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Multi-scale laser speckle contrast imaging of microcirculatory vasoreactivity

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Abstract

Laser speckle contrast imaging is a robust and versatile blood flow imaging tool in basic and clinical research for its relatively simple construction and ease of customization. One of its key features is the scalability of the imaged field of view. With minimal changes to the system or analysis, laser speckle contrast imaging allows for high-resolution blood flow imaging through cranial windows or low-resolution perfusion visualization of perfusion over large areas, e.g. in human skin. We further utilize this feature and introduce a multi-scale laser speckle contrast imaging system, which we apply to study vasoreactivity in renal microcirculation. We combine high resolution (small field of view) to segment blood flow in individual vessels with low resolution (large field of view) to monitor global blood flow changes across the renal surface. Furthermore, we compare their performance when analyzing blood flow dynamics potentially associated with a single nephron and show that the previously published approaches, based on low-zoom imaging alone, provide inaccurate results in such applications.

© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

Over the past two decades, laser speckle contrast imaging (LSCI) became one of the major blood flow imaging tools in basic and clinical research. It applies to not only various organs like the brain [1,2], the retina [3,4] and the skin [5,6], but also to different pathologies such as strokes [7], migraines [8,9], and diabetes [10]. A key feature that makes LSCI broadly adaptable is the scalability of the imaged field of view. With minimal, if any, changes to the system or analysis complexity, LSCI allows for e.g. high resolution blood flow imaging through 1–5 mm$^2$ cranial windows in animal models [1] and visualizing perfusion on human skin over a surface larger than 10000 mm$^2$ [5,6]. Inevitably, acquiring LSCI images in high spatial and temporal resolution limits the field of view and vice versa. Yet, some experimental conditions can gain critical information from simultaneous imaging of large and small field of view. While the prior provides global hemodynamics for a tissue of interest, the latter enables local high-resolution information.

One such application of simultaneous multi-scale imaging is in the study of vasoreactivity and neurovascular coupling in rodent stroke models. The large field of view (e.g. 100 mm$^2$ capturing the whole brain) enables observation of an entire hemisphere to assess whole organ blood flow phenomenons like cortical spreading depression. Meanwhile, high-resolution imaging (e.g. covering less than 5 mm$^2$) can precisely capture local vascular dynamics, which cannot be resolved with the aforementioned large field of view.

Another application is in observing blood flow dynamics in organs, like the kidneys, with strong local vasoreactivity or autoregulatory mechanisms [11]. The kidneys have morphologically discernable units called nephrons that self-regulate blood flow. Yet, recent studies have shown that nephrons form communication networks and synchronize their auto-regulatory oscillations [1214]. Collaborating neprhons can form large and small scale functional clusters that modulate different components of autoregulation. Due to this shift in paradigm of renal autoregulation, the research field demands an imaging system that fulfills the trade off between resolution quality and field-of-view scale. This enables observation of changes in individual nephron’s blood flow dynamics to query local networks and the scope of their contribution to global hemodynamics which remain unclear. While LSCI has been previously applied to the kidneys, its ability to scale to various resolutions has not been hitherto exploited.

LSCI was first applied to study nephron blood flow dynamics by Holstein-Rathlou [15]. They utilized a commercial LSCI system with 576x768 pixels to perform temporal contrast analysis and record blood flow over the whole kidney surface of about 11.5x15.4 mm at one frame per second. They injected lissamine green into the kidney to identify vessel locations and then represented a single afferent arteriole blood flow by averaging time series from 15x15 pixel region (approximately 300x300 $\mu$m) per identified vessel. The same system and similar field of view was then used in several follow-up studies that explored large-scale changes of blood flow in the renal cortex [16,17].

Some proposed that the spatial and temporal resolution used in those studies were insufficient to study all aspects of renal hemodynamics. Scully and Mitrou [18,19] utilized the same system, but in spatial contrast analysis mode to record blood flow with 113x152 pixels over 5x7 and 4x5 mm regions at 25 frames per second. In their analysis, rather than selecting specific regions of interest, they overlaid a 27x37 grid, where each cell corresponded to the blood flow from about 150x150 $\mu$m region. However, as a typical diameter of arterioles is less than 20 $\mu$m, these studies failed to identify individual mircrovessels in their results. More recently, we introduced high-resolution laser speckle contrast imaging of renal microcirculation: 1024x1024 pixels were used to record blood flow over 1.5x1.5 mm area of the renal surface [20]. We segmented individual arterioles and venules on the renal surface and accurately measured their blood flow dynamics. But the system is unable to visualize large scale blood flow changes or estimate nephron cluster sizes expanding outside the field of view.

In this paper, we introduce multi-scale laser speckle contrast imaging (msLSCI) system and study vasoreactivity in renal microcirculation. We combine high resolution (small field of view) to segment blood flow in individual vessels, with low resolution (large field of view) to monitor global blood flow changes across the renal surface. We explore limitations of low resolution renal imaging by performing wavelet analysis and by comparing spectral characteristics of tubuloglomerular feedback (TGF) activity. We also use our low resolution data to simulate the approaches suggested by Holstein-Rathlou [15] and Mitrou [19] and bridge the gap in interpreting laser speckle contrast images of renal microcirculation.

2. Methods

2.1 Multi-scale laser speckle contrast imaging

To enable multi-scale laser speckle contrast imaging (msLSCI), two CMOS sensor cameras (Basler acA2040-90umNIR, 5.5 $\mu$m pixel size, 8bit mode) were each attached to video zoom lenses (VZM 450, Edmund optics) and set to 1x and 4.5x magnifications, corresponding to object space resolution of approximately 40 and 144 lp/mm, for low zoom and high zoom respectively. Zoom lenses were mounted to a beamsplitter cube (CCM1-BS014/M, Thorlabs) at an orthogonal angle as shown in Fig. 1(A). The beamsplitter cube was fixed to a height adjustable vertical z-stage mounting plate (Thorlabs). A linear polarizing filter was attached to the bottom of the cube and adjusted to reduce the amount of recorded specular reflections and the first order scattering events. To illuminate the tissue surface, we used Volume Holograpic Grating stabilized diode (LP7850-SAV50, 785 nm Thorlabs) which was shown to provide optimal signal quality for LSCI applications [21]. Raw laser speckle images were acquired for 1 hour at 50 frames per second with an exposure time of 5 msec, the optimal exposure time for perfused small vessels in tissue [22]. Iris on the low-zoom arm was adjusted to achieve similar amount of light on both cameras. Positions of the tube lenses were adjusted to match their focus planes and the images obtained in real-time from the high-zoom arm served as a reference while adjusting the z-stage and focusing the system. Ambient light was dimmed and all of the system settings were kept consistent across experiments.

 figure: Fig. 1.

Fig. 1. Multi-scale laser speckle contrast imaging. A) Diagram of experimental setup to image a rodent kidney. Examples of raw image (B) and blood flow map (C) recorded with high-zoom arm with image resolution of 1.2 $\mu$m per pixel. Corresponding images (D, E) acquired with the for the low-zoom arm with image resolution of 5.5 $\mu$m per pixel. Blood flow index (BFI) maps were obtained by performing all data processing steps, including motion compensation, calculation of blood flow index (BFI), and averaging images over the observation time.

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2.2 Animal preparation

All rodent experiments were approved by Danish National Animal Experiment Inspectorate. Male Long Evans rats (N=3, RJOrl:LE, Janvier Labs, France) between 275-325 grams were imaged. Surgical procedures are described in Refs. [15,20]. Here we provide a brief summary. Animals were induced into anesthesia (Sevorane, Abbvie) in a chamber at 8% and maintained at 1.5 % throughout the surgery and experiment. Body temperature was maintained at 37 degrees on a servo-controlled heating table. Two polyethylene tubes (PP25, Smith Medicals) were placed into the right jugular vein for constant administration of saline (0.5 % NaCl) and muscle relaxant (0.5 mg/ml) (Nimbex, Apsen) at an infusion rate of 20 $\mu$l per minute each. One polyethylene tube was inserted into left carotid artery for continuous blood pressure measurement. Tracheotomy was performed for artificial ventilation at 60 strokes per minute. After laparotomy, the left kidney was isolated and immobilized with a 1.5 % (w/v) agarose solution in a 3D printed kidney shaped well. The kidney was covered with a glass cover slip to improve the imaging quality and to prevent the agarose and the kidney surface from drying. The glass did not touch the printed well and moved with the kidney whenever some motion was present. A myograph wire (40 $\mu$m in diameter) was bent at an acute angle and placed on top of the glass to serve as reference mark for post acquisition image registration . Left ureter was cannulated for free urine flow. Animal preparation was considered successful only when the intra-arterial blood pressure had stabilized between 100–130 mmHg for the duration of the experiment.

2.3 Data analysis

2.3.1 Motion compensation

To compensate for lateral motion (imaged in the x-y plane) caused by breathing and movements of the surrounding organs, a custom image alignment protocol was implemented. The myograph wire positioned on the glass surface was segmented in every raw image acquired with the high-zoom arm (Fig. 1(B)). The segmented images were used to calculate geometrical transformation required to register all frames to the first frame. The transformation was then applied to every raw image of the high-zoom data, which were then used for any consecutive analysis. Low-zoom data did not require motion compensation as myograph wire displacements were consistently below 5 $\mu$m and deemed negligible.

2.3.2 Data simulation

To better understand the differences caused by using low resolution imaging and to bridge the new results and earlier studies, we simulated approaches and resolutions used by Holstein-Rathlou [15] and Mitrou [19]. We resized raw low-zoom images by factors 0.225 and 0.14 respectively, to match the image resolutions of the contrast data in these studies. All the elements of the resulting datasets of floating point precision images were rounded to the nearest 8-bit integer values and used as raw laser speckle data in consecutive analysis. It is important to note that such simulation approach reduces (i) the effects of noise and (ii) the speckle-to-pixel size ratio compared to the original data. The latter consequence is beneficial for our study, as the speckle size of our low-zoom arm is about 3 pixels, while the system used by Holstein-Rathlou [15] and Mitrou [19]operated at sub-pixel speckle size values.

2.3.3 Blood flow index

Blood flow index is proportional to linear flow velocity in laser speckle contrast imaging [22]. To obtain blood flow index (BFI) from raw data sets, we first applied temporal laser speckle contrast analysis defined by

$$K=\frac{\sigma}{\mu} \; ,$$
where $\sigma$ and $\mu$ are standard deviation and mean values of the pixel intensity over the temporal kernel. Similar to the previous studies [15,16,20] we used temporal kernel of 25 frames. Subsequently, blood flow index was calculated as
$$BFI=\frac{1}{K^2} \; .$$

Examples of the average blood flow images are presented in Fig. 1(C,E). Differences in absolute BFI values caused by the mismatch of the speckle to pixel size ratios in different data sets does not affect the consequent analysis which is focused on comparing dynamics rather than average BFI values.

2.3.4 Extraction of nephron blood flow dynamics

To utilize and compare all data sets, we introduce three approaches to extract blood flow dynamics from the vessels of interest. Approach 2 (Region-of-interst) and approach 3 (Grid) are simulations of approaches used by Holstein-Rathlou [15] and Mitrou [19] respectively to measure nephron blood flow dynamics. Prior to this step, we calculated geometrical transformations required to align high-zoom data with the low-zoom data and its simulated modifications. These transformations were only applied to the binary masks and did not affect the data itself.

Masking approach 1: Segmentation. High-zoom BFI map averaged over time was used to segment individual blood vessels on the renal surface as described by Postnov [20]. Briefly, an adaptive intensity threshold was used to create an initial binary mask separating the blood vessels from the background. Initial mask was denoised, missing pixels were filled in with the erosion-dilation sequence and objects that were either too small or too large were removed: this left vessels of about 8-30 $\mu$m in diameter. The segmented image was manually corrected to remove falsely labeled blood vessels. The segmentation mask was applied to the high-zoom data and blood flow dynamics was found by averaging BFI over all pixels belonging to the vessel. In the consequent methods and result description, the corresponding data was labeled as "$HZ_{segm}$".

Masking approach 2: Region-of-interest. Followed by the segmentation technique described above, we identified centroid coordinates for each of the segmented vessels. The coordinates were converted to the low-zoom simulated data (Holstein-Rathlou [15], 0.225 resize factor) using the corresponding geometrical transformation. 15x15 pixels region of interest (ROI) was masked around each of the centroids. The ROIs masks were then converted to the high-zoom and low-zoom data using the corresponding geometrical transforms. The pixel values within the ROIs were averaged to provide the respective blood flow dynamics. In the consequent methods and result description, the corresponding data were labeled as "$HZ_{ROI}$", "$LZ_{ROI}$" and "$sLZ_{ROI}$" for the high-zoom, low-zoom, and the simulated low zoom approach (Holstein-Rathlou [15], respectively.

Masking approach 3: Grid. Each image in the simulated low-zoom data (Mitrou [19], 0.14 resize factor) was filtered with a Gaussian spatial filter of 8-pixel width and down-sampled by a factor of 4, therefore resulted in a grid where each cell corresponded to BFI in about 150x150 $\mu$m of the renal surface. As we were interested in comparing location-specific blood flow dynamics, we identified the cell that had the highest overlap with the vessel mask for each segmented vessel in the high-zoom data. In the consequent methods and result description, the data was labeled as "$sLZ_{grid}$" for simulated low zoom approach (Mitrou [19]).

2.3.5 Characterization of blood flow dynamics

To quantify the blood flow dynamics, we applied wavelet analysis to all segmented vessels, region-of-interest, and relevant grid cells. We then identified the dominant activity within tubuloglomerular feedback oscillation frequency range (0.018–0.035 Hz) [23,24] and extracted dominant activity frequency and phase for every point in the time series. We also calculated Pearson correlation coefficient to measure similarity among various blood flow dynamics derived from different approaches.

3. Results

Exemplary data shown in Fig. 2 highlights the importance of individual vessels segmentation allowed by high-resolution laser speckle contrast imaging of renal microcirculation. As for vessels 1-3 outlined in Fig. 2(A) their blood flow dynamics substantially vary depending on the moment in time, despite their juxtaposition and situated inside an ROI of 300x300 $\mu$m. Specifically, red rectangular selections in Fig. 2(B) show mismatch in the oscillations phase or visible even without wavelet analysis, while the black selections highlight moments of potential synchronization.

 figure: Fig. 2.

Fig. 2. Blood flow dynamics in individual segmented vessels. A) Segmentation of microvasculature obtained from high-zoom data. B) Time series from three micro-vessels show distinct periodic changes in blood flow that correspond to TGF. Colors correspond to the vessels outlined in the panel A. Black and red rectangles respectively highlight matching and mismatching periods of TGF oscillation in between the vessels.

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The mismatch in dynamics of neighbouring vessels shown in Fig. 2, brings concern to how this affects interpretation of results from low-resolution data. To investigate, we compared the masked region and the blood flow dynamics in segmented high-zoom data to the low-zoom data and approaches from previous publications [15,19]. Figure 3(A) displays examples of presumably single vessel masking for all listed approaches. As expected, microcirculatory vessels that are clearly visualized in the high-zoom data are not visible in the low-zoom data. The ROI and grid approaches, that are respectively centered on or maximally overlap with the chosen vessel, extended far out of its actual borders and can even include other vessels of similar caliber. Corresponding time series are shown in Fig. 3(B). Depending on the masking approach, phase and shape of oscillations can also vary, as highlighted with red rectangles. The amount of noise is visibly different among approaches, which is mainly caused by the difference in number of pixels used for signal averaging and is not critical for the problem in question.

 figure: Fig. 3.

Fig. 3. Comparison of images with different magnifications and masking approaches. A) 2-dimensional mean blood flow images of high zoom (I and II), low zoom (III), and simulated approaches from earlier studies (IV and V). Red outlines represent shapes of masks used to calculate average blood flow in respective approaches: segmentation (I), region-of-interest (II and III, and IV), and the most overlapping grid cell (V). B) Blood flow time series of 900 seconds extracted from corresponding masked regions shown in (A). One can see a difference in noise level, phase and form of oscillations despite the blood flow originating from the same vessel as exemplified in red rectangles.

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To quantify the mismatch and extract parameters critical for the analysis of synchronization and inter-nephron communication, we applied continuous wavelet transform and identified frequency and phase of the dominant TGF activity at every moment of time. Figure 4 shows examples of a good (panels A,C) and a poor (panels B,D) agreement among the TGF peaks identified by different approaches. Panels A and B show wavelet spectrum at a chosen moment of time; panels C and D display phase of the TGF activity within 5 minutes of the corresponding time. From the panel B it is clear that ROI approach did not accurately identify vessel specific TGF frequency. Regardless of proximity, neighboring nephrons can carry different TGF frequencies. Masking approaches that cover vessels originating from different nephrons result in superposition of various frequencies. Alterations in the averaged blood flow dynamics cause the frequency peak to broaden and shift compared to the spectra recorded in individually segmented vessels. The frequencies mismatch is naturally accompanied by difference in phases. From the panel D, it is clear that the mismatch occurs over a substantial period of time, rather than being short-lived; it can be associated with changes in neighbouring nephrons activity rather than with different noise levels.

 figure: Fig. 4.

Fig. 4. Presence of tubuloglomerular feedback (TGF) oscillations in a single micro-vessel detected by various methods. Example plots demonstrating alignment (A,C) and misalignment (B,D) in peak and phase of TGF frequency of a single vessel.

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Deviation of ROI masking approaches from segmentation was further explored in Fig. 5 which shows how the spectra for $HZ_{segm}$ and $HZ_{ROI}$ change in time. While TGF activity observed in $HZ_{segm}$ remains mostly stable, $HZ_{ROI}$ displays broadening of the peak which is followed by separation into multiple peaks at different frequencies. In Fig. 5(D) the mismatch reaches the point where the original peak belonging to the segmented vessel cannot be detected in the ROI data. This means that the contribution of blood flow dynamics in the segmented vessel to the signal averaged over ROI becomes negligible — an imminent outcome considering that the segmented vessels on average only cover about 1.5% of the corresponding ROI area.

 figure: Fig. 5.

Fig. 5. ROI masking approach misrepresents TGF in individual vessels. Snapshots from a 240 second duration showing that while TGF of high zoom segmentation (red) remains stable, high zoom ROI (blue) fails to measure precise TGF signal. A) Main peaks of at TGF frequency starts out by matching for both masking approaches. B) TGF peak of of HZ ROI shows broadening. C) TGF peak of HZ ROI splits into two peaks. D) The original TGF peak for HZ ROI from T=1 is no longer apparent.

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The findings were averaged over 3 animals and summarized in Table 1. The difference values were averaged over time and thus appear to be lower than presumably expected from Fig. 4 and 5. Nevertheless, the table clearly shows substantial difference between segmentation and ROI mask approaches even within the high-zoom data: 0.0012 Hz and 0.78 rad, for the peak frequency and phase respectively. The low-zoom data and approaches used in previous publications deviate even further, but, at the same time, they follow the $HZ_{ROI}$ closely. The difference between $LZ_{ROI}$ and $HZ_{ROI}$ are only: 0.0004 Hz and 0.3 rad. As one would expect, the difference grows for the $sLZ_{grid}$, reaching 0.0012 Hz and 0.78 rad.

Tables Icon

Table 1. Comparison of different methods.

4. Discussion

We developed multi-scale laser speckle contrast imaging system and applied it to study local microcirculatory vasoreactivity in rat kidneys. The technique bridges the gap between high resolution small field of view (high-zoom) and low resolution large field of view (low-zoom) blood flow imaging. We verified that low-zoom LSCI does not allow accurate and robust measurements of renal blood flow at the level of a single nephron.

Nevertheless, low zoom LSCI provides meaningful blood flow information at larger scales (i.e. the whole kidney) which is unobtainable with just high zoom. Multi-scale imaging is especially relevant in experimental conditions when we want to capture both the reference region and the region affected by the condition. Some of these examples include but are not limited to: kidney during a drug infusion where the whole or a part of the kidney can be affected [17], ischemia or organ damage that is localized in space (i.e. cortical spreading depression triggered by stroke) that induces alterations to microvascular hemodynamics [7], and ongoing studies investigating the connections between population of neprhons and the whole kidney.

We showed that using the ROI approach centered around the vessel instead of segmentation approach shifts and broadens the frequency peak of dominant activity. The same is observed when comparing high-zoom segmented vessel blood flow ($HZ_{segm}$) to the low-zoom data ($LZ_{ROI}$, $sLZ_{ROI}$, $sLZ_{grid}$) and to the high-zoom ROI data ($HZ_{ROI}$). This can be explained by the fact that only around 1.5% of the ROI belonged to the arteriole-sized (10–20 $\mu$m in diameter) vessel and visualized in the high-zoom data. The other 98.5% pixels covered capillary mesh and other vessels which might belong to another nephron. Such deviations reduce the fidelity of fine spatial or temporal metrics obtained via ROI or low-zoom approaches. Conversely, low-zoom imaging is satisfactory when we assess large scale blood flow changes or when there is synchronization of blood flow among neighboring vessels: when blood flow dynamics recorded with the low-zoom arm agrees well with the $HZ_{ROI}$ data.

We acknowledge that there are alternative modalities that provide quantitative methods based on speckle dynamics, such as multi-exposure speckle imaging or dynamic light scattering imaging. But since the dynamical features we aim to measure (frequency and phase) are not dependent on the flow-contrast relationship, conventional laser speckle contrast imaging is the most optimal modality for our application.

Multi-scale laser speckle contrast imaging potentially unveils renal hemodynamics hidden from the previous high-zoom or low-zoom methods. It allows future study of phase wave phenomena that we reported previously [16,20], as well as investigation into simultaneous local and global real-time changes of renal vasoreactivity and blood flow induced by pharmacological interventions or renal injury. The potential application of Multi-LSCI extends beyond renal blood flow imaging. We expect broad utilization from cerebral blood flow research to other studies that require precise local measurements of blood flow while in parallel, monitoring blood flow over a large field of view.

Funding

Novo Nordisk Fonden (NNF18OC0052728, NNF17OC0025224); UCPH Data+ funding (Strategy 2023 funds).

Acknowledgments

Thank you to Karin Larsen for the training in animal surgeries.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

Supplemental document

See Supplement 1 for supporting content.

References

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Supplementary Material (1)

NameDescription
Supplement 1       Overview of the data processing, simulation, and analysis pipeline for laser speckle contrast imaging

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (5)

Fig. 1.
Fig. 1. Multi-scale laser speckle contrast imaging. A) Diagram of experimental setup to image a rodent kidney. Examples of raw image (B) and blood flow map (C) recorded with high-zoom arm with image resolution of 1.2 $\mu$m per pixel. Corresponding images (D, E) acquired with the for the low-zoom arm with image resolution of 5.5 $\mu$m per pixel. Blood flow index (BFI) maps were obtained by performing all data processing steps, including motion compensation, calculation of blood flow index (BFI), and averaging images over the observation time.
Fig. 2.
Fig. 2. Blood flow dynamics in individual segmented vessels. A) Segmentation of microvasculature obtained from high-zoom data. B) Time series from three micro-vessels show distinct periodic changes in blood flow that correspond to TGF. Colors correspond to the vessels outlined in the panel A. Black and red rectangles respectively highlight matching and mismatching periods of TGF oscillation in between the vessels.
Fig. 3.
Fig. 3. Comparison of images with different magnifications and masking approaches. A) 2-dimensional mean blood flow images of high zoom (I and II), low zoom (III), and simulated approaches from earlier studies (IV and V). Red outlines represent shapes of masks used to calculate average blood flow in respective approaches: segmentation (I), region-of-interest (II and III, and IV), and the most overlapping grid cell (V). B) Blood flow time series of 900 seconds extracted from corresponding masked regions shown in (A). One can see a difference in noise level, phase and form of oscillations despite the blood flow originating from the same vessel as exemplified in red rectangles.
Fig. 4.
Fig. 4. Presence of tubuloglomerular feedback (TGF) oscillations in a single micro-vessel detected by various methods. Example plots demonstrating alignment (A,C) and misalignment (B,D) in peak and phase of TGF frequency of a single vessel.
Fig. 5.
Fig. 5. ROI masking approach misrepresents TGF in individual vessels. Snapshots from a 240 second duration showing that while TGF of high zoom segmentation (red) remains stable, high zoom ROI (blue) fails to measure precise TGF signal. A) Main peaks of at TGF frequency starts out by matching for both masking approaches. B) TGF peak of of HZ ROI shows broadening. C) TGF peak of HZ ROI splits into two peaks. D) The original TGF peak for HZ ROI from T=1 is no longer apparent.

Tables (1)

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Table 1. Comparison of different methods.

Equations (2)

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K = σ μ ,
B F I = 1 K 2 .
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