microfluidics (colin/microfluidics) || physiological microflows

Chapter 4

Physiological Microflows

Physiological fluids, such as blood, lymph, bile, sweat or air, keep the body in good functioning order thanks to the microscopic vessels carrying them to different kinds of cells in order to nourish them and eliminate waste. For blood and lymph, the connection is made via a microvessel network so dense that the mean distance between two vessels is of about 10 microns. If you prick yourself with a needle, a drop of blood will appear immediately in any part of your body (except for the eye ball).

We will now explain the microcirculatory blood network, underlining the “tricks” that optimize flow, i.e. allowing the carrying of blood to any point in the network with minimum dissipation of energy.

4.1. Description of the microvascular network

4.1.1. Topology

The microcirculatory network is made up of interconnected vessels – arterioles, capillaries, venules – the diameter of which is less than 300 µm. In the human body, the total length of these vessels is estimated to be several thousand km. If we try to visualize the aorta section to the total section of the terminal capillaries, the ratio is 1 to 800. Microscopic observation of a muscle section shows a density of 100 to 1,000 capillaries per cm2: the external surface exchange of the network is estimated to be 7,000 m2 [KRO 19].

Chapter written by Jacques DUFAUX, Marc DURAND, Gérard GUIFFANT and Kristine JURSKI.

122 Microfluidics

It is impossible to observe the microcirculatory network by means of non-invasive techniques in the human body, except in the case of the retina, the conjunctiva and the lunula. Most of that vast network is unknown territory, although it contains half of man’s blood volume. Nevertheless, studies on small anesthetized mammals have allowed scientists to collect the information needed to comprehend microcirculation. These experiments are mainly carried out on thin, transparent slices of tissue taken from the mesentery and the rat cremaster (in other words the thin, richly irrigated envelope of the testicle), the bat wing, guinea pig cheek, rabbit or mouse ear. Special devices have also been inserted into the back of a rat or into a rabbit’s ear. Three dimensional studies are extremely rare.

Figure 4.1. A rat cremaster (X 10) [SMA 70]

In spite of the aforementioned difficulties and of the incomplete collection of data, it is at present reasonable to think that the main characteristics of microcirculation in mammals are only slightly different from that of humans and can thus be applied to man.

Observation of the microcirculatory network shows the complexity of its organization. However, some networks – such as in the case of the rat cremaster – present a treelike structure: one vessel splits into two vessels on the arteriolar side and the reverse on the venule side. The capillary level can sometimes be shunted by a thoroughfare vessel either temporally or permanently if the network is overloaded.

It is difficult and indeed can be impossible, to associate structure and function [PRI 98]. Let us quote the case of polar animal paws: their arterioles and venules are parallel and very close to each other, achieving a counter-flow heat exchange with minimum loss to the external medium.

Physiological Microflows 123

4.1.2. Blood

4.1.2.1. Composition

4.1.2.1.1. Red blood cell

Without deformation, the red blood cell (RBC) or erythrocyte has a biconcave shape. Its diameter is about 8 µm, its thickness is 2 µm at its external border and 1 µm at its center, its volume is about 90 µm3 and its surface 140 µm2. It is like a water bag that is two-thirds full, so its ability to be deformed is very high. The dispersion of the values is very small. It is impossible to find such a well-calibrated suspension of particles on the market. The concentration of the particles is high: 4-5 million per mm3.

Figure 4.2. A human red blood cell (photo LBHP)

Surrounded by polysaccharides, the RBC consists of a membrane composed of two layers of lipids in which proteins are embedded. These molecules play an important role in aggregation.

Other molecules form a network of microfilaments under the inner surface of the membrane. This so-called skeleton and the asymmetric distribution of lipids in the double layer determine RBC deformability characteristics. The inside of the RBC is filled with an aqueous solution of hemoglobin (Hb). Hb transports oxygen and some carbon dioxide. Let us note the rheological importance of Hb encapsulation. The viscosity of Hb solution is 8 mPas, but for the same Hb concentration is 4 mPas when the Hb molecule is encapsulated. Another beneficial effect of encapsulation is oxygenation. The capacity of 1cm3 of human RBCs to carry oxygen is 2 cm3. If RBCs are hemolyzed, the Hb is dissolved in the plasma and its oxygen-carrying capacity falls to 0.05 cm3.

124 Microfluidics

4.1.2.1.2. White blood cell

In blood circulation the white blood cell’s (WBCs or leucocytes) shape is generally a sphere of about 10 µm in diameter in a passive state. In a non-pathological situation (5,000 WBCs per mm3) the WBCs do not disturb the hydrodynamics of circulation, although they are much less deformable than RBCs and their transit time in the capillaries is much higher than for RBCs.

In a pathological situation, when their number increases dramatically WBCs can cause a more or less permanent blockage of flow in the network. Moreover, the WBC membrane can be activated, mainly at the venule level. The WBC sticks to the vessel wall, loses its spherical shape and emits pseudopods before passing through the wall. These modifications can cause a temporary blockage of blood flow. This is why some doctors recommend deleucocyted blood transfusion to reduce flow perturbation during major operations [SCH 75].

Hemorheology is unable to distinguish the rheological differences between various kinds of WBCs. The only technique allowing us to determine WBC deformability uses a micropipette (the internal diameter of which is 1 µm). The micropipette is used to draw up a small part of the WBC membrane under a microscope and follow the evolution of its deformation in a given time. However, this technique is extremely tricky.

4.1.2.1.3. Platelets

The small dimension of platelets (or thrombocytes) means that they are unable to disturb blood flow even if they are relatively abundant (250,000 per mm3). If the vessel wall deteriorates, they stick to it and aggregate, forming a small clot (or “plug”) that cannot be broken by hydrodynamic forces alone. An emergency will occur if too many clots form in the microcirculatory network or if one of them moves to a vital area (such as the brain or heart).

4.1.2.1.4. Plasma

Plasma is the fluid in which RBCs, WBCs and platelets are suspended. Its viscosity is slightly higher than that of water. Its composition plays an important role in determining the rheological properties of blood, mainly for RBCs. For example, when its pH (approximately 7.4) varies slightly this causes a modification of RBC shape – they can become spheroids – thus their deformability decreases. Their concentration in protein (mainly fibrinogen) determines the aggregability of the RBCs.

4.1.2.2. Some remarkable properties of RBCs

4.1.2.2.1. Aggregability

When shear rate is low, even though they are negatively charged RBCs have the ability to form aggregates – or rouleaux – of 10 to 20 cells. This phenomenon is reversible: if the shear rate increases, the aggregate disintegrates and the blood cells become unstuck. In pathological cases, the aggregate is much more compact. It can be formed by hundreds of cells. In this case, its shape is spheroid and its diameter can reach 200-300 µm. RBCs are much harder to dissociate in this form. The increase in aggregation can be due to:

– an abnormal concentration of plasmatic protein, mainly fibrinogen and globulin;

– an alteration of the membrane structure;

– an abnormal concentration of Ca+ ions in the plasma;

– a modification of the pH of the medium.

Figure 4.3. Different levels of aggregation for human RBCs. Dextrans of different molecular weights have been used instead of fibrinogen (photo LBHP)

Several theories have been proposed to analyze the phenomenon of aggregation. At present the most widely accepted implies a lower concentration of protein molecules in the gap between two RBCs (when both are close to each other). The difference in the osmotic pressure tends to make them draw closer. This is called aggregation by depletion [SNA 96a, SNA 96b].

The aggregation rate can be measured with more or less sophisticated devices. The simpler, less accurate device measures the sedimentation rate of a blood sample. The interaction forces between the cells can be determined by means of a device called an erythro-aggregometer. The principle is the following: the blood sample

126 Microfluidics

(with added anticoagulant) is lit by a thin laser beam and subjected to decreasing shear rates. As the back-scattered light is a function of the aggregation rate, it is easy to obtain the shear rate for which the aggregates are completely disintegrated: γdisag. The value depends on the hematocrit (or packed cell volume). It is better to use the disaggregating shear stress τdisag, which is independent, but it is also necessary to know the viscosity of the sample at that specific shear rate. The measurements obtained with this device are accurate. Important differences do exist between the values obtained for control subjects and those obtained for patients suffering from a disease.

4.1.2.2.2. Deformability

RBCs are produced in the bone marrow. An RBCs nucleus is rapidly lost, which explains why its inner volume is reduced but not its surface. This so-called “bag” with two-thirds of its volume filled can change shape without any difficulty when passing through a vessel, the diameter of which is smaller than its own. After 120 days, RBC deformability decreases and the cell is trapped in the spleen or bone marrow, where it is destroyed. The measurement of the transit time of a RBC suspension through a filter (the pores of which have a diameter smaller than the RBC’s) gives an estimation of RBC deformability. These results are blurred, however, for a number of reasons. For example, it is difficult to take RBC interaction with the pore wall into account.

Figure 4.4. Passage of a RBC through a 5 µm diameter capillary [COU 9]

At present, the most reliable results are obtained with a system measuring the passage through one pore. The filter divides a basin filled with physiological serum into two parts. The RBC suspension is put into the first one and a voltage difference is applied between them in order to make the RBCs migrate from one side to the other. The electric resistance of the basin is measured. Considerable change is observed during the passage of a RBC through the pore. The duration of the passage gives an index of deformability.

4.1.2.3. Rheological properties of blood and of RBC suspensions

The rheological properties of blood are governed by the RBCs, their number (4 to 5 x106 per mm3) being far greater than that of WBCs and platelets. For 5 L of

blood, there are 2 L of RBCs, two tablespoons of WBCs and one tablespoon of platelets!

In a stationary state, blood viscosity η (with added anticoagulant) is a function of the hematocrit (RBC concentration), of the shear rate and of the plasma composition. Blood is a non-Newtonian fluid. Let us note that the concentration level for maximum packing is extremely high: over 90% to reach infinite viscosity. High-level sportsmen therefore have a long way to go!

RBSs in plasma serumRBCs in physiological serumHardened RBCs

110 -2 110-1 10 102 103

Aggregation Deformation+ Orientation

γ (s -1) Figure 4.5. Comparative viscosity related to plasma viscosity for

various RBC suspensions (adapted from [CHI 70])

The three suspensions have the same hematocrit: H = 45%; the suspension fluids have the same viscosity cp2.1p =η . The preparations are the following:

– normal RBCs in suspension in the plasma in which the globulin and especially the fibrinogen increase (reversible) RBC aggregation;

– normal RBCs in a saline solution (in which there is no RBC aggregation);

– hardening of the RBCs caused by glutaraldehyde.

The comparison of the three rheograms allows us to highlight the distinct effects of RBC aggregation and RBC deformability on blood viscosity. At a low shear rate, the effects of aggregation are predominant and dramatically increase the viscosity. At a high shear rate, the deformability and orientation effects are the same for the

128 Microfluidics

first two preparations. This is not the case for the third preparation, the viscosity of which is much higher [CHI 70, SKA 72].

The complex rheological behavior of the blood suspension delays the quantitative representation of its rheological properties by means of a simple behavioral law that can be used routinely [CAS 59, QUE 78], and not just in the aforementioned idealized cases.

4.1.3. Blood vessels

Blood vessels are made of different kinds of fibers. These give them longitudinal and radial deformability when their internal pressure varies, leading to an increase or decrease of the volume of blood they contain. Vessel compliance C is defined by:

∆S is the increased surface of the vessel lumen for a ∆P increase of the internal pressure. Compliance increases in line with the elasticity of the wall. When C = 0, the vessel is rigid.

Pressure ( mm Hg)

10 3020 6040 500 80

Small artery

Meta-arteriolePrecapillary arteryl

Physiological zone

Figure 4.6. Variation of the diameter of the frog arteriole with internal pressure variation. Adapted from [SMA 70]

4.1.3.1. Microvessel wall components

Elastin sheets and collagen fibers are the main elastic constituents of the arteriole wall. At rest, collagen fibers form a loose sheath around the vessel, which stiffens in

the case of a sudden increase in blood pressure, thus avoiding too great a distortion. The permanent shape of the vessel also depends on a third constituent: the smooth muscle. The terminal capillaries are made up of a single layer of endothelial cells surrounded by a base membrane. They are barely deformable, if they can be deformed at all. Cells called pericytes cover the outer wall at some points which, under the influence of local factors, can act as sphincters, opening and closing the capillary. The capillaries do not all open simultaneously. They vary in number according to the organ in the body: there are many at brain level and a few at the dermis and epidermis level.

Endothelium Elastic tissue Smooth muscle Fibrous muscle

Aorta 2 45 23 30 Small arteries 3 19 55 23 Pre-capillary arteries 10 8 67 15

Capillaries 100 0 0 0 Venules 17 83 0 0 Cave vein 2 23 37 38

Table 4.1. Different components of the wall vessels represented in %. Taken from [BUR 65]

Rat mesentery mmHgcmC /10 27×

Capillary 6-10 mμ 1.8 Venule 25-40 mμ 28-148 Skeletal muscle of the rat Arteriole 45 mμ 33 Capillary 6 -10 mμ 0.45-1.4

Table 4.2. Some compliance values [LBHP]

The data were obtained with observations on the rat. Studies were also done on the frog [SKA 86, SMA 70]. As all the techniques are invasive, there are unfortunately no results on microcirculation in humans.

4.1.3.2. The role of compliance

Microvessel compliance varies according to its function. Venule compliance is very high, so the venules are very maliable and the venule network plays the role of a reservoir, modulating the volume of blood. When the reservoir fills up it only

130 Microfluidics

modifies the local pressure slightly. Arteriole compliance is lower. Controlled by the sympathetic nerve network, the arteriole network plays the role of a pressure tank, toning down the pulsatile component of the heart pressure wave, thus regulating microcirculation blood flow.

The Marey experiment illustrates the role of wall elasticity: the flow rate in elastic tube 2 is steady and higher than in rigid tube 1 (whose length and diameter are identical). Flow in tube B can be regular, providing the rhythm of successive flow interruption is rapid enough.

water tap

handle Distensible duct (2)

Test-tube

Rigid duct (1) Distensible ductA

Figure 4.7. The Marey experiment

4.2. Blood flow: an unusual means of transportation

Poiseuille [POI 40] and Hagen [HAG 39] were among the first people to study the flow of different fluids in rigid tubes and to characterize them by their viscosity. They simultaneously carried out experiments on the microcirculatory network of frogs lungs. Although frog RBCs do not aggregate and are nucleated, the observations published in a report written for the Science Academy in 1835 are very similar to those obtained in small mammals. The RBCs flow through the core of the blood vessel and a thin layer of liquid is situated near the wall. Poiseuille understood that this layer could facilitate the flow and exchanges with the external medium. Poiseuille’s works are an essential moment in the history of haemorheology [POI 40]. For rigid tubes, the Hagen Poiseuille formula is given by:

)L8/()Pr(Q 4 ηΔπ= [4.2]

where PΔ is the pressure drop between the entrance and the end of the microvessel, Q is the flow rate, r the radius of the vessel, L its length, and η the fluid viscosity.

Unfortunately, the application of this the law governing the flow rate in small vessels leads us to disregard blood variation in viscosity and to favor variations in radius because the radius is at power 4. Scientists have only over the past 20 years understood that blood viscosity depends on flow conditions and that these must be taken into account in pathological situations for humans.

Figure 4.8. An RBC aggregate in a venule (diameter = 50 µm) (photo LBHP)

Ever since Poiseuille’s works over 100 years ago, many experiments have confirmed that the physical characteristics of blood are very different in the microcirculatory network and the macrocirculatory network [CAR 78, FUN 97a, FUN 97b, WAY 82]:

– blood flow velocity decreases from the arterioles to the venules: from 1 mm/s to less than 0.1 mm/s, with shear rates varying from 1,000 s-1 to 1 s-1;

– shear rate in the capillaries can reach 1,000 s-1;

– blood viscosity depends on the shear rate: the non-Newtonian behavior of blood is remarkable;

– hematocrit is lower than systemic hematocrit: this is what is called the Fårhaeus Lindquist effect [FAR 31];

132 Microfluidics

– the pulsatile component of the cardiac pulse gradually disappears in the arterioles and disappears totally in the capillaries and venules [LIP 80, WIE 64];

– blood flow is nonturbulent;

– viscous forces are greater than inertia forces: the flow is sluggish.

Use of the Poiseuille Hagen law is satisfying for the interpretation of phenomena occurring in the arterioles and venules. This law allows us to introduce the notion of hydraulic resistance [NEL 77, ZWE 81]: P/QR ΔΔ= . Experimental data are scarce, however, and this approach has only allowed us to model very small networks (with a surface of about 2 mm2) in the laboratory [POP 87, SCH 88].

4.2.1. Description of blood flow

4.2.1.1. Arteriovenular circulation

When the vessel diameter is greater than the RBC’s, flow is governed by the importance of the plasmatic layer, which can be considerable in the venules where its thickness can reach 10 µm.

Figure 4.9. Variation of the apparent viscosity of different RBC suspensions in a capillary tube (diameter 100 µm) with shear rate

Aggregating suspensions

ηa (cp)

s( 1−γ

0 20 40 60 80

ηa(cp)

γ·(s-1)

The most aggregated suspension shows a pronounced decrease in apparent viscosity in the region where the aggregation is very high and the plasmatic layer thickness is greatest. At very low shear rate, the aggregate size is higher than the tube diameter and causes an important increase in viscosity [DUF 82].

0.5 1.5 2.5

Capillary radius (mm)

Figure 4.10. Variation of the viscosity of a RBC suspension according to the diameter of the capillary in which it flows (adapted from [HAY 60])

The presence of the plasmatic layer has multiple physical consequences:

– It acts as a lubricant layer: ( )app core tube1 2 / D ;η = η − δ where δ is the thickness

of the plasmatic layer and D the tube diameter. The apparent viscosity of the suspension appη is lower than its core viscosity coreη . This is what is called the

Fårhaeus effect [FAR 29].

– The radial velocity profile is blunted. The shear rates are lower in the center of the vessel and as a consequence the aggregation is higher than it would be in an ordinary flow. The RBCs flow as a clot (or plug) faster than the serum in the plasmatic layer, therefore the main variation in shear rate occurs in the plasmatic layer.

After branching from one to two vessels, RBC distribution does not often remain uniform. The two phases split and the RBCs flow in the vessel where velocity is the highest. This is called plasma skimming [SCH 80].

As we have already mentioned, modification of the aggregation rate can be beneficial and lead to a decrease in the viscosity of the RBC suspension, but only if the hydrodynamic forces remain high enough to destroy the aggregates if necessary.

134 Mi

If the shvessel, tblockage

Figure 4

The excessivreturn toflow rateregulatin

– theopening

– thewall, mothe role o

– somproblem

– lastcan be pand chan

The hcapillarie

icrofluidics

hear rate is nthe diameter oe can occur. T

4.11. RBCs pass

dysfunctions ve aggregationo normality cae (achieved vng mechanism

e progressiveor to pressure

e local vasomodifying its diaof a local pum

me vessels cas when the oth

tly, at the venushed against

nge shape.

hydrodynamices. They are s

ot sufficient of which is sm

This is what is

sing through a bplasma s

of this part n rate. As RBan be obtainedvia the action

ms, such as:

e activation oe differences b

motion phenomameter in a ra

an locally shher mechanism

nule level, whet the wall. In a

c characterististrongly depen

to break the maller than itcalled the sta

branch. It is easskimming (phot

of the netwoBC aggregatiod without extof a drug lik

of ineffectivebetween differ

menon causesange that can r

hunt a part ofms are ineffici

ere RBC aggra pathological

ics of the phendent on the R

aggregate whts own, a tem

asis phenomen

sy to observe thto LBHP)

ork are therefon is a reversternal intervenke aspirin) or

e microvesselrent sections o

periodic conreach 100%. t

f the capillaryient;

regates can bel situation, the

enomena haveRBC aggregat

hen it flows tmporary or a dnon.

heir parachute s

fore mainly dsible phenomention by incre

through the

ls thanks to of the network

nstriction of ththis phenomen

y network, pr

e the largest, they can stick to

e been verifiedte size and are

through a definitive

shape and

due to an enon, the

easing the action of

sphincter k;

he vessel non plays

reventing

he WBCs o the wall

d in glass e not only

caused by chemiotactism. If the number of WBCs is abnormally high, in the case of an infection for example, the vessel can be partially or completely occluded. This can cause a huge increase in hydraulic resistance.

4.2.1.2. Capillary circulation

The blood flow in vessels with a diameter smaller than RBCs is much more complex.

The RBCs are identified and are not uniformly distributed, in space and time, in this part of the network. Their velocity varies rapidly. Some of the vessels empty their cells quickly and then fill up again with long rows of cells that are relatively close to each other. At one given moment it is estimated that only 20-30% of the capillaries are in activity in an organ at rest. This percentage increases when the organ becomes active: this is what is called the recruitment phenomenon.

On this scale, the notion of viscosity has no meaning and blood cannot be considered a homogenous suspension. The movement of each RBC must be studied. The cell must change its shape in order to pass through vessels that have a diameter that smaller than its own because the capillary walls are only slightly deformable. In a capillary under the effect of shear stress, the cell shape is similar to a parachute. There is a narrow space full of plasma near the wall [LEW 69]. This lubricating layer enables the RBCs to slide though the capillary easily. Of course, the transit time depends on the cell deformability. In some diseases (e.g. sickle cell disease and congenital spherocytosis) the capillary network can be blocked for a relatively long period.

1 3 2 0 4 5

R H / R 0

Hardened RBCs (%)

Blockage of the network

H = 0,40

Figure 4.12. Variation of the hydraulic resistance versus the proportion of hardened RBCs added to a normal RBC suspension (H = 0.40) adapted from [DUF 82]

136 Microfluidics

There are no irreversible effects on the body as long as the number of blocked vessels does not reach a threshold value. Beyond this value, the increase in hydraulic resistance can cause blood pressure to jump considerably, and consequently lead to added heart work.

As we know, WBCs are normally inactive in the blood flow and are almost spherical. Their low deformability means their transit time in the capillaries is higher than for RBCs. However, their low number means that they do not disturb the functioning of the microcirculatory network. On the other hand, in a pathological situation their number can drastically increase and the hydrodynamic disturbance can be significant.

As is the case for arterioles and venules, the effect of the platelets is only considerable when vessel wall lesions occur.

4.2.2. The mechanisms maintaining blood flow in the circulatory network

4.2.2.1. At the macrocirculation level

4.2.2.1.1. The heart

The heart, the main motor, can be assimilated to two pumps connected in series. A reversal pump – the left ventricle – at high pressures, from 110-140 mm Hg above atmospheric pressure, propels the blood throughout the network. At low pressures, from 25-30 mmHg above atmospheric pressure, the right ventricle (a suction pump) drains the blood from the inferior and superior venae cavae. The heart has remarkable qualities; as big as a fist, its mass is about 300 g (0.5% of the body mass). It is very richly and delicately irrigated, allowing the blood to run through thousands of km of blood vessels of various diameters.

Let us give a rough estimation of the work T, it performs during a cycle. Let us consider the left ventricle.

For an ejected volume dv of 100 cm3 per second:

T Pdv 1.2 J= = or a power P = 1.2 W

If the powers of the three other cavities are added, the heart power can be estimated to be 1.5 W.

In a 24 hour study, the heart at rest could fill a tank of almost 10,000 liters! To obtain a water pump with a similar performance, the catalog proposes a 400 W pump. This power must be compared to the power used by the heart muscle to work itself, 10 W, i.e. an efficiency of 15%. In fact, as the body metabolism provides a power of 100 W, the heart is nevertheless a great consumer of energy [SUG 79] considering its mass.

During effort (for example when exercising), power consumed by the heart can be multiplied by a factor of five to 10.

4.2.2.1.2. Secondary mechanisms

The action of the reversal pump is reinforced by different mechanisms that help the body to carry the blood back to the heart, particularly on the venular side:

– the right ventricle suction pumps, due to depression and lung movement during the inspiration phase of breathing;

– flattening of the arch of the foot: its venule circulatory network is richly irrigated, thus the walker periodically crushes this sort of sponge propelling the blood it contains towards the heart;

– the calf muscle pump: its contraction and release provokes a reversal cycle towards the right side of the heart because of the considerable pressure it exerts on the veins.

For the latter two mechanisms, valves, which are cups located regularly along the veins, prevent the blood from returning to the capillary network.

4.2.2.2. At the microscopic level

4.2.2.2.1. Vasomotion phenomena

The microscopic observation of microcirculatory flow shows a comparatively periodic and spontaneous contraction, according to the diameter of the arterioles. For the terminal arterioles (diameter = 20 µm), the frequency is about 20 cycles per minute, with an amplitude of 100%. For the 100 µm arterioles, the frequency and amplitude decrease: one to three cycles per minute for the frequency and 10-20% diameter for the amplitude. The profile of the velocity wave or pressure associated with it can vary from an almost perfect sinusoid to a very complex profile. These two situations can be observed in areas that are very close to each other [INT 73a].

Vasomotion is more active in pathological situations. Thanks to the modulation of local pressure, variations in diameter improve the flow and exchanges with the external medium.

138 Microfluidics

Figure 4.13. The vasomotion phenomenon for a 100 µm arteriole [INT 89]

4.2.2.2.2. Chemical mediators

Flow rate is also locally controlled by the smooth muscles surrounding the arterioles, which are controlled by nerve endings. The reactions depend on the chemical composition of the blood and the interstitial liquid. Let us list some of the chemical mediators that intervene in microcirculatory regulation:

– adrenalin shuts the sphincters;

– histamine loosens them;

– acetylcholine tenses the arterioles;

– serotonin tenses the venules.

4.2.3. Exchanges with the external medium

The greater part of exchanges between blood and external medium occur are at the level of the capillaries, post-capillariy venules and metarterioles. The variety of molecules crossing the wall is remarkable: there are water molecules, oxygen, carbon dioxide, ions with low molecular mass, sodium ions, HCO3 ions, urea,

organic molecules (molecular mass M < 60) and even lipoproteins (M = several million).

Different mechanisms are involved, of which we will present three:

– transport by diffusion;

– transport occurring through spaces located in the vessel wall; and

– transport carried out by vesicles.

4.2.3.1. Transport by diffusion

The wall of the microvessels participating in the filtration process is generally made of a unicellular layer and can be considered to be a semi-permeable membrane. At this level, the hydrostatic pressure and osmotic pressure inside and outside the vessel are of the same order. According to their respective values, the nutritive products will be filtered from the blood towards the tissues surrounding the vessels. The waste produced will be adsorbed and removed via the microvessel. The balance is governed by the Starling law:

)()( tissueplasmatissuecap ppp Π−Π−−=Δ

where pcap and ptissue are respectively the hydrostatic pressure in the capillary and in the tissue, and πplasma and πtissue are the osmotic pressure in the plasma and in the tissue.

Figure 4.14. Values of osmotic and hydrostatic pressure in a capillary in normal conditions (left) and in pathological conditions (right), adapted from [FRI 71]

140 Microfluidics

If 0p >Δ , the exchange occurs towards the outer environment and the opposite takes place if 0p <Δ . Some values are given in Figure 4.14. The balance is not exactly zero because a small portion of the filtrated molecules has been collected by the lymph network.

It is worth having an estimation of the time needed for the molecules to reach the cells they have to supply. The mean distance l to travel varies from 5 to 10 µm. The mean diffusion time is D2/t 2= where D is the diffusion coefficient of the molecule. With 125 scm10D −−= in water, diffusion time becomes:

2ot 0.04 s.= As

the length of a capillary is about 1 mm and the velocity of blood about 1 mm/s, the lifetime of a RBC in the capillary is 1 s. This is long enough to allow the exchange.

4.2.3.2. Transport occurring through spaces located in the vessel wall

Transport through spaces located in the vessel wall is the preferred way for water molecules and molecules soluble in water. The interval between spaces is about 10 mm.

4.2.3.3. Transport carried out by vesicles

The transport mechanisms carried out by vesicles involve big molecules. The probability that such large molecules will pass through the vessel is low but can be increased if a vesicle surrounds the molecule. A vesicle is made up of one or two bilipidic layers enclosing a portion of the liquid medium bathing them. Their diameter can reach °A15 .

Mechanisms of fusion or rupture of the membrane of the microvessel govern passage through the external medium: as soon as the rupture occurs, the molecules included in the vesicle can diffuse in the external medium.

Each dysfunction from these exchanges will produce an accumulation of waste in the tissues and may form, for example, edema due to water. In a tumor the hydrostatic pressure is higher than the pressure in a healthy, irrigated network. Drugs enter such areas with more difficulty. With a microemulsion injection of calibrated particles near tumor tissue, we have shown a disruption of the flow rate and verified that a greater quantity of the anticancer drug reached the tumor tissue [DEB 95].

The exchanges are not only exchanges of substances: the microcirculatory network is a place that allows heat exchange, with appropriate topology.

4.3. Instrumentation

The last decade has seen increased interest in the mechanical interaction between the heart and the vascular system. It has been suggested that several pathologies are closely related to vascular dysfunction. An early prediction may contribute to early diagnostics of several angiopathological diseases, such as diabetes mellitus and glaucoma. The microcirculatory perfusion and regulation measurements are therefore relevant for the assurance of normal function and metabolism. In order to characterize and quantify the blood flow, several methods, some invasive, some non invasive, have been developed. However, they have to satisfy definite criteria: they must be applicable from several µm up to 300 µm networks, to the animal model as well as humans in clinical practice.

In the following section, methods of blood pressure and flow characterization are described. We also give a review of tissular perfusion indicators and metabolic parameter monitoring devices as complement to diagnostics.

4.3.1. Intravascular pressure determination

Blood vascular pressure (as small as 20 µm vessel diameter) techniques are divided into two groups: passive and active methods.

In passive methods, the intravascular pressure is measured by a micropipette impaled in the vessel and connected to a micropressure measuring system. Using a pressure transducer-micropipette system with minimal volume displacement, Rappaport et al. [RAP 59] recorded pulsatile pressures in arterioles as small as 30 µm in diameter. However, because of the time response (several minutes), this method is not suitable for rapid pressure change measurements [INT 73].

In active methods, microvascular pressures are measured with the servonulling device and micropipettes filled with an electrolyte [LAN 26] (see Figure 4.16). A change in electrolyte composition within the tip of the micropipette will give rise to a marked change in its electrical impedance. Similarly, when the tip enters the lumen of a blood vessel, the higher pressure in the lumen tends to force serum into the tip, thus changing the impedance. If a counter pressure exactly equal to the external pressure is applied within the pipette, entrance is prevented and the pipette impedance remains constant. This counter pressure is generated by a servo system, which senses a change in pipette impedance and translates it into a pressure change within the micropipette.

Wiederhielm et al. [WIE 64] incorporate the micropipette in one arm of a Wheatstone bridge (see Figure 4.15).

142 Microfluidics

Figure 4.15. Micropipette and Wheatstone bridge, as defined by Wiederhielm et al. [WIE 64]

When the impedance of the pipette is increased, the bridge becomes unbalanced, giving rise to an error voltage. A current, proportional to the error signal, passing through the coil determines the force applied to the bellows and, thus, the pressure generated within the bellows. With rapid time response, this system makes it possible to record pulsatile pressure within the microcirculation for prolonged periods of time.

By using a similar procedure with two bridges and two sensors, the method can be extended to differential pressure measurements.

Figure 4.16. Pressure recorded in frog mesentery vessels as a function of vascular diameters (from Landis [LAN 26] and Wiederhielm [WIE 64])

4.3.2. Blood flow determination

There are many means of monitoring by which blood flow can be characterized and quantified, ranging from qualitative local observations to sophisticated quantitative global systems.

4.3.2.1. Optical methods

4.3.2.1.1. Cutaneous capillaroscopy

Capillary microscopy is the most direct method of observing blood flow. In normal skin, the structure of capillary loops and scattering properties of the dermis are such that only the apex of the loop can be observed (see Figure 4.17). By observing these loops, it is possible to estimate the speed of the red blood cells by assessing the transit time while in the field of view.

This technique involves placing a finger on the microscope stage, to position the area of skin at the junction of the cuticle and nail under the low-power objective (1 to 2 cm between the skin and the objective). The light is adjusted so that it is focused to strike the region at about 45°.

Figure 4.17. Periungual capillarography

Video capillary microscopy was the natural development of this method.

4.3.2.1.2. Photoplethysmography

This non-invasive technique consists of a constant infrared light source directed at the capillary bed. Some light is absorbed; the rest is reflected and detected by a photoconductive cell [JON 84]. The amount of light absorbed is supposed to be dependant on the increase of in local blood volume alone. Strictly speaking, this hypothesis is not exact [ROB 82].

144 Microfluidics

The method seems accurate; however its interpretation is relatively complex. Moreover, the technique can not be used for a depth greater than 2 mm.

4.3.2.1.3. Oximetry

This method is based on the red and infrared light absorption characteristics of oxygenated and deoxygenated hemoglobin. Oxygenated hemoglobin absorbs a greater proportion of infrared light and allows more red light to pass through [LIN 91]. Deoxygenated (or reduced) hemoglobin absorbs a greater proportion of red light and allows more infrared light to pass through. The system contains two diodes (red and infrared) and a photodetector that receives the light that passes through the measuring site (see Figure 4.18). The pulse is absent in an arterial occlusion. The oxygen saturation decreases in venous occlusion.

Figure 4.18. Oximeter (from CHU Nord Amiens)

4.3.2.2. Thermal methods

4.3.2.2.1. Thermocouple probe

Skin temperature measurement is frequently used to estimate blood flow. Skin temperature depends on the blood flow provided that heat loss is constant and the temperature of circulating blood reflects a constant deep-body temperature.

The technique consists of using two matched thermocouples as skin thermometers: a difference of 2°C between the two indicates circulatory impairment (arterial occlusion, for example, if the temperature decreases locally [MAY 83]). The material is cheap and the technique easy to use, but the method is invasive and measurements are not accurate.

4.3.2.2.2. Thermal clearance

Thermal clearance has been used as a measure of tissue blood flow for over 60 years. The method measures the variation of skin thermal conductivity, assuming modifications only arise from fluctuations in blood flow.

The device measures the rate of heat removal from the tissue volume under the probe. A relation exists between the flow rate and the rate by which heat dissipates from the tissue volume under study. The sensing unit is usually designed around a central metal disk and a concentric outer ring, between which a temperature difference is established. The two rings are thermally isolated and both are in contact with the skin. The temperature between the two rings is a measure of the blood flow under the probe.

These methods have not been extensively used because of their extreme non-linear properties and the highly variable thermal characteristic of the skin [DIT 82].

4.3.2.2.3. Thermography

Thermographic devices measure the infrared radiation that is constantly radiating from the surface of the human skin. According to the Stefan-Boltzmann law, infrared radiation is emitted by all objects based on their temperatures. Thermographic cameras detect radiation in the infrared range of the electromagnetic spectrum. The amount of radiation emitted by an object increases with temperature. The infrared imaging process allows us to visualize a local cartography of superficial temperatures [LOV 82].

4.3.2.3. Magnetic or electric methods

4.3.2.3.1. Electromagnetic flowmetry

Described by Fabre in 1932, this method consists of the detection of blood flow in a vessel by electromagnetic induction. The technique requires that a flow probe containing an electromagnetic field be placed around a vessel (see Figure 4.19). The strength of the electrical potential generated by a column of blood passing through the magnetic field potential is proportional to the volume of blood flow (several ml/mn).

Measurements are reliable, but the requirements for successful installation of the probe and calibration of the flowmeter are difficult and imply invasive action.

146 Microfluidics

Figure 4.19. Electromagnetic flowmetry

4.3.2.3.2. Irrigraphy

The irrigraphy method gives a topographic indication of blood circulation. From electric resistance or impedance measurements at different arteriovascular stages, variations in blood volume are detected.

4.3.2.4. Isotopic and fluorescence methods

4.3.2.4.1. Radioisotopic methods

The measurement of clearance of various radioisotopes from skin gives an accurate estimate of blood flow. This method is accurate and non-invasive. The blood flow is deduced from the exponential decrease in radioactivity as a function of time [FOR 80].

4.3.2.4.2. Fluorometry

When injected intravenously, fluorescein diffuses out of the capillaries and into the interstitial fluid, where it stains the skin yellow. This is visualized with ultraviolet illumination. Failure of an area of skin to stain implies inadequate perfusion. Staining takes 12 to 18 hours to clear after injection, however, restricting its application as a monitor of bloodflow [GAN 96].

4.3.2.5. Method for measuring the transcutaneous partial pressure

4.3.2.5.1. Transcutaneous gas measurement (oxygen) 2otcp

This method can be defined as the measurement of the reduction current produced by surplus oxygen molecules that have been available under the condition of maximal (

2otcp at 45°C) or slight hyperemia (at 37°C) for diffusion from the

cutaneous capillary loops to the skin surface [BEL 88]. It requires a small heated gas-sensing electrode to be connected to a compact gas analyzer, providing accurate

transcutaneous arterial pressure 2otcp . The method shows a good correlation

between 2otcp and blood circulation and a rapid response of

2otcp to clamping and

release of the supplying artery or vein. The method is invasive, however, and only weakly accurate.

4.3.2.5.2. Transcutaneous gas measurement (carbon dioxide) 2cotcp

Similar concept is achieved by the transcutaneous carbon dioxide measurement. This technique is similar to the

2otcp technique, and also has advantages and

drawbacks.

4.3.3. Velocity determination

4.3.3.1. Dual slit method

Using the dual slit measurement system, Wayland and Johnson [WAY 67] determined the velocity of RBCs in microvessels.

Two optical windows (slits) implemented through photodetectors are positioned parallel to the vessel axis in order to capture a time variant signal for each window caused by the passage of RBCs flowing through the vessels.

The separation between detectors causes the downstream signal to be delayed with respect to the upstream signal. RBC velocity is given by the ratio of the slit separation and delay between the two signals.

In the rat mesentery network, Lipowsky [LIP 82] measured blood flow velocity (several mm/s) in arterioles and venules with diameters of 10 µm.

4.3.3.2. Method for particle imaging

Particle imaging velocimetry (PIV) appeared in the early 1980s. Over the past two decades this measurement technique and the hard- and software have been improved continuously, so that PIV has become a reliable and accurate method for “real life” investigations.

The method is a non-intrusive optical measurement technique that consists of capturing several thousand velocity vectors within large flow fields instantaneously, based on imaging the light scattered by small particles in the flow illuminated by a laser light sheet. A detailed description of PIV and an overview of the improvements in PIV technique are given by Riethmuller [RIE 93, RIE 96].

148 Microfluidics

This method remains dedicated to in vitro velocity measurements in flow containing few diffusing particles.

4.3.3.3. Ultrasound Doppler velocimetry

Christian Doppler was the first to describe the frequency shift that occurs when sound or light is emitted from a moving source and the effect now bears his name. For the velocity measurement of blood, see Reid et al. [REI 69], Sigelman and Reid [SIG 73] and Shung et al. [SHU 76] for further details.

With this technique, ultrasound is transmitted into a vessel and the sound that is reflected from the blood is detected. Because the blood is moving, the sound undergoes a frequency (Doppler) shift that is described by the Doppler equation: by measuring the frequency shift between the ultrasonic frequency source, the receiver and the fluid carrier, the relative motion is measured. The technique is commonly used to estimate the blood flow rate in the macrocirculation [PER 91].

The frequency of systems using standard ultrasound techniques in medical situations ranges from 2-20 MHz. They are however limited to several mm for the spatial definition and to several mm/s for the velocities. In microcirculation, the velocity in the capillaries can be much lower than 1 mm/s, so the ultrasound Doppler velocimeter is inadequate [HAR 81].

4.3.3.4. Laser Doppler velocimetry

The first biomedical application of laser Doppler velocimetry (LDV) was the atraumatic measurement of blood flow in rabbit retinal vessels by Riva et al. [RIV 72]. Improving the method, Tanaka et al. [TAN 74] demonstrated that the same technique can be used in human retinal vessels. In the early 1970s, laser Doppler microscopes were developed to measure the velocity of RBCs in small vessels [EIN 75, MIS 74] as well as in various networks, like rabbit or rat mesenteries [BOR 75, LEC 76].

The first application of the laser Doppler technique to microcirculation was reported by Stern [STE 75]. In 1978 Powels [POW 78] developed a new LDV where optical fibers were used to guide the light to the measurement point. Since then, a number of medical applications of LDV with optical fibers have been implemented.

The LDV is a well-known, coherent technique that measures flow-induced Doppler frequency shifts by moving particles. The interpretation of the Doppler shift enables us to determine the velocity of particles in the flow. The LDV gives real time measurements and is suitable for short velocity variation measurements. The dependence between the shift frequency and diffuser velocity is simple to establish when the concentration of diffusing particles is low.

In vivo, photons generally suffer several collisions with somatic cells, connective tissue, blood vessel walls, etc., before interacting with a blood cell. The effect of such interaction is to randomize the direction of light that is incident upon the moving erythrocytes. Thus, only mean parameters can be determined.

4.3.3.4.1. LDV in flow with low particle concentration

When the particle concentration is very low, the signal can easily be interpreted: the velocity inside the probe volume is proportional to the frequency shift of the back-scattered light (the Doppler effect). The Doppler signal is linearly related to the velocity of seeded flow. For these applications, the LDV technique has lead to commercial 2D or 3D LDV devices, including the phase Doppler velocimeter.

A LDV system consists of a laser, a beam-separating device, a transceiver (which emits the laser beams and also collects the reflected signals), a photoreceptor unit to convert the optical signal to an electrical signal, a signal processor, and software to analyze the results. The most common LDV uses the description of fringes. When two coherent, collimated laser beams intersect, they form a fringe pattern. This process can be illustrated by two “beams” in parallel lines that intersect. At the point where the beams intersect, the wave fronts interact with each other constructively or destructively and form a pattern of horizontal lines that were not present before the beams intersected. This is the fringe pattern.

As a particle travels, it alternately reflects the light (as it passes through a fringe), and does not reflect light (as it passes between fringes). A signal detector that is focused on the beam crossing can pick up these minute flashes of light and determine their frequency. Light intensity versus time is processed by a Doppler burst signal processor. A temporal or spectral laser Doppler processing technique can extract the Doppler frequency of the signal (see Figure 4.20).

Figure 4.20. Spectral representation of the Doppler signal recorded in a seeded flow (low particle volume concentration)

150 Microfluidics

Each signal processing method has a typical application domain and conditions. Notice that the wavelet transforms, localized in both time and frequency, expand the signal in terms of wavelet functions.

4.3.3.4.2. LDV in turbid flow

In vivo there are a number of complications in applying this technique to the measurement of blood flow in tissue. The fibrous structures around the blood vessels tend to randomize the angle between the light and moving blood. A given photon may experience a number of collisions with moving scatterers, giving a complex dependence on the concentration of scatterers. The result of these effects is that only a mean flow can be described by Doppler spectra, which is dependent on the mean speed and number of moving particle interactions (see Figure 4.21). Other parameters also have to be considered: pigmentation, skin thickness, the local vessel structure and RBC quantity.

Figure 4.21. Spectral representation of the Doppler signal commonly recorded in vivo blood flow microcirculation

Even if the method is well established, the LDV does not completely meet the requirements of an ideal method for measuring skin blood flow, since it remains an arduous way of providing a quantitative measure of blood flow itself.

Therefore, various attempts have been made to improve the laser Doppler technique and Doppler spectra analysis. Bonner and Nossal [BON 81] developed a theoretical model for photon scattering within tissue in which they allow for RBC scattering, multiple Doppler shifting scattering events and a random angle of incidence. They demonstrate the non-linear manner in which frequency weight varies with increasing concentration of RBCs. They supported this with an in vitro demonstration of the predicted response.

Due to hematocrit (up to 20%), microcirculation is not strictly involved, implying that changes need to be made to the method in order to model the light diffusion in concentrated media [BIC 91, MAR 87, PIN 90]. From the mean free path, isotropization length and asymmetric optical factor, diffusing particles can be characterized in terms of light diffusion.

Using a Monte Carlo method, [SNA 95] simulated the trajectory of each photon in a diffusive media and calculated the corresponding frequency shift. When the medium became denser, because of multiple light scattering on moving or static particles, optical information was restricted to superficial layers. The interpretation of the spectral measurements required a theoretical effort taking the structure of vascular tissue, the phenomenon of multiple diffusion and velocity field of the particles into account.

The full Doppler frequency spectrum S(ω) of the scattered light emerging at the detector point is obtained by summing all scattering paths:

( ) ( ) dnnSPS n∫∞

⋅⋅=1

,ωω [4.4]

where the probability Pn of n diffusion paths is sensitive to boundary conditions (size and shape of the sample), detection geometry and optical properties of scatterers.

Snabre [SNA 98] and Chazel [CHA 99] propose an analytical model where light is considered as a collection of discrete photons. Each photon performs quasi-elastic scattering on an anisotropically diffusing particle in motion or at rest. The frequency shift of the light can be seen as the superposition of individual frequency shifts due to each scattering event:

– Simple shear flow (Poiseuille, Couette):

15exp1

ω⎟⎟⎠

⎞⎜⎜⎝

⋅⋅−−

⋅= [4.5]

with k the wave vector modulus and * the isotropization length;

152 Microfluidics

– Random flow:

Vk83exp1

ω⎟⎟⎟

⎜⎜⎜

⎛ −−

⋅= [4.6]

with 212V : mean quadratic velocity.

From Doppler spectra, Bonner and Nossal defined a blood flow indicator M1/M0, based upon the calculation of the first-order (M1) and zero-order (M0) moments of the Doppler spectrum. It characterizes a mean frequency that may be expected to linearly relate to the mean velocity in the test section, especially in a biological condition where the laser beam has multiple diffusions and interacts with a static structure, such as the skin.

( ) ωωω dSM ⋅⋅= ∫∞

∞−1 [4.7]

( )∫∞

∞−⋅= ωω dSM0 [4.8]

Snabre [SNA 98] and Chazel [CHA 99] highlighted that non-aggregated RBCs in a shear flow exhibit similar results to those obtained for latex micron beads.

4.3.4. Combined methods

Some authors have developed combined methods that present a simultaneous performance of several techniques at the almost identical sensing site under the same measuring conditions. For example, Francek et al. [FRA 94] suggest the combination of

2otcp measurements with complementary methods, like

capillaroscopy and LDV. A “triple-probe” device allows them to perform simultaneous skin oxygen pressure measurements, the evaluation of capillary morphological characteristics, capillary density and microangiodynamics by dynamic video microscopy at the oxygen-sensing site. All this is in addition to the continuous estimation of superficial microvascular skin blood flow by LDV.

4.3.5. Some examples of clinical application

In many cases a light probe is the only way to obtain information on blood flow in a tissue.

4.3.5.1. Plastic and reconstructive surgery

The aim here is to develop a device able to continuously give instantaneous information on the perfusion rate of a graft used to repair the human face in order to monitor the chance of a graft taking. The autologous graft can originate from different parts of the body (forearm, bone, etc.).

In this surgical procedure the flap is transferred from the donor site to the defect, and then the blood flow is re-established by microvascular anastomosis. For each surgical stage, the success of the transplant is linked to the quality of revascularization. Regarding the monitoring, a precise time representation of the modification of blood flow is required at the microcirculation level.

The detection of disturbance is essential at an early stage, because it allows a corrective re-operation to be carried out without delay, before irreversible damage occurs to tissues.

The graph in Figure 4.22 shows the results obtained by LDV with a forearm flap before, during and after a maxillofacial reconstruction operation [TES 00].

Figure 4.22. Blood flow evolution in a flap used for maxillofacial reconstruction, measured by LDV

154 Microfluidics

There are also applications in dermatology, where simple observations of skin blood flow are replaced with non-invasive measuring methods for assessing skin susceptibility to irritant trauma.

4.3.5.2. Ophthalmology

The conjunctive is the site where the microcirculatory network is most accessible and pathological situations can be observed. Non-invasive methods provide direct information about perfusion of the retina or optic nerve head. Early diagnostics of functional and morphological alterations of capillaries and several angiopathological diseases, such as diabetes mellitus and glaucoma, can be achieved [BER 94, HAM 94].

4.3.5.3. Ischemia

Ischemia is defined as an inadequate blood supply to a local area due to blockage of the blood vessels to the area. The vascular alterations before the critical ischemia impact the vascular blood flow and perfusion pressure [BAR 94, FRA 89].

Many methods for establishing a diagnostic can be considered: irrigraphy, 2otcp

measurements, and Doppler devices.

4.3.5.4. Microembolism detection

The blood flow network may be perturbed in a number of ways. Mechanical obstructions of blood flow may be caused by blood clots in the main blood circulation [SPE 90], gas microbubbles, fat droplets, etc.

Vessel obstruction can lead to different pathological issues, such as blood stasis and ischemia.

4.4. Description of flows and microcirculatory networks

4.4.1. Fluid flow in a duct, stationary conditions, non-stationary condition and Marey’s experiment

4.4.1.1. Low Reynolds number flow of a Newtonian fluid in a slightly deformable duct

The aim of this section is to give a simple description of a Newtonian fluid’s viscosity η and density ρ , flowing with low Reynolds number conditions in a cylindrical deformable duct. Moreover, it is supposed that the pressure variations are weak that they only permit small deformations along the length of the duct compared to the duct radius.

Using this hypothesis, it is clear that the flow solutions are not very different from the Poiseuille flow solutions. Nevertheless, because of the deformation of the duct, it is not possible to assume symmetry of translation, as in a Poiseuille flow: the rate of flow and pressure gradient are now functions of x and time t.

Let us consider a section of the duct of length xΔ and section S and SS Δ+ (see Figure 4.23).

a a + Δ

S S + Δ

Figure 4.23. Geometry and notations

PΔ denoting the variation in pressure along the length xΔ , it is possible to write the volumetric flow rate in the following way from the Poiseuille’s solution:

where 48a

η= is the hydraulic resistance. For an interval xΔ leading to zero, we

obtain the following expression for the volumetric flow rate:

∂∂1

The objective is now to express the mass conservation, taking the deformation of the duct into account. The flow and pressure being functions of x and time t, two conditions will be considered: one between t and tt Δ+ and the other between x and

xx Δ+ .

At the abscise x, for a variation QΔ of the flow rate during a time interval tΔ , the mass variation can be written: tQmt ΔΔ= .ρδ .

156 Microfluidics

The mass variation associated with variation SΔ of the section between x and xx Δ+ is xSmx ΔΔ= ρδ .

The whole mass conservation implies: 0=+ tx mm δδ . Then dividing by txΔΔ :

0=ΔΔ

The space and time intervals leading to zero, we obtain a relation between Q and S:

∂∂

[4.10]

We now have to express the response of the duct wall to variations in pressure. In other words, the basic idea is to associate a variation of the section with a pressure variation:

∂∂

For a wall with elastic properties and small deformations: steCPS

=∂∂

. Then we

C∂∂

[4.11]

Parameter C is called the compliance. In the SI system compliance is expressed in m2 Pa-1.

Equations [4.9], [4.10] and [4.11] give a complete description for the flow and pressure in a distensible duct following the hypothesis formulated by Gross et al. [GRO 74]. It is then easy to obtain an equation for the pressure:

P∂∂

∂∂

[4.12]

The pressure (or flow) is the solution of a Fourier equation that is formally identical to a diffusion equation. It is a partial differential equation whose solution necessitates initial and boundary conditions.

It is easy to verify that the term RC1

is expressed in m2s-1:

[ ][ ]

[ ][ ]2

⎥⎦

⎤⎢⎣

⎡= , then: 121 −=⎥⎦

⎤⎢⎣

⎡ tLRC

4.4.1.2. Attenuation of a pressure wave along a distensible cylindrical duct

Denoting χ=RC1

, this parameter is representative both of the properties of the

fluid (viscosity) and properties of the duct (diameter, elasticity). Let us suppose a pressure variation of frequency f ( 1−s ) localized at a position x along the duct. Then dimension of )( 12 −tLχ means that this modification will be sensible at a distance δ given by:

[4.13]

where δ is representative of a penetration depth. This result shows that the low frequencies have a greater penetration depth than high frequencies.

As an example, in the blood microcirculatory flow: χ ≈ 0.1 cm2s-1. Then for a frequency, f = 50 s-1, δ ≈ 4x10-2 cm.

4.4.1.3. Flow in a cylindrical distensible duct of finite length for an inlet step of pressure

We consider a cylindrical distensible duct of finite length L. The fluid is at rest at a time taken as the initial state and a step of pressure is applied at the entrance of the duct. The initial condition is at: 0=t , PsxP =)0,( . The boundary conditions at the entrance and at the exit of the duct are: PetP =),0( and PstLP =),( . It is useful to use non-dimensional quantities here in the form:

PePsPPsp

−−= , *x.Lx = , *t.RCLt

2= , *1

Q es −=

158 Microfluidics

Here the quantities with * are dimensionless but in those following the * will be omitted for simplicity of presentation.

The Fourier equation [4.12] is now:

∂∂

∂∂ =

[4.14]

With the conditions: 0)0,( =xp , 1),0( =tp and 0),1( =tp .

The solution is classic and leads to expressions in the form of Fourier series. The solution in non-dimensional form for the pressure is:

∑ −−−=n

xntnen

xp πππ sin

[4.15]

From the solution for p it is easy to obtain the solution for the flow:

∑ −+=n

xntneQ )cos(22

21 ππ

[4.16]

It is then possible to write the solutions for the physical quantities:

⎥⎥⎦

⎢⎢⎣

⎡−−−−−= ∑

n hess L

PPPP )sin()exp(121)( 2

22ππ

⎥⎥⎦

⎢⎢⎣

⎡∑ −+

nCRLtn

es )cos()exp(211

22ππ

Figure 4.24 shows the spatial variation of the pressure for three successive instants. Contrary to the situation with a rigid duct, the pressure field is not installed instantaneously. The stationary repartition of pressure is achieved after a relaxation

time depending on the value of RC1

Figure 4.24. Variation of p as a function of x

Figure 4.25 shows the time dependence of the flow for two positions in the duct 1=x (exit) and x = 0.2 (near the entrance). In a similar way for the pressure, the

flows reach a stationary value after a relaxation time. Near the entrance we observe a jump in flow resulting from an inflation of the duct submitted to a step of pressure at the entrance. Then the flow reaches a permanent value.

Figure 4.25. Variation of as a function of x

From the preceding analytical results (see relations [4.15] and [4.16]) it is possible to express the instantaneous local hydraulic resistance R*, defined by:

t = .01 t = .05 t = .2

0.2 0.4 0.6 0.8 1

0.1 0.2

2 x = 0.2

160 Microfluidics

22)cos()exp(21

⎥⎥⎦

⎢⎢⎣

⎡−+= ∑

n h Lx

nCRLtn

RR ππ

Figure 4.26 gives the repartition of R* in the duct for two successive times. During installation of the flow, the effective hydraulic resistance is lower in the entrance region. This corresponds to an increase in the instantaneous flow above the stationary flow (see Figure 4.25). The local variations of flow (see Figure 4.25) and hydraulic resistance (see Figure 4.26) are accompanied with local variation of the wall shear stress. In the capillary bed these effects can be of importance in the complex succession of actions and feedback, permitting the existence of intermittent blood flow.

Figure 4.26. Variation of R* as a function of x/L

4.4.1.4. Marey Experiment

The preceding expressions show that when applying a step of pressure at the entrance of a distensible duct of finite length L, part of the flow is used to distend the duct during the establishment of the flow. To quantify this effect we can determine the volumes of fluid flowing in and out of the duct. Let us denote eV the fluid volume entering the duct during a time delay t:

∑∫∫ +=⎥⎥⎦

⎢⎢⎣

⎡∑ −+== =

nunedtQtV 22

2221)(

In the same way we can evaluate the fluid volume sV flowing out of the duct during the same time delay:

0.2 0.4 0.6 0.8 1

* R 1 . 0 RC L t

∑∫∫−

+=⎥⎥⎦

⎢⎢⎣

⎡∑ −+== =

nttxvs

nnunedtQtV 22

)1(2)cos(

2221)(

πππ

The volume balance is different from zero and is then:

))1(1(1222 =−−=− ∑ n

We can now express the amount Vδ of fluid entering the duct in terms of the physical parameters:

LCPPLCPPVVV sesese )(41

))(( −=−−=δ

This volume of fluid is evidently restored when suppressing the pressure at the entrance. In other words the out flow is not instantaneously set to zero, but releases to zero evacuating the amount Vδ .

The experiment conducted by Marey [MAR 01] enabled us to verify this effect. Two cylindrical ducts, one rigid and the other distensible but with the same geometrical properties, are fed with the same reservoir. At the entrance of the two ducts, a mechanical device enables us to impose a succession of steps. The experiment shows that the fluid flowing through the distensible duct is greater than the fluid volume flowing through the rigid duct because of the possibility the distensible duct can stock part of the fluid entering the duct.

Both relaxation and storage effects have important implications in the understanding of different processes observed in the framework of blood microcirculation.

The interaction between a distensible duct and a flow has many other applications with peristaltism. A wave shear rate on a distensible duct can carry the fluid along the duct. This effect have many applications in biological flows (waves in the urethra, for example) or in practical devices, such as peristaltic pumps.

4.4.1.5. Description of Newtonian low Reynolds number flow in a distensible duct

The description of a Newtonian low Reynolds number flow in a distensible duct can be achieved in a similar way as in section 4.4.1.1, without the hypothesis of low deformations, and starting with the equations of movement:

162 Microfluidics

Q∂∂

[4.17]

∂∂

[4.18]

Introducing the deformation ε given by ⎟⎟⎠

⎞⎜⎜⎝

⎛−= 1

0SSε , where 0S is the

section at rest, it is easy to obtain a set of three equations for P, Q andε :

( )xPS

Q∂∂ε

8+−=

∂∂ε

∂∂

02−=

[4.19]

)(εPP =

Here the relation )(εPP = is representative of the rheological properties of the wall of the duct. For an elastic response: εα.=P . We can introduce this expression in [4.19] and obtain, for example, a nonlinear partial derivative equation for the pressure [SCH 89]:

x ∂∂

=⎥⎥⎦

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ +∂∂

απη

∂∂

2 1621

[4.20]

The solution of such nonlinear equation necessitates particular, original methods [GAB 92, GAB 94]. Figure 4.27a shows an example of simulation, giving the variation VΔ in volume for a distensible duct as a function of the variation PΔ of pressure applied at the entrance [GUI 98]. The results of Figure 4.27a are obtained for a duct of length L = 200 µm, radius R = 4 µm and elasticity parameter α = 400 mmHg. The dotted line shows the linear part of the curve. For comparison, Figure 4.27b shows similar experimental results. Such results are useful in order to determine the field of linear response for ducts submitted to impulses of pressure. Figure 4.28 gives estimations of compliances obtained in the framework of this

approach and using natural intermittent flows. Figure 4.28a gives results obtained from a chicken embryo (7-8 days). Figure 4.28b gives results obtained from the rat mesentery for two populations: normal (N) and diabetic (BB). The arteriolar side is noted (A) and the venular side (V).

20 40 60 80 100

= 400mmHg α

Δ P mmHg ( )

Δ V cm × 10 9 3

Figure 4.27. a) Theoretical;and b) experimental variations of volumes VΔ as a function of pressure variations PΔ in a microvascular duct

Figure 4.28. Estimation of the compliances: a) for a chicken embryo (7-8 days); and b) for the rat mesentery for normal (N) and diabetic (BB) populations

20 40 60 80 0

20 40 60

( A ) ( V )

C cm mmHg× 10 7 2 /

40 30 20 10 10 20 30 40 0

( A ) ( V )

C cm mmHg × 10 7 2 /

μm μm

C x 107cm2 /mmHg C x 107cm2 /mmHg

164 Microfluidics

4.4.2. Simulation of a network and simulation of flows in a network

4.4.2.1. Newtonian low Reynolds number flow in a dichotomic network of distensible ducts in the hypothesis of low deformations

The observation of some microcirculatory networks shows a structure where the vessels can be classified in successive levels with the same geometrical characteristics (length and diameters). Each vessel in a given level is divided into two vessels in the following level. Figure 4.29 shows a schematic representation of such a network.

1 R C i i

Figure 4.29. Schematic representation of a dichotomic microcirculatory network

Each level i [GRO 74] is characterized by a set of in segments with length i , hydraulic resistance iR and compliance iC . The pressure and flow in one segment in level i is governed by the following set of equations:

ii ∂

∂1−=

[4.21]

CRtP i

∂∂

[4.22]

We present a particular method for solving system [4.21] and [4.22] leading to analytical solutions. This procedure is important as it gives estimations of the different time scales that are of interest in many clinical applications [MON 00].

The basic idea of the method is to start with a particular form of the non-dimensional equations [GOS 78]. Taking the first level as a reference level we can transform the set of equations:

t 2111

1* = ,

⎥⎥⎦

⎢⎢⎣

⎡= , for

11*0 iii

x⎥⎥⎦

⎢⎢⎣

⎡≤≤

[4.23]

With Pδ being a representative interval of pressure, we can express flow in the

form qHP

= , where H is a global conductance and q the non-dimensional form

of the flow. With θ as the non-dimensional form of the pressure, the set of equations can be written in the form:

xt ∂θ∂

∂∂θ

xbq i ∂

∂θ−=

iex ≤≤ *0

111 ⎥⎥⎦

⎢⎢⎣

iiii CR

⎥⎥⎦

⎢⎢⎣

[4.24]

The method of solution is particularly well adapted to the case where the network is submitted to a step of pressure at one extremity, namely: 1),0(1 ±=tθ and 0),( =tennθ .

The solutions are then taken in the form ),( txS iii ϕθ += , where iS is the permanent solution in segment i. Then 0),0(1 =tϕ and 0),( =tennϕ . The boundary conditions between two levels imply continuity of pressure and flow:

),0(),( 11 tte iii ϕϕ =−− , for ni →= 2

bniexx

⎥⎦

⎤⎢⎣

⎥⎦

⎤⎢⎣

⎡−

−− ∂∂θ

∂∂θ , for ni →= 2 [4.25]

166 Microfluidics

It is convenient to state that iii bn=β and iii qnQ = .

We seek solutions for iϕ in the form iii gf=ϕ :

[ ]uxBuxAe iitu

i sincos2

+= −ϕ

[ ]uxBuxAeuQ iitu

ii cossin2

−= −β

Then the solutions are expressed in a matrix form:

sincos( , ) (0, )( , ) (0, )

sin cos

iii i i

ii i i

ueuee t tu

Q e t Q tu ue ue

ϕ ϕβ

⎡ ⎤−⎡ ⎤ ⎡ ⎤⎢ ⎥=⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦⎢ ⎥⎣ ⎦

[4.26]

the boundary conditions between two levels lead to:

sin sincos cos( , ) (0, )......

( , ) (0, )sin cossin cos

ue ueue uee t tu u

Q e t Q tu ue ueu ue ue

ϕ ϕβ βββ

⎡ ⎤ ⎡ ⎤− −⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎣ ⎦⎣ ⎦ ⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦

( , ) (0, )( , ) (0, )

n n n n

e t C D tQ e t E F Q tϕ ϕ⎡ ⎤ ⎡ ⎤ ⎡ ⎤

=⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦ ⎣ ⎦ [4.27]

The boundary conditions for ϕ at the entrance and exit of the network lead to:

0=nD [4.28]

Equation [4.28] is a transcendental equation whose solutions ku lead to the series of characteristic time scales kτ :

[4.29]

The solutions for iϕ and iQ are then given by:

kii ,ϕϕ

kii QQ ,

[4.30]

Taking expression [4.27], it is possible to re-write the expressions for iϕ and

iQ at a position ix in level i:

2, 1, 1,

, 1, 1, 1, 1

sin 0cos( , )( , )

sin cosk

kki k i k i k

k n u ti k i k i k k k

k i k k

u xu xx t C Du

Q x t E F B u eu u x u x

− −−

− −

⎡ ⎤ ⎡ ⎤−⎡ ⎤ ⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ −⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎢ ⎥⎣ ⎦ [4.31]

The solution for ki,ϕ is now in the form:

⎥⎥⎦

⎢⎢⎣

⎡−−= −−

kkikki

tukkki u

xuFxuDeButx k

ββϕ sin

cos),( ,1,1,11,2

taking: ik

kkikkiki u

xuFxuDX

cos ,1,1, −− −= for ni ,2=

then kitu

kkki XeButx k ,,11,2

),( −−= βϕ .

It is possible to demonstrate that kiX , verifies an orthogonality relation:

,, =∑ ∫i

dxXXβ , if ≠k

Because of the form of the solution, we can verify that it is only necessary to determine kB ,1 :

168 Microfluidics

∑ ∫

∑ ∫=

dxXxSb

The results being obtained for one step of pressure at the entrance, it is clear that the solution for a succession of steps can be achieved by taking the solution obtained at the end of the preceding step as the initial condition.

As an example, we consider a simplified network with three successive levels from the arteriolar level (level 1) to the capillary level (level 3). For convenience, the values (in cgs unit) of the resistances and compliances are taken for the rabbit omentum [GRO 74]. Using the preceding procedure, we can evaluate the first time scale 1τ for a step of pressure at the entrance of the network for “normal” or pathological situations: lowering of the compliance for diabetic situations (see Figure 4.28b) or increasing resistance due to hyper aggregability of the RBCs [DUR 98].

(cm) 1.5 0.5 0.5

810−×R 15 800 3,500

1410×C 6,300 470 200

Table 4.3. Resistances and compliances [GRO 74]

Figure 4.30 shows variation of 1τ for a set of decreasing values of compliance

1C of the arteriolar entrance level (diabetic situation, see Figure 5.28b). Figure 4.31 shows the variation in 1τ for a set of increasing values of the terminal resistance (for example hyper-aggregability). In all cases, the time scale exhibits two types of behavior: low variations for resistance and compliance variations not exceeding a ratio of 10, then exponential variation. The time scale for the response of a microcirculatory network is a key parameter for the occurrence of intermittent lows that are necessary for the functioning of the microcirculatory networks.

1C 100

0.08 1 τ ( s )

Figure 4.30. Relaxation time scale as a function of compliance

10* 3R 3 R 3R * 100

Figure 4.31. Relaxation time scale as a function of resistance

4.4.2.2. Low Reynolds number flow of a Newtonian fluid in an arbitrary, weakly distensible network

It well known that many microcirculatory networks exhibit complex topological structures with many anastomosis. The observation of such networks shows structures with polygonal patterns where each node is the meeting point of three vessels i, j and m in a set of n vessels (see Figure 4.32).

170 Microfluidics

The procedure is the same as in the preceding case for studying the response to a step of pressure at the entrance of the network. The time-dependent part of the solution for a vessel i is:

[ ]uxBuxAe iitu

i sincos2

+= −ϕ

[ ]uxBuxAeubq iitu

ii cossin2

−= −

Figure 4.32. Microcirculatory network

The boundary conditions at the limits of the network and for each node (i, j, m) are:

0cos sincos sin

( sin cos )

cos sin 0

i i i i m

i i i i j

i i j j m m

n n n n

AA ue B ue AA ue B ue A

A ue B ue B BA ue B ueβ β β

=+ =+ =

− = − −

[4.32]

The network being made of n segments, the set of connecting boundaries necessitates 2n coefficients )( ii,BA . 1A being determinate, relations [4.32] are a system with (2n – 1) equations and (2n – 1) unknown quantities in the form:

⎟⎟⎟⎟⎟⎟⎟⎟

⎜⎜⎜⎜⎜⎜⎜⎜

In order to obtain a non-trivial solution, we need 0=DetM . This is a transcendental equation with solutions that are a series ......,u 21 kuu leading to a similar solution, as in the preceding case.

As an example, we can consider an elementary network with one anastomosis (see Figure 4.33).

The test network is characterized by seven segments with resistances iR , compliances iC and lengths i ( 7,1=i ). The application of the preceding method leads easily to a matrix M ( 1313 × ) and then to the solution of 0=DetM .

Figure 4.33. Elementary network with one anastomosis

Figure 4.34 shows the solution for the first time scale as a function of compliance, taking the values given in Table 4.3 as a reference. The results are similar to those given in Figure 4.30.

Figure 4.34. Time scale variation as a function of compliance

172 Microfluidics

The elementary network represented in Figure 4.33 can often be seen as the simplest representation of elementary network with one anastomosis [GUI 91]. Such a representation is useful for the illustration of some aspects of the generation of intermittent flows in the microcirculation [GUI 98]. Figure 4.35 shows a numerical solution obtained for flow )2( 44q calculated at the middle of segment 4. If we consider a network with identical segments, then: = 0. At an arbitrary time

1t a modification of compliance in segment 3 induces a flow in segment 4 (see Figure 4.35). Further modifications of resistances and/or compliances in other segments of the network lead to flows in the anastomosis whose direction and intensity are dependent on the variations of resistances and/or compliances in the other segments.

3 C 3C 1 . 0 ×

6 R 6R 10 ×

2C 2C 10 ×

⎟ ⎠ ⎞

⎜ ⎝ ⎛

2 q 4 4

1 t 2t 3t

Figure 4.35. Variation of the flow in segment 4 (anastomosis) for different modifications of resistances or compliances of the other segments of the network

4.4.3. An alternative method for the study and simulation of a microcirculatory network

A microcirculatory network is seen as a set of vessels with n nodes including r boundary marks (extern connections with other networks) [THI 02a, THI 02b]. The boundary marks can be related to external networks by one or two vessels. Each node is the intersection of three vessels.

The starting point from which to describe the topology of the network using a square (n x n) is connection matrix T defined by 1T j,i = if nodes i and j are

)2( 44q

connected and 0T j,i = if nodes i and j are not connected. T is a symmetrical matrix and . The connection between the network with n nodes and r boundaries is

then represented by a rectangular (n x 2) matrix L, defined by: 0,1 ,2i iL L= = if the node i is not a boundary; ,1 1iL = and ,2 0iL = if node i is simple; and ,1 1iL = ,

1,2iL = if the node i is double.

It is then easy to define matrices for lengths, diameter or any geometrical of physical properties of the vessels. Defining R matrix for the resistances and H matrix for conductance, such as:

( ) [ ]2n,1j,i ∈∀

0 if 0

1 if 0

i j i j

i j i ji j

= =⎧⎪≠ −⎨ ≠⎪⎩

⎪⎩

⎪⎨

== ∑=

i,i R1Hji

the pressure P at each node is represented by a vector ( ) ,i iP P P= is the pressure at the node i and the values are obtained by solving the linear system;

Where the vector E is representative of the boundary conditions:

– 0Ei = – node i is not a boundary;

– ,1 ,1/i i iE P R= – node i is a simple boundary mark;

– ,1 ,1 ,2 ,2/ /i i i i iE P R P R= + – node i is a double boundary mark.

Such a procedure is very useful for describing a wide variety of microcirculatory networks [GON 04, GON 08] because it enables us to generate networks corresponding to measured distributions of parameters: repartition of lengths, diameters, mean number of intersections, etc. The adaptation of the method to take

0T j,i =

174 Microfluidics

compliant effects into account is not difficult – it introduces complex impedances for resistant and compliant effects.

4.5. The microcirculatory system: an optimized transport network?

4.5.1. Early works

It has long been suspected that the arrangement of vessels in the body is influenced by general physical laws as well as by specific physiological requirements. Optimality principles have been applied to the diameters and branching angles of arterial junctions since Murray’s work in the 1920s [MUR 26]. However, the analysis of these structures from principles of optimization and selection is still the subject of intense scientific activity and controversy.

In his earlier works, Murray used the principle of minimum work to predict the diameters of blood vessels. The basic argument was that the diameters of blood vessels are such that the work required to circulate and maintain the blood supply are a minimum. Murray derived a relation [FUN 97a, FUN 97b, MUR 26, SHE 81], hereafter referred to as Murray’s law, which states that the cube of the radius, 0r , of the parent vessel should equal the sum of the cubes of the radii, 1r and 2r , of the daughter vessels (see Figure 4.36):

3 3 30 1 2r r r= + [4.33]

Murray assumed that physiological vascular systems, subjected to natural selection, must have achieved an optimum arrangement so that in every segment of vessel, flow is achieved with the least possible biological work. He assumed that two energy terms contribute to the cost of maintaining blood flow in any section of any vessel: a) the energy required to overcome viscous drag in a fluid obeying Poiseuille’s law, and b) the energy metabolically required to maintain the volume of blood and vessel tissues involved in the flow. The energy terms are related to the radius of the vessel, but in opposite ways: the larger the radius, the smaller the power required for flow, but the larger the power required to maintain the blood and vessel wall tissue.

To minimize the total power required, the vessel cannot be too large or small. Minimization of the total flow rate is achieved when the flow rate is proportional to the cube of the radius. Finally, conservation of flow at each bifurcation leads to Murray’s law [4.33].

Figure 4.36. Murray’s law [4.33] and Zamir’s law [4.34] define the optimal geometry of a dichotomy branching

As for Murray’s law, local optimization principles have also been proposed in order to describe the branching angles of blood vessels: minimization of channel volume, channel surface area, dissipated power, and drag force on the walls [ZAM 76a, ZAM 76b, ZAM 78]. All of these approaches consist in varying the position of a given junction, while the positions of the other junctions, channel cross-sectional areas, and flow rates through every channel remain fixed. For a single bifurcation, each of the four optimality results in an equation for angle θ between the child segments take the form:

2 2 20 1 2

w w w,

w w− −

[4.34]

where iw is the weight computed according to the optimality principle selected, subscript 0 refers to the parent segment, and subscripts 1 and 2 refer to the two child segments. iw is a function of the flow rate and the diameter of segment i . Equation [4.34] is the triangle of forces law that describes the resulting angle of three strings tied together in a plane under respective tensions, or weights, 0w , 1w and 2w .

Although Murray and Zamir’s laws are general physical principles of great utility in the description of biological bulk transport systems, these local optimization approaches present some mathematical weaknesses. Specifically, two main criticisms need to be addressed:

– Optimization is achieved for a fixed incoming flow rate. As a consequence, flow and channel cross-sectional area are functionally related: an optimal cross-sectional area is found for a given flow and not for all levels of total flow. In fact,

176 Microfluidics

changing the diameter of a vessel or the angle of a bifurcation must change the flow-rate distribution in the entire network (think of an electrical network: changing one resistance in the network implies a re-distribution of the currents in any elements).

– The topology of the network is assumed (for instance, the node coordination is imposed), and not a consequence of optimization. It would be rather pleasant if the topology of the network could be deduced from optimality principles in the same way as its geometry.

To overcome these criticisms, we need to study the structure of the network in the context of global optimization. That is to say, we need to take account of the fact that a local change in geometry or topology implies a change in the flow-rate distribution of the entire network.

It must be noted that the global optimization of a transport network is a problem addressed in many other areas. We can cite, among others, water, natural gas and power supply to a city, telecommunications networks, rail, road traffic and more recently the design of labs-on-chips or microfluidic devices. Thus, global optimization of transport networks sheds light on the structure of natural networks (vascular systems of plants and animals, river basins, etc.), but presents some industrial and economical benefits too.

In the following section we show that this analysis leads to general laws similar to those of Murray and Zamir, and provides new features of the topology of the optimal network.

4.5.2. Network optimization

4.5.2.1. Definitions and notations

Before establishing some results on the architecture of optimal transport networks, we need to introduce some useful terminology. We call the entrances and exits by which the flow comes in and out of the network sources and sinks, respectively. The other junctions between conducts are referred to as additional nodes (see Figure 4.37). Each source, sink and additional node is referenced with an indice i, and the pipe-connecting nodes i and j are referenced by the pair of indices ( )i , j . Initially, every pipe ( )i , j may have an inhomogeneous curvature and non-

uniform cross-sectional area ( )ijs l along its curvilinear coordinate l (see Figure 4.38).

Figure 4(in whit

In thand the sjunctions

Figureinhomoge

By aresistanc

4.37. An exampte). The network

that pipe

he following, shape of each s and their spe

e 4.38. Schematenous curvature

analogy with ece ijr of pipe (

ple of a networkk also contains es can be curve

network geompipe, while ne

ecific arrangem

tic of the pipe (e and a non-unif

electrical circ( )i , j as:

k of pipes conneother junctions

ed and have non

metry refers tetwork topoloment.

)i , j connectiniform cross-sec

coordinate l

uits of resisto

Physiol

ecting the sourcs, referred to asn-uniform cross

to the knowleogy refers to th

ng nodes i and jctional area ijs

or elements, w

logical Microflo

ces (in grey) to s additional nods-sections

edge of node he number of p

j. The pipe can ( )l along its c

we can define

ows 177

the sinks des. Note

locations pipes and

have an curvilinear

e the flow

178 Microfluidics

( )( )0,ijl

dlrs l

ρ= ∫ [4.35]

with ρ being the resistivity of the pipe material, and m a positive constant characterizing the flow profile. For most flows encountered in physics, 1m ≥ . For instance, the 1m = case corresponds to the electrical current in wires, liquid flow in porous ducts, mass or heat diffusion in bars (provided that for the latter, the lateral surface of the bar is insulated). The 2m = case corresponds to the laminar Poiseuille flow in hollow pipes.

Finally, we define the dissipation rate over the network as:

,ij iji j

U r q=∑ [4.36]

where the summation is over all the links that constitute the network. In the case of an electrical network, U represents the Joule heat.

From the perspective of evolution and natural selection, the minimization of dissipation rate is an appealing hypothesis. Moreover, this optimality principle is supported on a physiological basis: it is well established that cells – parietal cells especially – are sensitive to stress and strain. Indeed, the minimum dissipation rate coincides with the uniformity of shear stress (or shear strain) on the pipe walls [FUN 97a] for a Poiseuille flow (which is the flow profile observed in most of the circulatory system).

4.5.2.2. Problem: least dissipative network

Consider a set of sources and sinks embedded in a 2D or 3D space, their respective number and locations being fixed. The rate of flow iQ into the network from each source and out of the network through each sink is also given. The problem is how to interconnect the sources and sinks via possible additional nodes in the most efficient way. That is, to minimize the dissipation rate

,ij ij

U r q=∑ .

Which parameters can be adjusted in order to minimize U? Obviously the structure of the network can be adjusted – its topology and its geometry – but also the flow rate distribution { }ijq in the network. Two types of constraints must be taken into

account to adequately settle this optimization problem:

– Constraint on the flow rates: flow rates in a network are not independent but must satisfy a conservation law at every source, sink and additional node. That is, the sum of algebraic flow rates at each site must satisfy the equation:

at every additional nodeq

Q at source or sink i⎧⎪⎪= ⎨⎪⎪⎩

∑ [4.37]

where the summation is over all of the nodes j connected to node i , and iQ is the fixed inflow (outflow) at the source (sink) i ( 0iQ > for a source, 0iQ < for a sink). This is in fact the second Kirchhoff law learned in basic electricity lectures. Since the flow rate is conserved through the network, we have:

,i isources i sinks i

Q Q Q=− =∑ ∑ [4.38]

where Q is the total flow rate through the network. Note that conservation laws [4.37] alone do not usually uniquely determine the flow in each pipe in the network. The flow-rate distribution can still be adjusted to minimize U for given network topology and geometry.

– Constraint on the pipe cross-sections: minimization must also be done with some constraints on the pipe cross-sections (otherwise the optimization problem would be trivial: any network connecting the sources to the sinks with infinitely large pipes would have zero dissipation, and thus would be a solution). We shall consider a global constraint on the total volume or total surface area. Such a global constraint is less restrictive than fixing every pipe cross-section to a given value, and more realistically mimics the architecture of vascular networks, for which adaptation of the diameters are usually observed.

For the sake of clarity, we use the notation nC to represent the total surface area (n=1/2) or the total volume (n=1) of the pipe network:

( )( )( ) 0,

n iji j

C s l dl=∑∫ [4.39]

A fixed lateral surface area is the relevant constraint if we want to save the material required to build the hollow pipes. A fixed total volume is the relevant constraint if we want to preserve the amount of liquid flowing through the network.

180 Microfluidics

4.5.2.3. Optimal flow-rate distributions

4.5.2.3.1. Thomson’s principle

We start the optimization analysis by adjusting the flow-rate distributions. We consider a network with fixed geometry and topology (so the resistance values are fixed) and we look for the flow-rate distribution that satisfies the conservation equations [4.37] and minimizes dissipation rate U (equation [4.36]).

The solution to this problem was given more than a century ago by Sir Thomson [THO 79]. This result is now known as Thomson’s principle: among all of the possible flow distributions that satisfy the equations of conservation [4.37], there is a unique flow distribution that makes the function 2

ij ijU r q= an absolute minimum. This flow-rate distribution is the one that derives from a potential function: i.e. the flow-rate in the pipe ( ),i j is:

,i jij

= [4.40]

where iv is a function defined at every source, sink and additional node i of the network. Indeed, there is a unique potential function (up to an additive constant), and hence a unique flow-rate distribution { }ijq that satisfy equations [4.37] and

[4.40].

It is worth noticing that in many situations in physics, the flow-rate distribution { }ijq derives from a potential function (electrical potential, pressure, concentration,

temperature, etc.) so that the potential difference i jv v− , flow rate ijq , and resistance

ijr of pipe ( ),i j are related by equation [4.40]. This equation is similar to Ohm’s law for an electrical resistor.

4.5.2.3.2. Cohn’s theorem

How does the dissipation rate change when the value of a resistance is altered? As it has already been pointed out, the variation in resistance value of a given pipe ( ),i j implies a change in the whole flow rate distribution (since the optimal flow rate distribution is a function of the resistances). Using the formula for the derivative of a product of two functions, the corresponding variation in dissipation rate can be expressed as:

klij kl

k lij ij

qU q rr r

∂∂= +

∂ ∂∑ [4.41]

Usually the second term of the right hand side of equation [4.41] cannot be simplified: this term depends on the particular values of the resistances and flow rates (i.e. on the topology and geometry of the network). Nevertheless, when the flow rates derive from a potential function, this term vanishes and the variation of the dissipation rate is simply given by [DOY 84]:

∂ [4.42]

This result, sometimes known as Cohn’s theorem, is very useful in characterizing

the geometry and topology of the least dissipative network. Note that ij

∂∂

is always

positive: the dissipation rate increases with the resistance value. However, when some resistances in the network are increased and others are decreased, the variation in dissipation rate generally cannot be predicted.

4.5.2.4. Geometry of optimal networks

We can further minimize the dissipation rate by adjusting the geometry and topology of the network, with the assumption this time that flow-rate distribution is always derived from a potential function for a given network architecture.

4.5.2.4.1. Optimal shape of a conduct

We want to characterize the shape of pipes in the least dissipative network. Consider a particular channel ( ),i j of the optimal network. By definition, any small change in its cross-sectional area or length, which is compatible with the constraints, must lead to an increase in dissipation rate. According to Cohn’s theorem (equation [4.42]), U is a monotone function of the individual resistance ijr associated with pipe( ),i j . Thus, in an optimal configuration, any small shape variation in the pipe

( ),i j leaving its volume/surface area unchanged leads to an increase in ijr . Considering the definition [4.35] of ijr , the channel length ijl must be as small as possible and its cross-sectional area as uniform and large as possible. Since the reasoning can be applied indifferently to any channel in the network, we thus

182 Microfluidics

conclude that each channel must be straight with a uniform cross-sectional area1 [DUR 06]. As a consequence, we can restrict our study to networks made of straight and uniform pipes only. The geometry of such networks will be characterized by the pipe cross-sections and locations of the nodes, only.

4.5.2.4.2. Diameter conditions at branching (Murray’s law revisited)

We now establish relations between diameters and angles in an optimal network for a fixed topology (meaning that no junction or channel can be added or removed from the network, but the channel lengths and cross-section areas are free to vary). In such a network, channels are straight with uniform cross-sectional areas, as we just showed in the section above. Then, for a given topology, the network architecture is entirely determined by the knowledge of independent variables ijs and ( ), ,i i i ir x y z= , respectively, the channel cross-sectional areas and additional node locations. For a fixed value of nC , however, these variables can no longer vary independently. Therefore, we will use the Lagrange multiplier technique and try to minimize the function nU U Cλ= + (where λ is a Lagrange multiplier) with

respect to the variables ( ){ }, , ,ij i i i is r x y z= , which are now considered to be

independent.

The condition of extremum with respect to the cross-sectional areas reads:

ij ij ij

CU Us s s

λ∂∂ ∂

= + =∂ ∂ ∂

. Using Cohn’s theorem and the expression of resistance for

a straight pipe ijij m

= , we obtain 21

lU m qs s

∂=−

∂ and 1nn

ij ijij

C ns ls

−∂=

Therefore, in the network with adjusted pipe cross-sections, the flow rate in pipe

( ),i j scales with its cross-section as: 2 m nij ij

+= . Finally, conservation of the

flow rate at each additional node i ( 0ijj

q =∑ ) yields:

2 2m n m n

ij ijin flows j out flows j

s s+ +

=∑ ∑ [4.43]

1 It is also worth noticing that a circular cross-sectional area has the specific property of minimizing both the pipe surface area for a fixed volume (or equivalently maximizing the pipe volume for a fixed surface area) and the dissipation rate in the channel for a fixed incoming flow rate in the case of a Poiseuille-flow regime (m = 2).

This relation is the generalization of Murray’s law [4.33] to any flow profile and different constraints (Murray’s law was originally derived for the particular case m = 2, n = 1). Moreover, relation [4.43] results from the global optimization of the network structure, while the original derivation of Murray’s law was based on a local optimization.

4.5.2.4.3. Angle conditions at branching (Zamir’s law revisited)

The conditions of extremum with respect to the junction positions are:

0i i i

U U Ux y z

∂ ∂ ∂= = =

∂ ∂ ∂. Consider the first equality: 0n

CU Ux x x

∂∂ ∂= = =

∂ ∂ ∂. The first

term can be rewritten as ij

ji ij i

rU Ux r x

∂∂ ∂=

∂ ∂ ∂∑ with ij ij i jm m

i iij ij ij

r l x xx xs s l

ρ ρ∂ ∂ −= =

∂ ∂. The

second term yields: ij i jn nnij ij

i i ij

l x xCs s

x x l∂ −∂

= =∂ ∂

. Using Cohn’s theorem again, it

finally becomes [DUR 06]:

2 0i jnij ijm

j ij ij

x xq s

λ⎛ ⎞ −⎟⎜ ⎟⎜ + =⎟⎜ ⎟⎜ ⎟⎝ ⎠

∑ [4.44]

We can use the same procedure for the equalities 0i

∂ and 0

∂. The

three equations can then be summarized by the vectorial equilibrium:

2 ,nij ijm

⎛ ⎞⎟⎜ ⎟⎜ + =⎟⎜ ⎟⎜ ⎟⎝ ⎠∑ ije [4.45]

where ije is the outward-pointing unit vector along the channel ( ),i j . When the

cross-sections are also adjusted, then 2 ,m nij ij

+= and the equation above

simplifies to:

s 0=∑ ije [4.46]

The optimal branching geometry described by equation [4.46] is similar to the one obtained by Zamir using local the optimization approach [4.34].

184 Microfluidics

It must be noted that equations [4.43] and [4.46] are necessary conditions in order to reach the minimum of U with respect to the geometrical parameters.

4.5.2.5. Topology of optimal networks

4.5.2.5.1. Tree-like structure of the optimal network

It can be shown that the optimal network contains no loop (such a network is called a tree). We briefly summarize the idea of the demonstration here. Readers interested in the full demonstration should read [DUR 07].

Suppose that the network contains loops. It can be shown that, from this original network, a new loopless network with a lower dissipation rate (and the same value of nC ) can be built. Consider an arbitrary loop in this network. To go from a given junction A to another junction B of this loop, there are two different paths, noted (α) and (β), as depicted in Figure 4.39.

Let us shift the material in such a way that flows in path (α) tend to be strengthened in one direction (say A to B) and flows in path (β) tend to be strengthened in the opposite direction (B to A). That is, the new cross-sectional areas

ijs ′ in the loop are defined as: 2 2 20

m n m n m n

ij ijs s s+ + +

′ = ± for path (α), with a plus sign if the flow rate in pipe ( ),i j is in direction A B→ and a minus sign if the flow rate is in the opposite direction, while signs are inverted for path (β) (see Figure 4.39). 0s is a positive number smaller than any cross-sectional area ijs of the original loop. Cross-sections outside the loop remain unaltered ( ij ijs s′ = ).

Such a variation in cross-sectional areas implies a redistribution of flows in the entire network. It can be shown that such transformation implies a decrease of both U and nC . In a further step, the total volume/surface area can be increased up to its original value nC by increasing any cross-sectional areas in the network. According to Cohn’s theorem, this will imply a further decrease in U . Thus, we find a small perturbation of the cross-sections where dissipation is reduced for a fixed value of

nC . The reasoning above can be applied with increasingly large values of 0s , until eventually one of the pipes in the loop has a zero cross-sectional area, and so one of the paths is cut off. Possible dead branches can be removed, the equivalent material being shifted to the rest of the network by increasing any other cross-sectional areas again, so that the constraint stays at its initial value while the dissipation rate is subject to a further decrease.

Finally, the whole procedure can be repeated to eliminate all duplicate paths until there are no loops in the network. The argument holds even in case of overlapping

loops (that is, loops having pipes in common), and more generally for any topology of the original network. Therefore, the architecture of the network that minimizes U is loopless.

Figure 4.39. The transfer of material from pipes to others in the loop implies a decrease in dissipation rate

An important consequence of this result is that it limits the number of possible topologies for the optimal network. Indeed, it can be shown [DUR 07] that the number of additional nodes is at most N – 2 in the optimal network, where N is the total number of sources plus sinks. This upper bound on the number of additional nodes restricts the number of possible topologies for the optimal network(s). These results make it possible to conceive efficient algorithms for determining the optimal pipe network.

4.5.2.5.2. Maximal node connectivity

The results derived in the previous sections were valid for both 2D and 3D networks. An additional topological feature can be established for 2D networks [DUR 06]: an upper bound on the number of adjoining channels at each junction of a 2D minimal resistance network. Suppose the network contains a junction connecting a number 4N ≥ of pipes. We can split this N-fold junction to a (N − 1)-fold junction plus a three-fold junction, with the creation of a new pipe of infinitesimal length 3dl , as depicted in Figure 4.40. If such a topological change can be made with a fixed value of nC and decreasing value of U, the new topology will be preferred. This way, it can be shown that, depending on the flow profile (value of m) and constraint considered (value of n), no more than three or four channels meet at every junction of the least dissipative network [DUR 06].

( ) ( ) ( )/2 /2 /2, , 0

m n m n m ni j i js s s+ + +′ = ±

186 Microfluidics

Figure 4.40. Splitting of an N-fold junction to a (N – 1)-fold junction plus a three-fold junction. If this elementary change of topology implies a decrease of U, the new

topology will be preferred

4.6. Conclusion

The geometrical and topological features of the least dissipative network are found to be similar to those of the microcirculatory network. In particular, Murray and Zamir’s laws are met within this approach. Although these features do not uniquely define the architecture of the optimal network, they limit the number of possible configurations for this network (the different configurations that can be enumerated). As for other well-known network problems (Steiner tree problem, travelling salesman problem, etc.), the solution probably cannot be found without an exhaustive search of all possible configurations.

4.7. Bibliography

[ARO 70] AROESTY J., GROSS J.F., “Convection and diffusion in the micro-circulation”, Microvasc. Res., vol. 2, pp. 247-267, 1970.

[BAR 94] BARBIER A., BOISSEAU M.R., BRAQUET P., CARPENTIER P., TACCOEN A., “Micro-circulation et rhéologie”, La Presse Médicale, vol. 23, pp. 213-224, 1994.

[BEL 88] BELCARO G., RULO A., VASDEKIS S., WILLIAMS M.A., NICOLAIDES A., Combined Evaluation of Postphlebitic Limbs by Laser Doppler Flowmetry and Transcutaneous po2/Pco2 Measurements, Vasa, Band 17, Heft 4, 1988.

[BER 94] BERTHAULT M.F., OTHMANE A., GUILLOU J., COUNORD J.L., KTORZA A, DUFAUX J., “Hemorheological abnormalities in rats with experimental mild diabets: improving effect of troxerutine and alpha-tocopherol”, Clinical Hemorheol., vol. 14, no. 1, pp. 83-92, 1994.

[BIC 91] BICOUT D., AKKERMANS E., MAYNARD R., “Dynamical correlations for multiple light scattering in laminar flow”, Journal de Physique I, vol. 1,pp. 471–491, 1991.

[BON 81] BONNER R.F., NOSSAL R, “Model for laser Doppler measurements of blood flow in tissue”, Applied Optics, vol. 20, pp. 12, 1981.

[BOR 75] BORN G.V.R., MELLING A., WHITELAW J.H., “Laser Doppler microscope for blood velocity measurements”, Biorheology, vol. 15, pp. 163–172, 1978.

[BUR 65] BURTON A.C., Physiology and Biophysics of the Circulation, Year Book Medical, Publishers, Inc, Chicago, 1965.

[CAR 78] CARO C.G., PEDLEY T.J., SCHROTER R.C., SEED W.A., The Mechanics of the Circulation, Oxford University Press Oxford, 1978.

[CAS 59] CASSON N., Rheology of Disperse Systems, London, Pergamon Press, 1959

[CHA 99] CHAZEL E., Diffusion de la lumière et vélocimétrie Doppler laser dans les suspensions en écoulement, Application à la micro-circulation sanguine, Third cycle thesis, University of Paris 7, 1999.

[CHI 70] CHIEN S., USAMI S., DELLENBACK R.J., GREGERSEN M.I., “Shear dependent interaction of plasma proteins with erythrocytes in blood rheology”, Am J Physiol, vol. 219, pp. 143-153, 1970.

[COU 93] COUNORD J.L., DUFAUX J., “Réalisation d’un capillaire sténosé permettant la mesure de la déformabilité des hématies”, ITBM, vol. 14, no. 6, pp. 689-695, 1993.

[DEB 95] DE BAERE T., DUFAUX J., ROCHE A., COUNORD J.L., BERTHAULT M.F., “Circulatory alterations induced by intra arterial injection of iodized oil and emulsions of iodized oil and doxorubicin”, Radiology, vol. 194, pp. 165-170, 1995.

[DIT 82] DITTMAR A., MARICHY J., GRIPPARI J.L., DELHOMME G., ROUSSEL B., “Measurement by heat clearance of skin blood flow of healthy, burned, and grafted skin”, in: Biomedical Thermology, Alan R. Liss, Inc., New York, pp. 413-419, 1982.

[DOY 84] DOYLE, P. G., SNELL, J. L.. “Random walks and electric networks”, inL T. M. America (Ed.). The Carus Mathematical Monograph Series. The Mathematical Association of America, 1984.

[DUF 82] DUFAUX J., Short thesis, University of Paris 7, 1982.

[DUR 98] DURUSSEL J.J., BERTHAULT M. F., GUIFFANT G., DUFAUX J., “Effects of red blood cell hyper aggregation on the rat microcirculation blood flow”, Acta Physiol Scand, vol. 163, no. 25, pp. 25-32, 1998.

[DUR 06] DURAND, M.. “Architecture of optimal transport networks”, Physical Review E, vol. 73, pp. 016116 1-6, 2006.

[DUR 07] DURAND, M.. “Structure of optimal transport networks subject to a global constraint”, Physical Review Letters, vol. 98, pp. 088701 1-4, 2007.

[EIN 75] EINAV S.H., BERMAN J., FUHRO R.L., DIGIOVANNI P.R., FRIDMAN J.D., FINE S., “Measurement of blood flow in vivo by laser Doppler anemometry through a microscope”, Biorheology, vol. 12, pp. 203- 205, 1975.

188 Microfluidics

[FAR 29] FÅRHAEUS R., “The suspension stability of the blood”, Physiological Reviews, vol IX, no. 2, pp. 241-274, 1929.

[FAR 31] FÅRHAEUS R., LINDQVIST T., “The viscosity of the blood in narrow capillary tubes”, Am J Phys, vol. 96, pp. 562-569, 1931.

[FOR 80] FORESTER D.W., SPENCE V.A., BELL I., HUTCHINSON F., WALKER W.F., “The preparation and stability of radiodinated antipyrine for use in local blood flow determination”, Eur J Nucl Med, vol. 5, pp. 145, 1980.

[FRA 89] FRANCO A., CARPENTIER P.H., MAGNE J.L., GUIDICELLI H., “La réanimation du membre ischémique”, in: Le Sauvetage des Membres en Ischémie Critique, Masson, Paris, pp. 61-80, 1989.

[FRA 94] FRANCEK U.K., HUCH A., ZIMMERMANN A.R., LEU A.J., HUCH R., HOFFMANN U., BOLLINGER A., “A triple electrode for simultaneous investigations of transcutaneous oxygen tension, laser Doppler flowmetry and dynamic fluorescence video microscopy”, Int J Microcirc, vol. 14, pp. 269-273, 1994.

[FRI 71] FRIEDMAN M.H., “Microcirculation”, in: Selkurt (ed), Physiology, Little Brown and Co., Boston, pp. 259-273,1971.

[FUN 97a] FUNG, Y. C., Biomechanics: Circulation, 2nd ed. Springer, New-York, 1997.

[FUN 97b] FUNG, Y. C., Biomechanics: Mechanical Properties of Living Tissues, 2nd ed, Springer, New-York, 1997.

[GAB 92] GABET L., Esquisse d'une théorie décompositionnelle et application aux équations aux dérivées partielles, PhD thesis, Ecole Centrale, Paris 1992.

[GAB 94] GABET L., “The theoretical foundation of the Adomian method”, Computers Math Applic, vol. 27, no. 12, pp. 41-52, 1994.

[GAN 96] GANDJBAKHCHE A.H., GANNOT I., “Quantitative fluorescent imaging of specific markers of diseases tissue”, IEEE J., vol. 2, no. 4, 1996.

[GOL 67] GOLDSMITH H.L., MASON S.G., Rheology: theory and applications, Eirich 4, New York London Academic, pp. 82-250, 1967.

[GON 04] GONG D., JURSKI K., GUIFFANT G., TESTELIN S., DUFAUX J., “Caractérisation bi-dimensionnelle quantitative de la topologie et de l’hydrodynamique de réseaux microcirculatoires”, Congrès SB, September 8-10, Paris, 2004.

[GON 08] GONG D. Caractérisation du comportement mécanique des microvaisseaux sanguins – application à l'estimation indirecte de la compliance des microvaisseaux chez des rats sains et diabétiques. Doctorate thesis, University of Paris 7, Denis Diderot, 15 July 2008.

[GOS 78] GOSS J., “Sur la conduction thermique variable dans un mur composite à propriétés constantes", C R Acad Sc Paris, vol. 286, pp. 303-306, 1978.

[GRO 74] GROSS J., INTAGLIETTA M., ZWEIFACH B.W., “Network of pulsatile hemodynamics in the micro-circulation of the rabbit omentum”, Am J Physiol, vol. 225, no. 5, pp.1110-1116, 1974.

[GUI 91] GUIFFANT G, DURUSSEL J.J., DUFAUX J., ROUBAUD P., “Topological analysis of microcirculatory networks and rheological blood properties from two teleost fishes. Comparison with a sea mammal and a terrestrial mammal”, Biorheologie, vol. 28, pp. 589-594, 1991.

[GUI 98] GUIFFANT G., GABET L., DUFAUX J., “Theoretical and experimental study of intermittent blood flows in the micro-circulation: Application to the in-vivo determination of compliance”, ASME, Journal of Biomechanical Engineering, vol. 120, pp. 737, 1998.

[HAG 39] HAGEN G.H.L., “Uber die Bewegung des Wassers in engen cylindrischen Rohren”, Poggendorffs Annalen der Physik and Chemie, vol. 46, pp. 423, 1839.

[HAM 94] HAMARD P., HAMARD H., DUFAUX J., QUESNOT S., Optic nerve head blood flow using a laser Doppler velocimeter and haemorheology in primary open angle glaucoma and normal pressure glaucoma, Br J Ophtalomol, vol. 78, pp. 449-453, 1994.

[HAR 81] HARRISON D.H., GIRLING M., MOTT G., ENG T., “Experience in monitoring the circulation in free-flap transfers”, Plast Reconstr Surg, vol. 68, no. 4, pp. 543-555, 1981.

[HAY 60] HAYNES R.H., “Physical basis of the dependence of blood viscosity on tube radius”, Am J Physiol, vol. 198, pp. 1193-1199, 1960.

[INT 73a] INTAGLIETTA M., ZWEIFACH B.W., “Micro-circulatory basis for fluid exchange”, Adv Bio Med Phys, vol. 15, pp. 111-159, 1973.

[INT 73b] INTAGLIETTA M., “Pressure measurement in the micro-circulation with active and passive transducer”, Microvasc Res, vol. 5, pp. 317-324, 1973.

[INT 89] INTAGLIETTA M., “Vasomotion et modulation du flux dans la micro-circulation”, in: Progrès en Micro-circulation Appliquée, Karger 1989.

[JAI 88] JAIN R.K., “Determinations of tumor blood flow: a review”, Cancer Res, vol. 48, no. 10, pp. 2641-2658, 1988.

[JON 84] JONES B.M., “Monitors for the cutaneous micro-circulation”, Plastic and Reconstructive Surgery, vol. 73, no. 5, pp. 843-850, 1984.

[KRO 19] KROGH A., “Number and distribution of capillaries in muscle with calculation of oxygen pressure head necessary for supplying the tissue”, Am J Physiol, vol. 52, pp. 409-415, 1919.

[LAN 26] LANDIS E.M., “The capillary pressure in frog mesentery as determined by micro-injection”, Am J Appl Physiol, vol. 75, pp. 548-570, 1926.

[LEC 76] LECONG P., Development of a laser Doppler velocimeter and its applications to micro-circulation study, Dissertation, University of California, San Diego University, micro-film order no. 77-522, 1976.

190 Microfluidics

[LEW 69] LEW H.S., FUNG Y.C., “The motion of the plasma beetween the red blood cells in the bolus flow”, J Biorheology, vol. 6, pp. 109-119, 1969.

[LIN 91] LINDSEY L.A., WATSON J.D., QUABA A.A., “Pulse oxymetry in postoperative monitoring of free muscle flaps”, Br J Plastic Surg, vol. 44, pp. 27-29, 1991.

[LIP 80] LIPOWSKY H.H., USAMI S., CHIEN S., “In vivo measurement of "apparent viscosity" and micro-vessel hematocrite in the mesentery of the cat”, Microvasc Res, vol. 15, pp. 93-101, 1980.

[LIP 82] LIPOWSKY H.H., “Determinants of micro-vascular blood flow”, The Physiologist, vol. 25, no. 4, 357-363, 1982.

[LOV 82] LOVE T.J., “Thermography as an indicator of blood perfusion”, Ann NY Acad Sci, vol. 335, pp. 429-437, 1980.

[MAR 87] MARET G., WOLF P.E., “Multiple light scattering from disordered media, the effect of brownian motion of scatterers”, J Phys B-Condensed Matter, vol. 65, 1987.

[MAR 01] MAREY E., “D’Arsonval, Garel, Chauveau”, Traité de Physique Biologique, Masson, 1901.

[MAY 83] MAY J.W., LUKASH F.N., GALLICO G., STIRRAT G.R., “Removable thermocouple probe micro-vascular patency monitor: an experimental and clinical study”, Plast Reconstr Surg, vol. 72, no. 3, pp. 366-379, 1983.

[MIS 74] MISHINA H., KOYAMA T. ASAKURA T., “Velocity measurements of blood flow in the capillary and vein using the laser Doppler microscope”, Applied Optics, vol. 14, pp. 2326-2327, 1974.

[MON 00] MONSUEZ J.J., DUFAUX J., VITTECOQ D., “Hemorheology in asymptomatic HIV-infected patients”, Clinical Hemorheology and Micro-circulation, vol. 23, pp. 59-66, 2000.

[MUR 26] MURRAY, C. D., “The physiological principle of minimum work. I. The vascular system and the cost of blood volume”, Proc. Natl. Acd. Sci. U.S.A, 12, pp. 207-214, 1926.

[NEL 77] NELLIS S.N., ZWEIFACH B.W., “A method for determining segmental resistances in the micro-circulation from pressure-flow measurements”, Circ Res, vol. 40, pp. 546-556, 1977.

[PER 91] PERRONEAU P., Vélocimétrie laser, Application en Pharmacologie Cardiovasculaire Animale et Clinique, Inserm, 1991.

[PIN 90] PINE D.J.,WEITZ D.A., MARET G., WOLF P.E., HERBOLZHEIMER E., CHAIKIN P.M., Dynamical correlations of multiply scatertered light, Scattering and Localization of Classic Waves in Random Media, Ping Sheng, World Scientific, 1990.

[POI 40] POISEUILLE J.M., “Recherches expérimentales sur le mouvement des liquides dans les tubes de très petits diamètres”, Comptes rendus hebdomadaires de l’Académie des Sciences, vol. 11, pp. 961-967, 1041–1048, 1840.

[POP 87] POPEL A.S., “Networks models of peripherical circulation”, In: Skalak R, Chien S (eds) Handbook of Bioengineering, McGraw-Hill, New York, 20.1-20.24, 1987.

[POW 78] POWELS E.W., FRAYER W.W., “Laser Doppler measurement of blood flow in the micro-circulation”, Plast Reconstr Surg, vol. 61, no. 2, pp. 250-255, 1978.

[PRI 98] PRIES A.R., SECOMB T.W., GAEHTGENS P., “Structural adaptation and stability of microvascular networks: theory and simulations”, Am J Physiol, vol. 275: pp. 349-360, 1998.

[QUE 78] QUEMADA D., “Rheology of concentrated disperse systems and minimum energy dissipation principle”, Rheologica Acta, vol. 17, pp. 643-653, 1978.

[RAP 59] RAPPOPORT M.B., BLOCH E.H., IRWIN J.W., “A manometer for measuring dynamic pressure in the micro-vascular system”, J Appl Physiol, vol. 14, pp. 651-655, 1959.

[REI 69] REID J.M., SIGELMAN R.A., NASSER N., BAKER D., “The scattering of ultrasounds by human blood”, Proc Int Conf Med Biol Eng, pp. 10-17,1969.

[RIE 93] RIETHMULLER M.L., “Vélocimétrie par image de particule”, Actes du séminaire Européen: Le Laser: Outil de Diagnostic en Milieu Industriel, Paris, October 12-13, 1993.

[RIE 96] RIETHMULLER M.L., “Vélocimétrie par images de particules ou PIV: Synthèse des travaux récents”, Actes du 5ème Congrès Francophone de Vélocimétrie Laser, pp. ESJ 1 - 13, Rouen, September 24-27, 1996.

[RIV 72] RIVA C.E., BOSS B., BENEDEK G.B., “Laser Doppler measurement of blood flow in capillary tubes and retinal arteries”, Invest Ophtalmol, vol. 11, pp. 936-944, 1972.

[ROB 82] ROBERTS V.C., “Photoplethysmography: fundamental aspects of the optical properties of blood in motion”, Trans Instr MC, vol. 4, pp. 101-106, 1982.

[SCH 75] SCHMID-SCHÖNBEIN G.W., FUNG Y.C., ZWEIFACH B., “Vascular endothelial-leucocyte interaction: sticking shear force in venules”, Circ Res, vol. 36, pp. 173-184, 1975.

[SCH 80] SCHMID-SCHÖNBEIN G.W., USAMI S., SKALAK R., CHIEN S., “Cell distribution in capillary networks”, Microvasc Res, vol. 19, pp. 18-44, 1980.

[SCH 88] SCHMID-SCHÖNBEIN G.W., “A theory of blood flow in skeletal muscle”, J Biomech Ing, vol. 110, pp. 20-26, 1988.

[SCH 89] SCHMID-SCHÖNBEIN G.W., LEE S., Y., SUTTON D., “Dynamic viscous flow in distensible vessels of skeletal muscle micro-circulation: application to pressure and flow transient”, Biorheology, vol. 26, pp. 215-227, 1989.

[SHE 81] SHERMAN, T. F., “On connecting large vessels to small, the meaning of Murray’s law”, J Gen Physiol, vol. 78, pp. 431-453, 1981.

[SHU 76] SHUNG K.K., SIGELMAN R.A., REID J.M., “Scattering of ultrasound by blood”, IEEE Trans Biomed Eng, vol. 23, pp. 460-467, 1976.

192 Microfluidics

[SIG 73] SIGELMANN R.A., REID J.M., “Analysis and measurement of ultrasound backscattering from an ensemble of scatters excited by sine wave bursts”, J Acoust Soc Am, vol. 53, pp. 1351, 1973.

[SKA 72] SKALAK R., CHEN P.H., CHIEN S., “Effect of hematocrit and rouleaux on apparent viscosity in capillaries”, Biorheology, vol. 9, pp. 67-82, 1972.

[SKA 86] SKALAK T.C., SCHMID-SCHÖNBEIN G.W., “Viscoelastic properties of microvessels in rat spinotrapezius muscle”, J Biomech Eng, vol. 108, pp. 193-200, 1986.

[SMA 70] SMAJE L., ZWEIFACH B., INTAGLIETTA M., “Micropressures and filtration coefficients in sigle vessels of the cremaster muscle of the rat”, Microvasc Res, vol. 2, pp. 96-110, 1970.

[SNA 95] SNABRE P., ARHALIASS A., “Multiple light scattering in random systems. Characterization of granular media by analysis of backscattering spot image”, ASME Heat Transfer Fluid Eng, vol. 1037, pp. 511-518, 1995.

[SNA 96a] SNABRE P, MILLS P., “I - Rheology of weakly flocculated suspensions of rigid particles”, J de Phys III, vol. 6, pp. 1811-1834, 1996.

[SNA 96b] SNABRE P., MILLS P., “II - Rheology of weakly flocculated suspensions of viscoelastic particles”, J de Phys III, vol. 6, pp. 1835-1855, 1996.

[SNA 98] SNABRE P., ARHALIASS A., “Anisotropic scattering of light in random media: incoherent backscattered spotlight”, Applied Optics, vol. 37, pp. 18, 1998.

[SPE 90] SPENCER M.P., THOMAS G.I., NICHOLLS S.C., SAUVAGE L.R., “Detection of middle cerebral artery emboli during carotid endarterectomy using transcranial Doppler ultrasonography”, Stroke, vol. 21, pp. 415-423, 1990.

[STE 75] STERN M.D., “In vivo evaluation of micro-circulation by coherent light scattering”, Nature, vol. 254, pp. 56-58, 1975.

[SUG 79] SUGA H., “Total mechanical energy of a ventricular model and cardiac oxygen consumption”, Am J Physiol, vol. 236, pp. H498-H505, 1979.

[SUT 92] SUTTON D.W., SCHMID-SCHÖNBEIN G.W., “Elevation of organ resistance due to leukocyte perfusion”, Am J Physiol, vol. 262, pp. H1646-H1650, 1992.

[TAN 74] TANAKA T., RIVA C., BEN SIRA I., “Blood velocity measurements in human retinal vessels”, Science, vol. 186, pp. 830-831, 1974.

[TES 00] TESTELIN S., Incidence du flux micro-vasculaire sur le comportement des tissus transplantés, Exploration par vélocimétrie Doppler Laser, Third cycle thesis, University of Picardie Jules Verne, 2000.

[THI 02a] THINEY G., GUIFFANT G., RICHERT A., BERTHAULT M.F., DUFAUX J. “Rat microvascular compliance measured by laser Doppler velocymetry technique”, XXVII Congrès de la Société de Biomécanique, Valenciennes, September 12-13, 2002.

[THI 02b] THINEY G. Relation pression-débit dans les réseaux microcirculatoires. Application à la mesure in vivo de la compliance. Doctorate thesis, University of Paris 7 Denis Diderot, December 18, 2002.

[THO 79] THOMSON W., Tait P. G., Treatise on Natural Philosophy, Cambridge University Press, 1879.

[WAY 67] WAYLAND H., JOHNSON P.C., “Erythrocyte velocity measurement in micro-vessels by a two-slit photometric method”, J Appl Physiol, vol. 22, pp. 333-337, 1967.

[WIE 64] WIEDERHIELM C.A., WOODBURY J.W., KIRK S., RUSHMER R.F., “Pulsatile pressure in micro-circulation of the frog's mesentery”, Am J Physiol, vol. 207, pp. 173-176, 1964.

[ZAM 76a] ZAMIR M., “Optimality principles in arterial branching”, J Theor Biol, vol. 62, pp. 227-251, 1976.

[ZAM 76b] ZAMIR M., “The role of shear forces in arterial branching”, J Gen Physiol, vol. 67, pp. 213-222, 1976.

[ZAM 78] ZAMIR M., “Nonsymmetrical bifurcations in arterial branching”, J Gen Physiol, vol. 72, pp. 837-845, 1978.

[ZWE 81] ZWEIFACH B.W., LIPOWSKY H.H., “Pressure-flow relations in blood and lymph microcirculation”, In: Renkin EM, Michel CC (eds), in: Handbook of Physiology, Bethesda, MD, Am Physiol Soc, pp. 251-307, 1981.

microfluidics (colin/microfluidics) || physiological microflows

Documents

dna microfluidics forensic flow - royal society of chemistry...

introduction to microfluidics - jacobs university bremen...

microfluidics and microencapsulation · 2013. 6. 18. ·...

microfluidics hands-on workshop guide...

group r14300 – digital microfluidics

low cost microfluidics

microfluidics design & chip application

programmable microfluidics - mit...

gas microflows in the slip flow regime: a critical review

bringing microfluidics to life

microfluidics system catalogue

hydrodynamics and heat transfer in gaseous microflows ... ·...

microfluidics: introduction sami.franssila@aalto.fi

henrikbruus microfluidics lectures

microfluidics thesis

2010_gomez_bioanalytical applications in microfluidics

towards programmable microfluidics

ibidi microfluidics

microfluidics profile brochure, minacned

electrokinetic instabilities in ferrofluid microflows