiological parameters of the fluid and vessel wall by:

Ri= 8l

r4, Li =

l

r2, Ci=

2r2l

Eh . (4)

where

E = Young modulus, l= length of vessel, r= radius of vessel

h = wall thickness, = blood viscosity, = blood density

The vascular network parameters required inA and Bare given in Table 1:

2.1 Network structure and model equations

While modeling arm arteries (with and without anastomosis), we adopt domaindecomposition (DD) approach in which the domain of whole cardiovascular systen isdecomposed into number of subdomains of cardiovascular segments. Each segment

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Nodes E l d h R C Lunits kgm2s2 m m m kgs1m4 m4s2kg1 kgm4

105 102 103 104 106 1011 106

1 4 6.1 7.28 6.2 3.539 7.454 1.5392 4 5.6 6.28 5.7 5.868 4.778 1.8983 4 6.3 5.64 5.5 10.15 4.035 2.6484 4 6.3 5.32 5.3 12.82 3.514 2.9765 4 6.3 5 5.2 16.43 2.974 3.3696 4 4.6 4.72 5 15.10 1.9 2.767 8 7.1 3.48 4.4 78.90 0.667 7.8388 8 7.1 3.24 4.3 105 0.531 9.0429 8 7.1 3 4.2 142.9 0.448 10.5510 8 2.2 2.84 4.1 55.11 0.1207 3.64711 8 6.7 4.3 4.9 31.94 1.067 4.844

12 16 7.9 1.82 2.8 1173 0.0834 31.8813 8 6.7 4.06 4.7 40.19 0.9366 5.43414 8 6.7 3.48 4.6 50.22 0.80 6.07515 8 3.7 3.66 4.5 33.60 0.3958 3.693

Table 1: Numerical values of parameters for each node of the arm arteries(shownin Fig. 2). The value of terminal resistance (Z) on three terminal nodes is 3.24 109 kgs1m4, = 0.004kgs1m1 and = 1050 kgm3 [4].

of the arm arteries in a network structure as given in Figure 2 is represented by anelectrical circuit as shown in Figure (1, top).To built a cardiovascular network, relations between the arterial segments need to

be defined like at bifurcation flows diverge i.e. q1

= q2

+q3

and at anastomosis flowsmerge i.e. q3 = q1+ q2. Pressure at bifurcation and at anastomosis remain same(see Figure 1, bottom).

2.1.1 Model equations without anastomosis

In analogy to Kirchhoffs current and voltage law (Ohms law of hydrodynamics),the arterial structure given in Figure 2, leads to the following system of coupledordinary differential equations for pressure and flow.Flow equations:

q1 = pin p1 R1q1

L1, q2=

p1 p2 R2q2L2

, q3=p2 p3 R3q3

L3

q4 =

p3 p4 R4q4L4 , q

5 =

p4 p5 R5q5L5 , q

6=

p5 p6 R6q6L6

q7 = p6 p7 R7q7

L7, q8 =

p7 p8 R8q8L8

, q9= p8 p9 R9q9

L9

q10 = p9 p10 R10q10

L10, q11=

p6 p11 R11q11L11

, q12=p11 p12 R12q12

L12

q13 = p11 p13 R13q13

L13, q14=

p13 p14 R14q14L14

, q15= p14 p15 R15q15

L15(5)

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Pressure equations:

p1 = q1 q2

C1, p2=

q2 q3C2

, p3= q3 q4

C3, p4=

q4 q5C4

p5 = q5 q6

C5, p6=

q6 q11 q7C6

, p7=q7 q8

C7, p8=

q8 q9C8

p9 = q9 q10

C9, p10=

q10 (p10 pout)/Z1C10

, p11=q11 q12 q13

C11

p12 = q12 (p12 pout)/Z2

C12, p13=

q13 q14C13

, p14= q14 q15

C14,

p15 = q15 (p15 pout)/Z3

C15(6)

2.1.2 Model equations with anastomosis

It was shown we have the AUR anastomosis in the model of arm artery (see Figure2), the flow will split at nodes 3 and 6, and will merge at node 11. the inlet pressureat node 11 is same as nodes 6 and 33. As we add three additional nodes so we havenow 5 new equations for pressure and flow. Additional equations for pressure andflow of the ulnar recurrent artery are,Anastomosis flow equations:

q3 = p2 p3 R3q3

L3, q31 =

p3 p31 R31q31L31

q32 =p31 p32 R32q32

L32

q33

= p32 p6 R33q33

L33(7)

Anastomosis pressure equations:

p3 =q3 q4 q31

C3, p31 =

q31 q32C31

, p32 =q32 q33

C32, p33 =

q33+q6 q11C6+C33

(8)

p33 andp6 are same according to the conservation of momentum.

2.2 Diverging and merging flows at junctions

Blood flow at junctions plays an important role in normal and pathological con-ditions of the cardiovascular system. In this section we briefly discuss pressureand flow modeling at junctions for both diverging (at bifurcations) and merging

blood flows. In the arterial system merging flow conditions appear in the contextof anastomosis.

2.2.1 Diverging blood flow

In the above model structure diverging flows occur at the bifurcation of the vesselsat nodes 3, 6 and 11 (see Figure 2). Flows at junction mainly depend on angle andradius of the daughter vessels. In this paper we limit our study to symmetric andasymmetric bifurcations with respect to radius of the daughter vessels. For diverg-ing flows the mass conservation equation at node 3 is,q3= q31+q4. For linearized

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Figure 2: Simplified anatomy of the arm arteries (left) and model geometry ofbrachial, anterior ulnar recurrent, ulnar and radial arteries, with number of nodesNs = 15 (without anastomosis), Nas = 18 (with anastomosis) and number of ter-minal nodes Nt = 3 (right).

system (eqns. 9, 10) the total pressure through bifurcation remain unchanged, i.e.the output pressure at node 3 is the input pressure for both nodes 31 and 4.

2.2.2 Merging blood flow

To model the merging flows at junctions has a great importance to understand theeffect of anastomosis in bypass in the cardiovascular system. These type of flowsoccur also in vascular grafting and arteriovenous fistula (AVF). According to theconservation of mass, the flow at node 11 is, q33+ q6= q11 and according to the lawof conservation of momentum, the total pressure remains continuous at nodes 11.

2.3 State-space representation

The state-space representation is a compact way to model a physical system as a setof input, output and state variables related by first order differential equations [17].In state-space form, we have a system of two equations: an equation for determiningstatextof the system (state equation), and another equation to describe the outputyt of the system (observation equation). The matrix form can be written as

xt = Axt1+But, (9)

yt = Cxt+Dut. (10)

Here xtis the state vector of the system, ut the input vector andyt the observation

vector. The dynamics of the system is described by the state dynamics matrixA M(nn). The input matrixB M(ni) specifies the time dependency of thein- and outflow boundary values and the observation matrixC M(mn) definesthe observation locations within the state-space system, i.e. the nodal locationin the network. Here m denotes the number of observations. Finally the inputto observation matrix D M(m i) adds the influence of the input vectors tothe observation vectors. Besides the computational advantage the state-space formallows the integration of experimental measurements (observations) into the modelbuilding process. This step is essential for the adjecent model parameter estimationfrom experimental measurements, that are planned in a future study.

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The state vector xt contains the flow and pressure functions at all network

locations, whereas the output vector yt contains the flow and pressure at selectednodes i. For a m = 4 dimensional observation vector, the output vector is e.g.y(t) = (q5(t), p5(t), q6(t), p6(t))

T where y Rm at nodes 5 and 6. The state-spacesystem for the arm artery given in Figure 2, using eqn. (5) and eqn. (6) is definedby

Aij =

R i+12

L i+12

i= 1, 3, 5,...29, j= i

1

Ci2

i= 2, 4, 6,...30, j= i 1

1

L i+12

i= 1, 3, 5,...29, j= i+ 1

1

L i+12

i= 3, 5, 7,...29, j= i 1 and i= 21, 25

also for i= 21, j= 12 and i= 25, j= 221

Ci2

i= 2, 4, 6,...30, j= i+ 1 and i= 20, 24

also for i= 12, j= 21 and i= 22, j= 251

ZkCi2

k= 1, 2, 3, i= 20, 24, 30, j = i

0 otherwise

Bij =

1

Lii= j = 1

1

Zj1Ci2

i= 20, j= 2

1

Zj1Ci2

i= 24, j= 3

1

Zj1Ci2 i= 30, j= 4

0 otherwise

, Cij =

1 i= 1, 2, 3, 4, j=i+ 8

0 otherwise

and Dij = 0.For AUR anastomosis we expand our state-space model by using eqns. (7, 8). Thesystem is solved using the MATLAB built in solvers ODE 45 and lsim.

3 Methods of local sensitivity analysis

Cardiovascular parameters which have influence on output variables like, pressureand flow are identified by sensitivity analysis. To understand the interdependence ofthe state variables and the parameters of the cardiovascular model, it is imperative

to quantify the sensitivity of the state variables with respect to model parameters. Inthis work we study the local sensitivities at different nodes in the vascular network.Mathematically the sensitivity coefficient can be calculated as,

Sij = yij

(11)

Where yi is i-th model output and j is the j -th parameter.

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3.1 Sensitivity by finite difference method

In local sensitivity analysis, parameters are varied segmentwise by some portionaround a fixed value and the effects of individual perturbations on the observationsare studied [18]. Using differential calculus the sensitivity coefficients are

Si = yi

= lim

0

yi(+ ) yi()

, (12)

where yi is the i-th model output, is the model input parameter and is thechange in model parameters. There are various methods to compute the sensitivitycoefficients in eqn. (12), within this work we use the method offinite difference:

Si = yi

yi(+ ) yi()

(13)

Equation (13) produces a set of two sensitivity time series Si(t) (one for pressureand one for flow) per parameter and per network node (see figure 4).Here we discuss three different scenarios of sensitivity analysis, (i) sensitivity anal-ysis with respect to structural parameters ELdh, (ii) electrical parameters RCLand (iii) location dependent sensitivities. In this context we also discuss the timedependence of the sensitivities and compare the results of the two scenarios withand without anastomosisThe sensitivity results are finally compared to the 2-norm of the distance vector ofthe state variables of two time series.

3.1.1 Sensitivity with respect to E , l , d and h

To study the effects ofEldh on pressure and flow, we first solve our model with

both 10% variation ofEldhfrom its nominal values (see table 1). The sensitivitiesare computed using eqn. (13). In this case is eitherEld or h. The sensitivities forpressure and flow w.r.t Eldhwere calculated for node 7 (radial artery) and plottedin figure 3.

3.1.2 Time dependent sensitivity

Sensitivity time series with respect to R, C and L, can be used to find sensitiveregions in pressure and flow waves at each location of the arm arteries. For thiswe first solve eqn. (9) and eqn. (10) with Matlab built in solvers, then calculatesensitivity time series for both pressure and flow at nodes 5 and 7 by using eqn.(13). Results are presented in figure 4

3.1.3 Sensitivity with respect to R, C and LTo find the sensitivity of the electrical parameters R, C and L on cardiovascularpressure and flow, we first solve eqn. (9) and eqn. (10) numerically using theCVODES solver, which is a part of SUNDIALS software suit [19, 20] and thencalculate the sensitivities using eqn. (13). Results are summarized in figure 5

3.1.4 Network structure and sensitivity

To study the influence of the vascular network structure, we use a non-physiologicalnetwork structure with identical node parametrization, i.e. the parameters Ri, Ci

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and Li are identical for each node. This allows us to analyze the influence of the

network structure onto sensitivity values at different node locations.

3.1.5 Mean sensitivity over time

For section 3.1.3 and section 3.1.4, we calculate mean of sensitivity time series forall arm arteries nodes. Therefore we compute the nodal sensitivity time seriesSi forall nodal parameters as described in previous section. For each node, n, we obtaintwo sensitivity time series (one for the pressure and one for the flow) per parametervariation. The total number of possible parameter variations is 3n2, so we end upwith a set of 6n2 time series. To reduce the complexity, we average out the timedependency by computing the mean of the absolute value of every time series. The6n2 real values are used for further analysis in section 4. They are displayed into 6matrices that characterize the sensitivity of pressure and flow (sensitivity nodes) inthe network structure based on changes in the electrical parameters RC L(selectednodes). Each cell in Figure 5 and 6, represents the mean absolute value of time series(Si). Due to the fact that we use the pressure as an input and output boundarycondition, the change in pressure with any parameter at all inlet and terminal nodeswill be zero.

3.1.6 Sensitivity within anastomosis

Blood flow in AUR anastomosis (collateral circulation) depends on size and mainlyon diameter of anastomosis, smaller the diameter lesser the flow in anastomosis andvice versa. In this paper we limit our study to end-to-side AUR anastomosis andshow the sensitivity of anastomosis structure on pressure and flow by changing bloodresistance Ra and Rb and terminal resistance (Z) [1114]. Here we consider Ra =R31+ R32+ R33 andRb= R4+ R5+ R6, where Ra and Rb are the total resistancethrough AUR anastomosis and through its parallel brachial artery respectively.Further more we discuss 5 different scenarios (a) when Ra Rb, (b) Ra Rb, (c)Ra Rb, (d) Z is small and (e) Z is large. The sensitivity is computed as earlierstated.

3.2 Sensitivities by using norms

To obtain a measure to validate the sensitivities, we calculated the mean Euclideandistances of the observations made in a model with different parameter sets 1, 2.Here 1 is the nominal parameter set and 2 is10% change in 1.

1, 2:=meantT

yi(2, t) yi(1, t) 2yi(1, t)2

i= 1, 2, 3...2Ns,

The results are compared to sensitivity measures in table 2 and table 3.

4 Results and discussion

In this section we discuss the results obtained from section 3.

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.81

1.5

2

2.5

3

3.5

4

4.5

5x 10

6 E sensitivity on flow at node 7

time [s]

flow[ml/s]

10%

+10%

Nominal values

Norm sensitivity=0.36

Phase andamplitude shift

A1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80.8

0.9

1

1.1

1.2

1.3

1.4

1.5x 10

4 E sensitivity on pressure at node 7

time [s]

pressure[Pa]

10%

+10%

Nominal values



A2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.81

1.5

2

2.5

3

3.5

4x 10

6 l sensitivity on flow at node 7

time [s]

flow[ml/s]

10%

+10%

Nominal values


Phase and

amplitude shift

A3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80.8

0.9

1

1.1

1.2

1.3

1.4

1.5x 10

4 l sensitivity on pressure at node 7

time [s]

pressure[Pa]

10%

+10%

Nominal values



A4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.81

1.5

2

2.5

3

3.5

4

4.5

5x 10

6 d sensitivity on flow at node 7

time [s]

flow[ml/s]

10%

+10%

Nominal values


A5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6x 10

4 d sensitivity on pressure at node 7

time [s]

pressure[Pa]

10%

+10%

Nominal values


A6

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.81

1.5

2

2.5

3

3.5

4

4.5

5x 10

6 h sensitivity on flow at node 7

time [s]

flow[ml/s]

10%

+10%

Nominal values



A7

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80.8

0.9

1

1.1

1.2

1.3

1.4

1.5x 10

4 h sensitivity on pressure at node 7

time [s]

pressure[Pa]

10%

+10%

Nominal values



A8

Figure 3: Pressure and flow for a 10% change inE,l, d and h in the radial arteryat node 7. The results reveal that dand l are sensitive parameters while Eand hare less sensitive, for the reason of linear appearence Eandh have same effects oncardiovascular pressure and flow.

4.1 Sensitivities with respect to E, l, d and h

The sensitivity for pressure and flow are obtained by a variation of the cardiovascularparametersE , l , d andhof arm arteries by 10%. From figure 3 it is directly evident,that diameter and length are most sensitive parameters, while the elastic modulusand wall thickness are comparatively less sensitive parameters.

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4.2 Time dependent sensitivity

Time dependent sensitivity is very important specially when we want to estimatecardiovascular parameter from pressure and flow waves. It further guide us to findoptimal positions or regions (systole and diastole), from where we can get moreinformation for parameters. In this section we study +10% change in R, C and Lin all locations. Time series w.r.t R,Cand L were calculated at nodes 5 and 7, byusing eqn. (13). The results are summarized as,(a). R sensitivity time series:According to the hydrodynamics form of Ohms law, the flow resistance R can becalculated from the ratio of pressure gradient and flow, i.e. R= p

q . It is evident

from the figure 4 (B1, B2)) that blood resistance R has an inverse relationship toblood flow q and has linear relationship with the pressure gradient. When arteriesconstrict the resistance increases and when arteries dilate the resistance will de-creases. Due to this fact resistance is most sensitive in the diastolic region of theflow wave, while it is less sensitive in the the systolic region. On the other handresistance is sensitive at diastolic region of pressure waves, which shows clear agree-ment with Ohms law of hydrodynamics.(b). C sensitivity time series:Compliance C, is the change in arterial blood volume V , due to the change inarterial blood pressure p, i.e. C= V

p. It is thus the slope of the pressure-volume

curve, so it depends on the pressure level at which the compliance or elastance iscalculated. The results show, that the compliance has a larger effect on diastolicparts of both pressure and flow waves (see Fig. 4 (C1, C2)).(c). L sensitivity time series:Blood inertia L relates pressure drop with flow rate i.e. L = p

q . Blood inertia

plays a role in acceleration (in systole) and deceleration (in diastole) the blood flow

in cardiovascular vessels. From Figure 4 (D1, D2) it is clearly shown that L is mostsensitive in diastolic parts of pressure and flow waves as compared to systolic parts.

4.3 Sensitivities with respect to R, C and L

The sensitivities w.r.t. R, C and L were calculated by finite difference approacheqn. (13), using the SUNDIALS software. The sensitivity pattern obtained byvariation of the viscous flow resistance R in any segment of brachial artery (seeFigure 5 (top)) indicates a strong local (within brachial artery) influence on flowand has significant global influence on all following nodes of the brachial, ulnar andradial arteries. In contrast, changing R in the parallel association of the ulnar andradial arteries have negligible local and global effects, because in parallel arteriesthe total flow resistance is given by the fraction 1

Rtotal= 1

Rulnar+ 1

Rradial, i.e. due

to the increment in total diameter the flow resistance reduces. Physically a changeofR in one branch redirects the flow into the other branch while the overall flowis maintained. The sensitivity of flow resistance in parallel branches is thus smallerthan in series connections.

In contrast to the flow resistance, the sensitivity of arterial compliance C inthe brachial part has small influence (locally and globally) on pressure and flow(see Figure 5 (middle)). This can be explained by the fact, that in series segmentsthe total compliance is given by the sum of all segmental compliances in seriesCtotal =

6

i=1Ci. The total compliance is larger than the individual compliances,

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0 100 200 300 400 500 600 700 8002

1.5

1

0.5

0

0.5

1x 10

15 sens v y on ow a no es an

Time ms

Node 5

Node 7B

1

R Sensitivity

0 100 200 300 400 500 600 700 8002

1

0

1

2

3

4x 10

7 sens v y on pressure a no es an

Time [ms]

Node 5

Node 7

B2

R sensitivity

0 100 200 300 400 500 600 700 8002

1.5

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Figure 4: Flow and pressure waves (A1, A2) and sensitivity time series with respectto R, C and L (B1, B2, C1, C2, D1, D2) at nodes 5 and 7. Results shows that R issensitive at diastolic and systolic part of flow and pressure wave respectively. WhileC and L are most sensitive at systolic part as compared to the diastolic part ofpressure and flow waves at respective nodes.

thus a change ofCin any node has a small effect on pressure and flow in the wholearm artery. In contrast a variation of the arterial complianceC in the ulnar andradial arteries have a large (local) effect on pressure and large (global) effect on flow

especially in the brachial artery.From eqn. (4) it is obvious that viscous resistance and blood inertia are inverselyrelated to r4 andr2 respectively. Which means in large arteries blood inertia playsan important role, while in small arteries viscous resistance is more important. Avariation of blood inertia in the first node of the brachial artery has large influence onpressure and flow of all following nodes of the arm arteries (see Figure 5 (bottom)).However, we observe only minor local (within brachial artery) effects on flow whenwe change L in each segment of the brachial artery. Further, due to the fact thatthe total inductance 1

Ltotal= 1

Lulnar+ 1

Lradialreduces at the furcation, the flow and

pressure in the ulnar and radial arteries are less sensitive with respect to L.

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Figure 5: Effects of viscous flow resistance R (top), vessel compliance C (middle)and blood inertia L (bottom) on pressure and flow in the arm arteries. Fromthe results changing flow resistance, R and blood inertia, L in brachial artery hasstrong local and global effects on flow and pressure waves respectivily. While flowin brachial artery is sensitive w.r.t. changing vessel compliance,C in all nodes ofthe arm aretry.

4.4 Network structure and sensitivity

In order to know important locations in arm arteries (without anastomosis), weconsider identical node parameter values in whole arm artery. Results show thatRis sensitive for flow in brachial artery while other locations in radial and ulnar areless sensitive. On the other hand changing R in brachial artery has influence on

pressure for all following nodes. Cand L sensitivities also have some local impactin brachial artery, while distal nodes are less sensitive (see Figure 6). Locationdependent sensotivity along with computational model of anastomosis, indicates afeasible location for creating anastomosis (vascular grafting) and AVF, e.g. one cancreate anastomosis from the sensitive parts of the arm arteries.

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R mean absolute sensitivity for flow

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Figure 6: Effects of viscous flow resistance R (top), vessel compliance C (middle)and blood inertiaL (bottom) on pressure and flow in the arm artery, with identicalnodes. From the results R is sensitive parameter for flow in brachial artery, whilechangingR in brachial have both local and global effect on pressure. CandL alsohave some local sensitivities in brachial part of arm artery.

4.5 Sensitivity within anastomosis

In this section we consider 5 different scenarios as,(a). Ra Rb:To study the blood flow through AUR anastomosis, we first consider identical valuesofRCLof nodes 31, 32, 33 to nodes 4, 5 and 6 respectively. As it is mentioned ear-lier, the diameter of vessels plays an important role in pressure and flow distribution

of cardiovascular system, so we limit our study in changing the blood viscosity inAUR anastomosis and its parallel brachial artery, i.e. Ra Rb, where the valuesofRa andRb are given in section (3.1.6). the Results show that pressure and flowdistribution in AUR and its parallel brachial artery are the same (see Figure 7 A,B).(b). Ra Rb:Here we consider small diameter of AUR anastomosis as compare to its counterpartbrachial artery (ideal case). The values ofRa =

Rb50

. Due to decrease in diameteror equivalently, decrease in total blood resistance through AUR anastomosis, the

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flow decreases. Results show some back flow at node 33 relate to the pressure drop

resulting from the increased blood resistance through the AUR anastomosis (seeFigure 7 C, D).(c). Ra Rb:A similar behavior of pressure and flow distribution can be seen by decreasing thediameter of nodes (4 , 5 and 6) of brachial artery. Figure (7 E, F) shows back flowat node 6 due to the pressure drop by setting Rb= 50Ra.(d). Z is small:Physical activities lead to a reduction in terminal resistance (Z), which increasesthe blood flow and mean cardiovascular pressure. As a result the cardiac outputwill increases. These are the temporary changes which appear only when we dosome physical exercise. Another artificial reason for low terminal resistance is theimplantation of arteriovenous fistula (AVF), which is abnormal connection betweena peripheral artery and vein. The decrement in terminal resistance increases blood

pressure and flow and finally increases cardiac output. In this study we considersmall terminal resistance, Zs =

Z10

(see Figure 7 G, H).(e). Z is large:In this section we consider large terminal resistance (Z) at terminal node 15 ofthe ulnar artery i.e. ZL = 10Z, which appears when we have no physical activity.Figure (7 I, J) shows that if the person is at rest, we have large value ofZ, thenflow will reduce (Ohms law of hydrodynamics). Results are obtained at differentlocations of the arm artery, which reveal that by increasing Zthe flow will reduceand pressure will increase near the terminal.

4.6 Sensitivities by using norms

Finally we compare the results obtained by sensitivity analysis with those obtainedby using norms. We found that the diameter and length of vessel are most influentialparameters and that the norm computed for the wall thickness and elastic modulushas identical values (see table 2 and 3).

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band large Z at Node 15

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ressure

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Figure 7: Effects of viscous flow resistance Rand terminal resistance Zon pressureand flow at different locations of the arm arteries (with anastomosis). From theresults, it is seen that increasing flow resistance decreases the flow and producessome back flow and flow will increase by decreasing terminal resistance.

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N0 N1 N2 N3 N4 N5 N6 N7 N8 N9 N10 N11 N12 N13 N14q0 q1 q2 q3 q4 q5 q6 q7 q8 q9 q10 q11 q12 q13 q14

E 0.45 0.44 0.42 0.36 0.34 0.27 0.24 0.36 0.39 0.19 0.18 0.18 0.11 0.24 0.14 l 0.54 0.53 0.47 0.43 0.39 0.32 0.31 0.95 0.86 0.48 0.46 0.25 0.13 0.34 0.19 d 1.96 1.76 1.75 1.55 1.43 1.16 1.06 1.34 1.19 0.80 0.78 0.87 0.45 0.10 0.63 h 0.45 0.44 0.42 0.36 0.34 0.27 0.24 0.36 0.39 0.19 0.18 0.18 0.11 0.24 0.14

Table 2: 10% change in E, l, d and h at node 7 of the arm artery (see Figure 2, without anastomosis) and percentage change in flow at each node.

N0 N1 N2 N3 N4 N5 N6 N7 N8 N9 N10 N11 N12 N13 N14 Np0 p1 p2 p3 p4 p5 p6 p7 p8 p9 p10 p11 p12 p13 p14 p

E 0 0.02 0.03 0.05 0.06 0.08 0.08 0.13 0.13 0.14 0 0.09 0 0.09 0.10 0l 0 0.03 0.05 0.06 0.08 0.09 0.10 0.32 0.32 0.37 0 0.11 0 0.11 0.12 0d 0 0.13 0.15 0.21 0.27 0.33 0.36 0.58 0.58 0.63 0 0.38 0 0.40 0.44 0h 0 0.02 0.03 0.05 0.06 0.08 0.08 0.13 0.13 0.14 0 0.09 0 0.09 0.10 0

Table 3: 10% change in E, l, d and h at node 7 of the arm artery (see Figure 2, without anastomosis ) and percentage change in pressure at each node. Change in pressure with respect to any parameter is zero, because boundary condition.

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5 Conclusion

In this work we have applied different methods of sensitivity analysis to a lumpedparameter Windkessel model of the arm arteries. The results indicate a strong de-pendence of the pressure and flow state variables onto a variation in vessel diameterand length. According to the elastic properties and the thickness of the arterialwall, a much lower sensitivity was found. Alternatively we can say that blood vis-cosity and compliance of vessel are sensitive parameters while blood inertia is lesssensitive parameter.

Results from time dependent sensitivity show that R is sensitive at diastole andsystole of the flow and pressure waves, alsoCandL are sensitive at systolic part ofthe flow and pressure waves, While in location dependent sensitivities, R is sensitivefor flow in the brachial artery and changing R in brachial artery have both localand global influences on the pressure.

Further we have presented a computational mathematical model for anastomosiswhich can predict pressure and flow through anastomosis. Results reveal that byincreasing flow resistance or equavalently decreasing diameter of the anastomosiswill decrease the flow and produces some back flow near end-to-side anastomosis,while flow near terminal nodes will increase by decreasing terminal resistance andvice versa.

Finally, we have used the concept of norms to quantify the results and to comparethe variation in state variables according to parameter changes. We found a goodagreement to the results obtained by sensitivity analysis.

6 Future work

The methods applied, give satisfactory results if the cardiovascular parameters areindependent, in the real scenarios however, they are often interdependent like e.g.the observation of a high correlation between the extension of the elastic walls andthe tangential tension caused by transmural pressure. To study these type of effectsin a more general way, we plan to apply global sensitivity analysis to a closed loopand open loop cardiovascular system models, which deal with variations in manyparameters at a time.

7 Acknowledgments

The author would like to thank the HEC/DAAD for their financial support also toRudolf Huttary for fruitful discussions and encouragement.

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