computational biomechanics of blood flow at macro- and micro-scales

7/30/2019 Computational Biomechanics of Blood Flow at Macro- And Micro-scales

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COMPUTATIONAL BIOMECHANICS OF BLOOD FLOW AT

MACRO- AND MICRO-SCALES

TAKAMI YAMAGUCHI1)*, TAKUJI ISHIKAWA2), YOHSUKE IMAI2)

1)Department of Biomedical Engineering, Tohoku University

6-6-01 Aramaki Aza Aoba, Sendai 980-8579, Japan

2)

Department of Bioengineering and Robotics, Tohoku University6-6-01 Aramaki Aza Aoba, Sendai 980-8579, Japan

Due to the rapid growth of computational power, computational biomechanics is now

thought of as the most rapidly growing engineering theme in the biomedical engineering

field. We have been investigating cardiovascular diseases over the microscopic and

macroscopic levels by using conjugated computational mechanics to analyze fluid, solid

and bio-chemical interactions. In this article, we present a novel hemodynamic index for

predicting the initiation of cerebral aneurysms, and a numerical model of blood flow in

malaria infection.

1. Introduction

Biological systems of the human body are always under an integrated nervous

and humoral control, i.e., in homeostasis. Multiple feedback mechanisms with

mutual interactions among systems, organs, and even tissues provide integrated

control of the entire body. These control mechanisms have different spatial

coverage, from the micro-scale to the macro-scale, and time constants, from

nanoseconds to decades. We think that these variations in spatial and temporal

scales should be taken into account in discussing human physiology and

pathology.

With this in mind, we have been investigating the biomechanics of the

human body, particularly the biomechanics of physiological flows over micro to

macro levels, by using conjugated computational mechanics to analyze fluid,

solid and bio-chemical mechanics. Recent examples include mass transport to

cerebral aneurysms [1], a model of aneurysmal development [2], and a

numerical model of thrombus formation [3]. Here, we review a novel

hemodynamic index for predicting aneurysm initiation [4], and the modeling of

blood flow in malaria infection [5].

* Takami Yamaguchi is the Tohoku University Global COE Leader.

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2. A Hemodynamic Index for the Initiation of a Cerebral Aneurysm

Saccular cerebral aneurysms are generally formed at arterial bifurcations and

arterial bends in the Circle of Willis. Rupture of an aneurysm can lead to

subarachnoid hemorrhage with high mortality and morbidity. It is crucial to

understand why an aneurysm initiates, grows, and ruptures. Structural changes

in the arterial wall, induced by a biological response to hemodynamic stress,

have been thought to play a key role in aneurysmal pathogenesis [6].

Identification of specific hemodynamic factors may provide better prediction of

the initiation, growth, and rupture of aneurysms. Wall shear stress (WSS) has

been investigated as a candidate responsible for aneurysm initiation [7]. Meng

et al. [8] proposed the spatial wall shear stress gradient, which represents non-

uniformity of shear stress on the arterial wall, as a hemodynamic index. In

experimental studies by Wang et al. [9], the disruption of actin cytoskeleton of

endothelial cells (ECs) by cyclic mechanical stretching was observed. This

implies that not only the spatial variation of shear stress, but also the temporal

fluctuation, has unfavorable effects.

We have proposed a novel hemodynamic index, the gradient oscillatory

number (GON), that emphasizes the temporal fluctuation of SWSSG [4]. Here,

we show the relationship between GON and aneurysm location using

computational fluid dynamics with a patient-specific geometry of the human

internal carotid artery.

2.1. Methods

Let f= (fp, fq) be the viscous drag force per unit area acting tangentially to the

wall surface (i.e. WSS), where thep-direction corresponds to the time-averaged

direction offover a flow cycle and the q-direction is perpendicular to p. Thespatial gradient offp andfq may generate a tension/compression force on ECs

[10], and it may oscillate greatly in the case of pulsatile blood flow. To quantify

the degree of oscillating tension/compression force, we formulated the gradient

oscillatory number (GON) as

T

T

dt

dt

GON

0

01

G

G, (1)

q

f

p

f qp,G , (2)

where Trepresents one pulse duration of pulsatile blood flow.

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Patient-specific geometry data was used for a human internal carotid artery

with an aneurysm [11]. Using this data, we reconstructed an artery geometry

model before aneurysm formation. First, a segment of artery around theaneurysm was artificially removed from the original geometry (Fig. 1a). The

segment was then interpolated and smoothened such that the elastic energy of

the arterial wall was minimized (Fig. 1b and c).

We performed three-dimensional pulsatile blood flow simulations using an

in-house CFD solver to obtain temporal variations in SWSSG vectors, and then

to calculate the GON index. Our CFD solver employed a boundary-fitted

coordinate method. The fluid was assumed to be a Newtonian fluid, and the

continuity and Navier-Stokes equations were solved by a finite volume methodthat included a rigid wall boundary condition. The Womersley number and inlet

peak Reynolds number were 2.6 and 300, respectively.

a b

c

Figure 1. Reconstruction of arterial geometry before aneurysm formation. The segment of arterynear the aneurysm is (a) removed, (b) interpolated, and (c) smoothened.

2.2. Results and Discussion

Temporal variations of the spatial wall shear stress gradient (SWSSG) vector

were calculated at the wall surface, and the GON index was evaluated from a set

of time-varying SWSSG vectors, as shown in Fig. 2. The location of high GON

corresponds well with the aneurysm location, which indicates that the GON can

be a good indicator for the initiation of cerebral aneurysm. We also examinedother indices proposed in previous studies, including WSS at peak systole, time-

averaged WSS, oscillatory shear index [12], SWSSG at peak systole, time-

averaged SWSSG, and aneurysm formation indicator [13]. Among these

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indices, only GON showed a significant correlation with the location of

aneurysm formation (Fig. 3).

a b

Figure 2. Distribution of GON: (a) side view; (b) top view. Original geometry is superimposed.

a b c

d e f

Figure 3. Distribution of a variety of indices: (a) WSS at peak systole; (b) time-averaged WSS; (c)

oscillatory shear index; (d) SWSSG at peak systole; (e) time-averaged SWSSG; (f) aneurysm

formation indicator.

The reason why GON becomes large in this region is because the SWSSG

vector rotates very strongly over the flow cycle at the location of aneurysm

formation. Since the term, Gdt0

T

, in Eq. (1) approaches zero where strong

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temporal variations of the SWSSG occur, the value of GON at the location

approached its maximum value (=1). The SWSSG would generate a

tension/compression force (flow-induced force) on ECs. Thus, these resultsindicate that before aneurysm formation, a strongly disturbed force was imposed

on the ECs in that region. Recently, Jamous et al. [14] reported that such

disturbed blood flow might precipitate early morphological change of ECs that

leads to the formation of cerebral aneurysm, and we have also demonstrated that

disturbed blood flow, i.e. a high GON index has a significant correlation with

the location of aneurysm formation. Although the reliability of GON requires

further examination in other clinical cases, these results suggest that it will be

useful as a hemodynamic index for the initiation of cerebral aneurysms.

3. Modeling of Blood Flow in Malar ia Infection

Malaria induced by Plasmodium falciparum is one of the most serious

infectious diseases on earth. There are about five hundred million clinical cases

with two million deaths each year. When a malaria parasite invades and

matures within a red blood cell (RBC), the infected RBC (Pf-IRBC) becomes

stiffer and sticks to healthy RBCs (HRBCs) and endothelial cells (ECs). These

changes are postulated to be linked to microvascular occlusion. Several

researchers have investigated the cell mechanics of Pf-IRBCs using recent

experimental techniques, such as micropipette aspiration [15], optical tweezers

[16], and microfluidics [17]. However, these experimental studies are still

limited to a single Pf-IRBC. Microvascular occlusion may be a hemodynamics

problem, involving hydrodynamic and chemical interactions among Pf-IRBCs,

HRBCs and ECs, but these current experimental techniques have several

limitations in this regard. Microvasculature in human body consists of a very

complex network of circular channels, but we cannot create such complex

microchannels for experiments. It is also difficult to measure the three-

dimensional velocity and stress fields simultaneously. Instead, numerical

modeling can be a strong tool for further understanding the pathology of malaria.

In this section, we present a numerical model of blood flow in malaria infection

[5,18].

3.1. Methods

In our method, all the components of blood are represented by a finite number

of particles [5,18]. The Lagrangian description of the fluid equations are solved

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at the position of each particle, and each particle is moved by the advection

velocity at every time step. The RBC membrane is modeled by a spring

network of membrane particles. Since a malaria parasite is much stiffer than aPf-IRBC, we model the malaria parasite as a rigid object. The Adhesive

property of a Pf-IRBC is also modeled by springs [18]. This model was

confirmed to simulate well the deformation of Pf-IRBCs in stretching with

optical tweezers (Fig. 5) [16], deformation in shear flow [19], and flow into

narrow channels (Fig. 6) [17].

Cytoplasm

Malariaparasite

RBCmembranePlasma

Figure 4. Particle-based modeling of blood flow.

Stretching force [pN]

Normalizeddiameter[-]

0.0 50.0 100.0 150.0

0.5

1.0

1.5

2.0

2.5

3.0

Axial (Si mulation)

Axial (Ex perimen t)

Transverse (Simulation)

Transverse (Experiment)

Figure 5. Stretching of a Pf-T-IRBC by optical tweezers: comparison between numerical and

experimental results [16].

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a b

Figure 6. Flow of a Pf-T-IRBC into a narrow channel: (a) 4m; (b) 6m.

3.2. Results and Discussion

A Pf-IRBC can tether and roll on ECs, and then adhere to the ECs. In order toestablish contact with ECs, the Pf-IRBC must migrate toward the wall while in

circulation; however, RBCs undergo axial migration and generate a cell-free

layer near the wall under physiological conditions. To investigate the

mechanism of margination, we simulated flows in a straight circular channel

with a diameter of 12m [21]. We considered blood flow at 11% and 27%

hematocrit (Hct) conditions, where Pf-IRBCs at the trophozoite stage (Pf-T-

IRBCs) were included.

Significant margination is observed in the case of Hct = 27%. Figure 7shows the typical visualization of the configuration of RBCs for Hct = 11% and

Hct = 27%. Since the velocity of the Pf-T-IRBC is lower than the surrounding

HRBCs, the HRBCs catch up and follow the Pf-T-IRBC, mimicking "train"

formation in the RBCs-leukocyte interaction [20]. One or two HRBCs leading

the train lean against the Pf-T-IRBC, pressing the Pf-T-IRBC to the wall of the

channel (Fig. 7b). Since HRBCs in the back are tightly packed, even when one

of the leading HRBCs passes away, the next HRBC presses against the Pf-T-

IRBC. This continuous interaction results in the margination of the Pf-T-IRBC,

in which it exists near the wall for a long time period. Figure 8 shows the time

histories of the distance between the Pf-T-IRBC membrane and the wall surface,

where a distance of around 0.4 m indicates contact with the wall, due to the

spatial resolution of the simulation. The Pf-T-IRBC is in contact with ECs

almost all the time in the high Hct condition.

By contrast, the Pf-T-IRBC in Hct = 11% can flow freely in the channel as

shown in Fig. 8. The Pf-T-IRBC and HRBCs form a short train (Fig. 7a), so the

Pf-T-IRBC can marginate and contact instantaneously with ECs in the same

manner as the high Hct condition. This interaction, however, does notcontinuously occur in the low Hct condition. The surrounding HRBCs move

away from the Pf-T-IRBC, and then the Pf-T-IRBC immediately comes back to

the center of the channel (Fig 7a). Until when the following HRBCs catch up to

the Pf-T-IRBC, the Pf-T-IRBC flows around the center of the channel.

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a

b

Figure 7. Flow of RBCs in a circular channel with a diameter of 12m: (a) Hct = 11%; (b) Hct =

27%. The purple cell indicates a Pf-T-IRBC.

Time [s]

Distancefromwall[m]

0.0 0.5 1.0 1.5 2.00.0

1.0

2.0

3.0

4.0

Hct=27%Hct=11%

Figure 8. Distance between Pf-T-IRBC membrane and wall surface.

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4. Conclusions

In this article, we have reviewed our recent studies on computational

biomechanics. In considering clinical applications, however, we needs to

consider biological complexities in the analysis of blood flow, especially with

respect to disease processes. A disease is not just a failure of machine. It is an

outcome of complex interactions among multi-layered systems and subsystems.

They mutually interact across the layers in a strongly non-linear and multi-

variable manner. It is also noteworthy that a living system, either as a whole or

as a subsystem, such as the cardiovascular system, is always under the

integrated nervous and humoral control of the whole body, i.e., in homeostasis.

Multiple feedback mechanisms with mutual interactions between systems,

organs, and even tissues provide integrated control of the entire body. These

control mechanisms have different spatial coverages, from the micro- to the

macro-scale, and different time constants, from nanoseconds to decades.

Though it has not been fully acknowledged, much longer time scale phenomena

such as evolution and differentiation of the living system must also be paid full

attention if we are to understand the living system per se. In future analyses,

therefore, these biological phenomena need to be included in discussing

physiological as well as pathological processes. We expect this to be

accomplished in the future by integrating new understanding of macro-scale and

micro-scale hemodynamics, in addition to advances in related sciences and

technologies.

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