4.1.1.

Cross-correlation as a function of the interspot distance

We used the gold nanorod suspensions as a test of the invariance of the flow speed measurement on the distance between the two spots. The actual value of the speed, $V_{\exp }=280\pm 50\mu \text{m}/\mathrm{s}$, was obtained by directly measuring the volumetric rate of the hydraulic system. Cross-correlation functions were computed at different distances between the excitation spots in the range of $3\mu \text{m}\le |\mathbf{R}|\le 17\mu \text{m}$ [Fig. 2(a)]. The functions can be well described by Eq. (5) [see Fig. 2(a), solid lines], and the separate fits of the four CCFs give best fit drift times ${\tau }_{VR}$ that depend linearly on the interspot distance up to $|\mathbf{R}|\cong 20\mu \text{m}$ [see Fig. 2(b)]. The slope of the linear trend corresponds to an average drift speed $V_{\mathrm{fit}}=280\pm 10\mu \text{m}/\mathrm{s}$ and diffusion time obtained from the best fit of the CCFs is ${\tau }_{D,\mathrm{fit}}=3\pm 0.5\mathrm{ms}$ (the uncertainty corresponds to the standard deviation computed over different interspot distances). A global fit to the whole set of CCFs leads to very similar results (best fit ${\tau }_{D,\mathrm{fit}}=3\mathrm{ms}$, $V_{\mathrm{fit}}=270\pm 80\mu \text{m}/\mathrm{s}$). The drift speed is very well retrieved from the analysis of the CCFs in a wide range of the ${\tau }_{VR}/{\tau }_D$ ratio [$2.5\le {\tau }_{VR}/{\tau }_D\le 17$; upper inset of Fig. 2(b)]. The best fit slope in the inset of Fig. 2(b) is $\partial ({\tau }_{VR}/{\tau }_D)/\partial |\mathbf{R}|=(1.3\pm 0.1){\mu \text{m}}^{-1}$, in good agreement with the expected value of the ratio $8D/V{\omega }_0^2=1.0\pm 0.2{\mu \text{m}}^{-1}$ [${\omega }_0=0.6\mu \text{m}$, $V=270\mu \text{m}/s$, $D=(12\pm 2)\times {10}^{-12}{\mathrm{m}}^2/\mathrm{s}$]. It is noteworthy that, according to the numerical simulations reported above, the range of drift times investigated corresponds to ${\mathrm{\Gamma }=\tau }_{VD}{/\tau }_D=0.4$ to 0.75, leading to a maximum 5% correction to ${\tau }_{\max }$ with respect to ${\tau }_{VR}$ [see Eq. (6)].

Fig. 2

Experiments on gold nanorods ($\approx 16\mathrm{nm}\times 48\mathrm{nm}$) in square microcapillary. (a) Normalized CCFs acquired from spots at distances $|\mathbf{R}|=3$, 4, 14, 17 μm (squares, circles, up and down triangles, respectively), collinear to the flow in the microcapillary. Solid lines are best fit of Eq. (5) to data (${\omega }_0=0.6\mu \text{m}$; ${\lambda }_{\mathrm{exc}}=0.8\mu \text{m}$). (b) reports the best fit values of the flow speed obtained from ${\tau }_{VR}$ as a function of $|\mathbf{R}|$. The solid line is the best fit with slope $3.59\pm 0.04\mathrm{ms}/\mu \text{m}$. Inset: ratio of the best fit drift over diffusion times ${\tau }_{VR}/{\tau }_D$ as a function of the interspot distance. The solid line is the linear trend ${\tau }_{VR}/{\tau }_D=|\mathbf{R}|(8D/V{\omega }_0^2)\approx |\mathbf{R}|(1.3\pm 0.1)$. (c) Normalized cross-correlation signal from gold nanorods (circles, $|\mathbf{R}|=12\mu \text{m}$), and rhodamine 6G (squares, $|\mathbf{R}|=24\mu \text{m}$) in water solution (beam waist ${\omega }_0=0.6\mu \text{m}$). Solid lines are the best fit to Eq. (5). Fitted diffusion coefficients are $(310\pm 30)\times {10}^{-12}{\mathrm{m}}^2/\mathrm{s}$ for rhodamine 6G and $(13\pm 1)\times {10}^{-12}{\mathrm{m}}^2/\mathrm{s}$ for gold nanorods. The best fit drift speeds are $V=310\pm 10\mu \text{m}/\mathrm{s}$ and $V=195\pm 2\mu \text{m}/\mathrm{s}$ for the rhodamine 6G and the nanorods suspension, respectively.

Limitations on the maximum interspot distance are set by the microscope objective $F$ number and magnification. For the $20\times$ objective used here, the maximum interspot distance is $|\mathbf{R}|\approx 50\mu \text{m}$. Very small interspot distances (${|\mathbf{R}|<2\omega }_0$) cause leakage of signals between the two spots leading to autocorrelation components in the cross-correlation functions and a larger uncertainty on the best fit drift parameters. These spatial limitations (${|\mathbf{R}|\ge 4\omega }_0$) couple to the time requirement given by the CCD (minimum image sampling time ${\tau }_s=4\mathrm{ms}$). In fact, the CCF trend at small lag times is approximately Gaussian, $G(m)\propto S\exp [-(|\mathbf{R}{|}^2/{\omega }_0^2){|1-(m{\tau }_S/{\tau }_{VR})|}^2]$, and its amplitude ($S$) is typically 10 times the noise ($S\approx 10N$). For the sake of simplicity, we measure the drift time in terms of the sampling time, ${\tau }_S$, as ${\tau }_{VR}=M$ ${\tau }_S$. If we now require that the CCF peak is well above the noise level for at least three lag time points, the maximum and two adjacent ones [i.e., $G(M\pm 1)>N$), we come to an analytical estimate of the minimum measurable drift time that we write as $M\ge (R/{\omega }_0)/\sqrt{\ln (S/N)}$. Therefore, for $S/N=10$, $(R/{\omega }_0)=4$, and ${\tau }_S=4\mathrm{ms}$, we cannot reliably measure drift times, ${\tau }_{VR}$, $<10\mathrm{ms}$. These limitations lead to maximum measurable values of flow speed of about $V_{\max }\cong 6000\mu \text{m}/\mathrm{s}$ for our setup.

4.1.2.

Cross-correlation as a function of the diffusion coefficient

Cross-correlation analysis can provide information on the diffusion coefficient of the fluorophore beyond the time resolution of the detector by exploiting the weak dependence of the cross-correlation peak width on the diffusion coefficient.¹⁵ For this purpose, we compared CCFs measured on rhodamine 6G solutions to those measured on suspensions of gold nanorods [see Fig. 2(c)]. In Fig. 2(c), we report two CCFs acquired with slightly different parameters ($|\mathbf{V}|$ and $|\mathbf{R}|$) in order to visually obtain a good superposition of the cross-correlation peak. The data fit to Eq. (5) provides best fit drift speeds of $V=310\pm 10\mu \text{m}/\mathrm{s}$ and $V=195\pm 2\mu \text{m}/\mathrm{s}$ for the rhodamine 6G and the nanorods suspension, respectively. The best fit diffusion coefficients ($T=22°\mathrm{C}$) are $(310\pm 30)\times {10}^{-12}{\mathrm{m}}^2/\mathrm{s}$ for rhodamine 6G and $(12\pm 1)\times {10}^{-12}{\mathrm{m}}^2/\mathrm{s}$ for gold nanorods. Both results are compatible with literature and our independent estimates.^15,27,32 It must be noted that rhodamine 6G diffusion time on a 0.6-μm beam waist is $\cong 75\mu \text{s}$, which is much smaller than our sampling time. Finally, we notice that, according to Eq. (6), the peak lag time of the cross-correlation underestimates the drift time ${\tau }_{VR}$ only by 0.1% for gold nanorods. For rhodamine 6G, instead, the drift time underestimation rises to 6%.

4.1.3.

Cross-correlation as a function of the drift speed

We computed fluorescence CCF for fluorescent microbeads dispersed in water and flowing in a borosilicate square section capillary. The drift speed was changed by microregulating the height, $H$, between two vessels (insulin syringes) connected with a 800 μm inner diameter Teflon tube to the capillary. We have varied the relative height of the two vessels from $H=-3500\mu \text{m}$ to $H=+5000\mu \text{m}$ (sketch in Fig. 3). The analysis of the CCFs acquired as a function of $H$ reveals three regimes. At negative values of $H$, ($-3500\mu \text{m}\le H\le -1000\mu \text{m}$), we find a linear dependence of the drift speed on $H$, with slope $-(0.55\pm 0.08){\mathrm{s}}^{-1}$, as we should expect in a Poiseuille flow regime.⁶ At intermediate values of the relative height we could not measure any cross-correlation between the spots. In this case, wall adhesion and capillary effects^33–35 interfere severely with the viscous flow and the fluid speed effectively vanishes. The estimate of the capillary pressure to be overcome, due to adhesion between water and the wall of the syringe, is $\approx 4({\gamma }_{sg}{-\gamma }_{sl})/d_S$, where ${\gamma }_{sg}$ and ${\gamma }_{sl}$ are the surface tensions between plastic (solid) and air (gas), and between plastic and water (liquid) and $d_S$ is the reservoir diameter ($d_S\approx 4000\mu \text{m}$). For poly(methyl methacrylate) (PMMA),³⁵ $({\gamma }_{sg}{-\gamma }_{sl})\approx 0.02\mathrm{J}/{\mathrm{m}}^2$ and the pressure difference amounts to $\sim -20\mathrm{N}/{\mathrm{m}}^2$. This value corresponds to 2000 μm of water height that compares well with the height gap in which we cannot measure any water flow ($\approx 3000\mu \text{m}$).

Fig. 3

Drift velocity measured on 30 nm fluorescent nanosphere suspensions flowing through a square capillary ($\mathrm{size}=720\pm 20\mu \text{m}$) as a function of the difference in height (H) between the two reservoirs. The distance between the two laser spots (hourglass symbols in the sketch) is $|\mathbf{R}|=16\mu \text{m}$. The CCFs were fit to Eq. (5). The solid lines correspond to the linear fit to the drift speeds in the two regions $H>0$ $(\text{slope}=0.42\pm 0.12){\mathrm{s}}^{-1}$ and $H<0$ $(\text{slope}=-0.55\pm 0.08{\mathrm{s}}^{-1})$.

In the third region, at ($2500\mu \text{m}\le H\le 5000\mu \text{m}$), we can measure a clear cross-correlation peak in the reverse CCF $G_{X21}(\tau )$ and the fluid speed again scales linearly with height difference, with a slope of $(0.42\pm 0.12){\mathrm{s}}^{-1}$. These values can be compared to the prediction made according to Poiseuille law for a square capillary of size $d$,³⁵ connected to two reservoirs. The flow is determined by the pressure gradient due to the relative height, $H$, of the two reservoirs, $\mathrm{\Delta }P=\rho gH/L$, where $\rho$ is the density of the fluid, here taken as that of water, $g$ is the gravity acceleration, and the tubing length is $L\approx 0.69\mathrm{m}$ in our setup. The estimated maximum speed in the capillary is then $V_{\max }(H)=H[(\rho g/L\eta )0.084]d^2$. For the present case (the real size of the capillary is $(d=720\pm 40\mu \text{m})$, the prediction $V_{\max }(H)/H=(0.60\pm 0.06){\mathrm{s}}^{-1}$ is in reasonable agreement with our experimental findings $V_{\max }(H)/H=(0.51\pm 0.07){\mathrm{s}}^{-1}$.

4.1.4.

Speed profile across the capillary section

The benefits of the EM-CCD dual-beam cross-correlation setup for parallel acquisition can be better estimated when extended excitation patterns are employed. To provide an example of this application, we have combined line scanning with our dual-beam setup. For this purpose, the line scanning rate is implemented at $200\text{lines}/\mathrm{s}$ and synchronized with the image acquisition rate. In this way, we have the possibility to perform, on a single image, cross-correlation along the flow direction over a range of radial positions in the capillary section that is limited only by the field of view. The only spatial limitation in this case is posed by the size of the pixel on the CCD chip that corresponds to 7.4 μm on the sample. Since the line scanning is performed synchronously with the image acquisition, our configuration is equivalent to shining an irradiation pattern composed of two parallel laser lines perpendicular to the capillary axis. We describe the two scanned lines as a set of pixels, $\{{\{P_1[i_1]\}}_{i_1=1\dots N}{\{P_2[i_2]\}}_{i_2=1\dots N}\}$, where the subscripts indicate the line and the index $i_1$ or $i_2$ the order of the pixel along each line, composed of $N$ total pixels. The CCFs were computed between two pixels of the two lines along the flow, $G_{X1,2}^{(i1.i2)}(\tau )=\langle I_{1,i1}(\tau )I_{2,i2}(0)\rangle -\langle I_{1,i1}(0)\rangle \langle I_{2,i2}(0)\rangle$. With a $20\times$ microscope objective we have sampled almost half of the section of the square capillary (Fig. 4) over $N=42\text{pixels}$.

Fig. 4

Characterization of the speed profile of the section of the glass square capillary. The flow speed was obtained directly from the lag time of the maximum of the CCFs $G_{X1,2}^{(i1=i2)}(\tau )$. The CCFs over almost half of the capillary section were computed in parallel on two corresponding spots lying along the flow direction that were scanned in a direction perpendicular to that (see sketch). The solid line is the best fit to the data of Eq. (10) with best fit values of $z0=48\pm 2\mu \text{m}$, $d=720\pm 20\mu \text{m}$, and $V_{\max }=464\pm 5\mu \text{m}/\mathrm{s}$. The dashed line is the best fit of the data to Eq. (10) with $d$ kept fixed at 800 μm. The inset reports the data superimposed on the extrapolation of the best fit profile over the whole capillary width.

We measured the flow speed directly from the lag time of the maximum of the CCFs $G_{X1,2}^{(i1=i2)}(\tau )$. The flow speed values as a function of the radial distance ($z$) from the center of the capillary (Fig. 4) have then been fit to the following trial function (see Fig. 4, sketch):³⁵

In this equation, which is an extension of the known parabolic profile observed in circular cross-section capillaries, $d$ and $V_{\max }$ are the size of the capillary and the maximum speed of the fluid (at the center of the capillary). The position of the center of the capillary section is not known and is used as a best fit parameter $z_0$, which is the radial coordinate of the maximum speed in the capillary. The best fit of Eq. (10) to the data reported in Fig. 4 (continuous line) gives $d=720\pm 20\mu \text{m}$, $V_{\max }=464\pm 5\mu \text{m}/\mathrm{s}$, and $z_0=48\pm 2\mu \text{m}$.

This fit offers an estimate of the actual internal size of the microcapillary that is 10% lower than nominal data. The forced fit of the data in Fig. 4 to Eq. (10) while keeping $d$ fixed at the nominal size $d=800\mu \text{m}$ gives clearly unsatisfactory results (Fig. 4 dashed line).

We finally notice that we could have fit each $G_{X1,2}^{(i1=i2)}(\tau )$ function to Eq. (5) to measure drift times and flow speeds. However, this time-consuming procedure leads to drift times that are underestimated with respect to the maximum CCF lag time by only 0.3% (0.9%) at the maximum (minimum) of the flow speed profile.

4.2.1.

Blood flow can be followed by cross-correlations along the vein

We applied CCD cross-correlation spectroscopy to study blood flow in zebrafish embryos vessels. For this purpose, we have employed transgenic embryos whose blood red cells carry dsRed protein. dsRed fluorescence is primed by two-photon excitation at ${\lambda }_{\mathrm{exc}}=0.9\mu \text{m}$ and the emission is collected at ${\lambda }_{\mathrm{em}}=0.6\mu \text{m}$ through a $\mathrm{HQ}600/40$ filter (Chroma, Bellows Falls, Vermont). The two spots of the infrared laser were set at variable distances along the vessel axis in the middle of the vessel cross-section and along the blood flow direction. Coarse adjustment of the spots along the flow lines was reached by moving the embryo and finer tuning was obtained by moving one of the spots in the focal plane by means of the interferometer mirrors.

We tested at first the accuracy of the blood flow speed measurements on a single embryo by performing repeated measurements on the embryo’s vein at increasing distances between spots. All the CCFs have been analyzed according to Eq. (5) in the range of $6\mu \text{m}\le |\mathbf{R}|\le 24\mu \text{m}$ (Fig. 5), and in this range, the average value is determined with $<3\%$ uncertainty on the single embryo. The data reported in Fig. 5 correspond to a 4 d.p.f. embryo and $|\mathbf{V}|=640\pm 20\mu \text{m}/\mathrm{s}$ (3% uncertainty). The diffusion time in the analysis of all the CCFs is kept at ${\tau }_D\approx 0.1\mathrm{ms}$. As discussed previously in the text, this parameter has to be taken, for sparse samples with rare fluorescence bursts, as an effective value that describes the random superposition of a few, distinct, cross-correlation peaks [Figs. 1(c), 1(d), and 1(e)].

Fig. 5

CCFs computed on vein blood vessels of a zebrafish embryo (four days postfertilization) at increasing values of the distances between the laser spots. The different symbols refer to $|\mathbf{R}|=6\mu \text{m}$ (open square), 11 μm (filled circles), 16 μm (open up-triangles), 20 μm (filled down-triangles), and 24 μm (open diamonds). The error bars correspond to an average over three different measurements on the same embryo. The dashed lines are the best fit curve to Eq. (5). The inset reports the best fit values of the drift speed obtained from fitting the cross-correlation to Eq. (5). The dashed line indicates the speed average value.

4.2.2.

Discrimination of systolic and diastolic phases

We move to the study qualitative and quantitative differences in the blood flow along the caudal vein and the dorsal aorta. For this purpose, we should evaluate the biological variability of the data. We have taken averages of the flow parameters over three to four different embryos at the same developmental stage.

From the fluido-dynamics point of view, the major difference between the two vessels is the presence of a double peak in the CCF measured on the artery. In the vein, far from the tail, we find a single peak that corresponds to a drift speed of $V_{\text{vein}}=630\pm 70\mu \text{m}/\mathrm{s}$ [Fig. 6(a)]. On the contrary, in the artery, two peaks in the CCFs are found [Figs. 6(b) and 6(c)], suggesting the presence of two phases in the blood flow. According to the literature,^22,25 these phases can be ascribed to the systole (faster phase) and diastole (slower phase) phases. We have then analyzed the data taken in the embryo artery according to a drift model [Eq. (5)] with two drift components (the same diffusion time was taken for both the drift components of the CCFs) at different distances from the tail capillary bed.

Fig. 6

CCFs measured in vessels of different zebrafish embryos at four days postfertilization (d.p.f.) ($|\mathbf{R}|=20\mu \text{m}$). Examples of CCFs measured in the vein are reported in (a) together with their best fits [Eq. (5), solid lines]. The average value of the drift speed $V_{\text{vein}}=630\pm 70\mu \text{m}/\mathrm{s}$. Examples of the CCFs measured in the artery close (20% tail-heart distance, see image) to embryo’s tail are reported in (b). The average value of the best fit systolic and diastolic speeds are $V_{\mathrm{syst}}=980\pm 120\mu \text{m}/\mathrm{s}$ and $V_{\mathrm{diast}}=420\pm 40\mu \text{m}/\text{s}$, respectively. Examples of the CCFs computed in the artery at half way between embryo heart (50% tail-heart distance, see image) are reported in panel (c). The average value of the best fit systolic and diastolic speeds are $V_{\mathrm{syst}}=1420\pm 200\mu \text{m}/\mathrm{s}$ and $V_{\mathrm{diast}}=380\pm 20\mu \text{m}/\mathrm{s}$, respectively. Inset in panel (c) report the comparison of the CCFs $G_{X12}(t)$ (thin line) and $G_{X21}(t)$ (thick line). The latter function does not show a correlation peak, confirming the direction of the flow. The image reports the transmitted light image of a zebrafish embryo at 4 d.p.f. ($\text{size}\approx 3.5\mathrm{mm}$).

The CCFs collected at half way (50% distance, see image in Fig. 6) of the heart-tail distance are reported in Fig. 6(c). The analysis of these functions gives $V_{\mathrm{syst}}=1420\pm 200\mu \text{m}/\mathrm{s}$ and $V_{\mathrm{diast}}=380\pm 20\mu \text{m}/\mathrm{s}$. When we perform the same experiment at 20% of the tail-heart distance, we still obtain two drift components with much more closely spaced speed values [see Fig. 6(b)]. The average values of the systole and diastole speeds are, in this case, $V_{\mathrm{syst}}=980\pm 120\mu \text{m}/\mathrm{s}$ and $V_{\mathrm{diast}}=420\pm 40\mu \text{m}/\mathrm{s}$. The systolic component shows a larger decrease when approaching the tail, compared to the diastolic component that is almost constant. The averages taken over three to four different embryos at the same developmental stage show larger uncertainty (10%) than those obtained from repeated blood flow speed measurements on the same embryo. The figure of 10 to 12%, therefore, represents the biological uncertainty on the blood flow measurements.

4.2.3.

EM-CCD cross-correlation estimate strain rate in the microvessels

Plasma shear stress is a relevant parameter for hemodynamics since it can trigger the response of a number of proteins on the vessel epithelium.^22,36,37 In the literature, various methods have been applied to monitor the flow speed in microfluidic setups and even in animal models.^38–41 Beside a few applications of laser speckle imaging,⁴² fluorescence detection in cytometry⁴³ and nuclear magnetic resonance (NMR),⁴⁴ PIV^{26,37,39–41} has been widely applied to follow injected nanospheres and platelets and/or blood red cells observed in transmission optical microscopy and tomography.⁴⁵ Microinjection of particles needed for PIV could alter physiology due to unavoidable spheres’ aggregation,³⁸ and correlation methods applied to cells flowing in the blood stream should be preferred for physiologically relevant insights.⁴⁵

We measured CCFs of red blood cells fluorescence both in the dorsal vein and in the caudal aorta at half way between the zebrafish tail and heart, far enough from the pulsation origin (the heart) and the tail capillary bed, which has a large hydraulic compliance. The edges of the vessel were detected by following the abrupt fall of the value of the CCF amplitude $G(0)$, and the parallel increase in the relative uncertainty on the drift speed [${\epsilon }_V/V$ in the inset of Fig. 7(a), where ${\epsilon }_V$ is the standard deviation of blood flow speed as from the best fit of the CCFs], that occurs when we focus the laser spots outside the vessel (see Fig. 7(a), vertical arrows). For each pixel, the CCFs were fit to Eq. (5) with one [for the vein, Fig. 7(a)] or two [for the artery, Fig. 7(c)] drift components. We have checked that the blood flow was perpendicular to the two scanned lines across the vessel by computing CCFs of adjacent pixels, $G_{X1,2}^{(i·i\pm 1)}(\tau )$ and $G_{X1,2}^{(i·i\pm 1)}(\tau )$. The amplitude of the CCF decreased by increasing distances between the pixels with $G_{X1,2}^{(i·i)}(0)=0.08\pm 0.01$, which is larger than $G_{X1,2}^{(i·i-1)}(0)\cong G_{X1,2}^{(i·i+1)}(0)=0.042\pm 0.005$ and $G_{X1,2}^{(i·i+2)}(0)\cong G_{X1,2}^{(i·i-2)}(0)=0.009\pm 0.004$.

Fig. 7

Speed profile across veins and arteries measured at 50% of heart-tail distance in zebrafish embryos (developmental stage: 4 d.p.f., $|\mathbf{R}|=40\mu \text{m}$). (a) reports two exemplary CCFs measured in zebrafish veins at a distance $\approx 0.25d_V$ (open squares) and $\approx 0.125d_V$ (filled squares) from the vessel wall ($d_V$ is the estimated vessel diameter). Solid lines are the best fit to Eq. (5) with a single drift component. Inset: plot of the CCF amplitude, G(0), and the drift speed uncertainty, ${\epsilon }_V/V$, obtained from the data fitting as a function of the radial distance across the vessel. The vertical arrows indicate the estimated position of the vessel wall. (b) Profile of the best fit speed values in veins measured on two (4 d.p.f.) zebrafish embryos. Each set of data is the average of three different scanning on the same embryo. The solid line is the best fit of the data to the trial function $V(r)=V(r_0)-[V(r_0)-V(r_0\pm d_V/2)]{[2(r-r_0)/d_V]}^2$ (see text). (c) Two exemplary CCFs measured in zebrafish arteries at $\approx 0.25d_V$ (open squares) and $\approx 0.125d_V$ (filled squares) from the vessel wall. Solid lines are best fit to Eq. (5) with two drift components corresponding to the systole (higher values) and diastole (lower values). (d) Profile of the best fit speed values (open squares, systole and filled squares, diastole) measured on a 4 d.p.f. zebrafish embryo. The data are the average of three different scanning on the same embryo. The solid line is a parabolic fit to the data according to the same trial function as in (b).

The radial profile of the speed is well reproduced in embryos of similar ages [Fig. 7(b)]. We assume a linear relationship between the shear stress and the strain rate. In this way, we can obtain the shear stress^46,47 from the measurement of the steepness of the flow speed profile in the vessels of zebrafish embryos. We measure here an effective shear stress since we are following the motion of red blood cells (RBCs) that are comparable in size with the vessel diameter. The maximum effective strain rate in the vein, $\gamma =|(\partial V/\partial r)[r_0\pm (d_V/2)]|=4\{V(r_0)-V[r_0\pm (d_V/2)]\}/d_V=15\pm 3{\mathrm{s}}^{-1}$, is intermediate to those relative to the systole and diastole pulses measured in the artery, $|(\partial V/\partial r)[r_0\pm (d_V/2)]|=34\pm 6{\mathrm{s}}^{-1}$ (systole) and $|(\partial V/\partial r)[r_0\pm (d_V/2)]|=8\pm 3{\mathrm{s}}^{-1}$ (diastole). This can be due to the fact that the heart pulsation is much more effective in the artery than in the vein²⁶ and that the volumetric flow rate is much lower in the systole than in the diastole phase.²⁴ The values of the strain rate found here, when converted to shear stress (RBC $\mathrm{viscosity}\approx 5\times {10}^{-3}\mathrm{Pa}·\mathrm{s}$),⁴⁶ give values $\approx 0.05$ to 0.15 Pa that are lower than those ($\approx 1\mathrm{Pa}$) derived for rabbit arterioles (larger than zebrafish vessels) by Tangelder et al.³⁷ and those measured in the aortic arch in zebrafish 4 d.p.f. by Chen et al. by $\mu$-PIV.²⁶ These studies correspond, however, to higher pressures and reproduced the speed profile more closely than was done here since the tracked object (platelets or microbeads) is much smaller than the vessel diameter. Moreover, one should take into account that the wall shear stress measurement based on RBC tracking is an effective value.⁴⁷

As shown in Figs. 7(b) and 7(d), the measured speed does not vanish at the edge of the vessel and the ratio of maximum velocity to the marginal velocity is $\chi =V(r_0)/V(r_0\pm d_V/2)$$\approx 1.2\pm 0.05$ for both the vessels. This has been observed earlier^37,47–49 in similar cases. Zhong and coworkers⁴⁸ reported detailed speed profiles in 32- to 120-μm-diameter retina veins obtained by tracing the motion of red blood cells on optical scanning microscopy images, finding $\chi \approx 1.36$ for 32 μm diameter capillaries and $\chi \approx 2$ for very large capillaries (the values should tend to infinity in a Poiseuille regime). A model developed by Yen et al.⁵⁰ and generalized by Zhong et al.⁴⁸ relates $\chi$ to the vessel diameter $d_V$.

In our case, the vessel diameter is $d_V\approx 12\pm 2\mu \text{m}$ for the vein and $d_V\approx 13\pm 3\mu \text{m}$ for the aorta at midway between the tail and the heart, leading to $\chi \approx 1.25$ from Eq. (11), which is very close to our experimental result.

The simplest rationalization of the reduced value of the speed ratio $\chi$ is related to the finite size of the RBCs. The size of the vessels is, on the average, only twice the size of the red blood cells (${\delta }_{\mathrm{RBC}}\approx 5\mu \text{m}$): the red blood cells are, therefore, most of the time rolling over the epithelium along the vessel. The minimum distance from the epithelium at which we can measure the flow speed is then only half the red blood cell size, $\approx {\delta }_{\mathrm{RBC}}/2$. For a circular capillary of diameter $d_V$ in Poiseuille regime, the speed profile is $V(r)-V_{\max }[1-{(2r/d_V)}^2]$ and the minimum speed value measurable due to the finite size of the red blood cells should then be $V_{\min }=V_{\max }\{1-{[(d_V-{\delta }_{\mathrm{RBC}})/d_V]}^2\}$. For the present case, these estimates lead to $\chi \approx 1.6$, a value that, though not so low as the one found experimentally, is much less than the one corresponding to pure Poiseuille flow in no-slip boundary conditions. In larger capillaries, additional hydrodynamic effects (reduced effective viscosity close to the vessel walls) can partially contribute to the reduction of $\chi$.^37,48,51

In the above measurements, we exploited dual line scanning coupled with CCF computation on the EM-CCD frames. This technique can also be used to simultaneously monitor flows in closely running vessels. Such a situation is found in zebrafish close to the capillary bed (5% from tail, see Fig. 6, image) at the transition between artery and vein in the embryo tail. We have scanned two parallel lines (40 μm apart) close to the zebrafish tail capillary bed and derived the drift velocity from the analysis of the CCFs (Fig. 8). We also notice here the presence of a systolic and a diastolic phase in the vein CCFs.

Fig. 8

CCFs acquired simultaneously on zebrafish at heart-tail 5% distance from the tail (see zebrafish embryo picture in Fig. 6). (a) CCFs measured in artery at two radial positions corresponding to 3 (open triangles) and 12 μm (filled triangles) along the scanning line in (c) and (d) (open triangles). (b) CCFs measured in vein at two radial positions corresponding to 24 (open squares) and 33 μm (filled squares) in (c) and (d) (open squares). In (a) and (b), the solid lines are best fit of the CCFs to Eq. (5) with two drift components. (c) and (d) report the best fit drift speed, obtained from the analysis of the CCFs, as a function of the position across the vessel. (c) Diastolic component (open triangle, artery; open squares, vein). (d) Systolic component (open triangle, artery; open squares, vein). The solid lines are the best fit to the parabolic function $V(r)=V(r_0)-[V(r_0)-V(r_0\pm d_V/2)]{[2(r-r_0)/d_V]}^2$. The image shows a blow up of a section of a zebrafish embryo close to the tail (5%, artery above, vein below). Green and red colors indicate the epithelium emission (GFP) and the red blood cells (dsRed). The white horizontal lines represent the two lines scanned during the fluorescence correlation spectroscopy measurements.

From the speed profiles, we can also derive in this position the relevant parameters such as the velocity ratio, $\chi \approx 1.18\pm 0.03$ for the artery and $\chi \approx 1.12\pm 0.05$ for the vein, and the strain rate that is larger in the artery, both for the systole and the diastole phases, $\gamma =19\pm 6{\mathrm{s}}^{-1}$ (diastole) and $\gamma =40\pm 15{\mathrm{s}}^{-1}$ (systole), than in the vein, $\gamma =11\pm 5{\mathrm{s}}^{-1}$ (diastole) and $\gamma =12\pm 9{\mathrm{s}}^{-1}$ (systole).