In vivo whole-field blood velocity measurement techniques

The geometry of the cardiovascular system is strongly three-dimensional (e.g. the heart and aortic arch) and involves multiple scales which range from the aorta (with a diameter of about 2 cm) to the arteries and veins (with diameters in the millimeter range) to arterioles, venules, and capillaries with diameters that are much smaller than a millimeter (see Table 1). The flow ranges from strongly pulsatile (in the aorta and arteries) to nearly stationary in the smallest capillaries. The flow is primarily characterized by two dimensionless numbers: the Reynolds number Re and Womersley number α, defined as where d is a characteristic length (e.g., the diameter of a blood vessel), \({\bar{V}}\) the bulk velocity, ρ the density, η the dynamic viscosity, and f the characteristic frequency. In general the Reynolds number is below 2000 (Table 1), which implies that the flow in most blood vessels is laminar. For low Womersley numbers the pulsatile flow is dominated by viscous effects rather than momentum effects. The Reynolds number and Womersley number can be defined for the flow in blood vessels which have a cylindrical geometry, but for the constantly varying geometry and flow patterns in the heart it is more difficult to define these numbers.

Blood is a complex fluid that consists of a continuous fluid, or plasma, containing complex molecules (e.g. proteins, nutrients, hormones) and larger particles (e.g. red and white blood cells, platelets). Among technically relevant fluids, blood exhibits the most intricate behavior. Blood as a whole reacts mechanically as well as chemically with its environment. At the same time the components of blood react mechanically and chemically with each other. The mechanically most relevant components of blood are plasma and red blood cells (erythrocytes).

Figure 2 shows photographs of unprocessed, human blood at a scale of 0.16 μm per pixel. The left photo shows a layer of about 20 μm thickness. The tendency of the erythrocytes to stick together in aggregates is clearly visible. Those aggregates are widely referred to as rouleaux, because of there resemblance with stacks of coins. The right photo shows erythrocytes in a layer of less than 8 μm thickness. Their diameter ranges around 7 μm. The rim of the flat, bi-concave shaped cells has a thickness of about 2 μm. The volume fraction of erythrocytes (hematocrit) is about 45% but varies with age and gender (Schmidt and Thews 1987). The ability of erythrocytes to form aggregates and their high deformability are responsible for the non-Newtonian viscosity of blood (Chien 1970). Figure 3 illustrates this effect by comparing the shear dependent viscosity of normal blood with the viscosity of blood where the cells do not aggregate (addition of albumin) and blood with cells that can also not deform (hardened cells). The increased apparent viscosity of normal blood at very low shear rates is, therefore, attributed to the agglomeration of cells, while the lowered viscosity at high shear rates is attributed to the deformability of the cells (Chien 1970).

Blood contains a small number (4 × 10³ to 10 × 10³ per μl) of white blood cells (leukocytes), that are an essential part of the immune system (Schmidt and Thews 1987). Leukocytes are larger than erythrocytes. Of fluid mechanical interest is their ability to roll over and to stick to the surface of blood vessels (Ley 1993; Sperandio et al. 2006).

Another essential component of blood are platelets (thrombocytes). Thrombocytes have a length of 1–4 μm and a thickness of 0.5–0.75 μm (Schmidt and Thews 1987). Because of their small size and large number (150 × 10³ to 350 × 10³ per μl) they can be used as tracer particles for velocity measurements (Slaaf et al. 1984; Tangelder et al. 1985, 1986, 1988). They are particularly useful for measurements in the plasma rich layer near the vessel wall. This flow regime is generally erythrocyte-depleted due to the Fåhraeus-Lindquist effect (Barbee and Cokelet 1971; Albrecht et al. 1979; Pries et al. 1992; Shul’man et al. 1995; Bishop et al. 2001b). This effect also indicates that the erythrocytes do not follow exactly the motion of the blood plasma, which implies that they may not always be suitable as “natural” tracer particles for blood flow measurements. Uijttewaal et al. (1993, 1994) showed that platelets experience a reversed Fåhraeus-Lindquist effect and concentrate in the erythrocyte-depleted layer.

Plasma consists mainly of water. A mass fraction of about 8% of different proteins raises the viscosity of plasma to almost twice the viscosity of water (Harkness 1971). Among the plasma-proteins, fibrinogen can be of fluid mechanical relevance. The clotting-ability of blood rests on this protein that activates and attaches blood platelets (thrombocytes). Plasma contains a number of electrolytes. These are mainly the ions of dissolved sodium chloride that alone make up 96% of the osmotic pressure of about 745 kPa (Schmidt and Thews 1987).

Reviews about hemorheology and hemodynamics appear regularly (Copley et al. 1942; Copley and Blair 1962; Wayland 1967; Bugliarello 1977; Liepsch 1986, Copley 1990; Liepsch 2002; Lipowsky 2005; Popel and Johnson 2005). The treatment of blood as a continuous or two-phase fluid and the consideration of non-Newtonian properties depends on the studied problem and the scale. At macro scale, when the ratio of blood cell size and characteristic length of the flow domain is small, one might treat blood as a liquid of continuous phase. Low shear rates in the center region of large vessels, however, lead to a radial viscosity gradient. This results in a flattened velocity profile, if compared to a Hagen-Poiseuille profile (Bishop et al. 2001a). Flow through intermediate blood vessels can be simplified further. Arterioles and venules are still much larger than blood cells. Typical shear rates range between 800 and 8,000 1/s and belong to the highest shear rates in the human vasculature (Charm and Kurland 1974) (Table 1). The low shear region in the center of the blood vessel is narrow, when compared to the typical size of erythrocyte aggregates. The radial viscosity gradient becomes less pronounced, the blood viscosity might even be assumed constant. Bishop et al. (2001a) measured parabolic blood velocity profiles in venules of 45–75 μm diameter when the shear rate is higher than 40 s⁻¹. Charm and Kurland (1966), Charm et al. (1968), and Charm and Kurland (1974) suggest to treat blood as a Newtonian fluid when the dimensionless group \({C / \left(\bar{V}^2 \rho \right)}\) is smaller than 5 × 10⁻⁴, the cell hematocrit ϕ is greater than 0.4, and the vessel diameter d is between 155 and 2,000 μm. The yield stress of blood, C, is the critical stress that must be applied to start blood to flow. Values are published by Charm and Kurland (1967). Further, \({\bar{V}}\) is the mean velocity and ρ the density. At small scale, when blood cells and flow domain diameter are of the same magnitude, blood flow is generally treated as two-phase flow. At even smaller scale (e.g., the erythrocyte depleted layer near the vessel wall) we might treat blood cells and blood plasma domain separately. For the smallest capillaries the flow becomes irregular and is determined by the transport of individual blood cells. A lubrication theory for the movement of single erythrocytes through capillaries that are smaller in diameter than the cells is developed by Feng and Weinbaum (2000).

The transport of blood through the cardiovascular system also includes complex and scale-dependent boundary conditions. Not only are we dealing with instationary or pulsatile flow, but also with flows through tubes with deformable elastic walls. The wall deformation interacts with the flow. A detailed discussion of the fluid–structure interaction as a result of pulsatile flow in flexible tubes would fall outside the scope of the present paper, and for a further discussion of this topic we refer to Zamir and Ritman (2000). At small scales the walls of the blood vessels have a very complex structure. The vessels are lined with endothelial cells that are covered with a gel-like permeable layer of sugar-like molecules, which is referred to as the glycocalyx layer (van den Berg et al. 2003). The glycocalyx layer has an estimated thickness between 10 and 400 nm (Fig. 4). The function of the glycocalyx is unknown, but it may be involved in the regulation of the total blood flow and protect the endothelial cells against high flow shear rates that may rupture the cells. The glycocalyx may also play a role in reducing the friction of the transport of RBC through very small capillaries (Feng and Weinbaum 2000).

Laser Doppler velocimetry, in its original form, is a single point measurement technique with extraordinary temporal resolution. A detailed review of this technique was published by Tropea (1995). Systems that measure up to three velocity components are commercially available. Laser Doppler Velocimetry for spatial measurements is primarily implemented in three ways: scanning (Sect. 3.1.1), simultaneous single point measurements with multiple probes (Sect. 3.1.2), and differentiation of tracer particle position within the measurement volume (Sect. 3.1.3). Full field laser Doppler systems generally provide only one velocity component.

The measurement principle of laser Doppler velocimetry is based on the optical Doppler effect: if a light source (or a light scattering tracer particle) is moved into the direction of a light-detector, the frequency of the reflected light is increased. If the light source moves away, the frequency is decreased. In practice this frequency shift is very small (10–100 kHz) when compared to the high frequencies of light (100 THz) and impractical to measure directly. The frequency shifted light is, therefore, combined with a non-shifted reference beam. The resulting beat frequency equals the frequency shift and is directly related to the velocity of the light scatterer.

A different way of looking at the concept of the two intersecting beams is to study the interference pattern within the measurement volume. The wave-fronts of the laser beams form interference fringes parallel to the optical axis of the focusing lens system (represented by the Moiré pattern in Fig. 5). When a tracer particle moves through the dark and bright spaces of the fringe system, it will reflect light at a frequency that depends on the fringe spacing and the particle velocity component, v perpendicular to the fringes. Both perceptions lead to an identical relationship between particle velocity and measured frequency-shift, Δf: Here, λ is the wavelength of the incident light. The penetration depth of light in tissue depends on its wavelength. Near infrared light (λ ≈ 1 μm) penetrates several mm, red light (λ ≈ 650 nm) up to about 2 mm and green light (λ ≈ 540 nm) hardly at all (Briers 2001).

It should be mentioned that real laser beams are slightly diverging when they leave the laser cavity. This is related to the curved (focusing) cavity mirrors at the ends of the laser rod (Hecht 2002). Lenses focus the diverging beams in the measurement spot to reduce the size of the measurement volume. The Moiré pattern in Fig. 8 represents a more realistic fringe system than the one in Fig. 5. The smallest diameter of the laser beam is called the beam waist. The distance over which the cross section of the beam doubles is referred to as the Rayleigh length (Hecht 2002). In Sect. 3.1.3 the presence of a short Rayleigh length is exploited for measuring a velocity profile.

Laser speckle is a grainy, stationary interference pattern (Merzkirch 1987). It is produced when coherent light is scattered at a diffuse surface or a number of individual scatterers (like particles in a fluid). Interfering wavelets of different optical pathlenghts fill the surrounding space with a random pattern (Goodman 2000). Individual speckles are very small close to the diffuse surface and grow in size at larger distance. When scatterers move in a uniform manner, the speckle pattern moves at the same rate. Small relative displacements of scattering particles slightly change the speckle pattern. Large relative displacements alter the pattern entirely.

Particle image velocimetry relies on the visualization of flow by means of small tracer particles. In general, the movement of the particles is recorded in two consecutive, digital images. The displacement of the particles in the second image, relative to the position of the particles in the first image, divided by the time delay between the exposures, is a measure for the velocity of the fluid. The displacement of the particles is calculated by means of a two-dimensional cross-correlation (Keane and Adrian 1992). Therefore, a small interrogation window of the first image is correlated with different sections of the second image until the maximum correlation magnitude between the image areas is found. This position is the most probable displacement of the particle pattern in the interrogation window (Keane and Adrian 1990, 1991; Adrian 1991; Raffel et al. 1998). The local velocity is approximated by dividing the displacement of the particle pattern by the given time difference between the two images. In contrast to speckle methods it is not the grade of decorrelation that contains the velocity information. Decorrelation only occurs due to particles leaving the measurement volume or due to shear within the interrogation area, but not due to the movement itself. Particle image velocimetry, therefore, does not rely on the correlation peak height but rather on the correlation peak position to obtain the velocity magnitude.

For typical macroscale particle image velocimetry applications, a two-dimensional measurement plane is formed by illuminating a thin plane of the flow with a narrow laser light sheet, whereas for microscale applications the strongly limited depth-of-focus of the microscope objective is used to sample a thin plane in which the particles are sharply imaged (Santiago et al. 1998; Meinhart et al. 1999, 2000a; Olsen and Adrian 2000; Bourdon et al. 2004a, b).

Closely related to particle image velocimetry are particle tracking techniques. Here the movement of individual particles is tracked manually or by appropriate computer programs. In contrast to particle image velocimetry, particle tracking velocimetry requires the distance between the tracer particles to be larger than the displacement (Westerweel 1997). Otherwise it is not possible to unambiguously identify matching particle pairs. This restriction limits the resolution of pure particle tracking velocimetry. The combination of a pre particle image velocimetry evaluation and sub interrogation window particle tracking, however, reaches utmost spatial resolution (Keane et al. 1995), only exceeded by single pixel ensemble correlation methods (Westerweel et al. 2004). Ji and Danuser (2005) and Danuser (2005) use an ensemble correlation method like introduced by Meinhart et al. (2000b) in combination with sub-interrogation-window particle tracking to measure actin flow in an epithelial cell at a spatial resolution in the order of 100 nm.

Another closely related term is optical flow. In general, optical flow algorithms (Lucas and Kanade 1981; Horn and Schunck 1981), aim to calculate the shift of intensity gradients between two images for every pixel. The displacement of a single pixel cannot be calculated independently of neighboring pixels (Horn and Schunck 1981). This leads to comparable spatial averaging effects as in particle image velocimetry, where blocks of pixels are correlated with each other.

Single camera set-ups measure the two velocity components parallel to the image plane. Two camera (stereo) systems can also measure the out-of-plane velocity component (Arroyo and Greated 1991; Prasad 2000; Lindken et al. 2006).

Several research groups used particle image velocimetry or related particle tracking techniques to study blood flow. The probably first quantitative, optical measurement of erythrocyte velocities in vivo is reported by Basler (1918). Basler used a self-made light conductor for back-illumination of a straight blood vessel in the sartorius muscle of a frog. A microscope was used to project the erythrocyte movement onto a film that was driven perpendicular to the erythrocyte movement (Fig. 11). The erythrocytes produce streaks on the film. The angle of the streaks relative to the film is determined by the speed of the film and by the velocity of the erythrocytes. Developments based on this technique are discussed by Monro (1965). Wayland (1967) developed a technique that is based on the passage time of erythrocytes between two lines of photosensitive elements. The signals of two elements are correlated with each other, respectively, to determine the velocity profile of erythrocytes over the blood vessel diameter (Gaehtgens et al. 1971). Spatial averaging, however, leads to strongly blunted velocity profiles (Baker and Wayland 1974). Schmid-Schoenbein and Zweifach (1975) used a high speed film camera to analyze the erythrocyte motion in arterioles, venules, and postcapillaries in the omentum of a rabbit. A series of more than 100 instantaneous cell velocity profiles were determined as shown in Fig. 12. Tangelder et al. (1986) enhanced the spatial resolution of the tracking technique by labeling blood platelets with a fluorescent dye to measure flow velocities in arterioles of the rabbit mesentery (17–32 μm diameter). Dual flash video images were used to capture the platelet motion. The velocity profiles were assembled by a number of individual measurements at different radial positions of the vessel. The illumination flashes were triggered my means of an electrocardiogram (ECG). Smith et al. (2003) and Long et al. (2004) adapted this technique by using artificial, fluorescent tracer particles of 470 nm diameter to further enhance the spatial resolution. The velocity gradient in the near wall region of a mouse cremaster muscle venule could be estimated by manually tracking the movement of tracer particles at irregular distances from the flow boundary. A typical particle tracking result is shown in Fig. 13. Velocity data can be acquired at high spatial resolution by this method. The manual evaluation is highly dependent on the operator and limits the amount of processable data. Sub-pixel accuracy is not achieved.

Singh and Prakash (1994) used optical flow algorithms to determine velocity profiles in 180 μm vessels of frogs and mice. Hitt et a (1996) applied a correlation technique to video images of the venous flow in the hamster cremaster muscle. Tsukada et al. (2000), Sugii et al. (2002a, b), and Jeong et al. (2007) used particle image velocimetry to measure red blood cell velocity profiles in mesentery vessels of rats (Fig. 14). Hove et al. (2003) and Hove (2004) followed the course of small groups of erythrocytes through the heart of a zebrafish embryo (Fig. 15). These works have in common that they use erythrocytes as tracer particles, so the physiological disturbance is limited to a minimum. The measured erythrocyte velocity, however, might differ from the plasma velocity (Sect. 2.2). Conclusions on the wall shear stress distribution are, therefore, difficult to deduce. Debaene et al. (2005) showed how to derive wall shear rate fields in artificial vessels from a single particle tracking measurement. They determined the depth position of the tracer particles from their brightness in a hazy fluid.

Vennemann et al. (2006a) used fluorescent liposomes (Woodle and Lasic 1992; Allen 1996; Singh et al. 2001) of 400 nm diameter to resolve the velocity distribution in the beating heart of a chicken embryo. The use of small tracer particles that also penetrate the red cell-depleted layer close to the wall enables the automated determination of wall shear stress. Sufficient spatial resolution, however, has not been achieved due to low seeding density. Measurements were carried out at nine different cardiac phase angles. Figure 16 shows the measured velocity distribution in the developing ventricle at its maximum dilation. The velocity has a maximum magnitude of 26 mm/s and peaks off-center at the side of the inner curvature wall. The eccentricity of the flow profile can be explained by the low Dean number, Dn =  δ^1/2 Re, of the flow. The curvature of a curved tube, δ, is defined as the ratio of tube and coil radii, a/R. The Dean number is proportional to the ratio of centrifugal forces and viscous forces. Centrifugal forces shift the velocity peak to the outer curvature wall. Viscous forces lead to a homogeneous pressure distribution over the cross-section of the vessel. This results in a higher pressure gradient along the inner curvature wall and shifts the peak velocity to the inner curvature wall (Dean 1927, 1928; van Dyke 1978; Hung et al. 1979, 1980; Wang and Bassingthwaighte 2003). The curvature of the heart-coil is about 1/2 for the case shown in Fig. 16. The Reynolds number is about unity. Three-dimensional information can be acquired by extending the measurements to parallel planes (Vennemann et al. 2006b).

Particle image velocimetry is not restricted to optically acquired images. Lin et al. (2003) used ultrasonic imaging to measure the velocity distribution in the left ventricle of a pig. Kim et al. (2004a, b); Zheng et al. (2005) and Zheng et al. (2006) determined flow profiles in artificial models of arteries and in a rotating flow apparatus. In contrast to optical imaging, an ultrasound image is composed by scanning the field of interest. Scattering tracers, in general small gas bubbles, reflect echoes. The intensity of the echo depends on the reflective property of the tracers. The time between sound emission and detection (time of flight) is used to determine the scanning depth. The time difference between two images in optical particle image velocimetry is replaced by the time difference between identical beam positions in two successive echo scans. Due to high seeding densities and small tracer diameters, when compared to the utilized wavelength, ultra sound images rather show a speckle pattern than single particle images. The speckle pattern, however, stays fairly stable and moves with the flow (Lin et al. 2003). Figure 17 shows an exemplary result. The spatial resolution of ultrasound particle image velocimetry is effectively about three orders of magnitude lower than with optically acquired images (Table 2). Zheng et al. (2006) report a spatial resolution of up to 0.4 mm and a temporal resolution of up to 0.5 ms.

Nucleons have an angular momentum which leads to a quantized magnetic momentum or spin. In case of a hydrogen atom the nucleus is composed of a single (unpaired) proton. The magnetic moment is thus not counterbalanced. The potential energy of the magnetic momentum in an homogeneous, external magnetic field depends on the direction of the momentum. The potential energy is minimized when momentum and external field vectors point into the same direction and is maximized when they point into opposite directions. In case of the hydrogen nucleus, quantum mechanics permits just two possible orientations for the magnetic momentum: parallel and anti parallel with the outer magnetic field. The magnetic momentum of a hydrogen atom of low potential energy (parallel momentum vector and field vector) can be flipped into the high energy state (anti parallel vectors) by absorbing a photon of exactly the missing energy difference. Flipping a nuclear spin into this excited state is called nuclear magnetic resonance.The nucleus flips back into its normal state by emitting a photon. The frequency of this photon depends on the energy difference between the normal and the excited state which is determined by the strength of the outer magnetic field.

Hydrogen atoms are usually bound into molecules. The atoms in the immediate vicinity of a hydrogen atom interfere with the outer magnetic field and in this way change the energy difference between normal and excited state. The frequency of a resonance photon can, therefore, identify different chemical environments.

In nuclear magnetic resonance tomography, a second, inhomogeneous magnetic field is superimposed on the homogeneous magnetic field. The inhomogeneous field is shaped in a way that small resonance frequency bands can be assigned to a specific volume element of the measured object.

First investigations to exploit the nuclear magnetic resonance technique for measuring blood velocities date back to Singer (1959). Blood flow influences the nuclear magnetic resonance signal in several ways. The signal from fast flowing blood, for example, vanishes. This is explained by the displacement of the protons between excitation and detection and is referred to as spin wash-out. This effect can be exploited for velocity measurements by measuring the signal amplitude at varying time periods between excitation and detection (Singer and Crooks 1983). A wash-out curve is produced that can be related to the flow velocity.

Phase velocity cine magnetic resonance imaging exploits the phase shift of radio waves emitted by hydrogen in flowing blood (Powell and Geva 2000). Photons emitted by stationary tissue, in contrast, show no phase shift. The magnitude of the phase shift can be related to the flow velocity. The velocity component perpendicular to the measurement plane and the direction of the flow is determined. Measurements in orthogonal planes allow for the determination of all three velocity components (Markl et al. 2003). Frydrychowicz et al. (2006a, b) quantify the three dimensional flow in a Dacron graft repair (Fig. 18) and show the hydrydynamic influence of pathological geometric alterations of the thoracic aorta (Frydrychowicz et al. 2007).

Numerous other techniques for extracting flow velocity information from nuclear magnetic resonance imaging are reviewed by Smith (1990). Bradley and Waluch (1985) describe the effect of different flow phenomena on standard magnetic resonance images. Bauer et al. (2001) demonstrated myocardial perfusion measurements in isolated rat hearts at a spatial resolution of 140 × 140 × 1,500 μm and a temporal resolution of 40 s. Cuocolo et al. (2005) review the application of magnetic resonance imaging on the visualization of myocardial perfusion. Magnetic resonance imaging can also be used for the visualization of skin blood flow (Zhao et al. 2002).