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PMC12408821
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Mapping the arterial vascular network in an intact human kidney using hierarchical phase-contrast tomography.
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Step 4: Using the tSO from Step 3 the network can be split into larger calibre (tSO ) and smaller calibre vessels (tSO ).
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PMC12408821
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Mapping the arterial vascular network in an intact human kidney using hierarchical phase-contrast tomography.
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The larger calibre vessel can then be smoothed as shown in insets, orange arrows show the points where smoothing has noticeably acted on regions of larger vessels.
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PMC12408821
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Mapping the arterial vascular network in an intact human kidney using hierarchical phase-contrast tomography.
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Step 5: tSO vs mean radius is plotted for every segment (blue circles); potential collapsed vessels (red crosses) are flagged for all larger vessels and identified in smaller calibre segments by those as having a radius below the 90% percentile for their tSO.
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PMC12408821
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Mapping the arterial vascular network in an intact human kidney using hierarchical phase-contrast tomography.
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Step 6: The segments identified as outliers are presented to an annotator in an interact pop-up window, which allows the annotator to visualise the segment and manually confirm if it should be corrected for collapse.
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PMC12408821
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Mapping the arterial vascular network in an intact human kidney using hierarchical phase-contrast tomography.
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Step 7: For vessels which are confirmed as collapsed, planes which are normal to the centreline of the vessel (indicated by orange arrows) are created at every point along the centreline, these are presented in pop-up windows to the annotator, as are the 2D image for each orthogonal plane (lower panels).
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PMC12408821
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Mapping the arterial vascular network in an intact human kidney using hierarchical phase-contrast tomography.
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From these 2D planes, the collapsed vessel is identified (red cross) and the perimeter (yellow dashed line) is extracted.
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PMC12408821
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Mapping the arterial vascular network in an intact human kidney using hierarchical phase-contrast tomography.
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The perimeter is used to calculate an equivalent radius and assigned as the new radius of the segment.
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PMC12408821
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Mapping the arterial vascular network in an intact human kidney using hierarchical phase-contrast tomography.
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Step 8: The new radii at each point are plotted, and outliers are removed to reduce the effect of any remaining tortuosity in the centerlines.
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PMC12408821
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Mapping the arterial vascular network in an intact human kidney using hierarchical phase-contrast tomography.
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At this stage, the annotator can also manually remove planes that are visibly affected by residual tortuosity.
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PMC12408821
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Mapping the arterial vascular network in an intact human kidney using hierarchical phase-contrast tomography.
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Our pipeline first (Fig. 2, Step 1) assesses validation of the segmentation.
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PMC12408821
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Mapping the arterial vascular network in an intact human kidney using hierarchical phase-contrast tomography.
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By aligning segmentations from scans taken at 13 µm per voxel, with VOIs captured at 50 µm per voxel, the higher resolution scans served as ‘ground truth’ for the lower resolution scanning.
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PMC12408821
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Mapping the arterial vascular network in an intact human kidney using hierarchical phase-contrast tomography.
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We used the cl-DICE metric to quantify the overlapping vessel portions finding that 97% of vessels with a vessel lumen radius greater than 50 µm are detectable at 50 µm per voxel.
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PMC12408821
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Mapping the arterial vascular network in an intact human kidney using hierarchical phase-contrast tomography.
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Our next step (Fig. 2, Step 2) comprises the optimisation of the skeletonization algorithm.
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PMC12408821
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Mapping the arterial vascular network in an intact human kidney using hierarchical phase-contrast tomography.
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We applied three different skeletonization algorithms, and utilised the recently developed skeleton super-metric to determine the most suitable algorithm and its parameter optimisation.
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PMC12408821
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Mapping the arterial vascular network in an intact human kidney using hierarchical phase-contrast tomography.
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We found that the Centerline Tree algorithm (Amira-Avizo v2021.1) was the best candidate algorithm, as indicated by its lower super-metric value in comparison to other skeletonization algorithms (Fig. 2, Step 2).
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PMC12408821
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Mapping the arterial vascular network in an intact human kidney using hierarchical phase-contrast tomography.
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Thereafter, several steps were implemented to correct the skeleton for HiP-CT specific challenges (Fig. 2, Steps 3–8), namely the multiscale nature of the vasculature, and the presence of collapsed vessels as a consequence of the ex vivo, label-free HiP-CT protocol.
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PMC12408821
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Mapping the arterial vascular network in an intact human kidney using hierarchical phase-contrast tomography.
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The challenge of multiscale vascular trees was corrected using a truncated Strahler ordering system, which partitions the network into larger or smaller calibre vessels.
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PMC12408821
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Mapping the arterial vascular network in an intact human kidney using hierarchical phase-contrast tomography.
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Smoothing was then applied to all large calibre vessels to reduce tortuosity in the vessel centreline, an artefact which occurs due to the sensitivity of skeletonization algorithms to noise along the vessel surface (Fig. 2, Steps 3 and 4).
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PMC12408821
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Mapping the arterial vascular network in an intact human kidney using hierarchical phase-contrast tomography.
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Following this multiscale smoothing approach, Fig. 2 Steps 5 and 6 involved the identification and manual verification of collapsed vessels.
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PMC12408821
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Mapping the arterial vascular network in an intact human kidney using hierarchical phase-contrast tomography.
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Initially, all large-calibre vessels were flagged as potentially collapsed.
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PMC12408821
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Mapping the arterial vascular network in an intact human kidney using hierarchical phase-contrast tomography.
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Additionally, smaller calibre vessels that were potentially collapsed were identified based on their categorisation below the 10th percentile for radius in their truncated Strahler order (Fig. 2, Step 5).
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PMC12408821
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Mapping the arterial vascular network in an intact human kidney using hierarchical phase-contrast tomography.
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Once identified, collapsed vessels were subject to a bounding box, automatically extracted and thereafter manually determined whether correction of the radius was required to account for collapse (Fig. 2, Step 6).
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PMC12408821
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Mapping the arterial vascular network in an intact human kidney using hierarchical phase-contrast tomography.
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For vessels requiring correction, cross-sectional planes along the vessel centreline were extracted, and the radius was calculated based on the cross-sectional perimeter (Fig. 2, Step 6).
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PMC12408821
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Mapping the arterial vascular network in an intact human kidney using hierarchical phase-contrast tomography.
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Finally, the identification of outlier radii in these cross-sectional planes was performed, using a 95th and 5th percentile windowing for radius along the vessel length.
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PMC12408821
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Mapping the arterial vascular network in an intact human kidney using hierarchical phase-contrast tomography.
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Additionally, an option was applied to manually flag planes that appeared compromised by residual tortuosity in the vessel centreline (Fig. 2, Step 8).
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PMC12408821
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Mapping the arterial vascular network in an intact human kidney using hierarchical phase-contrast tomography.
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The result of this novel pipeline, when applied to our HiP-CT data of human kidney, was the generation the first open-source spatial graph of the intact human kidney arterial vasculature, ranging from renal artery to interlobular arteries.
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PMC12408821
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Mapping the arterial vascular network in an intact human kidney using hierarchical phase-contrast tomography.
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We were able to identify 97% of vessels >50 µm radius across the whole intact human kidney.
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PMC12408821
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Mapping the arterial vascular network in an intact human kidney using hierarchical phase-contrast tomography.
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The final network consisted of 10,193 nodes, 376,603 points and 10,190 vessels.
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PMC12408821
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Mapping the arterial vascular network in an intact human kidney using hierarchical phase-contrast tomography.
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The total network volume was 1.68 × 10 µm, with a length of 2.3 × 10 µm.
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PMC12408821
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Mapping the arterial vascular network in an intact human kidney using hierarchical phase-contrast tomography.
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This spatial graph, which is provided in our Supplementary Information, captures the morphological features and connectivity of the human kidney arterial vasculature, which was then used for downstream analyses as described below.
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PMC12408821
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Mapping the arterial vascular network in an intact human kidney using hierarchical phase-contrast tomography.
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Having created a reproducible spatial graph of the human kidney arterial vasculature, we then performed topological generation and truncated Strahler ordering analyses.
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PMC12408821
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Mapping the arterial vascular network in an intact human kidney using hierarchical phase-contrast tomography.
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This resulted in nine truncated Strahler orders (Fig. 3A) and twenty-five topological generations (Fig. 3B).
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PMC12408821
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Mapping the arterial vascular network in an intact human kidney using hierarchical phase-contrast tomography.
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As the main artery supplying the kidney was cut during autopsy, we inferred that 10 truncated Strahler orders, representing 26 topological generations, were imaged over the intact human kidney with HiP-CT.Fig.
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PMC12408821
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Mapping the arterial vascular network in an intact human kidney using hierarchical phase-contrast tomography.
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3Ordering and branching ratio analyses of the human kidney vasculature.
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PMC12408821
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Mapping the arterial vascular network in an intact human kidney using hierarchical phase-contrast tomography.
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Rendering of the human kidney vascular network, with vessels coloured according to A Strahler order and B topological generation.
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PMC12408821
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Mapping the arterial vascular network in an intact human kidney using hierarchical phase-contrast tomography.
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Ci Plot showing the number of vessels per truncated Strahler order, with fit for the log plot to calculate branching ratio.
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PMC12408821
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Mapping the arterial vascular network in an intact human kidney using hierarchical phase-contrast tomography.
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Cii Truncated Strahler order against cumulative vascular volume fraction.
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PMC12408821
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Mapping the arterial vascular network in an intact human kidney using hierarchical phase-contrast tomography.
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D One of the VOIs with all glomeruli segmented.
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PMC12408821
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Mapping the arterial vascular network in an intact human kidney using hierarchical phase-contrast tomography.
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E A region within the VOI in (D) segmented at 2.6 µm per voxel, showing connection down to afferent glomeruli.
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PMC12408821
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Mapping the arterial vascular network in an intact human kidney using hierarchical phase-contrast tomography.
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Inset shows the six glomeruli that were connected back to the whole network.
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PMC12408821
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Mapping the arterial vascular network in an intact human kidney using hierarchical phase-contrast tomography.
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The five red arrows indicate glomeruli arising from non-terminal arteries, while the black arrowhead indicates a glomerulus arising from a terminal artery.
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PMC12408821
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Mapping the arterial vascular network in an intact human kidney using hierarchical phase-contrast tomography.
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Rendering of the human kidney vascular network, with vessels coloured according to A Strahler order and B topological generation.
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PMC12408821
|
Mapping the arterial vascular network in an intact human kidney using hierarchical phase-contrast tomography.
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Ci Plot showing the number of vessels per truncated Strahler order, with fit for the log plot to calculate branching ratio.
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PMC12408821
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Mapping the arterial vascular network in an intact human kidney using hierarchical phase-contrast tomography.
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Cii Truncated Strahler order against cumulative vascular volume fraction.
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PMC12408821
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Mapping the arterial vascular network in an intact human kidney using hierarchical phase-contrast tomography.
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D One of the VOIs with all glomeruli segmented.
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PMC12408821
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Mapping the arterial vascular network in an intact human kidney using hierarchical phase-contrast tomography.
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E A region within the VOI in (D) segmented at 2.6 µm per voxel, showing connection down to afferent glomeruli.
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PMC12408821
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Mapping the arterial vascular network in an intact human kidney using hierarchical phase-contrast tomography.
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Inset shows the six glomeruli that were connected back to the whole network.
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PMC12408821
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Mapping the arterial vascular network in an intact human kidney using hierarchical phase-contrast tomography.
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The five red arrows indicate glomeruli arising from non-terminal arteries, while the black arrowhead indicates a glomerulus arising from a terminal artery.
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PMC12408821
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Mapping the arterial vascular network in an intact human kidney using hierarchical phase-contrast tomography.
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Strahler ordering, and other approaches to classify vascular networks, have potential caveats (See Supplementary Note 4), for example, the Strahler (or truncated Strahler) order of any individual vessel depends upon the downstream network, and thus on the identification of the network endpoint.
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PMC12408821
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Mapping the arterial vascular network in an intact human kidney using hierarchical phase-contrast tomography.
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Ideally, this endpoint would correspond to the afferent arteriole entering the renal glomerulus.
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PMC12408821
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Mapping the arterial vascular network in an intact human kidney using hierarchical phase-contrast tomography.
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However, at 50 µm per voxel resolution, we were unable to detect afferent arterioles, and thus, the smallest vessels, defined as truncated Strahler order 1 vessels, were the interlobular arteries.
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PMC12408821
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Mapping the arterial vascular network in an intact human kidney using hierarchical phase-contrast tomography.
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Diameter-based statistical approaches to estimation of the Strahler order of the kidney vascular network’s terminal ends were not appropriate to correct for this due to the ex vivo and non-perfused nature of HiP-CT, as well as the connectivity of glomeruli relative to terminal vessel ends.
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PMC12408821
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Mapping the arterial vascular network in an intact human kidney using hierarchical phase-contrast tomography.
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Thus, we applied truncated Strahler ordering to our spatial graph and report how morphological features of vessels in the network vary with truncated Strahler order.
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PMC12408821
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Mapping the arterial vascular network in an intact human kidney using hierarchical phase-contrast tomography.
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We also mapped our truncated Strahler orders to known anatomical subdivisions of the arterial tree to give anatomical context.
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PMC12408821
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Mapping the arterial vascular network in an intact human kidney using hierarchical phase-contrast tomography.
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This resulted in the following classification: truncated Strahler orders 7–9 (n = 25 segments; mean radius = 929 ± 477 µm) mapped to the branches of the kidney artery entering the kidney hilum.
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PMC12408821
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Mapping the arterial vascular network in an intact human kidney using hierarchical phase-contrast tomography.
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Orders 5–6 comprised interlobar arteries (n = 219 segments; mean radius = 417 ± 247 µm), and orders 2–4 arcuate arteries (n = 4841 segments; mean radius = 78 ± 45 µm).
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PMC12408821
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Mapping the arterial vascular network in an intact human kidney using hierarchical phase-contrast tomography.
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Finally, interlobular arteries fell within orders 1–3 (n = 9430 segments; mean radius = 55 ± 23 µm).
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PMC12408821
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Mapping the arterial vascular network in an intact human kidney using hierarchical phase-contrast tomography.
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We further plotted the cumulative volume of the kidney vascular network (Fig. 3Cii), finding that over 20% of the volume of the network lies within Strahler orders 1–4, corresponding to segments from interlobular arteries and arcuate arteries.
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PMC12408821
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Mapping the arterial vascular network in an intact human kidney using hierarchical phase-contrast tomography.
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We found 5105 truncated Strahler order 1 segment and identified a logarithmic relationship between truncated Strahler order and segment number (Fig. 3Ci).
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PMC12408821
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Mapping the arterial vascular network in an intact human kidney using hierarchical phase-contrast tomography.
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Using this relationship, we determined the branching ratio within this subsection of the vascular tree to be 2.92, a value which is similar to that of the human pulmonary arterial tree (3.0) and the rat kidney vasculature (2.85).
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PMC12408821
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Mapping the arterial vascular network in an intact human kidney using hierarchical phase-contrast tomography.
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To provide further context to the truncated Strahler order and to investigate the small-calibre vessels within the human kidney, we leveraged the hierarchical capability of HiP-CT.
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PMC12408821
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Mapping the arterial vascular network in an intact human kidney using hierarchical phase-contrast tomography.
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Using high-resolution VOIs, we segmented and counted all glomeruli within each of the 3 high-resolution VOIs of the HiP-CT data (Fig. 3D).
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PMC12408821
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Mapping the arterial vascular network in an intact human kidney using hierarchical phase-contrast tomography.
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We extrapolated from these VOIs to the total of ~1.2 million glomeruli in the intact kidney, which aligned well with estimates for adult males within a similar age range.
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PMC12408821
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Mapping the arterial vascular network in an intact human kidney using hierarchical phase-contrast tomography.
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Given the 5105 truncated Strahler order 1 segments, the branching ratio of 2.921 and the total number of glomeruli, we estimated that there are a further 4–5 truncated Strahler orders between the end of our whole organ network and the afferent arterioles.
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PMC12408821
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Mapping the arterial vascular network in an intact human kidney using hierarchical phase-contrast tomography.
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To further evaluate this estimate, we assessed on high-resolution VOI, connecting six afferent arterioles of individual glomeruli back to the main vessel tree (Fig. 3E, Supplementary Movie 4 and Supplementary Fig. 8).
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PMC12408821
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Mapping the arterial vascular network in an intact human kidney using hierarchical phase-contrast tomography.
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Of the 6 glomeruli, 5 originated from non-terminal arteries and one from a terminal artery (Fig. 3E, red arrows and black arrows respectively).
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PMC12408821
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Mapping the arterial vascular network in an intact human kidney using hierarchical phase-contrast tomography.
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This supports recent findings, in the rat kidney, which similarly demonstrated the existence of non-terminal branch arterioles, with potential contributions to the synchronicity of blood flow within the kidney.
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PMC12408821
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Mapping the arterial vascular network in an intact human kidney using hierarchical phase-contrast tomography.
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The existence of non-terminal glomeruli also prevents the application of the statistical methods which have previously been used to estimate the true Strahler order of a truncated network.
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PMC12408821
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Mapping the arterial vascular network in an intact human kidney using hierarchical phase-contrast tomography.
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Given the presence of non-terminal glomeruli, statistical estimation of true Strahler order would necessitate connecting a larger number (~1000) of glomeruli back to the main tree to generate an accurate statistical representation of the proportion of terminal to non-terminal glomeruli.
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PMC12408821
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Mapping the arterial vascular network in an intact human kidney using hierarchical phase-contrast tomography.
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Such an estimation cannot be made with this dataset as, even with our highest resolution scans, the small vessels connecting the glomeruli to the main vascular tree could not be annotated reliably for a large number of cases.
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PMC12408821
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Mapping the arterial vascular network in an intact human kidney using hierarchical phase-contrast tomography.
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However, our estimation of total glomeruli number provides a data-driven estimate for the number of missing orders, and thus gives the context needed to support our use of the truncated Strahler order for our ongoing analysis.
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PMC12408821
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Mapping the arterial vascular network in an intact human kidney using hierarchical phase-contrast tomography.
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Vascular network geometric properties, including vessel diameters, lengths and branching angles, are key metrics for quantitative and objective comparison of vascular networks in health or disease.
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PMC12408821
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Mapping the arterial vascular network in an intact human kidney using hierarchical phase-contrast tomography.
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Thus, we extracted and reported the metrics for the human kidney vasculature.
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PMC12408821
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Mapping the arterial vascular network in an intact human kidney using hierarchical phase-contrast tomography.
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Data were grouped according to truncated Strahler order (Fig. 4, Table 1) to enable quantitative comparison to rat and other human organ data.
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PMC12408821
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Mapping the arterial vascular network in an intact human kidney using hierarchical phase-contrast tomography.
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The raw data for each segment, which may serve as inputs for computational models, have been provided as Supplementary Information.
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PMC12408821
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Mapping the arterial vascular network in an intact human kidney using hierarchical phase-contrast tomography.
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Fig. 4Quantitative branching metrics of the adult human kidney arterial network.
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PMC12408821
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Mapping the arterial vascular network in an intact human kidney using hierarchical phase-contrast tomography.
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A Schematic diagram of how the metrics in (B–E) are calculated.
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PMC12408821
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Mapping the arterial vascular network in an intact human kidney using hierarchical phase-contrast tomography.
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B The length:diameter ratio.
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PMC12408821
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Mapping the arterial vascular network in an intact human kidney using hierarchical phase-contrast tomography.
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C The branching angle between the child and parent segments.
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PMC12408821
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Mapping the arterial vascular network in an intact human kidney using hierarchical phase-contrast tomography.
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D The tortuosity of segments, E their radius, and F the inter-vessel distance as measured between the midpoint of each segment.
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PMC12408821
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Mapping the arterial vascular network in an intact human kidney using hierarchical phase-contrast tomography.
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Table 1Human kidney vascular branching metrics by truncated Strahler generation (means with standard deviation are shown)Truncated Strahler orderNo.
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PMC12408821
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Mapping the arterial vascular network in an intact human kidney using hierarchical phase-contrast tomography.
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of segmentsRadium (µm)Length µm × 10TortuosityLength:diameterVol.
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PMC12408821
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Mapping the arterial vascular network in an intact human kidney using hierarchical phase-contrast tomography.
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× 10 µmBranching angle IVD µm × 101510545 ± 52.6 ± 1.71.2 ± 0.2057.4 ± 360.18 ± 0.22129 ± 281.1 ± 0.42303056 ± 151.8 ± 1.41.1 ± 0.1632.3 ± 260.21 ± 0.46128 ± 290.9 ± 0.43129595 ± 371.8 ± 1.41.1 ± 0.1420.2 ± 186.7 ± 10.3128 ± 291.1 ± 0.44516165 ± 602.3 ± 1.91.1 ± 0.1314.6 ± 1225.5 ± 34.9135 ± 291.4 ± 0.55150294 ± 1103.3 ± 2.61.1 ± 0.0512.2 ± 9.411.3 ± 17.1142 ± 251.8 ± 0.7669684 ± 2504.3 ± 3.11.1 ± 0.066.5 ± 4.584.6 ± 104149 ± 212.9 ± 0.7720839 ± 2517.2 ± 4.91.1 ± 0.058.5 ± 4.8223 ± 302148 ± 246.1 ± 1.584877 ± 1888.1 ± 4.71.1 ± 0.079.6 ± 6.3227 ± 159141 ± 64.7 ± 1.59129236691.00.2180-- A Schematic diagram of how the metrics in (B–E) are calculated.
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PMC12408821
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Mapping the arterial vascular network in an intact human kidney using hierarchical phase-contrast tomography.
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B The length:diameter ratio.
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PMC12408821
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Mapping the arterial vascular network in an intact human kidney using hierarchical phase-contrast tomography.
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C The branching angle between the child and parent segments.
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PMC12408821
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Mapping the arterial vascular network in an intact human kidney using hierarchical phase-contrast tomography.
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D The tortuosity of segments, E their radius, and F the inter-vessel distance as measured between the midpoint of each segment.
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PMC12408821
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Mapping the arterial vascular network in an intact human kidney using hierarchical phase-contrast tomography.
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Human kidney vascular branching metrics by truncated Strahler generation (means with standard deviation are shown) Further quantitative analysis of the human kidney vascular network revealed that, as truncated Strahler order increased, there was a reduction in the ratio of vessel length:diameter (Fig. 4B).
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PMC12408821
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Mapping the arterial vascular network in an intact human kidney using hierarchical phase-contrast tomography.
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In contrast, the mean radius (Fig. 4E) and inter-vessel distance increased (Fig. 4F).
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PMC12408821
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Mapping the arterial vascular network in an intact human kidney using hierarchical phase-contrast tomography.
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Tortuosity did not vary significantly with truncated Strahler order (Fig. 4D); with most segments possessing tortuosity close to 1, thus implying limited deviation from a straight path.
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PMC12408821
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Mapping the arterial vascular network in an intact human kidney using hierarchical phase-contrast tomography.
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These findings are consistent with anticipated trends for a healthy tissue, wherein a vascular network is assumed to be a fractal structure, with branching pattern driven by optimised delivery of blood to the whole organ.
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PMC12408821
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Mapping the arterial vascular network in an intact human kidney using hierarchical phase-contrast tomography.
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Interestingly, within truncated Strahler orders 8–6, the mean branching angle was approximately 150°, decreasing to 130° for truncated Strahler orders 3–1 (Fig. 4C).
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PMC12408821
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Mapping the arterial vascular network in an intact human kidney using hierarchical phase-contrast tomography.
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Importantly, the latter value of 130° is the predicted optimal theoretical branching angle for volume-constrained vascular growth.
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PMC12408821
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Mapping the arterial vascular network in an intact human kidney using hierarchical phase-contrast tomography.
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Simulation of kidney haemodynamics has previously been performed using μCT data from the rat kidney.
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PMC12408821
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Mapping the arterial vascular network in an intact human kidney using hierarchical phase-contrast tomography.
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To facilitate comparison between existing rat data and our human HiP-CT results, we aligned our network based on the Strahler order allocated to the segmental arteries, thus aligning Strahler order 9 in the previously published rat dataset to truncated Strahler order 8 of our human data.
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PMC12408821
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Mapping the arterial vascular network in an intact human kidney using hierarchical phase-contrast tomography.
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We then related normalised vessel metrics from each species, matching anatomically defined vessel types.
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PMC12408821
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Mapping the arterial vascular network in an intact human kidney using hierarchical phase-contrast tomography.
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The expected increase in vessel radius with order followed a similar trend between human and rat kidney (Fig. 5A).
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PMC12408821
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Mapping the arterial vascular network in an intact human kidney using hierarchical phase-contrast tomography.
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However human kidney vessel radii increased to a greater extent across Strahler orders than in the rat kidney, evaluated based on a fit of log(radius) to Strahler order (Fig. 5B), (p < 0.0001 Sum-of-F test F (DFn, DFd) = 700.6 (2, 12)).
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PMC12408821
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Mapping the arterial vascular network in an intact human kidney using hierarchical phase-contrast tomography.
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To provide additional insights into this difference observed between human and rat kidneys, we extracted radial scaling exponents of the human vascular network.
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PMC12408821
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Mapping the arterial vascular network in an intact human kidney using hierarchical phase-contrast tomography.
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The radial scaling exponent provides insight into how the network has structurally developed with respect to its functions such as the efficiency of blood flow and nutrient delivery to meet metabolic demands and minimise flow resistance.
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PMC12408821
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Mapping the arterial vascular network in an intact human kidney using hierarchical phase-contrast tomography.
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Exponents of 0.33 and 0.5 each have theoretical bases in different models, (i) Murray’s law (expected exponent of 0.33 for all the whole network, derived from considering the energy balance between energy of flow and viscous drag), (ii) the West-Brown-Enquist (WBE) model which predicts 0.5 in larger vessels and 0.33 in smaller vessels, resulting from balancing the energy for metabolic distribution of blood across a fractal-like network.
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