vascular development
PROEFSCHRIFT
ter verkrijging van de graad van doctor aan de Technische Universiteit Delft,
op gezag van de Rector Magnificus prof. ir. K.C.A.M. Luyben, voorzitter van het College voor Promoties,
in het openbaar te verdedigen op woensdag 4 september 2013 om 10:00 uur
door
Astrid KLOOSTERMAN wiskundig ingenieur
Prof. dr. ir. J. Westerweel
Copromotor: Dr. ir. C. Poelma
Samenstelling promotiecommissie: Rector Magnificus, voorzitter
Prof. dr. ir. J. Westerweel, Technische Universiteit Delft, promotor Dr. ir. C. Poelma, Technische Universiteit Delft, copromotor Prof. dr. ir. L.J. van Vliet, Technische Universiteit Delft
Prof. dr. E.T. van Bavel, Universiteit van Amsterdam Prof. dr. J.F. Dijksman, Universiteit Twente
Dr. B.P. Hierck, Universiteit Leiden Prof. dr. Y. Ventikos, University College London
Prof. dr. ir. G. Ooms, Technische Universiteit Delft, reservelid
The work presented in this thesis was financially supported by Medical Delta Cover: Bas Kloosterman
Copyright © 2013 by A. Kloosterman All rights reserved.
ISBN 978-94-6186-187-0 Printed by DPP, Houten
Summary vii
Samenvatting ix
1 Introduction 1
1.1 Cardiovascular system . . . 1
1.2 Hemodynamics and vascular development . . . 2
1.3 Chicken embryo . . . 2
1.4 Flow characteristics . . . 4
1.5 In vivo blood flow measurements . . . 5
1.5.1 Velocity measurements . . . 5
1.5.2 Particle image velocimetry . . . 5
1.5.3 Depth of correlation . . . 6
1.5.4 Network characterization . . . 7
1.6 Network modelling . . . 8
1.7 Thesis outline . . . 9
2 Flow rate estimation in large depth-of-field micro-PIV 11 2.1 Introduction . . . 12
2.1.1 Depth-of-correlation . . . 12
2.2 Methods . . . 15
2.2.1 Setup . . . 15
2.2.2 Particle image diameter & depth-of-correlation . . . 17
2.2.3 Data acquisition and analysis . . . 18
2.3 Results . . . 22
2.3.1 Particle images . . . 22
2.3.2 PIV measurements . . . 25
2.4 Discussion . . . 29
2.5 Conclusions . . . 30
3 Accurate Blood Flow Measurements: Are Artificial Tracers Necessary? 33 3.1 Introduction . . . 33
3.1.1 Natural versus artificial tracers . . . 34
3.1.2 Resolution and measurement volume . . . 35
3.2 Methods . . . 37 v
3.2.1 Embryo preparation . . . 38
3.2.2 Artificial tracer measurements . . . 38
3.2.3 Red blood cell measurements . . . 39
3.2.4 Protocol and data analysis . . . 39
3.3 Medium magnification results . . . 41
3.4 High magnification results . . . 43
3.5 Discussion . . . 46
3.5.1 Implications for 3D reconstructions . . . 48
3.6 Conclusions and Outlook . . . 49
4 Quantification of the changing hemodynamics in the developing vitelline network 53 4.1 Introduction . . . 53 4.2 Methods . . . 56 4.2.1 Chicken embryos . . . 56 4.2.2 Experimental setup . . . 57 4.2.3 Data aquisition . . . 59
4.2.4 Time-averaged velocity fields . . . 60
4.2.5 Hemodynamic parameter extraction . . . 69
4.3 Results . . . 75
4.3.1 Modelled vitelline network . . . 75
4.3.2 Qualitative observations . . . 77
4.3.3 Data quality . . . 81
4.4 Conclusions . . . 85
5 Conclusions & outlook 89 5.1 General conclusions . . . 89
5.2 Outlook . . . 91
A In silico model for flow underestimation by micro-PIV 95
B Characterized networks 99
Bibliography 107
Acknowledgments 113
Quantification of hemodynamics during vascular development
Vascular remodelling is an important process in the developing and mature cardiovascu-lar system. Since fluid dynamics plays a critical role in vascucardiovascu-lar remodelling, and more general in cardiovascular development, quantification of the hemodynamics is crucial to gain more insight into this complex process. This improved insight can result in an en-hanced prediction of this process, and may eventually even be used to influence the final vascular structure, which can for example be applied to improve wound healing, drug delivery, or tumour control.
The work described in this thesis has been performed to obtain accurate quantitative in-formation about the changing hemodynamics in a developing vascular network. This has been realized in the developing vitelline network of a chicken embryo, a commonly-used model for cardiovascular research. This network is the extraembryonic vasculature of the chicken embryo, and serves as the connection between the embryo and the nutritious yolk sac. To study the development, relevant hemodynamic parameters are determined at two consecutive developmental stages using the corresponding measured blood flow velocities. These velocity measurements are obtained by in vivo microscopic Particle Im-age Velocimetry (micro-PIV). Micro-PIV, and more general PIV, is capable of measuring velocities in a plane, and is based on the most likely displacements of a group of tracer particles in the flow using cross-correlation.
In micro-PIV, the thickness of the measurement volume is defined by the depth of corre-lation. This parameter is a direct consequence of the used optics and the tracer particle size, and cannot be adapted without changing one of these two, and thus the measure-ment system. The depth of correlation is often, and also in the experimeasure-ments described in this thesis, considerable larger than what would be acceptable for the common assump-tion that measurements are obtained in a single plane (for example the centerplane). When velocity gradients in the viewing direction are present, this can lead to undesired averaging of the velocities. For the measurements in the vitelline network, a microscope with low magnification is preferred, as well as naturally-present tracer particles (red blood cells) to make is possible to perform multiple measurements at consecutive stages in the same network. These conditions will lead to a larger depth of correlation, and will hence increase the velocity averaging in the viewing direction. For a correct interpretation of the measured velocity results, this effect is first studied in this thesis. First, low-magnification
measurements are investigated in an in vitro model using a reference case: Poiseuille flow. From this study, the averaging effect turned out to be much larger than expected based on the existing theory. Further research showed particle images that deviated sub-stantially from the theoretical descriptions, and these particle images are also involved in the expression for the depth of correlation. This is a result from the complex optics in the specialized microscope which was used, meeting the requirements of the measurement setup. Nevertheless, a correct interpretation of the measured velocity can be obtained when the shape of the velocity profile is known, using the experimentally determined characteristics of the particle images for particles in-focus and out-of-focus.
The question whether it is possible to perform accurate micro-PIV measurements using red blood cells as tracer is related to the aforementioned study considering the enlarged depth of correlation. To verify this, an in vivo study is done where the velocities in the vitelline network resulting from experiments using artificial microparticles and red blood cells respectively are compared. The main conclusion relevant for the next and last part of this thesis is that both tracers will give comparable results for low-magnification blood flow velocity measurement, and a correct interpretation of the results can again be ob-tained. Now, accurate blood flow velocities can be obtained by PIV under the desired conditions, despite the resulting larger depth of correlation.
For seven embryos, the vitelline networks have been measured and characterized at two consecutive stages, varying from HH 13+ to HH 17+. The main observations include that vascular remodelling is present in all seven networks, and, moreover, the charac-ter of remodelling differs and can be non-intuitive. The characcharac-ter of remodelling is likely to depend on the different locations in the total vitelline network and the developmental stages. The non-intuitive and complex hemodynamical and structural changes indicate that measuring these changes is crucial to gain more insight into this process. Further, the changes in hemodynamic parameters, such as the vessel diameter, flow velocity, length, and flow direction can be related to the observed structural development for all networks. To conclude, the quality of the data sets is investigated by verifying the flow balances in the branching points, which shows that the quality of the obtained data sets from the seven embryos are comparable, and the data is suitable for further analysis with known accuracy.
Because significant large parts of vascular networks have been characterized at two con-secutive stages, the obtained data set is particularly suitable for validation of vascular de-velopment models. Further, the followed method for characterizing vascular networks can easily be applied for other two-dimensional networks having optical access. To conclude, the method can also contribute to quantitatively describing the effects of, for example, mechanical and chemical interventions when applied to the developing networks both before and after the interventions. One example is described in a preliminary study on the formation of collateral blood vessels in the vitelline network after ligating one of the vitelline arteries.
Kwantificeren van de hemodynamica tijdens vaatontwikkeling
Remodellering van bloedvaten is een belangrijk proces in het ontwikkelende en vol-wassen cardiovasculaire systeem. Omdat stromingsleer een belangrijke rol speelt bij de remodellering van bloedvaten, en in cardiovasculaire ontwikkeling in het algemeen, is het kwantificeren van de hemodynamica cruciaal om meer inzicht te krijgen in dit complexe proces. Dit verbeterde inzicht kan resulteren in een betere voorspelling van dit proces en kan eventueel zelfs worden gebruikt om de uiteindelijke vaatstructuur te beïnvloeden, wat bijvoorbeeld kan worden toegepast voor de verbetering van wondheling, medicijnafgifte of tumorbeheersing.
Het werk dat beschreven is in dit proefschrift is uitgevoerd om accurate kwantitatieve informatie over de veranderende hemodynamica in een vatennetwerk in ontwikkeling te krijgen. Dit is gerealiseerd in het ontwikkelende vitellinenetwerk van een kippenem-bryo, een veelgebruikt model voor het cardiovasculaire systeem. Dit netwerk is het ex-traembryonaal netwerk van het kippenembryo en dient als verbinding tussen het embryo en de voedzame dooierzak. Om de ontwikkeling te kunnen bestuderen worden rele-vante hemodynamische parameters bepaald voor twee opeenvolgende ontwikkelings-stadia met behulp van de bijbehorende gemeten stroomsnelheden van het bloed. Deze snelheidsmetingen worden verkregen met in vivo microscopic Particle Image Velocimetry (micro-PIV). Mirco-PIV, en PIV in het algemeen, is in staat om snelheden in een vlak te meten en is gebaseerd op de meest waarschijnlijke verplaatsing van een groep tracer deeltjes in de stroming op basis van kruiscorrelatie.
In micro-PIV wordt de dikte van het meetvolume gedefinieerd door de correlatiediepte. Deze parameter is een direct gevolg van de gebruikte optica en de grootte van de tracer deeltjes en kan niet worden aangepast zonder één van deze twee, en dus de meetop-stelling, te veranderen. De correlatiediepte is vaak, en ook voor de metingen beschreven in dit proefschrift, aanzienlijk groter dan wat acceptabel zou zijn voor de algemene aan-name dat metingen zijn verkregen uit een enkel vlak (bijvoorbeeld het middenvlak). Als er snelheidsgradiënten aanwezig zijn in de kijkrichting kan dit leiden tot een ongewild middelen van de snelheden. Voor de metingen in het vitellinetwerk heeft een microscoop met lage vergroting de voorkeur. Tevens wordt het gebruik van rode bloedcellen als tracer deeltjes, die al van nature aanwezig zijn, verkozen boven kunstmatige deeltjes zodat het mogelijk wordt om meerdere metingen tijdens opeenvolgende stadia in hetzelfde netwerk
uit te voeren. Deze voorwaarden leiden tot een grote correlatiediepte en zullen daarom de snelheidsmiddeling in de kijkrichting vergroten. Om de gemeten snelheidsresultaten juist te kunnen interpreteren wordt dit effect eerst onderzocht in dit proefschrift. Eerst worden metingen met een lage vergroting onderzocht in een in vitro model met een re-ferentiestroming: Poiseuille stroming. Uit deze studie blijkt dat het middelingseffect veel groter blijkt te zijn dan wat verwacht wordt op basis van de bestaande theorie. Verder onderzoek liet zien dat de afbeeldingen van de deeltjes aanzienlijk afwijken van de theo-retische beschrijvingen en deze afgebeelde deeltjes zijn ook betrokken in de uitdrukking voor de correlatiediepte. Dit is het gevolg van de complexe optica in de gespecializeerde microscoop die gebruikt werd die nodig was voor de metingen. Echter, de gemeten snelheden kunnen juist wordt geïnterpreteerd als het snelheidsprofiel bekend is, door gebruikt te maken van de experimenteel bepaalde karakterisering van de afgebeelde deeltjes, zowel in- als out-of-focus.
De vraag of het mogelijk is om nauwkeurige micro-PIV metingen met rode bloedcellen als tracer uit te voeren is verwant met de bovengenoemde studie naar de vergrote cor-relatiediepte. Om dit te verifiëren is een in vivo studie gedaan waar de snelheden in het vitellinenetwerk van experimenten, waar respectievelijk kunstmatige micro-deeltjes en rode bloedcellen werden gebruikt, worden vergeleken. De belangrijkste conclusie die relevant is voor het volgende en laatste deel van dit proefschrift is dat beide deeltjes vergelijkbare resultaten geven voor metingen van de bloedstroom als een lage vergroting wordt gebruikt en dat deze snelheidsmetingen juist kunnen worden geïnterpreteerd. Nu kunnen nauwkeurige snelheden van de bloedstroom worden verkregen met PIV onder de gewenste voorwaarden, ondanks de grotere correlatiediepte.
De vitellinenetwerken van zeven embryo’s zijn gemeten en gekarakteriseerd tijdens twee opeenvolgende stadia, variërend van HH 13+ tot HH 17+. De belangrijkste observaties zijn dat remodellering van bloedvaten is aanwezig bij alle zeven embryo’s en dat boven-dien het karakter van remodellering onderling verschilt en tegen de intuïtie kan ingaan. Het karakter van remodellering hangt vermoedelijk af van de locatie in het totale vitelline-netwerk en de ontwikkelingsstadia. De complexe hemodynamische en structurele veran-deringen, die bovendien tegen de intuïtie kunnen ingaan, geven aan dat het cruciaal is om deze veranderingen te meten om meer inzicht te krijgen in het proces. Verder kunnen de veranderingen van de hemodynamische parameters, zoals de diameter van het bloedvat, de stroomsnelheid, de lengte en de stroomrichting worden gerelateerd aan de zichtbare structurele ontwikkeling voor alle netwerken. Tenslotte is de kwaliteit van de netwerken onderzocht door de in- en uitgaande bloedstroom in de bifurcaties te verifiëren. Hieruit volgt dat de kwaliteit van de verkregen data set van alle zeven embryo’s vergelijkbaar zijn en dat de data geschikt is voor verdere analyse met bekende nauwkeurigheid.
Omdat een substantieel deel van een vatennetwerk is gekarakteriseerd voor twee opeen-volgende stadia is de verkregen data set zeer geschikt voor de validatie van modellen voor vaatontwikkeling. Verder kan de gebruikte methode voor het karakteriseren van vatennetwerken gemakkelijk toegepast worden voor andere tweedimensionale netwerken die optische toegang hebben. Tenslotte kan de methode ook bijdragen aan het kwanti-tatief beschrijven van de effecten die het gevolg zijn van bijvoorbeeld mechanische en
chemische interventies wanneer de methode zowel voor als na de interventie in het ont-wikkelende netwerk gebruikt wordt. Een voorbeeld hiervan wordt beschreven in een inleidende studie die de vorming van collaterale bloedvaten, na de afsluiting van één van de vitelline arteriën, in het vitellinenetwerk beschrijft.
Introduction
1.1
Cardiovascular system
The human blood circulatory system consists of the heart and blood vessels of different sizes. There are numerous textbooks describing the cardiovascular system extensively, including Fung (1997) and McDonald (1974). The beating heart pumps blood through the collection of blood vessels, which is an efficient hierarchical vascular tree. When blood leaves the heart, it flows from the large arteries through the smaller arterioles to the blood vessels of the smallest size: the capillaries. Their sizes range from 5 to 20 µm. In this part of the cardiovascular system takes place the exchange of oxygen, nutrients, carbon dioxide and other waste products. After this exchange, the blood flows back to the heart through the venules and veins, which are increasing in size. Since diffusion is the mechanisms that drives the exchange of products in the capillaries, a large exchange area in this part of the tree makes the design functionally efficient. On the other side, the larger blood vessels in the begin and at the end of the tree are mainly for transport. The larger diameters here make the design hemodynamically efficient and thus reduce the heart load.
Nowadays, cardiovascular diseases receive much interest, which is an obvious conse-quence of the fact that they are still the principal cause of death worldwide (Murray and Lopez, 1997) (see also Table 1.1). A well-established circulatory system is essential for proper functioning of the human body. More insight into the development of this com-plex system can help understanding problems associated with the cardiovascular system. When vascular development is better understood, the prediction of this process can be enhanced. This can then be used to influence the process and subsequently the final vascular structure, which can be useful when applied in the field of, for example, wound healing, drug delivery, or tumour control (Chaplain et al, 2006; Ferrara and Kerbel, 2005; Folkman, 1995; Mantzaris et al, 2004).
Table 1.1: The top 10 causes of death worldwide in 2008 by the World Health Organiza-tion.
Causes of death deaths in millions % of deaths
Ischaemic heart disease 7.25 12.8
Stroke and other cerebrovascular disease 6.15 10.8
Lower respiratory infections 3.46 6.1
Chronic obstructive pulmonary disease 3.28 5.8
Diarrhoeal diseases 2.46 4.3
HIV/AIDS 1.78 3.1
Trachea, bronchus, lung cancers 1.39 2.4
Tuberculosis 1.34 2.4
Diabetes mellitus 1.26 2.2
Road traffic accidents 1.21 2.1
1.2
Hemodynamics and vascular development
We know that hemodynamics plays an important role in cardiovascular development, as demonstrated in studies described by for example Hove et al (2003); Hogers et al (1999); Lucitti et al (2007), and Reneman et al (2006). Studying the development of a vascular network contributes to the overall understanding of the developing cardiovascular sys-tem. Quantification of the hemodynamics during developement is essential for improving the insight into the relation between hemodynamics and vascular development. From the studies described by for example Jones et al (2004); Pries et al (1994); Lee and Lee (2010a), and Sugii et al (2002), some quantitative information is already available for different types of networks. But for an effective study on the relation between the hemo-dynamic and structural changes, such data should preferably be available at multiple moments during developement for a large network. Until now, such data is still lacking.
1.3
Chicken embryo
The extraembryonic vasculature of the chicken embryo (vitelline network) is very suit-able for studies on the vascular network. It is not only a commonly-used model for the human circulatory system, but the network can also be accessed multiple times during developement while the embryo remains in the egg. The possible effects on the de-velopment of the embryo and vasculature is hence minimized. The dede-velopment of the chicken embryo can be divided according to certain stages described by Hamburger and Hamilton (1951). These HH stages range from 1, just after fertilization, to 46, which is a newly-hatched chick. This takes approximately 21 days. The development of the chicken embryo is also extensively described by Bellairs and Osmond (2005). At the beginning, the vitelline network is just a collection of blood ‘islands’ (Bellairs and Osmond, 2005), see also Figure 1.2. These blood islands consist of hematopoietic cells (primitive blood cells) surrounded by endothelial cells. In later stages, these endothelial cells cover the inner part of the formed blood vessels. The blood islands align, which results in a primary
vascular plexus, also called the capillary plexus. It consists of small blood vessels (see also Figure 1.2), and no hierarchical structure can yet be observed. At the end of day two, the heart starts beating. This causes blood starting to flow through this early em-bryonic vasculature, and remodelling into a hierarchical tree starts. The embryo has now approximately reached stage HH 12. Remodelling of the capillary plexus starts at the presumptive navel, which is located halfway the body, just outside the body (see Figure 1.1). At this location, the flow exits the embryo and enters the vitelline circulation. Re-modelling extends further outwards, until the entire capillary plexus has developed into a hierarchical tree. At stage HH 16, about 6 hours later, the developing network is sur-rounded by the sinus terminalis: a circular vein which sharply defines this vascular area. Now, blood leaves the embryo by the arteries at the presumptive navel and flows through the vascular tree, with decreasing vessel size, to the capillaries. Eventually it flows back to the embryo again through the veins located above and below the embryo, directly or by using the sinus terminalis. During the third day, some arteries are transformed into veins (arterial-venous differentiation, Le Noble et al (2004)), and an arterial and venous circu-lation is formed. At day four, about stage HH 20, the sinus terminalis begins to regress, and by the end of the fifth day, a dense vascular network has been established. A chicken embryo and surrounding vitelline network is shown in Figure 1.1.
tail heart head 2mm sinus terminalis
*
Figure 1.1: A chicken embryo and surrounding vitelline network laying on top of the yolk, during the third day after fertilization. Blood flows from the heart through the dorsal aorta in the embryonic body. A part of the blood flow leaves the body at the presumptive navel, indicated by *, and enters the vitelline network. A possible path for blood flowing through the vitelline network using the sinus terminalis is indicated by the black arrows.
blood islands
(primitive blood cells surrounded by endothelial cells)
primary vascular plexus hierarchical vascular tree
(including arterial-venous differentiation)
after heart starts beating
Figure 1.2: The development of a vascular network: from a collection of blood islands to a hierarchical vascular tree with seperate arterial and venous circulations (adapted from Al-Kilani et al (2008))
1.4
Flow characteristics
The blood flow in the vitelline network can be characterized using two dimensionless numbers: the Reynolds number (Re) and the Womersley number (α) (e.g. Fung (1997) and McDonald (1974)):
Re = V D
ν , α = R
r 2πf
ν (1.1)
whereV is the mean velocity, D the blood vessel diameter, ν the kinematic viscosity, R the blood vessel radius, andf the pulsatile frequency. Re represents the ratio between the inertial and the viscous forces. It can be used to characterize different flow regimes: for low Reynolds numbers the flow will be laminar while turbulence will be present in flow with high Reynolds numbers. For pipe flows,Re < 2300 generally indicates the laminar regime, andRe > 4000 the turbulent regime. In the vitelline networks studied in this thesis,Re ∼ 10−3
(typically: D = 75 µm, V = 50 µm/s, and ν = 3 × 10−6
m2/s), which indicates a laminar flow where viscous forces dominate. Under steady conditions, these blood vessels have a parabolic velocity profile. The pulsatility character of the flow is captured inα. α represents the ratio between the transient inertial forces and the viscous forces. In these networks, α = 0.07 (typically: f = 1.5Hz). For pulsating flows with α < 1, the frequency of the pulsations is sufficiently low that during the cardiac cycle, the parabolic velocity profile has time to develop. For higher values ofα, flattening of the parabolic velocity profile will occur. This phenomenon can be observed in for example the larger arteries in adult humans. Hence, in the studied embryonic networks, the velocity profile will be parabolic with varying magnitude throughout the cardiac cycle. Further, sinceRe ≪ 1, the entrance length Le(i.e. the minimum length required for establishing a fully developed laminar flow) can be calculated byLe/D = 0.06 Re (White, 2008). Since the vessel lengths are considerable larger thanLe, the oscillating parabolic velocity profile is already present just after leaving a bifurcation and extends throughout a blood vessel.
1.5
In vivo blood flow measurements
1.5.1
Velocity measurements
Quantification of the hemodynamics in the vitelline network starts with measuring local blood flow velocities in a plane. For most blood vessels, except the smallest ones (with a diameter less than 25-30 µm), the flow velocity is therefore known at multiple locations in their wall-normal cross-sections. These measured velocities are further used to define the local parabolic velocity profiles, generally at multiple locations along each blood ves-sel (in the steamwise direction) to improve measurement accuracy (see Section 4.2.5). The blood vessel’s maximum flow velocity is now represented by the maximum of the velocity profile, and the vessel diameter can also be extracted from the velocity profile (see also Section 1.5.4). With these two parameters, other hemodynamic parameters, such as the flow rate and wall shear stress, can easily be determined.
1.5.2
Particle image velocimetry
Different measurement techniques can be used to obtain quantitative in vivo information about the hemodynamics in the vitelline network. The review by Vennemann et al (2007) summarizes a number of possible measurement techniques. This review shows that in vivo micro Particle Image Velocimetry (micro-PIV) is suitable for blood flow measure-ments in the vitelline network. PIV is a optical flow velocity measurement technique and an extensive decription can be found in literature, such as Adrian and Westerweel (2010) and Wereley and Meinhart (2010). Optical access is achieved by making an opening in the egg shell, and the method allows the embryo to remain in the egg during the mea-surement procedure. Although this should be considered an intrusive operation, when performed carefully and under controlled circumstances this procedure mimimizes the effects on the development of the embryo and vasculature. This is supported by the long survival times in the in ovo studies mentioned by Korn and Cramer (2007) and the low survival rates in case of ex ovo development in for example Auerbach et al (1974). Fur-thermore, the natural shape of the yolk sac, and therefore also the surrounding stress distribution, will be better preserved. Unnecessary changes of these environmental con-ditions are undesirable since the vitelline network lays on top of the yolk sac, and the vascular developement is likely to be influenced by them.
PIV is based on the displacements of particles that are carried along by the flow. These particles are often artifical, but they can also be naturally present in the flow. PIV starts with recording a series of subsequent images capturing these particles. To extract the local velocities, the domain of the measurement area is divided into smaller subdomains, called interrogation areas or interrogation windows, each containing multiple particle im-ages. This subdivision also defines the spatial resolution of the result. Now, a local displacement, and eventually a local velocity, is obtained by applying cross-correlation on these interrogation windows. So, the resulting local displacement is not based on a sin-gle particle displacement (‘tracking’), but it is based on the most likely displacement for a group of particles. This statistical approach makes PIV a more robust method compared to other ‘tracking’-based measurement techniques.
Previously, in vivo microscopic Particle Image Velocimetry (micro-PIV) measurements were applied successfully in the chicken embryo’s heart (Vennemann et al, 2006; Poelma et al, 2010), as well as in the vitelline network (Poelma et al, 2008, 2012). In these stud-ies, artifical bioinert fluorescent particles were injected in the circulation. The amount of particles in the circulation diminishes some time after injection, which makes it more difficult to get reliable velocity results. This is worsened by particles attached on the wall and particle agglomerates. Moreover, the blood pressure will also increase due to the repeated injections. If micro-PIV is used for repeated measurements during the de-velopement, the previously used method should be slightly adapted such that adding artificial tracer particles is no longer necessary. The use of artifical tracer particles can be omitted by using red blood cells (RBCs) as tracer particles (see for example Lee et al (2007); Sugii et al (2002)). They are already present and they do not have the unwanted effects mentioned. With this modification, performing consecutive PIV-measurements at different stages becomes feasible. However, using RBCs also means that epifluores-cence microscopy can not be used. Images resulting form epifluoresepifluores-cence microscopy usually have a high contrast since particles will be visualised as bright dots on a dark background. Another consequence is an increased measurement plane thickness, rep-resented by the depth of correlation (see Section 1.5.3).
1.5.3
Depth of correlation
In PIV, the measurement plane thickness is defined by the thickness of the light sheet (Adrian and Westerweel, 2010). But micro-PIV has to make use of volume illumination instead of a thin light sheet, and the measurement plane thickness is hence not defined explicitly. To define the measurement plane thickness, the depth of correlation (DOC) was introduced by Olsen and Adrian (2000), and can be calculated by:
DOC= 2 " 1 −√ε √ ε f #2 d2p+ 5.95 (M0+ 1) 2 λ2f#4 M2 0 !#1/2 (1.2) wheref#
is the aperture number of the lens,dpthe particle diameter,M0the nominal magnification,λ the wavelength of light emitted by the particle, and ε a threshold value for the intensity of particle images that contribute to the measured spatial correlation (typ-icallyε is chosen to be 0.01; NB the DOC is a weak function of ε). The DOC describes the depth over which particles contribute significantly to the spatial correlation (Olsen and Adrian, 2000), see also Figure 1.3. When the DOC increases, also particles out-side the focal plane contribute to the correlation function, which influences the measured velocity. When using RBCs, measuring approximately 7 µm, as tracer in micro-PIV mea-surements instead of smaller artificial particles with a diameter around 1 µm, the DOC increases. As a consequence, particles further away from the focal plane will contribute to the measured signal and may influence the measured velocity since velocity gradients are present. Furthermore, the DOC is inversely proportional to the numerical aperture (NA) of the microscope objective; the larger the NA, the smaller the DOC. In general, objectives with a low magnification (resulting in a large field of view) also have a lower
NA. To study the developing vitelline network, low magnification measurements should be performed since a large field of view is desirable, which increases the DOC even further. Furthermore, the in vivo measurements require a specialized upright microscope with a long working distance and zooming capability. Due to the more complex arrangements, the optical properties can be quite different from conventional microscopes, which may also influence the measured results. Besides, the theoretical predictions are based on a single ideal lens, which is a very simplified model in comparison to the actual objective, as can be seen in Figure 1.4, which shows the cross section of a fixed magnification objec-tive. For a correct interpretation of the measured velocity results in the vitelline networks, these effects should be examined and quantified.
DOC
viewing
direction
Figure 1.3: Micro-PIV provides instant two-dimensional velocity fields of the blood flow in a group of blood vessels. The three-dimensional parabolic velocity profile of the blood flow in a blood vessel is shown in blue; the two-dimensional measured velocity field, per-pendicular to the viewing direction, is represented by the velocity vectors on the green centerplane. Because of the relatively large DOC (shown is purple), the measured ve-locities are not the flow veve-locities in the centerplane, but they are a weighted average of all flow velocities present in the viewing direction within the DOC. Due to the parabolic velocity profile and the alignment of the focal plane with the vessel’s center plane, the measured velocities are an underestimation of the actual blood flow velocities.
1.5.4
Network characterization
As described earlier, the vitelline network develops very rapidly from a collection of blood islands into a hierarchical tree with arterial and venous circulations: subsequent stages are seperated by only a few hours during this part of the development. To perform PIV measurements, the embryo should have reached a stage in which the amount of cir-culating RBCs will be sufficient. On the other hand, when a embryo gets older, the arterial-venous differentiation results in two circulations that may have overlapping parts.
Figure 1.4: A relatively simple objective with one fixed magnification is not just a single lens, but is already very complex. The shown cross section of an objective is taken from the website of Zeiss, www.zeiss.de
Overlapping vessels may obscure vessels of interest. Since PIV is an optical measure-ment technique, obtaining accurate velocity results will therefore become problematic. Furtermore, the vitelline network will have a two-dimensional structure when the embryo is younger, but when the embryo gets older, it will start extending in the third dimension. This also limits the range of suitable stages to study, since PIV is capable of obtain-ing a velocity field in two directions (prependicular to the viewobtain-ing direction) on a two-dimensional plane.
To study the relation between structural and hemodynamic changes, the structure of the developing network should also be characterized. This can also be established by post-processing the PIV results. The laminair velocity profile is known and is parabolic and the flow velocity will be zero at the vessel walls. The walls can hence be determined by finding the locations where the velocity goes to zero.
1.6
Network modelling
Modelling of the blood vessel structure and the influence of hemodynamics is discussed in studies by for example Pries and Secomb (2010); Nguyen et al (2006), and Hacking et al (1996). Accurate experimental data is desired to validate network modelling and will be useful to improve them. Certain models result from design principles assuming an optimal network. One way to define an optimal network is a network where the en-ergy required to maintain the blood flow is minimized resulting in a constant shear stress throughout the vascular tree. This results in a relation between the mother vessel’s di-ameterD0and daugther vessels’s diametersD1andD2in a bifurcation which was first
introduced by Murray (1926a,b) and is known as Murray’s law: D3
0 = D31+ D23 (1.3)
Much research has been done on the validation of Murray’s law (Kassab and Fung (1995); Sherman (1981); Zamir et al (1983), among others), also in the vitelline network of the chicken embryo (Taber et al, 2001; Lee and Lee, 2010b). They both show good agree-ment in the vitelline network; Taber et al (2001) for chicken embryos of HH 16 to 24, and Lee and Lee (2010b) for chicken embryos of HH 18.
1.7
Thesis outline
To improve insight into the relation between hemodynamics and vascular development, quantification of the hemodynamics in the developing vitelline networks of seven chicken embryos has been performed. To determine the accuracy of these measurements, they are preceded by velocity measurements in an in vitro reference model. This is described in Chapter 2. This reference study is extended to an in vivo study in Chapter 3: the accuracy of PIV measurements using both artificial tracers and RBCs is investigated. In this study, the interpretation of the measured velocities obtained in vivo is treated. After this, the accuracy of the obtained blood flow velocities in a developing vitelline netwerk without injected artificial particles is determined. With this, consecutive measurements during the developement have become possible. Chapter 4 contains the characterization of the seven developing vitelline networks at two consecutive stages. The quantitative results are then compared with qualitative observations. Additionally, the agreement with theoretical design rules is investigated.
Flow rate estimation in large
depth-of-field micro-PIV
1
Abstract
In micro-Particle Image Velocimetry, the requirement of a large field-of-view often results in a large depth-of-correlation, i.e. large depth of the measurement volume. When the velocity varies substantially over the depth-of-correlation, special attention should be paid to a correct interpretation of the measured velocities. When a specialized microscope is needed to meet the requirements of a setup, the resulting more complex optical arrange-ments can have additional effects on the measurement results. In order to determine flow parameters such as the flow rate, it is sufficient to have a robust estimate of the maximum velocity when the flow is Poiseuille flow. In this paper, an interpretation of the results from particle image velocimetry measurements with low magnification in a round capillary is given for two types of microscopes: a conventional and a specialized micro-scope. The measured velocity appears to be lower than the maximum velocity, yet is still above the average velocity. The interpretation of the measured velocity differs for the two types of microscopes. The under-estimation of the maximum velocity obtained from the conventional microscope remains small (within 6%) for low-magnification measure-ments, while the under-estimation of the maximum velocity obtained from the specialized microscope increases up to 25% for a large depth-of-correlation. The images of the in-and out-of-focus particles turn out to play a crucial role in this difference between the two microscopes. Validation of the optical properties of a microscope is important, es-pecially for specialized microscopes where particle images deviate substantially from the existing theory, and this theory is also used to derive the analytical expression for the depth-of-correlation. A procedure is recommended to obtain a correct interpretation of the measured velocity. This procedure is generally applicable, but mainly of importance for specialized microscopes.
1This chapter has been published in Experiments in Fluids (Kloosterman et al, 2011b)
2.1
Introduction
In micro-Particle Image Velocimetry (micro-PIV) the depth of the measurement volume is determined by the depth-of-correlation (DOC). The DOC is inversely proportional to the numerical aperture (NA) of the microscope objective; the larger the NA, the smaller the DOC. In general, objectives with a large magnification also have a higher NA. This implies for many practical situations where a small DOC is used (i.e., with a high NA and high magnification), that the field-of-view is very small. This can be a severe lim-itation in microfluidic applications where a large field of view is desirable, for example in complex microfluidic systems or in biological applications (e.g., the investigation of a blood vessel network). As the flow is often Poiseuille-like, a robust estimate of only the mean or maximum velocity can provide parameters such as the flow rate and wall shear stress, which are in many flows more important than the exact shape of the profile. Also, certain applications require specialized microscopes. For example, the investigation of microscale biological flows require upright microscopes, or microscopes with a long work-ing distance, or even microscopes that can be used with combinations of different optical configurations (e.g., the combined stereoscopic/upright single-view microscope used by Vennemann et al 2006). The optical properties of various microscopes can be quite dif-ferent, which means that microscopes with more complex optical arrangements, such as microscopes with a zooming capability, can be quite different from conventional micro-scopes. In this paper, we investigate the effect of the DOC on the measured flow velocity as a function of the DOC relative to the typical dimension of the flow. In other words, how should the measured displacement be interpreted when the velocity varies substantially over the DOC. It appears that the measured velocity cannot be considered as a simple volume average of the velocity over the measurement domain. This also depends on the optical characteristics of the microscope, and in particular on the way in which the particle image diameter varies with the distance from the (nominal) focal plane.
2.1.1
Depth-of-correlation
The DOC describes the depth over which particles contribute significantly to the spatial correlation (Olsen and Adrian, 2000). When the DOC increases, also particles outside the focal plane contribute to the correlation function, which influences the measured ve-locity. The DOC is thus an estimate of the thickness of the measurement volume. It can be calculated by (Olsen and Adrian, 2000):
DOC= 2 " 1 −√ε √ ε f #2 d2p+ 5.95 (M0+ 1) 2 λ2f#4 M2 0 !#1/2 (2.1)
wheref#is the aperture number of the lens,d
pthe particle diameter,M0the nominal magnification,λ the wavelength of light emitted by the particle, and ε a threshold value for the intensity of particle images that contribute to the measured spatial correlation (typ-icallyε is chosen to be 0.01).
The DOC is mainly sensitive for changes in the particle diameterdp and the aperture numberf#. In the studies described in Vennemann et al (2006) and Poelma et al (2008,
2010), artificial (bio-inert) tracer particles were added to the blood to obtain velocity pro-files from PIV measurements. PIV measurements are also possible without these artificial tracer particles, by using the red blood cells (RBCs) as tracer particles (Lee et al, 2007; Sugii et al, 2002). For typical experimental values, e.g.f#= 1.0,M
0= 20,λ = 590 nm, andε = 0.01, the DOC increases from 11 µm to 43 µm when RBCs with a diameter of 7 µm are used instead of particles with a diameter of 1 µm. While the use of RBCs has the advantage that the system is not affected, it also results in a fourfold larger DOC. Some applications demand large fields-of-view, i.e. low microscope magnification (e.g. studying a blood vessel network). With most microscopes, a reduction in the magnifica-tion results in a lower numerical aperture and thus a higher aperture numberf#, and eventually to a larger DOC. As a result, using low magnifications will not only influence the resolution and averaging characteristics in the directions of the measurement plane, but also in the direction normal to the plane. Taking the same values for the parameters in the formula of the DOC as above, the DOC will increase from 11 µm to 40 µm, when changing the magnification from the previous value of 20× to 10×, with a corresponding aperture numberf#= 2.0.
In Fouras et al (2007), the averaging problem due to volume illumination is treated for particles with constant contribution in the out-of-plane direction. Earlier work from e.g. Bourdon et al (2004a) and Rossi et al (2010) describe methods for modifying the effect on the measured velocity due to the depth-of-correlation by applying image preprocess-ing techniques. Fouras et al (2009) suggest to use decomposition of the correlation function in order to obtain the actual velocity data. Another procedure for decreasing the measurement depth is using selective seeding. This is described in Mielnik and Saetran (2006). This may be a solution for certain situations, but for in vivo measurements this method seems impractical. In this study, the focus will be on interpreting the measured velocity correctly.
The expression for the DOC in (2.1) is based on certain assumptions. For example, the size and intensity distribution of a particle image are modeled for both in- and out-of-focus particles under ideal conditions (e.g. a single thin lens and equally distributed light intensity). The particles image diameterdτ as a function ofz, where z is the distance to the object plane, is modeled as (Olsen and Adrian, 2000; Bourdon et al, 2004b):
d2 τ(z) = M02d 2 p+ 5.95 (M0+ 1)2λ2f#2+ M2 0z2D2a [(nf/nl) (s0+ z)] 2 (2.2)
whereDais the lens aperture diameter,s0the distance from the lens to the object plane, nf the refractive index of the fluid containing the particles, andnlthe refractive index of the lens immersion medium. The first two terms in expression (2.2) represent the particle image diameter for a particle located in the object plane. This is a combination of the geometric image (first term) and the diffraction (second term). The last term describes the geometric spreading of the particle image when a particle is not located in the object plane, but at a distancez to the object plane. The particle intensity distribution I (r, z) is
assumed to be Gaussian and is modelled as (Olsen and Adrian, 2000): I (r, z) = JpD 2 aβ2 4πd2 τ(s0+ z)2 exp −4β 2r2 d2 τ (2.3) wherer is the distance to the particle center, Jpthe flux of light emitted from the particle surface, andβ a constant that is often taken as β2= 3.67 (Adrian and Yao, 1985). For this value ofβ2, the Gaussian distribution is the best approximation for the Airy distribution. This model appears to be in good agreement with experimental data (Olsen and Adrian, 2000). In Figure 2.1, thez-dependency of a particle image is illustrated.
d τ(z) d τ(0) z r I
Figure 2.1: Schematic representation of the cross sections of particle images for differ-entz-positions based on expressions (2.2)-(2.3). The particle image diameter, dτ(z), increases for increasingz, while the peak intensity I(0, z) decreases. The dashed line represents the geometrical spreading of an image of an out-of-focus particle (last term in expression (2.2)).
Micro-PIV measurements in for example biological setups may require specialized mi-croscopes. An upright microscope with one or more additional features, such as a long working distance, stereoscopic imaging abilities, and a zoom function may be desirable. The first part of this study consists of the verification of the assumptions for the in- and out-of-focus particle images for two types of microscopes: the particle image diameter as a function ofz and the intensity distribution are determined experimentally and are com-pared with the expressions in (2.2)-(2.3). The two types of microscopes are an inverted epifluorescent microscope in combination with different interchangeable objectives to ob-tain different magnifications and an upright epifluorescent microscope in combination with one objective with long working distance and a zoom function to obtain different magni-fications in a specific range. Due to the differences in optics used in the microscopes, resulting from the different features, the way particles are imaged may be affected. This, in turn, influences the DOC and the measured velocity can therefore deviate from the ve-locity based on the calculated DOC by (2.1). A relatively simple objective (i.e. one fixed
magnification) is already very complex, with many more optical elements than the single thin lens that is used to model it.2
To gain a better insight into how measured velocities relate to the actual velocity in the presence of local gradients, the accuracy of the micro-PIV method will be investigated experimentally for two commonly used types of microscopes. In the second part of this study, an in vitro experiment is performed in a known reference flow: a glass capillary is used to model a blood vessel. With a micro-PIV system, measurements are performed for different magnifications. The measured maximum velocities are compared with the known maximum velocities to investigate the influence of the spatial-averaging charac-teristics, given the in-plane and out-of-plane (DOC) spatial resolution. The experimentally measured data fordτ(z) is used to provide a better understanding of the measured ve-locity.
In applications that require a large field-of-view (i.e. low image magnification), such as application of micro-PIV to biological flows, it is difficult or sometimes even unfeasible to have a sufficiently small DOC; note that in most studies it is implicitly assumed that the PIV measurement plane represents a thin slice coinciding with the center-plane of a blood vessel. When the DOC becomes larger than the diameter of the blood vessel, the measured velocity is often interpreted as an average of the velocity due to the depth averaging.
2.2
Methods
2.2.1
Setup
For our investigation we considered two microscopes: a conventional inverted epiflu-orescent microscope in combination with different interchangeable objectives to obtain different magnifications (Zeiss, Axiovert 200) and an upright epifluorescent microscope with zoom function in combination with a single objective to obtain different magnifications (Leica MZ 16 FA) (see Figure 2.2 for pictures of both microscopes). With the inverted mi-croscope three different objectives are used, with magnificationsM0= 20×, 10× and 5× (with numerical apertures NA = 0.5 (Zeiss Plan Neofluar), 0.3 (Zeiss Plan Neofluar) and 0.25 (Zeiss Fluar), respectively); see Table 2.1. In order to traverse the focal plane a piezo objective positioner (Piezo Jena) is combined with the inverted microscope. The upright combi-microscope can be used for stereoscopic and normal viewing. The microscope is also equipped with a long working distance. These additional features can be useful or are sometimes even necessary for application of micro-PIV to biological flows, e.g. in vivo measurements in a chicken embryo (Vennemann et al, 2006; Poelma et al, 2008, 2010). The upright combi-microscope has a zoom functionality and is used together with a single objective with magnificationM0= 5× and numerical aperture NA = 0.5 (Leica FluoCombi III). This combination provides magnifications in the range of 2.8-46× with corresponding numerical apertures in the range of 0.08-0.5. For this microscope, moving the focal plane
is accomplished by moving the built-in translation stage of the microscope, which is con-trolled by the recording software of LaVision DaVis. Furthermore, for both microscopes we use a PCO Sensicam QE camera (1,376×1,040 pixels) for image recording and a diode-pumped Nd:YLF laser (New Wave Pegasus) for illumination.3
Figure 2.2: Pictures of the two micro-scopes used in this study: (a) the inverted microscope from Zeiss (Axiovert 200) and (b) the upright combi-microscope from Leica (Leica MZ 16 FA).
(a) (b)
For the PIV measurement, a glass capillary with an inner diameter of 148 µm (TSP150375, Polymicro Technologies) is used, which is comparable in dimension to typical dimensions of commonly-used micro-channels and to the larger blood vessels of the vitelline network of a chicken embryo (Poelma et al, 2008). To allow optical access, part of the external coating is removed by treating it with sulfuric acid and heating it, not exceeding 130 °C, to preserve the glass quality. A 1-ml syringe is connected to the glass capillary, and the syringe is placed into a syringe pump (Model 101, KD Scientific). Fluid is pumped into the glass capillary at a flow rate of 5 µl/min. The fluid contains tracer particles and glycerin to increase the refractive index to 1.45, which closely matches the refractive index of the glass capillary. This minimizes the difference in refractive indices of the fluid and glass to reduce refraction effects that would otherwise distort the image. The section of the capil-lary that is observed is placed on a microscope slide and a cover glass is placed on top of it (see Figure 2.3). To further reduce any effects related to differences in refractive index, the space around the glass capillary, between the microscope slide and cover glass, is filled with the same fluid as inside the capillary, i.e. glycerin with a refractive index of 1.45. The entrance lengthLe, i.e. the minimum length that is required for establishing a fully developed laminar flow, for laminar flows can be calculated by (White, 2008): Le/d = 0.06 Re, where d is the inner diameter of the capillary. The prescribed flow rate
corre-3In our setups, a 1× coupling between the microscope and the camera is used. The large field-of-view can
be maintained for increasing magnifications when the coupling is lowered. The effective numerical aperture is likely to increase. As a side effect, this solution might cause vignetting problems.
sponds to a flow with a Reynolds number in the capillary of10−2, so that the entrance length is very small (≈ 10−7m). The flow will thus be fully developed at the measurement location (≈ 10 cm downstream of the syringe).
The tracer particles are polystyrene spheres (Microparticles GmbH) with a diameter of 1.28 µm. The tracer particles contain a fluorescent dye (Rhodamine B) and are coated with poly-ethylene-glycol (PEG), which avoids particle coagulation and also avoids that the tracer particles attach to the wall of the capillary. To check for unexpected results related to the particle size, the PIV measurements are repeated with similar particles, but with a diameter of 0.56 µm. These PIV measurements are only done on the upright combi-microscope. Due to the more complex optical configuration of the microscope and the larger range of magnifications that we investigated, the possible deviations are more likely to occur for this microscope.
Figure 2.3: Schematic representation of the setup for the PIV experiments (not to scale). The fluid, containing tracer particles and glycerin, is pumped through a glass capillary with a diameter of 148 µm. PIV images are recorded with a camera from below when the inverted microscope is used and from above when the upright combi-microscope is used.
.
2.2.2
Particle image diameter & depth-of-correlation
We first determined for both microscopes the particle image shape and diameter as a function of the distancez from the (nominal) focal plane. We then performed a series of PIV measurements of the flow in the capillary for different measurement planes with re-spect to the center line (i.e.,z=0) of the capillary, in order to determine the velocity profile atz=0. The measurements on the inverted microscope are done with different objectives with different magnification and numerical aperture. The upright combi-microscope has a zoom functionality and is used together with a single objective (M0=5, NA=0.5). The effective numerical aperture is not a fixed value, as it depends on the effective magnifica-tion. The aperture numberf#is used in (2.5) and (2.1), which is related to the numerical
aperture, and can be computed with (Meinhart and Wereley, 2003)4 f#= 1 2 n NA 2 − 1 1/2 (2.4) wheren is the refractive index of the immersion medium between the object and the lens (n = 1.0 for air).
The theoretical values for the particle image diameter in the focal planedτ(z = 0) can be computed with (Adrian and Yao, 1985; Adrian and Westerweel, 2010):
dτ= C q
M2d2
p+ 5.95(M + 1)2λ2f#2, with C=0.74 (2.5) The value ofC is defined such that the width of the particle image corresponds to the e−2
diameter of the intensity distribution, normalized by the peak intensity. The values for dτ can be found in Table 2.1.
For every magnificationM0and corresponding numerical aperture NA, the DOC is cal-culated using equation (2.1) withλ = 590 nm (this wavelength represents the range of emitted wavelengths by the fluorescent particles), dp = 1.28 µm andε = 0.01. For the measurements on the upright combi-microscope, the DOC is also calculated withdp = 0.56 µm. The results are given in Table 2.1 and are represented graphically in Figure 2.4.
2.2.3
Data acquisition and analysis
In order to determine the particle image diameters and intensity distributions, the focal plane has to be traversed with respect to tracer particles at a fixed location. To accom-plish this, a set of images is taken for differents0positions, i.e. distances between the lens and the particle. Fluid containing tracer particles is placed between a microscope slide and a cover glass. During the measurements there was no observable flow. The appropriate distance between subsequent parallel image planes depends on the magni-fication and varies between 1.5 and 2.5 µm for the combination inverted microscope and electric piezo objective positioner. For the upright combi-microscope, the distance be-tween subsequent parallel image planes varies bebe-tween 5 and 12 µm. These variations do not depend on the magnification, but are due to mechanical limitations of the micro-scope and are registered and taken into account. To reduce the signal-to-noise ratio, 200 images are recorded for everys0position. The average of these 200 images is used for further analysis. The image quality is improved by applying a mean filter when needed and the affected parameters are corrected for this.
The intensity distribution of a particle is assumed to be Gaussian (for particle images, see Figures 2.6,2.7), and the edge of the particle, defined as where the intensity has dropped toe−2of the peak value occurring in the center, has to be determined. This is done by fitting a second-order polynomial to the natural logarithm of the intensity. The diameter and the peak intensity of the particle can be expressed in terms of the fitting
Table 2.1: Magnifications and corresponding numerical apertures, particle image diame-ters, depths of correlation, exposure time delays∆t, and sizes of the interrogation win-dows (IW) for both microscopes (λ = 590 nm, dp = 0.56 µm anddp=1.28 µm and ε = 0.01). Inverted microscope M0 NA dτ (px) DOC (µm) ∆t (µs) IW (px) 20 0.50 4 10 375 40×64 10 0.30 3 27 750 20×64 5 0.25 2 42 1,500 10×64 Upright combi-microscope M0 NA dτ (px) DOC (µm) ∆t (µs) IW (px) 0.56 1.28 0.56 1.28 38 0.50 6 8 7 9 200 76×64 30 0.48 5 6 8 10 250 60×64 25 0.45 4 6 9 12 300 50×64 20 0.40 4 5 13 15 375 40×64 15 0.33 4 4 19 22 500 30×64 10 0.24 3 4 39 42 750 20×64 8 0.19 3 3 66 68 1,000 16×64 5 0.14 3 3 130 132 1,500 10×64 4 0.11 3 3 224 226 2,000 8×64 3 0.09 3 3 349 352 2,400 6×64
The numerical apertures for the upright combi-microscope are extracted from the specifications provided by Leica
parameters. This is repeated for all planes with consecutivez-positions. Another way to determine the average particle image diameter is by using the auto-correlation of the total image (Adrian and Westerweel, 2010). Due to the finite space between the microscope slices, particles with slightly different distances to the lens are imaged so that the particle images do not have equal diameters. As a consequence, the results might be biased toward a slightly larger particle image diameter with respect to a situation where all tracer particles are exactly in the focal plane. Therefore, the auto-correlation method may not be suitable in this study, while the current method can handle this.
The data acquisition of the PIV measurements in the capillary is done using single-exposure/double-frame recording. The flow rate is kept constant (within 1%) during all measurements. For each series of PIV measurements corresponding to one magnifica-tion the maximum pixel displacement is kept more or less constant at a value of approxi-mately 10 pixels, by adjusting the time between two frames (∆t). This implies a small ∆t for the highest magnification, and increasingly larger exposure time delays for decreasing values ofM0. In our measurements we thus varied the values of∆t between 200 and 2400 µs. The exact values for each magnification can be found in Table 2.1.
Figure 2.4: The theoretical values for the depth-of-correlation (DOC) relative to the capil-lary diameter for different magnificationsM0for both microscopes, given by the parame-ters in Table 2.1. To obtain different magnifications with the inverted microscope, different objectives are used. Because of the zoom functionality of the upright combi-microscope, different magnifications can be obtained with a single objective.
The maximum velocity occurs at the center of the capillary. To determine when the focal plane coincides with the center-plane of the capillary, scanning PIV is applied: multiple PIV recordings are done for different planes normal to thez-axis. The distance between subsequent parallel planes is 5 µm for the inverted microscope. For the upright combi-microscope, the distance between subsequent parallel planes varies and is less than 12 µm (the variation is due to the mechanical limitations of the microscope). The measure-ment plane containing the highest velocities is considered to be the center-plane of the capillary and will be used for further analysis. Assuming a parabolic flow profile, the error in the maximum velocity resulting from the inaccuracy of thez-position is estimated to be less than 1% for both microscopes.
In every measurement plane, 500 image pairs are recorded. For the analysis of the images, correlation averaging is applied: the local cross-correlation data is averaged to determine the displacement field (Meinhart et al, 2000). In fully developed laminar flows,
only the streamwise velocity component is non-zero. Hence, in our capillary the velocity in they-direction (v) dominates and ∂v/∂y = 0 and therefore, it is appropriate to use rect-angular interrogation windows (with respect to the coordinates defined in Figure 2.3). The analysis with rectangular interrogation windows is done in three passes with 50% overlap and grid refinement by a factor two for the second iteration is applied. The lengths of the interrogation windows in the direction of they-axis (streamwise direction) are fixed, but to exclude the effects of the in-plane gradients, the lengths of the interrogation windows in the direction of thex-axis are variable and depend on the magnification. The sizes of the interrogation windows are chosen in a way such that for every magnification ap-proximately 24 interrogation windows are used in thex-direction of the plane through the capillary. For example, the sizes of the interrogation windows used for the analysis for M0=25 for the three iterations are 100×128, 50×64, and 50×64. As a result, the final sizes of the interrogation windows vary from 76×64 for the highest magnification on the upright combi-microscope (M0=38) to 6×64 for the lowest magnification on the upright combi-microscope (M0=3), see Table 2.1. The PIV evaluation is done with an in-house Matlab code, which is also used by Poelma et al (2008, 2010).
From the PIV data, the displacement is computed by determining the peak location in the correlation function. This correlation function is the summation of the contribution of all particles imaged on the area corresponding with the interrogation window, i.e. the summation of the individual correlation functions resulting from all these particles. When gradients are present, the centers of the individual correlation peaks may no longer co-incide, and as a result, the measured displacement may be influenced, see Figure 2.5. For a better understanding, the experimental results are compared with a model: the re-constructed correlation function. In this model, the correlation function is rere-constructed by summing up the individual correlation functions. This can be done here because the shape of the velocity profile is known. An individual correlation function can be obtained by computing the auto-correlation of the particle image and centering this around the corresponding displacement. Assuming the particle image to be Gaussian, the auto-correlation is also Gaussian and can be expressed in terms of the diameter and peak intensity of the particle image. The particle image diameterdτ(z) and normalized peak intensity ˜Ipeak= I (0, z)/I (0, 0) as functions of z are obtained in two different ways: (a) The theoretical particle image diameters and normalized peak intensities computed by (2.2) and (2.3) are used (Olsen & Adrian-based reconstructed correlation function), (b) The measured particle image diameters and normalized peak intensities from the first part of this study (Section 2.3.1) are used (experiment-based reconstructed correlation function).
In combination with the corresponding centers of the individual correlation functions, known from the parabolic velocity profile, the individual contributing correlation functions are determined. This way, we can reconstruct the resultant correlation function incorpo-rating the contributions of all particles with differentz-positions.
depth-of- -field depth-of-correlation d (z) z x(z) R(s | z0+ z) R(s | z0) R(s | z0 z) R(s) s 0 s 0
Figure 2.5: The correlation function R(s) is the summation of the contributions of all particles within the depth-of-correlation (DOC):R(s) =R R(s|z)f(z)dz. When the DOC increases, the influence of the particles outside the focal plane becomes larger and this may effect the measured displacement. Note that in this figure the maximum velocity does not coincide with the focal plane, which is the case in this study. After: Adrian and Westerweel (2010).
2.3
Results
2.3.1
Particle images
For both microscopes, the particle image diameter and the peak intensity for different z-positions are determined experimentally for different magnifications, see Table 2.1. The peak intensities are normalized by dividing them by the maximum peak intensity, occur-ring atz = 0. For the upright combi-microscope, the diameters and peak intensities of the particle images could not be properly determined, because of the low contrast in the images for the three lowest nominal magnifications (M0=5, 4 and 3). For the in-verted microscope, the results forM0=10 are given in Figure 2.6 and for the upright combi-microscope the results forM0=25 are given in Figure 2.7; the results for the other magnifications show similar behavior and are therefore not shown. The outcome of the analysis of the measurements does not directly agree with the theoretical values com-puted by the expressions in (2.2) and (2.3) (see solid lines in Figure 2.6 and 2.7). Small corrections on the theoretical estimate for the particle image diameterdτ(z) are applied for both microscopes. The theoretical value for the particle image diameter for a parti-cle in the object plane (dτ(0), the first two terms in expression (2.2)) is replaced by the measured particle image diameter atz = 0. The last term in expression (2.2) remains unchanged (see dashed lines in Figures 2.6 and 2.7). This can be seen as incorporat-ing the imperfections of the microscopes into the expression fordτ. Good agreement is shown for both the particle image diameters and the normalized peak intensities for the inverted microscope. The theoretical values fordτ are corrected by multiplying them by 1.8×, 1.5×, and 2.3× for the nominal magnifications M0=20,M0=10, andM0=5
respec-tively.
For the upright combi-microscope, a correction of the theoretical prediction of the particle
(a) (b) (c) (d) (e) −10 −5 0 5 10 0 1 2 3 4 5 6 7 8 z [µm] dτ [px]
experiment inverted microscope theory Olsen&Adrian
theory Olsen&Adrian with adapted dτ
a b c d e −10 −5 0 5 10 0 0.2 0.4 0.6 0.8 1 z [µm] I peak (z)/I peak (0)
experiment inverted microscope theory Olsen&Adrian
theory Olsen&Adrian with adapted dτ
a e
c
d b
Figure 2.6: The measured and calculated particle-image diameters and the peak intensi-ties are shown for the inverted microscope withM0=10 and NA=0.3. When replacing the theoretical value fordτ by the measured value, the experimental and adapted theoretical values show good agreement. For illustration, particle images are shown for different z-positions, indicated by a-e
(a) (b) (c) (d) (e) −60 −40 −20 0 20 40 60 0 10 20 30 40 50 60 70 80 90 100 z [µm] dτ [px] experiment upright microscope theory Olsen&Adrian theory Olsen&Adrian with adapted dτ model by (6)−(7) a b c d e −60 −40 −20 0 20 40 60 0 0.2 0.4 0.6 0.8 1 z [µm] Ipeak (z)/I peak (0) experiment upright theory Olsen&Adrian theory Olsen&Adrian with adapted dτ model by (6)−(7) e d c b a
Figure 2.7: The measured and calculated particle-image diameters and the peak intensi-ties are shown for the upright combi-microscope withM0=25 and NA=0.45. New models are used to describe the particle-image diameter and normalized peak intensities as func-tions ofz. For illustration, particle images are shown for different z-positions, indicated by a-e
image diameter and normalized peak intensity is not sufficient to describe the observed behavior. For this microscope, new empirical models are used to describe their behavior.
To describe the particle image diameter: d2τ(z) = d2 τ(0) 1 − Cdz2 (2.6) and to describe the normalized peak intensity ˜Ipeak:
˜ Ipeak= I (0, z) I (0, 0)= 1 1 + Ciz2 (2.7) whereCdandCiare positive-valued constants. These models are included in Figure 2.7.
Severe deviation of the imaging characteristics from the theory can affect the results of the velocity measurements. Since this is the case for the upright combi-microscope, un-expected results for the measured velocity are more likely to occur for this specialized microscope. Moreover, Bourdon et al (2004b, 2006) have validated the expression for the DOC by inspecting the weighting functions (i.e. the individual contributions to the correlation function). It was shown that the expression holds for a large range of param-eters for a microscope which is comparable, concerning the complexity, to the inverted microscope used in this study.
2.3.2
PIV measurements
Velocity profiles at the center-plane of the capillary are determined for both microscopes for the different magnifications. The PIV measurements on the upright combi-microscope are repeated with particles with a diameter of 0.56 µm. The scaling coefficient is deter-mined by dividing the known capillary diameter (148 µm) by the diameter (in pixels) of the observed capillary image. Because the Reynolds number of the flow is small (10−2), the flow can be described as Poiseuille flow with a parabolic flow profile. The measured velocity profiles are compared with the theoretical Hagen-Poiseuille velocity profiles. The only parameter we use to compute the theoretical velocity profiles is the known fixed flow rate of 5 µl/min. This value is the prescribed flow rate for the pump and verified in a separate measurement. The fluid, which is pumped through the capillary during one hour, is captured in another syringe, so that no evaporation could occur, and weighted. Together with the measured density of the fluid, the pumped volume can be determined and the flow rate can be verified. The determined flow rate deviated less than 1% of the prescribed one. All velocities are normalized with the maximum theoretical veloc-ity of the theoretical profiles (9.7 mm/s), given the imposed flow rate. The position and distances in thex-direction are normalized with the diameter of the capillary (148 µm). The direction of the flow velocity does not exactly coincide with they-axis defined for the images. The error in the alignment is estimated to be less than 0.4 degrees. Therefore, the velocity data in subsequent Figures 2.8-2.11 are velocity magnitudes. In Figure 2.8 the measured and theoretical velocity profiles are shown forM0=10 obtained with the inverted microscope. The results for the other two magnifications show similar behavior and are therefore not shown. In Figure 2.9 the measured and theoretical velocity profiles are shown forM0=25, NA=0.45 resulting from the upright combi-microscope, using 1.28 µm particles. The results for other magnifications are similar and are therefore not shown.
