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MSc PhysicsPhysics of Life and Health

Master Thesis

Modeling transport of contrast agent through avascular network

Finding the correlation between blood flow and appearanceof contrast in a region of interest

by

Duncan de Graaf10061851

January 201630 EC

Research carried out from February 2015 to January 2016

Supervisors ExaminersProf. dr. E.T. van Bavel Prof. dr. E.T. van BavelDr. H.A. Marquering Dr. ir. G.J. Streekstra

Biomedical Engineering and Physics / AMC

Abstract

Critical limb ischemia (CLI) is the final stage in peripheral arterial disease(PAD) and might result in amputation of the foot. Perfusion angiography isdeveloped to image the transport of contrast agent in the vessels of the lowerextremities. The purpose of this technique is to assess the severity of thedisease. Perfusion angiography creates a time-density curve which reflectsthe appearance of contrast in a region of interest (ROI). In this researcha computer model is built of the vascular network to simulate the flow ofblood and transport of contrast agent. The aim of this study is to find acorrelation between the appearance of contrast and the total blood flow ina ROI.

The full model is written in Matlab. First a structure of the vascularsystem is created, with a predefined large artery network and a randomlygenerated network of smaller arteries. The transport of contrast agent issimulated by processes of advection and diffusion. After that the behaviorof different systems with atherosclerosis or different permeability coefficients,which reflects the leakage of the microcirculation, is analyzed.

The systems analyzed show ambiguous results for the correlation be-tween flow and appearance of contrast agent. If the main artery is narrowed,one can see a linear relation between the inverse of the flow and the time topeak and arrival time. If the ROI is also reachable via another route, whichis simulated by narrowing the artery of a side branch of the main artery, theblood flow remains constant while the time-density curve worsens. Increas-ing the permeability coefficient leads to a higher peak density of contrast inthe ROI, while the blood flow does not change.

The results show that the relation between total blood flow and appear-ance of contrast in a ROI is not so trivial. Not only atherosclerosis hasinfluence on the time-density curves, but also the permeability coefficientand the presence of collateral vessels.

Contents

1 Introduction 2

2 Theory 52.1 Hemodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.1 Calculating flow . . . . . . . . . . . . . . . . . . . . . 62.2 Contrast agent transport . . . . . . . . . . . . . . . . . . . . . 8

2.2.1 Advection-Diffusion equation . . . . . . . . . . . . . . 82.2.2 Advection . . . . . . . . . . . . . . . . . . . . . . . . . 112.2.3 Taylor dispersion . . . . . . . . . . . . . . . . . . . . . 122.2.4 Contrast agent diffusion into tissue . . . . . . . . . . . 12

2.3 Computer generation of vascular trees . . . . . . . . . . . . . 14

3 Method 153.1 Structure of vascular system . . . . . . . . . . . . . . . . . . . 15

3.1.1 Predefined large artery network . . . . . . . . . . . . . 153.1.2 Generation of small arteries . . . . . . . . . . . . . . . 163.1.3 Completion of the vascular network . . . . . . . . . . 18

3.2 Simulation of contrast agent transport . . . . . . . . . . . . . 183.3 Imaging of contrast agent transport . . . . . . . . . . . . . . 213.4 Analyzing behavior of the system . . . . . . . . . . . . . . . . 23

4 Results 25

5 Discussion 32

1

1 Introduction

Critical limb ischemia (CLI) due to peripheral arterial disease (PAD) is aserious condition. PAD is atherosclerosis or narrowing of the arteries inthe lower extremities. The prevalence of PAD is approximately 12% inthe US. Risk factors include hypertension, dyslipidemia (high low-density-lipoproteıne cholesterol, low high-density-lipoproteıne cholesterol), diabetesand cigarette smoking. Male gender, obesity and a family history of atheroscle-rosis are also potential risk factors. [1, 2] The spectrum of PAD is dividedin asymptomatic, claudication and CLI. Claudication is a manifestation ofexercise-induced reversible ischemia, similar to angina pectoris (chest pain).Patients with severe PAD may experience pain during rest, with or with-out ischemic tissue loss, which is a characterization of CLI. [3] Howeverwhat exactly is CLI? The Second European Consensus Document (ECD)on Chronic Critical Leg Ischemia defines CLI as ischemic rest pain for morethan two weeks, and/or tissue necrosis (ulceration or gangrene) and anklesystolic blood pressure less than 50 mmHg and/or toe systolic blood pres-sure less than 30 mmHg. [4] However, there are some problems with thisdefinition. First, these are not parameters for the severity of the disease.Second, it cannot be used to determine which patient needs treatment andwhich not. [5] Third, in patients who are diabetic, pressure measurementsare often unreliable. [6] Lastly, perfusion of the foot is not only dependenton the status of the macrocirculation (inflow of blood), but also on thestatus of the microcirculation. The micrcocirculation is important becausethis is where oxygen diffuses into the tissue. In conclusion, it appears thatthe current ECD definition gives an indication of CLI, but does not have apredictive value in CLI. Actually the most important parameter in CLI isoxygen, because CLI is basically shortage of oxygen in the foot. The lack ofoxygen will eventually lead to tissue necrosis. According to the Fick prin-ciple, oxygen extraction ratio of a whole organ, such as the foot, dependsprimarily on the ratio between blood flow F (units: volume per unit time)and metabolic demand QO2 (units: moles per unit time):1

[O2]a − [O2]v =QO2

F(1)

The term on the left is the arteriovenous (a-v) difference (units: molesper unit volume), which is the difference of oxygen concentration in thearterial inflow and venous outflow of that organ. The extraction ratio is justthe a-v difference normalized to [O2]a and is the relative amount of oxygenthat the tissue removes from the capillaries. Based on equation 1 one couldargue that, with oxygen consumption being constant, oxygen concentrationin the tissue is only dependent on blood flow (F) through the capillaries.

1W.F. Boron, E.L. Boulpaep, Medical Physiology, 2nd edition, page 485

2

Based on medical history and physical examination, additional diag-nostics, such as Doppler ultrasonography and angiography, may be carriedout. The latter provides details of the location and extent of arterial steno-sis or occlusion. Moreover, angiography is a prerequisite for percutaneoustransluminal angioplasty (PTA), which is currently the primary nonsurgicaltreatment for patients with CLI. [1, 2, 7]

Over the last years, a new technique is developed for imaging of thetransport of contrast agent in vessels of the lower extremities. This tech-nique, perfusion angiography, is described in the paper of Jens et al. [7]Because the first-line treatment for most patients with CLI is PTA, it offersthe opportunity for injection of a contrast agent for direct imaging of perfu-sion of the foot. Perfusion angiography creates a time-density curve duringthe first pass of a non-ionic contrast agent through the foot. This could intheory provide information about the status of the microcirculation. [7]

(a) Perfusion angiography ofthe foot

(b) Time density curve

Figure 1: Occlusion of the anterior tibial artery due to trauma2

Figure 1a is a typical foot perfusion image of the blood vessels in apatient with CLI. This figure is calculated from the image made with digitalsubtraction angiography (DSA). The image is pseudo-coloured: the colorsindicate the time of arrival of a contrast agent that was injected into themain artery in the upper leg. To assess the perfusion of the foot, a wideregion of interest (ROI) is drawn manually. After the ROI is determined, anoverall time-density curve, arrival time, and time to peak can be determinedby the perfusion software for this region.

2Reproduced from the article of Jens et al. [7]

3

Figure 1b is an example of a time density curve, with on the x-axis thetime in seconds and on the y-axis the total density of the contrast agent,which is a reflection of the amount of contrast in the ROI. The arrival timeindicates the time between injection of the contrast agent and arrival in theROI. Time to peak shows the time between injection and the maximumconcentration of the contrast agent. By comparing the time density curvesbefore and after treatment, some conclusions can be made about the differ-ence in perfusion of the foot. In the paper of Jens et al, a relation was foundbetween a successful PTA and a change in the time-density curve; reducedarrival time and increased peak density [7].

It is not known what the correlation is between the appearance of con-trast in the ROI and the total blood flow in that region. Because the severityof CLI is for most part dependent on the total blood flow, it is necessary tofind the relation between the concentration of contrast and the total bloodflow. For this reason I have, in collaboration with prof. dr. E. van Baveland dr. H. Marquering, built a computer model of the vascular networkto simulate the transport of blood and contrast agent. In short I set up acomputer model of the blood vessels, implemented routines for calculationof local pressures and flows and simulated the advection and diffusion ofcontrast agent. Eventually I aim to create time-density curves such as fig-ure 1b for different network configurations. That way, I might be able tofind a relation between the appearance of contrast on the one hand and thetotal blood flow on the other.

4

2 Theory

As mentioned in the introduction, the amount of oxygen in the tissue is pri-marily dependent on blood flow through the tissue and metabolic demand.In the case of Critical Limb Ischemia, the metabolic demand is not supplied,because the blood flow is reduced.

2.1 Hemodynamics

Flow (F), or the displacement of volume per unit time, between a high-pressure point (P1) and a low-pressure point (P2) is proportional to thepressure difference (∆P ) and to the inverse of the resistance (R) betweenthose points:3

F =∆P

R(2)

We can see directly the analogy to Ohm’s law for electrically circuits:

I =∆V

R(3)

The overall resistance (Rtotal) across a structure of blood vessels resultsfrom parallel and serial arrangements of branches and is governed by lawssimilar to those for the electrical resistance:

Rtotal = R1 +R2 +R3 + ... (multiple resistance in series) (4)

1

Rtotal=

1

R1+

1

R2+

1

R3+ ... (multiple resistance in parallel) (5)

In a well-defined system, it is possible to predict the flow from the ge-ometry of the vessel and the properties of the fluid. In the 19th century,Jean Poiseuille found the following relation for flow in a straight, rigid andcylindrical tube: [8]

F = ∆P · πr4

8ηl(6)

This is known as the Poiseille-Hagen equation, where r is the inner radius ofthe tube, η the viscosity of the fluid and l the length of the tube. As one cansee, the resistance to flow is due to both the geometry of the fluid (length andradius of the tube) and the internal friction of the fluid (viscosity). Viscosityis defined as the ratio between shear rate (velocity gradient perpendicularto direction of flow) and the associated shear stress (force per unit area).Because of cohesive forces between the inner surface of the tube and thefluid, we can assume that an infinitesimally thin layer of fluid closest tothe wall has zero velocity. The next layer of fluid does have a non-zerovelocity, but still moves slower than the next inner fluid layer. The resulting


5

velocity profile is a parabola with maximum velocity at the central axis.This velocity profile may lead to a phenomenon called Taylor dispersion,which I will explain in section 2.2.3.

2.1.1 Calculating flow

If the geometry in a system of blood vessels is well-defined and the absolutepressure at the in- and outflow is known, we can calculate the flow in everypart of the system. Suppose we have the following simple system, as shownin figure 2, with a pressure Pi at the beginning of the system and a pressure

Figure 2: Simple network

Po = 0 at the end of the system. Now the total flow through the system isgiven by, according to equation 2:

F =Pi − Po

Rtotal=

Pi

Ra + Rb·Rc

Rb+Rc+Rd

(7)

The flow through the system equals the flow through resistance Ra and Rd,so F = Fa = Fd. Now we can solve for the pressure points P1 and P2:

Pi − P1 = F ·Ra

P1 = Pi − F ·Ra

P2 − Po = F ·Rd

P2 = F ·Rd

(8)

Now we know the pressures P1 and P2 we can calculate the flow throughresistance Rb and Rc:

Fb =P1 − P2

Rb

Fb =Pi − F (Ra +Rd)

Rb

Fc =P1 − P2

Rc

Fc =Pi − F (Ra +Rd)

Rc

(9)

6

For simple networks like figure 2 it is possible to do calculations by hand.For more complex network, we can make use of Kirchhoff’s first law4. Theprinciple of conservation of volume implies that at any node the sum of flowsflowing into that node is equal to the sum of flows flowing out of that node:

n∑k=1

Fk = 0 (10)

Figure 3 shows a network consisting of four nodes, six conductors with con-ductance G = 1/R and a source and sink pressure. This is an analogy toa network of six blood vessels with one source (with pressure Psource) andone sink (with pressure Psink). We can write down Kirchhoff’s first law for

Figure 3: Complex network

every of the four nodes. This results in a system of 4 lineair equations:(−(G2+G3) G2 G3 0G2 −(G2+G4) 0 G4

G3 0 −(G3+G5) G5

0 G4 G5 −(G4+G5)

)·

(P1P2P3P4

)+

(G1 0 0 00 0 0 00 0 0 00 0 0 G6

)·

(Psource

00

Psink

)= 0

(11)or:

A · P +B · Ps = 0 (12)

which we can solve for P:

P = −A−1 ·B · PS (13)

When the pressure P in every node has been calculated, the flow throughevery vessel can be computed with equation 2.

4https://en.wikipedia.org/wiki/Kirchhoff27s-circuit-laws

7

2.2 Contrast agent transport

Under the influence of blood flow, injected contrast agent will travel throughthe vascular system. The transport of contrast agent through the systemis due to a combination of advection and diffusion. The advection-diffusionequation describes these phenomena in one dimension:5

∂c

∂t+ v · ∂c

∂x= D · ∂

2c

∂x2(14)

Here, ∂c∂t is the change of concentration with respect to time, v the aver-

age velocity of blood, ∂c∂x the concentration gradient and D the diffusion

coefficient of contrast agent in blood. In this chapter I will first explainthe advection-diffusion equation in more detail, using three dimensions andscale analysis for the simplification of this equation. After that I will exlainthe different regimes in the advection-diffusion equation: pure advection,Taylor dispersion and diffusion.

2.2.1 Advection-Diffusion equation

The advection-diffusion equation in three dimensions (cylindrical coordi-nates) is:6

∂c

∂t+ v(r) · ∂c

∂x=D

r

∂

∂r

(r · ∂c

∂r

)+D · ∂

2c

∂x2(15)

Here, the first term is change of concentration over time, the second termdescribes advection, the third term describes radial diffusion and the lastterm describes axial diffusion. The velocity profile in a cylindrical tube withradius R is given by:6

v(r) = 2v ·(

1− r2

R2

)(16)

One can imagine that for a different combination of parameters, the trans-port of contrast agent (or any solute) behaves differently. When blood flowdrops to zero for example, v(r) = 0 and the advection term drops out of theequation. A useful dimensionless number, which gives the ratio between theadvective and diffusive transport rate, is the Peclet Number:7

Pe =V R

D(17)

with V the flow velocity. To describe the different regimes in the advection-diffusion equation, I make use of scale analysis8 for the simplification ofequation 15:

5https://en.wikipedia.org/wiki/Convection-diffusion equation6MIT lecture about Taylor dispersion: http://video.mit.edu/watch/1961-lecture-7-

a8211-taylor-dispersion-2367/7https://en.wikipedia.org/wiki/Peclet number8https://en.wikipedia.org/wiki/Scale analysis (mathematics)

8

C

T+ V · C

L= D · C

R2+D · C

L2

C concentrationT timeV velocityL length tubeD diffusion coefficientR radius tube

The first regime is pure axial diffusion, where axial diffusion dominatesover advection:

D · CL2

>> V · CL

=⇒ D

L>> V

=⇒ V L

D· RR<< 1 =⇒ Pe <<

R

L

(18)

The second regime is that of pure advection, where advection dominatesover both radial and axial diffusion:

V · CL>> D · C

L2=⇒ V >>

D

L· RR

=⇒ Pe >>R

L

(19)

and:

V · CL>> D · C

R2=⇒ V

L>>

D

R2

=⇒ Pe >>L

R

(20)

But typically (and definitely in the case of blood vessels) L > R, so that thecondition for the second regime is Pe >> L

R . The last regime is the Taylordispersion regime, where advection and radial diffusion both dominate overaxial diffusion:

D · CR2

>> D · CL2

=⇒ L >> R (21)

The timescale is set by radial diffusion:

C

T∼ D · C

R2=⇒ T ∼ R2

D

L ∼ V · T ∼ V R2

D>> R

=⇒ Pe >> 1

(22)

The first condition for the Taylor regime is thus Pe >> 1. For the secondcondition we need to insert equation 16 into the advection-diffusion equation

9

and take out the axial diffusion term:

∂c

∂t+ 2v ·

(1− r2

R2

)· ∂c∂x

=D

r

∂

∂r

(r · ∂c

∂r

)(23)

Now we change the reference frame from a standing position to a movingreference frame with velocity v. This change in reference frame x′ = x− vtgives:

∂c

∂t+ v ·

(1− 2r2

R2

)· ∂c∂x′

=D

r

∂

∂r

(r · ∂c

∂r

)(24)

Now we make the assumption that the process is quasistatic or happensinfinitely slow in our new reference frame, which is valid if:

C

T<< D · C

R2=⇒ T >>

R2

D

T ∼ L

V=⇒ L

V>>

R2

D

=⇒ Pe <<L

R

(25)

Thus, the second condition for Taylor dispersion is Pe << LR . Figure 4

summarizes the three different regimes.

Figure 4: Different regimes in the advection-diffusion equation. On the left (redslashes) we see the pure axial diffusion regime, where Pe << R/L. The Taylordispersion regime is indicated with blue slashes. Here Pe >> 1 and Pe << L/R.On the right (red slashes) we see the pure advection regime, where Pe >> L/R.The white section in the middle of the figure indicates the combination of axial andradial diffusion.

10

2.2.2 Advection

The molar concentration is a measure of the concentration of solute (contrastagent) in a solution (blood). The molar concentration or molarity is definedas:9

C =n

V(26)

with n the amount of solute in moles and V the volume of the solution.To model the transport of contrast agent we divide every vessel in elementswith length dx, see figure 5. No contrast agent will leave the system, which

Figure 5: Advection

means that the change in amount of contrast agent per unit time betweenx and x+dx equals:

∆n

∆t= −F · [C(x+ dx)− C(x)] (27)

It follows that the change in concentration per unit time is:

∆C

∆t=

∆n

∆t · V= −F · [C(x+ dx)− C(x)]

V(28)

Because flow is the displacement of volume per unit time, flow is the productof the cross sectional area of a vessel and the average velocity of the blood:10

F = A · v (29)

Filling this in equation 28, making use of the fact that V = A · dx gives:

∆C

∆t= −v · [C(x+ dx)− C(x)]

dx= −v · dC

dx(30)

which is in accordance with equation 14 in the regime where D· ∂2c∂x2 << v· ∂c∂x .

9https://en.wikipedia.org/wiki/Molar concentration10W.F. Boron, E.L. Boulpaep, Medical Physiology, 2nd edition, page 434

11

2.2.3 Taylor dispersion

Within a vessel the blood has a velocity profile, meaning that blood nearestto the vessel wall has zero velocity and maximum velocity at the central axis,as mentioned in section 2.1. The result is that within the vessel wall, there isa concentration gradient of the contrast agent not only in the axial direction,but also in the radial direction. Figure 6 gives a schematic diagram of theprocess of Taylor dispersion. At the starting point t0 the bulk of contrast

Figure 6: Schematic diagram of the process of Taylor dispersion

agent has the shape of a solid cylinder. At t0 there is only a concentrationgradient in the axial direction. Because of the velocity profile within thevessel, contrast agent in the middle of the vessel will move faster due toadvection than contrast agent near the vessel wall as represented at time t1.This results in a concentration gradient in the radial direction. Because ofthe radial concentration gradient, the back of the contrast agent diffuses tothe center of the vessel and the front diffuses to the vessel wall, as shown att2. Due to advection the back of the contrast agent will speed up (becauseof higher velocity in the center) and the front will slow down. Eventuallythe solute patch is compressed and being hold together as shown at t3.

2.2.4 Contrast agent diffusion into tissue

During the first pass of contrast agent, after injection, there is a significantdiffusion of contrast into the interstitial space or tissue that surrounds themicrocirculation. This extraction of contrast agent is around 30% in coro-nary arteries. [7] According to Fick’s first law, the diffusion of contrast agentacross a capillary wall depends on both the permeability and the concentra-tion gradient:11

JX = PX · ([X]m − [X]is) (31)

JX is the flux or the amount of solute that crosses a particular surface areaper unit time, PX is the permeability coefficient and [X]m and [X]is arethe concentrations of contrast agent in the microcirculation and interstitialspace, respectively. Flux has units of [mol/(m2s)] and so equation 31 can


12

be written as follows:

∆n

Am ·∆t= PX · ([X]m − [X]is) (32)

where Am is the surface area of the microcirculation. Because C = n/V(equation 26), the change in concentration per unit time is:

∆C

∆t=

∆n

∆t · V=Am · PX · ([X]m − [X]is)

V(33)

13

2.3 Computer generation of vascular trees

In order to properly model the blood transportation system, it is necessaryto simulate both the larger arteries/veins and the smaller ones. The prob-lem with smaller arteries is that they are not visible in an angiogram. Tocreate small arteries, and thus a complete vascular system, we need to applyalgorithms for generating vascular trees. In the literature, three differenttypes of algorithms have been reviewed. The first one is based on con-strained constructive optimization (COO), where vascular networks needto satisfy certain physiological optimality criteria. In COO a binary treeis constructed by adding a vessel segment to an initial tree, each time in-troducing an optimal bifurcation. [9–12] The second approach is based onconstruction of self-similar or area filling objects, which is called determin-istic geometric construction. [13] This type of algorithm has the advantageof avoiding optimization problems, but lacks an overall natural structure.The last type of algorithm is the angiogenesis-based construction, which isthe actual process through which new blood vessel form from pre-existingvessels. In theory this last approach is the best method to construct vascu-lar system, yet it requires complex types of algorithms and more computingpower. [14]From the literature I conclude that the best method for my modelis the constrained constructive optimization.

A common topic in the different approaches is that vessels bifurcate intwo daughter segments and that at a higher level in the vascular tree thetotal cross-sectional area increases. [15] This increase in cross-sectional areais required to bring down the velocity of the flowing blood. The velocityof blood has to be low enough in the capillaries to permit the exchange ofoxygen and metabolic products. Because of conservation of volume, the flowin every level of the vascular system is the same, which means that accordingto equation 29 the velocity of blood is proportional to the inverse of the cross-sectional area. Another major topic is the discussion of Murray’s law, whichgives a constraint for the radius of the daughter segments in comparison tothe parent segment: [9–12,14–19]

r3p = r3d + r3d (34)

This is known as the bifurcation rule and is derived in 1926 by Murray andis based on the principle of minimum work for blood transport. [20]

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3 Method

In this chapter I will explain the working mechanism behind the differentprograms used in the model. The full model is written in MATLAB, whichis commonly used by engineers and physicists for matrix manipulations andplotting of data.

3.1 Structure of vascular system

In order to simulate the transport of contrast agent through a vascularnetwork, we first need to create a structure of arteries and veins.

3.1.1 Predefined large artery network

In the file ’configuration.m’ the configuration of the larger arteries is defined.The structure is divided into two types of vessels: source vessels and internalvessels. Each vessel has in common that it has a starting point u(x,y), aterminus v(x,y) and a radius. A source vessel is connected to one node andan internal vessel is connected to two nodes, as denoted by the blue dots infigure 7a. The source vessels have a certain source or sink pressure which isneeded to calculate the flow through the system.

(a) Configuration of larger arteries (b) Watershed of every large vessel is deter-mined

Figure 7

15

3.1.2 Generation of small arteries

The next task is to create smaller arteries which are connected to the largerarteries shown in figure 7a. As mentioned in section 2.4, the constrainedconstructive optimization [11,12] is the best method to do this. The overallstructure has to fulfill two constraints. First, according to equation 34 theradius of the two daughter segments is related to the radius of the parentsegment. Second, the sum of flows through the daughter segments is equal tothe flow through the parent segment. Before new segments are added to theoverall structure, it is necessary to determine to which part of the vascularstructure the new vessel eventually is connected. In other words, which ofthe larger arteries is closest to a certain point in the two dimensional grid.The program ’distancematrix.m’ determines for every point in a rectanglegrid which vessel is closest by. In order to do so, the program creates agrid of 1000x1000 points and calculates for every point the distance to eachof the larger arteries. The vessel which is closest by is then saved into the1000x1000 matrix. Figure 7b shows the result of this program, where everycolor correlates with the watershed of an larger artery.

The function ’structurerandom.m’ creates all the small arteries. First,the area of the watershed of an element is determined. This way, com-bined with a certain density of terminal segments, all watersheds are roughlyevenly perfused. Second, the proximal end of the first new segment is con-nected to a large artery, thereby creating an extra node. Third, a randomposition within the perfusion area of the artery is chosen as the distal endof the first new segment, see figure 8a. Now, the following algorithm is usedto generate the rest of the terminal segments:

• The coordinates for the distal end of a new terminal segment are se-lected. First a random point within the watershed is chosen. Nextthe distance to each of the already existing segments is calculated. Ifthe distance to the nearest already existing segment is smaller than acertain threshold value d, a new random point is chosen, see figure 8b.This process favors regions with low density of existing segments, thusleading to an evenly perfusion of the watershed. [11,12]

• If a point is accepted as the distal end of a terminal segment, thenearest already existing segment is split in half. Hereby, a new nodeand a new segment is created, see figure 8c.

• Because a new terminal segment is created, the total number of dis-tal segments changes for certain already existing segments. This isadjusted in the last step and is important to determine the radius ofevery segment in the configuration. The segment indicated with theblack arrow (figure 8d) has five distal terminal segments.

16

(a) Generate first new segment. (b) Chosing distal end next new segment.In this case, new coordinates have to bedetermined.

(c) Generate next terminal segment. (d) Generate more terminal segments.The black arrow indicates a segment withfive terminal segments, whereas the greenarrow indicates a segment with two termi-nal segments.

Figure 8

17

3.1.3 Completion of the vascular network

After creation of all segments, the radius of every segment has to be ad-justed according to equation 34. The result is shown in figure 9a. Themicrocirculation is simulated by the function ’micronetwork.m’. Microcir-culatory boxes with a certain volume and conductance are connected tothe terminal segments. To complete the vascular structure, the function’makevenoussytem.m’ adds a venous network to the existing structure. Thisis done by replicating the arteries, increasing the radius and connecting thewhole structure to the corresponding microcirculatory boxes. The vascularnetwork now consists of four different types of segments: 1.Large and smallarteries, 2. terminal segments, 3. microcirculatory boxes, which are repre-sented by a certain volume and resistance and forms the connection betweenthe arterial and venous network, and 4. small and large veins.

(a) Small arteries are added to the structure (b) Complete network

Figure 9

3.2 Simulation of contrast agent transport

In order to calculate the flow through every segment of the vascular structurewe need to solve equation 11 as stated in section 2.1.1. First we need the def-inition of the connectivity of the model. The function ’MakeNodeTable.m’makes a table of nodes with each node containing information on the con-nected segments. This function is written by prof. E. van Bavel. Next, the

18

conductance of every segment is calculated with the function ’calcconduc-tance.m’. Because the conductance G is the inverse of the resistance R, theconductance is given by:

G =πr4

8ηl(35)

With the connectivity of the model the function ’solvehemodyn.m’ creates aNxN matrix A (with N the number of internal nodes), a matrix B definingconnectivity to source pressures and a matrix PS with the correspondingsource pressures. The pressure in every node is now calculated with equation13 and the flow through every segment with equation 2. This function isalso written by prof. E. van Bavel.

The function ’injectcontrast.m’ simulates the transport of contrast throughthe vascular system. According to section 2.2.1 the transport of contrastagent behaves differently in different regimes of figure 4. For simplicity rea-sons I have assumed that D · ∂2c

∂x2 << v · ∂c∂x which implies that the followingequation holds for the transport:

C(t+ ∆t)− C(t)

∆t= −F · [C(x+ ∆x)− C(x)]

A ·∆x(36)

with A the cross-sectional area of the vessel. In other words, I have neglectedthe velocity profile within the vessel and assumed that blood flows withaverage velocity v. In the discussion (chapter 5) I will review whether thisis a reasonable assumption. The function creates, for every segment inthe structure, a MxN matrix with M the amount of time steps and N theamount of place steps. Every segment is thus divided into N elements,which implies that ∆x equals l/N with l the length of the segment. Thebolus concentration is set to 100 and bolus volume to 9000 mm3, which isthe same volume used in the research of Jens et. al. [7]. According to thespecific place (x) within the vessel, there are two options for the value ofC(x):

• x = 0, thus we want to calculate the change in concentration per timestep for the first element of a segment, which means that contrast agentis received from the last element of one or two segments upstream.

• x 6= 0, which means that contrast agent is received from the samesegment.

If x = 0 and the segment is the first segment of the vascular structure (thesegment where the contrast agent is injected into) C(x) has the value of 100.If the segment is any other segment in the vascular network than the first,the function uses the following algorithm to determine the value of C(x):

• Checks between which nodes the segment is located. If the flow F has apositive value, the first node of the two is the beginning of the segment.

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If F < 0 the second and last node is defined as the beginning of thesegment. Segment 3 in figure 10 is located between nodes C and D.Node C is the beginning of the vessel, so contrast agent is potentiallyreceived from segment 1 and 2.

• Now the function checks the direction D of the other segments con-nected to the first node. If the segment is directed towards the nodethen D = [1 0], otherwise D = [0 1]. Second, the function checkswhether the flow F has a positive or negative value. If F > 0, then

Q =

[10

], otherwise Q =

[01

].If now the inner product D ·Q = 1, blood

is indeed received from that segment. In figure 10 both segment 1 and2 are directed towards node C. This means that D1 = D2 = [1 0].

The flows in segment 1 and 2 are both positive, so Q1 = Q2 =

[10

].

• The value C(x) is now the weighted average of the concentration ofthe last element of the other segments:

C(x) =C1F1(D1 ·Q1) + C2F2(D2 ·Q2)

F1(D1 ·Q1) + F2(D2 ·Q2)(37)

Figure 10: Schematic diagram contrast agent algorithm. The flow in all threesegments is positive. Segment 1 is located between nodes A and C, segment 2between nodes B and C and segment 3 between nodes C and D.

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3.3 Imaging of contrast agent transport

To create an image, or rather a movie, of the transport of contrast agentthe function ’plotstructure.m’ plots the state of the system for N differentstates and concatenates those states into a movie. Figures 11 and 13 showsix different states of the system. The ellipsoid boxes in the figure indicatethe amount of contrast agent in the microcirculation and surrounding tissue.

(a) Time is 1 sec (b) Time is 5 sec

Figure 11

Figure 12: Simulation of contrast agent through a vascular network. The timestep∆t is 0.004 s and total running time is 30 s (7500 time steps). Every segment isdivided in 10 elements. The main artery has a radius of 3 mm and the radius of theterminal segments are 0.4 mm. The radius of every segment in the venous systemis 50% larger than the corresponding segment in the arterial system.

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(a) Time is 10 sec (b) Time is 15 sec

(c) Time is 20 sec (d) Time is 30 sec

Figure 13

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3.4 Analyzing behavior of the system

In order to draw conclusions about the correlation between blood flow andappearance of contrast agent in a certain region of interest (ROI), we needto analyze the behavior of the system. The program ’analyzerectangle.m’determines the amount of contrast agent as a function of time and the totalblood flow in a preset region of interest (ROI), see figure 14. After the ROIis determined an overall time-density curve is created and time to peak andarrival time is calculated. The arrival time in my model is defined as themoment of time at which the amount of contrast in the ROI exceeds 100moles. I will compare the time-density curves and total flow of different sys-tems, all with the same ROI. The difference in systems includes a decreasein radius of one segment (indicated with green in figure 14) in the configu-ration. This represents atherosclerosis which is the major cause of criticallimb ischemia. It also includes different permeability coefficients which rep-resents the degree of leakage of the microcirculation. Tables 1,2 and 3 givean overview of the different systems analyzed.

(a) Case I: Simulation of atherosclerosis ingreen segment

(b) Case II: Simulation of atherosclerosis ingreen segment

Figure 14

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Conductance Radius

Normal 100 % 2.21

Atherosclerosis 25 % 1.56



Atherosclerosis 0.1 % 0.39

Table 1: Case I, four different systems all with atherosclerosis. The permeabilitycoefficient in all systems equals 0.1.

Conductance Radius

Normal 100 % 2.78









Table 2: Case II, eight different systems all with atherosclerosis. The permeabilitycoefficient in all systems equals 0.1.

Permeability coefficient

Normal 0.1

Leakage 0.01

Leakage 0.05

Leakage 0.5

Leakage 1

Table 3: Case III, simulation of different degrees of leakage in the microcirculation.The radius of both green segments (figure 14) is 2.21 and 2.78 respectively.

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4 Results

Figure 15 shows the time-density curve for different systems with atheroscle-rosis in one segment in the vascular structure (case I). Table 4 gives thearrival time, time to peak and total blood flow in the region of interest.

(a) Conductance in one segment is 25%. (b) Conductance in one segment is 10%.

(c) Conductance in one segment is 1%. (d) Conductance in one segment is 0.1%.

Figure 15: The time-density curves for different degrees of atherosclerosis. Theamount of contrast (units: moles) is plotted against the time. The blue curveshows the normal time-density curve, whereas the red curve shows the amount ofcontrast in the case of atherosclerosis.

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Conductance Arrival time Time to peak Flow in ROI

100 % 3.82 s 11.41 s 122.54 mm3/s

25 % 3.72 s 11.60 s 122.42 mm3/s

10 % 3.84 s 12.85 s 122.27 mm3/s

1 % 4.60 s 18.53 s 121.88 mm3/s

0.1 % 5.08 s 15.70 s 121.77 mm3/s

Table 4: Case I, four different systems with atherosclerosis. Arrival time and timeto peak increases with lower conductance. The total blood in the ROI does notchange significantly.

Figures 16,17 and 18 show the time-density curve for different systemswith atherosclerosis in one segment in the vascular structure (case II). Table5 gives the arrival time, time to peak and total blood flow in the region ofinterest. The total blood flow in the ROI is determined as the sum of flowsof the microcirculatory boxes in that region. Figures 19 and 20 show therelation between total blood flow and arrival time and time to peak.

(a) Conductance in one segment is 1 %. (b) Conductance in one segment is 0.5 %.

Figure 16

26

(a) Conductance in one segment is 0.2 %. (b) Conductance in one segment is 0.1 %.

(c) Conductance in one segment is 0.067 %. (d) Conductance in one segment is 0.05 %.

Figure 17

27

(a) Conductance in one segment is 0.04 %. (b) Conductance in one segment is 0.02 %.

Figure 18

Conductance Arrival time Time to peak Flow in ROI Total blood flow

100 % 3.82 s 11.41 s 122.54 mm3/s 1755 mm3/s

1 % 3.52 s 11.36 s 116.41 mm3/s 1692 mm3/s

0.5 % 3.67 s 11.85 s 110.81 mm3/s 1635 mm3/s

0.2 % 4.13 s 13.3 s 97.04 mm3/s 1494 mm3/s

0.1 % 4.99 s 15.94 s 78.91 mm3/s 1308 mm3/s

0.067 % 5.71 s 18.17 s 68.05 mm3/s 1197 mm3/s

0.05 % 6.48 s 20.58 s 59.26 mm3/s 1107 mm3/s

0.04 % 7.24 s 23.06 s 52.43 mm3/s 1038 mm3/s

0.02 % 10.88 s 35.09 s 33.39 mm3/s 843 mm3/s

Table 5: Case II, eight different systems with atherosclerosis. In first instance, thearrival time and time to peak does not change much. Only from a conductance of0.2 % and lower, the arrival time and time to peak increase significantly. Overall,the flow in the ROI decreases with lower conductance.

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Figure 19: Relation between the total blood flow in the ROI and the arrival time.

Figure 20: Relation between the total blood flow in the ROI and time to peak.

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Figure 21 shows the time-density curve for systems with a different per-meability coefficient (case III). Table 6 gives the time to peak and maximaldensity. The arrival time is the same for the different systems.

(a) Permeability coefficient is 0.01 (b) Permeability coefficient is 0.05

(c) Permeability coefficient is 0.5 (d) Permeability coefficient is 1

Figure 21

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Permeability coefficient Time to peak Maximal density

0.01 11.16 s 50038 moles

0.05 11.27 s 50836 moles

0.1 11.41 s 51736 moles

0.5 12.65 s 56227 moles

1 12.98 s 58257 moles

Table 6

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5 Discussion

The different systems used to find a correlation between blood flow andappearance of contrast agent show ambiguous results. In case I, whereatherosclerosis was simulated in a branch of the main artery, the arrival timeand time to peak increased when the segment narrowed. However narrowingof the segment did not have much influence on the total blood flow in theROI. This can be explained as follows: in the normal case, blood flowsas indicated with the arrows in figure 22a. In the case of atherosclerosis,the higher the resistance in the narrowed segment, the lower the pressuredownstream of that segment. At some point the pressure downstream to thenarrowed segment is lower than the blue pressure point in figure 22. Thiswill lead to a change in direction of flow, as shown in figure 22b. Such a

(a) Black arrows give the direction offlow in normal case.

(b) Black arrows give the direction offlow in atherosclerosis.

Figure 22

circulatory arterial structure thus works as a safety net to avoid ischemia. Inthe human body such strucures also exist, with the Circle of Willis12 as mostimportant example. Still we do see something strange happening. In figure15c and figure 15d one can notice that despite the further descrease of theradius, an improvement of the time-density curve is seen. Figure 23 showswhat is happening is this case. In figure 23a contrast arrival in the ROI isdue to both the left and right branch of the main artery. However in figure23b one can see that contrast agent flows in only from the right branch ofthe main artery. In other words, from the perspective of the ROI the smallsegment is completely blocked. In figure 24 I have shown the contributions

12https://en.wikipedia.org/wiki/Circle of Willis

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from both side branches in the region of interest. If the contrast is coloredin red then the contrast comes only from the right branch whereas blueindicates contrast from the left branch. If the contrast is black, then thereis a contribution from both branches. It seems that the complete blockageactually improves the time-density curve, because contrast agent flows in theROI more effeciently via the other route. However more research is neededto explain this odd behavior of the system.

(a) (b)

Figure 23: Figure a: the red line shows contrast agent arrival from both the leftand right branch of the main artery, while the blue line only shows contrast agentarrival from the right branch of main artery. Figure b: both lines overlap. Thismeans that all the contrast in the ROI comes from the right branch of the mainartery

In case II, the stenosis is located in the main artery. In first instancethere is not much change in total blood flow, arrival time and time to peak.Arrival time and time to peak even decreases a bit until the conductanceis lower than 0.5 %. If the conductance is lowered even more, we do seesignificant changes in the arrival time, time to peak and flow in the ROI.Figures 19 and 20 show the relation between the blood in the region ofinterest and the arrival time and time to peak respectively. These figuresshow a postive correlation between appearance of contrast in the ROI andthe inverse of the blood flow in that region. This is what we exspected andhoped for, because in this case conclusions may be made about the bloodflow on the basis of a time-density curve.

In case III, we see that the value of the permeability coefficient hasmuch influence on the time-density curve. A higher permeability coefficientmeans an increase of endothelial leakage, which is a property of the micro-

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(a) (b)

Figure 24: The state of the systems at t=12 s. Conductance in one segment of thesystem on the left is 1/100. Conductance in one segment of the system on the rightis 1/1000.

circulation. It is important to note that the arrival time and slope of thetime-density curves does not change with different values of the permeabilitycoefficient. This seems logical, because the macrocirculation is not affectedin this case. Therefore the total resistance is unchanged, which results in anormal arrivial time and slope of the time-density curve.

In this model I made the assumption that blood flows with an averagevelocity v instead of taking the velocity profile into account. In section2.2.1 I gave an overview of the different regimes of the advection-diffusionequation and it’s implications. An important number in this discussion isthe Peclet Number, which gives the ratio between advective and diffusivetransport. The diffusion coefficient of iodixanol, used as contrast agent inperfusion angiography was found to be ∼ 2.5 · 10−4mm2/s [21]. In the mainartery of the computer model, this leads to a Peclet Number of:

Pe =V R

D=

62 · 32.5 · 10−4

= 7.4 · 105 (38)

It follows that Pe >> L/R , which means that we are operating in the pureadvection regime. There is a problem with the transport of contrast agentbeing in the pure advective region. Because of the velocity profile of bloodflow within a blood vessel, contrast agent would get stretched infinitely, if

34

there isn’t any radial diffusion. This means that some of the contrast agentwill stick to the vessel wall instead of travelling through the vascular system.This seems not the case in the imaging technique perfusion angiography,where all the contrast agent travels in the direction of blood flow. Howeverit is very hard to actual see small amounts of contrast agent that might stickto the wall, some some sticking of contrast may happen. As an extensionon this model one could divide every segment not only in small elementswith length ∆x, but also in small circular layers with thickness ∆r. Thiswould certainly increase computation time, but it offers the opportunityto include the velocity profile into the model. Moreover, the simulation oftransport agent will lead to concentration gradients in the radial direction.Including a certain value for the diffusion coefficient, even the process ofTaylor dispersion may be simulated.

In conclusion, the results in this study indicate that the relation betweenblood flow and appearance of contrast in a region of interest is not so trivial.Differences in both the micro- and macrocirculation lead to changes in thetime-density curve. Moreover, when there is an alternative pathway for theblood to reach the ROI there is no correlation between blood flow and ap-pearance of contrast agent. Interpretation of the time-density curves madewith perfusion angiography is therefore hard, as the vascular structure andthe state of the microcirculation is not known. However, the vascular struc-ture that I have used is not representative for any vascular structure in thehuman body. The results given here are to show that the model does indeedsimulate the transport of contrast through a vascular network and that thereis a correlation between blood flow and appearance of contrast agent. Thismodel offers the possibility to create more realistic vascular structures, toadd collateral vessels and even the non-newtonian properties of blood.

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[1] M.H. Beers, R.S. Porter, T.V. Jones, J.L. Kaplan, M. Berkwits, ”Pe-ripheral arterial disorders”, The Merck Manual of diagnosis and ther-apy, 747-753.

[2] M. Levi, C.D.A Stehouwer, ”Stollingsstoornissen, trombose, atheroscle-rose en vaatziekten”, Interne geneeskunde, 198-204.

[3] F. Violi, S. Basili, J.S. Berger, W.R. Hiatt, ”Antiplatelet therapy in pe-ripheral artery disease”, Handbook of experimental pharmacology, 547-563.

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[6] L. Potier, C. Abi Khalil, K. Mohammedi, R. Roussel, Use and Utilityof Ankle Bachial Index in Patients with Diabetes, Eur. J. Vas. Endovas.Surg. 41, 110-116.

[7] S. Jens, H.A. Marquering, M.J.W. Koelemay, J.A. Reekers, PerfusionAngiography of the Foot in Patients with Critical Limb Ischemia: De-scription of the Technique, Cardiovasc. Intervent. Radiol. 38 (2015),201-205.

[8] S.P. Sutera, R. Skalak, The History of Poiseuille’s Law, Annu. Rev.Fluid Mech. 25 (1993), 1-20.

[9] R. Karch, F. Neumann, M. Neumann, W. Schreiner, Staged growth ofoptimized arterial model trees, Annals of Biomedical Engineering 28(2000), 495-511

[10] R. Karch, F. Neumann, M. Neumann, W. Schreiner, A three-dimensional model for arterial tree representation, generated by con-strained constructive optimization, Computers in Biology and Medicine29 (1999), 19-38.

[11] W. Schreiner, Computer generation of complex arterial tree models, J.Biomed. Eng. 15 (1993), 148-150.

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