Blood flow modeling reveals improved collateral artery performance during the regenerative period in mammalian hearts

Collateral arteries bridge opposing artery branches, forming a natural bypass that can deliver blood flow downstream of an occlusion. Inducing coronary collateral arteries could treat cardiac ischemia, but more knowledge on their developmental mechanisms and functional capabilities is required. Here we used whole-organ imaging and three-dimensional computational fluid dynamics modeling to define spatial architecture and predict blood flow through collaterals in neonate and adult mouse hearts. Neonate collaterals were more numerous, larger in diameter and more effective at restoring blood flow. Decreased blood flow restoration in adults arose because during postnatal growth coronary arteries expanded by adding branches rather than increasing diameters, altering pressure distributions. In humans, adult hearts with total coronary occlusions averaged 2 large collaterals, with predicted moderate function, while normal fetal hearts showed over 40 collaterals, likely too small to be functionally relevant. Thus, we quantify the functional impact of collateral arteries during heart regeneration and repair—a critical step toward realizing their therapeutic potential. Anbazhakan et al. use whole-organ imaging and three-dimensional computational fluid dynamics modeling to define spatial architecture and predict blood flow through collaterals in neonate and adult mouse hearts after injury, and compare their findings to the functionality of collaterals in human adult and fetal hearts.

C ardiovascular disease, including coronary artery disease (CAD), is the leading cause of death worldwide 1 . CAD can result in decreased blood flow to the myocardium, jeopardizing cardiac muscle function. Current treatments involve invasive surgical procedures, but a substantial number are unsuccessful, especially in diffuse multivessel CAD 2 . Humans and other mammals can develop specialized blood vessels called collateral arteries that function as natural coronary bypasses. These are defined as artery segments directly bridging two artery branches without intervening capillaries, directly providing blood flow distal to a coronary blockage. Although only a minority of adult humans have functional collateral arteries, clinical observations indicate that they can successfully shunt blood around a stenosis to protect against myocardial ischemia and reduce the risk of cardiac death 3,4 . Thus, inducing collateral development could be a promising therapeutic approach for treating CAD 5 . However, a roadblock to this goal is the lack of knowledge about collateral development and their ability to restore blood flow.
While studies have characterized the presence or absence of native collateral arteries across different mammals 6 , mice are the most common model for investigating their function during cardiac injury, usually through surgically induced myocardial infarctions (MIs) [7][8][9][10] . Mice do not have preexisting collateral arteries, but by 7 d after MI, 6-10 collaterals (averaging 18 μm in diameter) are observed in adults through Microfil vascular filling 8 . Thus, mice are a useful model for exploring collateral biology.
We recently used a different technique to identify collateralswhole-mount immunofluorescence-coupled with lineage tracing and mouse genetics to identify mechanisms driving collateral development 7 . In regenerating neonate hearts, collaterals form after MI when arterial endothelial cells migrate into the infarct zone in response to hypoxia-induced CXCL12 and coalesce into collateral arteries 7 . This process was termed artery reassembly and did not occur in non-regenerative adult hearts, suggesting that the collaterals observed during vascular filling (described above) utilized a different mechanism. Exogenous CXCL12 application induced artery reassembly in adults to create collaterals up to 40 μm in diameter 7 . Although these collaterals were positively correlated with heart regeneration/repair, characterizing blood flow capacity is required to understand their functional capabilities and therapeutic potential.
How structure affects collateral hemodynamics remains unknown due to technical barriers of imaging blood flow. Clinical measurements rely on qualitative assessments from angiograms or indirect pressure measurements 11 . More accurate measurements in humans are currently invasive and somewhat unreliable since conclusive relationships cannot be made without quantifying all collaterals, many of which are undetectable in angiograms. Visualizing blood flow is extremely difficult in small experimental animals, and collaterals are quantified from ex vivo methods with the following limitations: (1) vascular filling (Microfil casting, micro-computed tomography (µCT) and fluorescent conjugates) 12 , which creates a Blood flow modeling reveals improved collateral artery performance during the regenerative period in mammalian hearts masked so that 3D rendering in Imaris created a map of every collateral artery ( Fig. 2b and Supplementary Video 1). The resolution of our method allowed us to annotate the precise collateral segments that bridged two artery branches (Fig. 2b(ii)). A collateral bridge was defined as the segment of continuous smooth-muscle-covered vessel that existed between two branch tips with opposing branch angles (Fig. 2c). Importantly, tracing did not detect collateral connections in non-injured neonate hearts (Extended Data Fig. 1).
Collateral arteries form faster in neonates than in adults 7 . When we injured adult mice, and assessed for collaterals 4 d after MI, we observed profound arterial pruning indicating robust injury, but found virtually no α-SMA + collateral vessels in most hearts (Extended Data Fig. 2a-c). Thus, we compared collateral metrics in injured hearts 4 d after MI for neonates and 28 d after MI in adults, a time point previously shown to possess the typical number of collaterals in adult hearts 8 .
To validate the functionality of α-SMA + collaterals, we injected rat anti-mouse CD31 intravenously 30 min before euthanasia to label perfused endothelial cells. Hearts were fixed and immunolabeled with α-SMA and donkey anti-rat antibodies before clearing. All α-SMA + collaterals arteries were patent because they were also CD31 + (Extended Data Fig. 2d-f and Supplementary Video 2).
To localize collateral bridges with respect to injured myocardium, we labeled all coronary vessels in the neonate with vascular endothelial cell marker Podocalyxin, and used the autofluorescence to define surviving muscle. Areas lacking autofluorescence, which were not present in uninjured hearts, delineated injured myocardium, which was confirmed by accompanying disrupted vasculature ( Fig. 2d and Supplementary Video 3). Injured regions were outlined and overlaid onto collateral models ( Fig. 2b(i)). Collateral bridges were usually located at the edge of the infarcted area, connecting regions of muscle and vascular death to unaffected sites in the heart (Fig. 2d). At this location, almost half of the collateral bridges were found spanning healthy and infarcted tissue, but many bridges were completely within either healthy or infarcted tissue (Extended Data Fig. 3a,b). Similar patterns were observed in adult hearts ( Fig. 2e-g).
We next quantified collateral connection type, numbers and relative sizes. Collateral connections were categorized based on which artery they connected (Fig. 2h), most often to the SpA. Neonate hearts formed more LCA-LCA and fewer RCA-LCA connections than adult hearts (Fig. 2h). We hypothesized that anatomic proximity could make the SpA more likely to connect with an injured LCA. Measuring the distances between the mid-collateral bridge and the aorta revealed that the SpA forms connections at points closer to its origin than the RCA (Extended Data Fig. 3c). The distal-most tips of the SpA in uninjured hearts are also closer (Extended Data Fig. 3d). Measuring the distance between SpA or RCA tips and those of the LCA showed only a mild trend of SpAs being closer (Extended Data Fig. 3e,f). Therefore, the sites where collateral bridges form might be dictated in part by hemodynamic factors resulting from the distance of the SpA to the aorta. Neonate hearts also formed approximately 40% more collaterals than adults (Fig. 2i), and their diameters were larger, both absolutely and relative to the proximal LCA ( Fig. 2j and data not shown). (Normalization to proximal LCA was performed to account for small differences in individual mouse body size). These data highlight the importance of advanced imaging methods for observing accurate vascular remodeling patterns, that is, those involving the SpA, and underscore the notable differences between young and old hearts.
Modeling coronary blood flow. To understand how these collaterals might restore blood flow in the presence of a vascular occlusion, an in silico approach was used to computationally estimate blood flow while manipulating different parameters in isolation, such as collateral number, size and location. First, an anatomically representative model of the native adult coronary tree was created using the open-source software, SimVascular (www.simvascular. org/) 33 , from a light-sheet image of a non-injured adult heart labeled with α-SMA (Fig. 3a). The light-sheet images (Fig. 3a(i)) were used as a guide for drawing path lines through every artery in the heart up to tertiary branches ( Fig. 3a(ii) and Methods). Special care was taken to ensure that the only vessels not modeled were downstream of the outlets to the 3D model. Arteries were then segmented by drawing a circle spanning the entire cross-section at even intervals along the vessel (Fig. 3a(iii)). The segmentations were lofted into a 3D model (Fig. 3a(iv)). We next measured tissue shrinkage during iDISCO by calculating heart volumes before and after clearing ( Fig. 3b(i)). Since shrinkage averaged 37% (Fig. 3b(ii)), the model was computationally uniformly scaled up by 1.58-fold ( Fig. 3b(iii) and equation (1)). The result was a model reflecting the realistic anatomic 3D architecture of an adult mouse coronary artery tree.
This model was then used to computationally estimate physiologically realistic blood flow parameters throughout the arterial network. Simulations first required setting boundary conditions. At the aortic inlet, a flow waveform was set based on experimentally measured blood velocities from the literature for the neonate 34 and adult 35 . Two outlet boundary conditions were set: (1) an RCR Windkessel model representing the systemic circulation at the aortic outlet 36 , and (2) a lumped parameter network (LPN) representing the coronary vessels downstream of the 3D model 37,38 (Fig. 3c). The lumped parameters included values accounting for vessel resistance at downstream arteries, capillaries and veins and intramyocardial pressure due to ventricle contraction (Fig. 3c) Fig. 1 | Whole-organ imaging of coronary arteries at cellular resolution. Neonate and adult hearts (atria removed) were subjected to tissue clearing and immunolabeling with α-SMA and imaged using a light-sheet microscope. a, Maximum intensity projection of entire neonate heart. b, Z-stack subset projections of light-sheet images (b(i)(ii)) or those captured using a confocal microscope (b(iii-v)). High-magnification view of septum shows the complexity of the SpA (b(iii)) and view of the LCA shows colocalization of α-SMA + branches with arterial marker connexin40 (purple arrowheads, similar results were found in n = 40 hearts; b(iii-v)). An α-SMA lo Connexin40 + vein (orange arrowheads) is also present in b(iii-v). c, 3D rendering of myocardial volume (red; c(i)) and main coronary artery branches: right, septal and left (c(ii)). d, MIP of entire adult heart. Upper-left corner shows 3D rendering of main coronary artery branches similarly as in c(ii). e,f, Z-stack subsets of indicated heart regions (e) and region-of-interest (ROI) images (f) reveal the high resolution and specificity of immunolabeling with this technique. LCA, left; RCA, right; SpA, septal. Scale bars, 300 μm (a-c) and 500 μm (d-f).
on initial estimated parameters (Methods) and were subsequently tuned to match expected flow splits between coronary branches to ensure our CFD simulation distributed flow proportionally. Flow splits were calculated based on perfusion territories for each of the three main branches. Each region of the myocardium was connected to its closest arterial end branch, and all the subregions were identified as belonging to branches of the LCA, RCA or SpA (Fig. 3d(i)(ii)).
The method estimated the LCA, RCA and SpA to perfuse 60%, 25% and 15% of the myocardium, respectively ( Fig. 3d(ii)). Using this information to tune outlet boundary conditions (Extended Data Fig. 4 and Methods) resulted in close agreement between estimated perfusion territory and simulated flow splits ( Fig. 3d(

Fig. 2 | Increased collateral arteries in neonate versus adult hearts after injury. a-d,
Whole-organ imaging of P6 neonatal heart labeled with α-SMA after myocardial infarction (MI). a, Maximum intensity projection (MIP) of entire heart. b, Collateral connections traced from downstream of the suture (red X) were assessed using 3D rendering and overlaid with infarct volume (b(i)) and collateral bridges (b(ii)). c, 3D reconstruction of a 100-μm Z-stack containing a representative collateral bridge within a traced vessel (red dotted line). d, MIP of a 35-μm Z-stack within c highlighting an α-SMA + collateral (d(i)) and its relation to Podocalyxin labeling all vessels (d(ii)) and autofluorescence labeling surviving myocardium (d(iii)). e-g, Adult (16-week-old) injured hearts labeled with α-SMA. e, MIP of entire heart. f, 3D rendering of collateral connections overlaid with infarct volume (f(i)) or collateral bridges (f(ii)). g, 3D reconstruction of representative collateral bridge (pink). h, Classification and distribution of collateral connections. i, Collateral numbers in neonate (n = 9 hearts) and adult (n = 8 hearts) after MI. j, Collateral diameters in neonate (n = 26 collaterals, n = 3 hearts) and adult (n = 55 collaterals, n = 8 hearts) after MI normalized to the proximal LCA. Scale bars, 300 μm (a, b, e and f) and 150 μm (c, d, g and h). Error bars are the mean ± s.d. **P = 0.0019 and ****P < 0.0001, two-sided Student's t-test.
to the stenosis, we are specifically modeling the immediate flow of collateral arteries that exist before a stenosis develops. In total, the adjustments to the model and boundary conditions provided a model with close concordance to native physiology.
Our next goal was to investigate collateral blood flow, and one benefit of a computational approach is that parameters, such as collateral number/size and stenosis severity, can be virtually modified and systematically tested (Fig. 3e). We placed virtual collaterals within the native coronary tree model described above, using post-injury imaging data to guide general placement (Fig. 2).
Computationally derived pressure values were then used to precisely adjust placement at each branch so that collaterals joined two regions of equal pressure. This minimized flow across collaterals without stenosis, which is important to establish a consistent baseline so that different configurations could be properly compared ( Fig. 3e and Extended Data Fig. 5). These guidelines were used to produce five different collateral configurations in the adult heart (Extended Data Fig. 5). We compared pressure difference, flow and shear stress in all collaterals from each configuration to Poiseuille's law, which analytically describes flow through a cylinder, to ensure  b, Scaling model to account for tissue volume reduction during iDISCO procedure. Measuring heart volumes (n = 10 hearts) before and after processing (b(i)) yielded an average reduction value (b(ii)) used to generate a scaling factor for models (b(iii)). c, Schematic of coronary simulation with a prescribed flow waveform at the inlet, RCR boundary condition at the aortic outlet, and coronary LPN at each coronary outlet. d, Determining perfusion territories required first utilizing the Voronoi algorithm to outline perfusion subvolumes for each individual outlet (colors in d(i)). Then, subvolumes were grouped by RCA, LCA and SpA coronary branches (d(ii)). Outlet boundary conditions were tuned by matching simulated flow splits to perfusion territories (d(iii)). e, Schematic depicting variations on collateral and stenosis parameters used in this study. Collaterals were placed to connect approximately equal pressure zones (white arrows). RCR, 3-element Windkessel model; P im , intramyocardial pressure. NA, not applicable.
our simulation results were reasonable (Extended Data Fig. 6 and Methods). The same workflow was performed for an uninjured P6 heart. Perfusion territories were similar, but a lower aortic inflow was prescribed for neonates to match published values 34,39 . Four collateral configurations were produced for neonates (Extended Data Fig. 5). Then, adult and neonate models were used to investigate reperfusion upon virtual stenosis.
Investigating flow recovery by collateral arteries. One way to quantify reperfusion is to sum the flows from all outlets downstream of the virtual stenosis and compare this to a normalized baseline flow with no stenosis (set at 100%). As expected, with no collaterals in adults, total flow downstream of stenosis decreases when the percentage occlusion increases, especially above 90% (Fig. 4a). When comparing all the configurations tested at all stenosis levels, the configuration with nine collaterals at 40 μm (9col, 40 μm) provides the most flow recovery, especially at 99% stenosis where it restores almost 25% of the non-stenotic flow compared to just 1% without collaterals (Fig. 4a). However, this extent of collateralization does not occur naturally with coronary artery ligation in adult mice ( Fig. 2i and refs. 7,8 ). We noted that configurations similar to those observed experimentally, that is, 6-12col, 20 μm, recovered very little flow as measured by this method (Fig. 4a). These data demonstrate that collateral arteries as they naturally form after adult coronary occlusion are not expected to appreciably recover blood flow, but that increasing diameters, which is a major factor in reducing overall resistance, could enhance their function.
In contrast, comparable collateral configurations in neonates performed well. The natural configuration (that is, 12col, 20 μm; Fig. 2i and ref. 7 ) is estimated to recover up to 60% of total flow downstream of a 99% stenosis (Fig. 4b).
Remarkably, the largest diameter tested (40 μm) only required one vessel to provide massive recovery (Fig. 4b). To compare adult and neonate flow recoveries, it is important to confirm that collaterals generally connect equal pressure zones (±10 mmHg) so that all configurations start with a similar collateral flow. This was further evident by the observation that adding collaterals did not change total downstream flow without stenosis and primarily increased flow only with increasing stenosis severity (Extended Data Fig. 5).
The above analysis calculated overall recovery of pre-stenosis levels, but clinical data indicate that myocardial tissue could be supported at approximately 30% of baseline flow 40,41 . Thus, we next explored a more nuanced perspective by considering individual outlet perfusion territories downstream of the stenosis, so that we could observe if certain regions were receiving sustainable reperfusion (that is, >30% reperfusion). First, we grouped all perfusion territories downstream of the stenosis to obtain the full volume-at-risk (VaR; Fig. 4c) and then plotted the percentage of that volume that is reperfused above a certain threshold ( Fig. 4c(i)). This revealed that while there were still not sustainably reperfused regions in the 12col, 20 μm and 6col, 28 μm configurations, the 3col, 40 μm and 9col, 40 μm configurations were able to sustain 10 and 25% of the VaR, respectively ( Fig. 4c(i-iii)). However, in the neonate, the 6col, 20 μm reperfused 80% of the VaR over the 30% threshold, while the 12col, 20 μm and 1col, 40 μm configurations reperfused the entire VaR ( Fig. 4d(i-iii)). These data emphasize that collateral configurations of the same size, and thus same resistance, function better in the neonatal heart.
One hypothesis is that hemodynamic conditions at specific vessels dictate where collateral connections form. To gain evidence for this, we modeled an injured neonate heart 4 d after MI to identify if hemodynamic forces correlate with real collateral connection sites. In total, 180 vessels of an injured neonatal heart were segmented (Extended Data Fig. 7a(i)). We then virtually removed the 9 injury-induced collaterals and fixed the occlusion to obtain a representative model of the coronary vasculature after the MI but before collaterals formed (Extended Data Fig. 7a(ii)). Calculating pressures at tips near real collateral attachment sites and comparing those to non-attached tips in the injury region did not reveal a difference (Extended Data Fig. 7b). Comparing pressure at tips at the end of either the SpA (most common connection site) or RCA (least common connection site; Fig. 2h) in non-injured hearts also revealed no difference (Extended Data Fig. 7c). These data show a lack of correlation between hemodynamic factors and attachment site, suggesting that other factors, such as interaction with infarcted tissue (Extended Data Fig. 3a,b), may play a more prominent role.
One interesting difference between the non-injured and injured models is that the volume perfused by the LCA was reduced due to vessel pruning downstream of the occlusion (Extended Data Fig. 7a(iii)). Due to this pruning, the LCA had a large pressure drop even after the occlusion was relieved (Extended Data Fig. 7a(iv)). Consequently, after injury, there was a persistent pressure difference between the two connection sites (Extended Data Fig. 7d), which began to equalize once collaterals were added back into the model (Extended Data Fig. 7e). These observations suggest that real collaterals connect vessels of different pressure, and this difference is equalized toward a homeostatic level upon collateral connection.
Using the same neonate MI model, we next compared the performance of real post-MI collaterals to the virtual pre-MI vessels modeled in Fig. 4. The real neonatal collaterals restored blood flow at slightly higher levels than predicted by our virtual scenario. Specifically, the 9 real collaterals with an average of 20 μm were most similar to the 12 virtual collaterals at 20 μm ( Fig. 4d(i)). This was also true for volumes above 30% reperfusion and total flow recovery (Extended Data Fig. 7f,g). This slight enhancement is likely due to the above describe vessel pruning causing a lower baseline flow being restored. In total, the similar magnitudes between real and virtual flow recoveries and volume reperfusions support our predictions that neonatal collaterals have greater reperfusion abilities.
We next explored whether a more favorable placement of collaterals could improve the poor performance seen in adults. We started with a 3col, 40 μm configuration (Fig. 5a) and moved each collateral to a more proximal location in the coronary tree (Fig. 5a). This manipulation almost doubled total flow recovery ( Fig. 5b) and approximately tripled the volume of myocardium reperfused above the 30% threshold ( Fig. 5c-e). Thus, variation in location can improve collateral function.
The above data suggest that fewer, larger collaterals are better than many, smaller ones ( Fig. 4). However, in those experiments, the total collateral resistance varied between the configurations. We tested this hypothesis by varying the number and size of the collaterals while keeping the total resistance equal. Simulations were performed on two configurations-16col, 20 μm and 1col, 40 μm (Fig. 5f). While total flow recovery was approximately equivalent (Fig. 5g), the 1col, 40 μm configuration was uniquely able to reperfuse 5% of the VaR above 30% (Fig. 5h-j). Streamlines from each collateral illustrate differences in flow distribution between the 16 collateral and 1 collateral configurations (Extended Data Fig. 8). This analysis shows that fewer, larger collaterals could be more beneficial because they at least protect a portion of the myocardium, while many, smaller collaterals distribute the reperfusion so that none reach protective levels.
Adult versus neonate coronary artery morphology. Given that collaterals of the same size and total resistance were predicted to proportionally recover more flow in the neonate, we sought to understand why and first investigated arterial pressures at collateral formation sites. To facilitate comparisons, Strahler ordering was used to classify branch segments into orders based on hierarchal position in the coronary tree and vessel diameter 42,43 . Order 13 represented the aorta, order 12 represented the most proximal    coronary artery segments, and subsequent orders represented downstream vessels until order 8, which were the most distal branches modeled (Fig. 6a). This was used to compare hemodynamic and anatomic quantities at similar points in the coronary tree in both the adult and neonate. While absolute aortic and proximal coronary (order 12) pressures were vastly increased in the adult, the pressures at the most distal coronary tips (order 8) were approximately equal (Fig. 6b,c). Quantification revealed that the pressure drop along the coronary tree was ~20 and ~50 mmHg in neonate and adults, respectively (Fig. 6c). This is also true when considering just the segments downstream of the stenosis, making the pressure difference downstream of a potential stenosis in the adult (ΔP Ad ) greater than in the neonate (ΔP Neo ; Fig. 6b,c). Thus, the collateral pressure difference (ΔP Col ) required to restore pre-stenotic flow downstream of the occlusion is higher in the adult. Specifically, the ΔP Col needs to be about twofold higher in the adult to restore the same flow. Given that we see similar distal pressures at both stages, this explains why, even though collaterals in both recover the same absolute flow, it is much lower than the baseline, non-stenotic flow in the adult. A sensitivity analysis was next performed to control for user variability in segmentation. We utilized zero-dimensional (0D) reduced-order models (ROMs) due to the computational cost of running full 3D simulations. While 0D simulations do not account for pressure losses at junctions, perturbations of parameters in 0D models have been shown to be highly correlated with 3D perturbations 44 . The 0D ROMs were created by extracting radius, length and connectivity of vessel segments using automated scripts and modeling the segments as simple resistors in a network 45 (Methods). To quantify the expected variability of vessel segmentation, five users with knowledge of vascular anatomy determined the diameter of 16 segmentations sampled across all orders of the coronary tree. The average coefficient of variation was 16.7% (Extended Data Fig. 9a). We tested the extremes by decreasing the adult radius and increasing the neonate radius by 20%. This confirmed differences in ΔP Ad and ΔP Neo , demonstrating robustness of our overall findings (Extended Data Fig. 9b).
Our next experiments probed why ΔP Ad was greater than ΔP Neo. Two factors critical for determining ΔP are flow rate and total resistance. First, we compared the flow rate at each Strahler order between neonate and adult. Literature values indicated that aortic flow in adults is approximately ten times more than in neonates, which was used as the inflow boundary condition for the computational model (Fig. 3c) 34 . Simulations revealed that flow was also tenfold greater for every vessel order modeled in the coronary tree (Fig. 6d). Shear stress was lower in neonates compared to adults, particularly in higher-order vessels (Fig. 6e). We confirmed this trend held true when increasing the mesh size from 1.8 to 10 million elements; there was less than 10% difference in average shear stress with increased mesh resolution. Flow values were in line with increases in myocardial volume over time, that is, volumes at P32 were more than tenfold greater than those at P0 (Fig. 6f). We also observed a linear increase in myocardial volume during the first 2 weeks of life, a plateau between P18 and P25, and a burst of growth from P25 to P32. Second, we used the simulated flows and pressures to calculate the total resistance of the 3D coronary model. Neonate total vascular resistance was threefold higher than that of adults (Fig. 6g). The neonate 3D coronary resistance is much higher due to a significantly lower number of branches (quantified below). Because flow was increased by tenfold, the threefold decrease in total resistance is not enough to offset flow increases. Thus, while the resistance of the coronary vasculature decreases in the adult, it is not able to lower the resistance enough to balance the much greater flow, manifesting in a greater ΔP in adults.
We next investigated what features contribute to the nonproportional decrease in resistance with respect to the flow increase from neonate to adult. Two factors critical for determining resistance are vessel diameter and the number of branches. Increases in these parameters work to lower total resistance, diameter being the most impactful. Surprisingly, we found that the diameters were the same across all Strahler orders in each model (Fig. 7a). We validated this by comparing diameters of the most proximal segments of the RCA, SpA, LCA and aorta in multiple replicates of neonatal and adult hearts (Fig. 7b). The coronary stem diameter remained virtually the same, while aortic diameter increased with age (Fig. 7b), a result we validated using an orthogonal method (Fig. 7c). Thus, coronary diameters do not grow proportionally to heart volume, which suggests that diameter expansion does not function to relieve vascular resistance in the face of increased flow demand in adults.
If arteries do not increase in diameter, additional branches must be added to at least partially offset the increased flow that accompanies heart growth. We next quantified branching during postnatal development. Comparing the Strahler ordering of the two stages revealed that the number of distal vessels (orders 9 and 8) were vastly increased (Fig. 7d), aligning with qualitative observations from imaging (Fig. 1). Because the 3D SimVascular models did not contain arterioles distal to tertiary branches, we further investigated morphometry by manually tracing all α-SMA vessels in a representative branch-the left circumflex (LCx; Fig. 7e). Imaris software filament tracing binned each segment of the LCx according to branching levels and quantified the number of arteriole tips (Fig. 7f). The number of branching levels spiked between P0 and P6 and then hit a plateau until another spike between P25 and P32 (Fig. 7g). The number of tips increased linearly up to P18 with another spike between P25 and P32 (Fig. 7h). To find a growth-halt point in adult mice, we quantified arterial tip numbers in 2-and 3-month-old mice, and found similar quantities to those in P32, suggesting a stop point after 1 month of age (Fig. 7h). The P6-18 plateau in number of branch levels compared to the linear increase in number of tips over the same time period indicated that the coronary arteries grow by adding branch segments along the entire length of existing branches. We also observed that the length of each segment was constant among all ages tested (Fig. 7i). This results in a coronary tree with many lateral branch segments of a set length.
Human fetal and adult coronary collateral arteries. A subset of human hearts contains collateral arteries, which are easily observed during an angiogram and are correlated with increased survival in individuals with heart disease patients 46,47 . We sought to identify how our computational modeling studies could help us better understand human collateral function. Thus, we compared the data available from human hearts to our mouse model data. We measured vessel diameters for the collaterals observable in angiograms from five individuals living with CTOs because these individuals' collaterals can sufficiently support myocardial perfusion downstream of the occlusion without exercise (Fig. 8a). To compare with mouse data, we normalized human diameters to the most proximal segment of the LCA. Collateral diameters were on average 15% of the LCA (Supplementary Table 1). These values were in between those observed in the neonate and adult mouse hearts (Supplementary Table 1). Diameters in angiograms are measured in 2D projections affecting the accuracy of absolute values. We also found an average of two collaterals per heart (Supplementary Table 1), but comparisons with mouse data using this parameter are less desirable because angiograms will only highlight a subset of the collaterals that immunostaining would label. These data provide a foundation to determine reperfusion benefit, but a very precise understanding in humans will need to consider the different pressure distributions resulting from human-specific morphology.
Using postmortem perfusions, studies from the 1960s reported the presence of coronary collateral arteries in infants and children 48,49 , but no one has reported whether collaterals develop during embryogenesis. Furthermore, using smooth muscle coverage to identify collateral connections in humans has not been done. We processed two fetal hearts aged 17 and 22 weeks with the same whole-organ immunolabeling method used for mouse hearts (Fig. 8b(i)(ii) and Extended Data Fig. 10), imaged at high resolution, and three additional hearts aged 14, 18 and 19 weeks, imaged at a lower resolution (Extended Data Fig. 10). Both 17-and 22-week-old hearts had visible collaterals on the dorsal and ventral sides (Fig. 8b(iii)(iv) and Extended Data Fig. 10). Remarkably, >17 collaterals were detected per side (Fig. 8c), which suggests that the whole human heart has at least 40 pre-existing, smooth-muscle-covered collaterals forming during (17-22 weeks) stages of embryonic development. On the ventral side, most connections bridged distal branches of the RCA and LCA, while the majority on the dorsal side connected two LCA branches (Fig. 8d). Collateral diameters were not significantly different across locations or between ages and were on average 7% of the most proximal LCA segment (Fig. 8e). Thus, unlike mouse, human hearts have mechanisms in place to form native collateral arteries as part of normal development, evident by the number of collaterals in younger, 14-week-old hearts-less than half that found in the older hearts (Extended Data Fig. 10d-f). We hypothesize that these mechanisms could be the precursors for collaterals that preserve myocardium downstream of an occlusion.

Discussion
This study applies 3D CFD to quantify hemodynamic forces in the adult and neonatal mouse coronary vasculature, the latter being naturally more restorative. The findings suggest the therapeutic benefit of promoting fewer, larger collaterals. Imaging the intact 3D arterial structure revealed its architectural complexity. Our methods also reveal a critical role for the SpA. Previous studies utilizing flattened hearts for whole-mount imaging failed to distinguish the SpA from the RCA 7 . Results point to the SpA as a primary collateral connection site. The SpA proximal attachment configuration was proposed to impact cardiac recovery 50 , and it will be important to assess the influence on collateral positioning. Collateral bridges preferentially formed within or near the infarct, further supporting local hypoxia as a trigger 7 , but tissue stiffness or disrupted blood flow could also play a role.
Because there is no automated method to generate 3D artery models 23 , we manually segmented over 300 vessels in an adult and over 200 vessels in a neonate to ensure a high model fidelity for SimVascular fluid simulations. Methods for the brain utilize machine learning to improve automation (for example, TubeMap) 51 . Future work will use or develop similar methods for cardiac vasculature.
One advantage of CFD modeling over ex vivo measurements of experimental samples is the capability to easily modify one feature, that is, collateral structure, while keeping all other parameters constant. We tested multiple collateral configurations within the  = 1 P60 hearts (a and d); n = 4 P6, n = 3 P25 hearts (b); n = 5 P6, n = 3 and n = 4 P60 hearts (c); n = 3 P0, n = 7 P6, n = 3 P11, n = 2 P18, n = 2 P25, n = 2 P32 hearts (g-i). Changes in green color tonality denote increments in age. n = 1 P60, n = 2 P90 hearts (h and i). Scale bars, P0, 100 μm; P6, 200 μm; P18, 400 μm; P32, 500 μm (f). Error bars are the mean ± s.d. **P = 0.0030 (g), *P = 0.0216 (h, P0-P6) and **P = 0.0017 (h, P25-32), two-sided Student's t-test. same model to understand the relationship of number, position and diameter on flow recovery, without potential secondary effects from mouse-to-mouse variations in coronary structure. We considered flow recovery above 30% of non-stenotic perfusion levels as being beneficial because previous studies detected cardiac dysfunction when flow levels dipped below 25-30% of baseline 40,41 . Simulations demonstrated that increasing diameters or positioning collaterals more proximally restored more tissue to this 30% reperfusion value, more than by increasing numbers of smaller collaterals. Our data didn't determine the reperfusion level required for cardiomyocyte viability, but they did provide general guidelines of how parameters affect collateral flow and reperfusion. Physical differences between phenotypes or conditions can be more confidently related to functional differences.
Surprisingly, the most effective collateral configuration modeled in the adult reperfused only ~20% of myocardium above the 30% threshold. This aligns with consistent scar formation after permanent coronary ligation in mice 8 and underscores the importance of understanding blood flow through experimentally induced collateral arteries when considering therapeutic implications. Conversely, virtual collaterals with the same characteristics reperfused most of the myocardium in neonates. This, combined with more numerous and larger collaterals in neonates, likely contributes to their greater recovery after MI 7 . We attributed the difference in collateral performance to a lower pressure drop in the neonatal coronaries because increased flows in adults are not compensated by an equal reduction in vascular resistance. Our two models were segmented to a similar extent because artery tip pressure was approximately 40 mmHg at both ages. The more gradual, steady decrease in pressure in the adult arises from more extensive branching, similar to porcine coronaries 52 . We estimated a mild pressure drop in native neonate coronaries. However, the pressure downstream of the 3D model is expected to abruptly drop to capillary levels. The reduction in total coronary resistance from neonate to adult is in general concordance with µCT measurements of coronary vessels > 40 µm, but the reduction in our data is quantitatively much greater 30,53 . While these studies described the coronary tree with quantitative scaling laws, they were not able to quantify small-diameter arteries, which were absent compared to our P6 hearts. Future work could investigate how quantitative scaling laws apply to vessels observed with organ clearing.
Since only a minority of adult human hearts have functional collateral flow, it was surprising that fetal hearts contained >40 native collaterals. Collaterals have been reported to decrease during adolescent years, suggesting they may regress over time 48,49 . If we can preserve and enlarge these collaterals in adulthood, they may improve cardiac perfusion in individuals who have CAD.
Overall, by combining advanced computational and imaging techniques, a critical connection between collateral flow, native morphology and pressure distributions was established. By bridging these fields, we uncovered how changes in fundamental coronary morphology from the embryonic stage to adulthood affect collateral flow. These findings provide insight into why coronary collateral arteries are better suited for recovering from injury in young hearts versus old.

Limitations
We did not measure subject-specific aortic flows and pressures for each mouse to generate boundary conditions, but instead used literature-derived averaged flows scaled to the model size. Our results are specific to LCA ligations and SpA-from-RCA coronary tree configurations. Virtual collaterals were assumed to be straight tubes, while the tortuosity index of in vivo collaterals was, on average, 0.1 (±0.07), a small difference when compared to the much more tortuous human collaterals 54 .
Due to small size and inaccessibility, coronary outlet pressures cannot be validated in vivo. Pressure measurements in rabbits and dogs show 50-70% pressure loss at the capillary bed 55,56 . Our outlet boundary conditions were 65% of the total coronary resistance to match literature values 55 . We changed outlet pressure by adjusting the outlet resistances and found relative differences between the collateral configurations remained the same.
Our modeling does not account for short-term or long-term effects of vasodilation or remodeling. Adult coronaries compensate for ischemia through autoregulatory vasodilation. Recent studies have accounted for autoregulation, enabling accurate predictions of fractional flow reserve [57][58][59] . However, autoregulatory effects remain uncharacterized in neonatal hearts. Once quantified in vivo, they could be incorporated into future modeling approaches. If autoregulatory effects are similar in adults and neonates, our conclusions on the relative differences in collateral performance are expected to remain unchanged. Vascular pruning and scar development following adult injury could impact downstream resistance, but requires in vivo experiments to obtain mouse-specific modeling parameters. Future studies will enhance our computational methods by using in vivo experiments to define and capture autoregulation and remodeling.

Methods
Animals. All mouse colonies were housed and bred in the animal facility at Stanford University in accordance with Institutional Animal Care and Use Committee (IACUC) guidance on a 12 h-12 h day and night cycle with water and food ad libitum.
The following mouse strains were used: pure C57BL/6J, (Jackson Laboratory, strain 000664) pure CD1 (Charles River, strain 022) and mixed C5BL6 and CD1. Both male and female mice were used for postnatal studies up to day 6. Males were exclusively used for studies of P11 to 3 months old.
All experiments were conducted in accordance with protocols approved by the IACUC of Stanford University.
Light-sheet imaging. Samples were imaged with ImSpector Pro 7.0.98 software and a LaVision BioTec Ultramicroscope II light-sheet microscope in a quartz cuvette filled with ethyl cinnamate. For imaging, we used an MVX10 zoom body (Olympus) with a ×2 objective (pixel size of 3.25 µm/x, y) at a magnification from ×0.63 to ×1.6. Up to 1,400 images were taken for each heart and the Z-steps were set to a 3.5-µm Z-step size, and the light-sheet numerical aperture was set to 0.111 NA. Band-pass emission filters (mean nm per spread) were used, depending on the excited fluorophores: 525/50 for autofluorescence; 595/40 for Cy3; 680/30 for AF647 and 835/70 for AF790. Exposure time was 10 ms for single-channel and 25 ms for multichannel acquisition.

Perfusion territory mapping.
To determine the approximate volume of myocardium that each outlet of the coronary model was responsible for perfusing, we used (1) a model of the myocardial tissue as the total volume to be perfused and (2) the outlet coordinates as the seed points for the subvolumes. We used the background signal from the staining to segment the model of the myocardial tissue and the cap centers for the outlet coordinates. Then, we used a Voronoi diagram algorithm to assign subvolumes of the myocardial tissue to each outlet of the coronary model such that every point in the myocardial mesh was assigned to the closest outlet. Distances to the closest outlet were determined using a Dijkstra algorithm. By integrating the subvolumes of every outlet on each of the three main branches (LCA, RCA and SpA), we were able to calculate the approximate percentage of the total myocardial volume that each main branch is responsible for. We used these percentages as targets for the flow splits when tuning the outflow boundary conditions for the fluid simulations.
We used outlet coordinates instead of centerlines because we were able to better resolve the small coronary arterioles compared to previous studies 60 . This allows us to be certain that myocardial regions close to an outlet are perfused by that outlet, rather than by a large artery nearby that has no outlet nearby. CFD simulation. We constructed 3D subject-specific models of the mouse vasculature using the SimVascular (2021.06 release) cardiovascular modeling pipeline 33 . Briefly, we created path lines for each vessel (about 349 vessels for the adult and 244 for the neonate). Vessels distal to the quaternary branches were ignored. For each path line, the image data were viewed in planes orthogonal to the tangent of the path line to segment the cross-section. Circles were used to approximate the cross-section, as some areas of the vasculature appeared collapsed or deformed. All segmentations were lofted to create a solid model of each branch, and the branches were then unioned together to form a complete geometric model. Finally, the lofted model was discretized into a linear tetrahedral mesh using the commercial meshing library, MeshSim (Simmetrix), resulting in a total of 600,000 and 1.8 million elements for the neonatal and adult models, respectively.
After obtaining the mesh, we uniformly scaled it to account for the shrinkage that occurs via iDISCO. We quantified the volume change due to our specific clearing protocol using water displacement before and after iDISCO and found that the heart shrank to about 63% of its original volume. So, we uniformly scaled the entire volumetric mesh by the inverse (1.58-fold) to ensure that our model faithfully matched the pre-iDISCO geometry according to equation (1).
Inlet boundary conditions were determined as follows. We first determined typical neonate and adult mean pressure and aortic velocity values from literature 34,35,61 (Supplementary Table 2). Using the mean aortic velocity and the aortic cross-sectional inlet area for each mouse used, a subject-specific aortic inflow was calculated and applied. For pulsatile flow simulations, we constructed representative flow waveforms for an adult mouse by digitizing, smoothing and scaling a waveform from the literature to match the mean inflow at both ages as calculated previously 62 . At the aortic outlet, we applied a simple RCR boundary condition 36 (Supplementary Table 2). At the coronary artery outlets, we applied a specialized LPN to represent the downstream coronary vasculature and the time-varying intramyocardial pressure due to the beating cardiac tissue 37,38,63 (Supplementary Table 2). The resistance of each coronary outlet was estimated using Murray's law and tuned such that each of the three main branches (LCA, RCA and SpA) had flow splits equal to the percentage volume they perfused. We further tuned the capacitances and resistances of the coronary outlet boundary conditions to match literature values of the proximal and distal resistance ratio 55,56 . To do this, we used a 0D surrogate model for increased efficiency (Extended Data Fig. 2).
We globally corrected the viscosity in our pulsatile simulations to 1.25 cP to account for the Fahraeus-Lindqvist effect; this is necessary because the apparent viscosity of blood decreases in very small tube diameters (<100 μm) 64 . While this may underestimate the shear stress in the aorta, the pressure drop in the coronaries was more representative and important for the findings presented here ('Limitations').
We ran blood flow simulations with rigid walls using the stabilized finite element svSolver code in the open-source SimVascular software package 33 to determine spatially and temporally resolved hemodynamic values, such as pressure, velocity and wall shear stress at every node in the computational mesh. Simulations ran for five cardiac cycles with timesteps of 0.0001 s, and hemodynamic values were determined based on the final cardiac cycle. Flow and pressure waveforms converge to time-periodic solutions with less than a 5% difference from cycle 5 to 6. Simulation time was approximately 40 h on 96 cores via XSEDE and 90 h on 96 cores via Sherlock. Paraview 5.9.1 software was used for visualization of the results.
Virtual collateral placement. Virtual collaterals were strategically added to native coronary vasculature to minimize the initial pressure difference of the two points that the collateral was connecting. Specifically, based on an initial simulation of the native vasculature (without any virtual collaterals), a pressure distribution was determined. Using this pressure distribution, virtual collaterals were placed such that each connected equal pressure zones. We replicated realistic connections as closely as possible given upstream native vessel diameter and pressure constraints (Extended Data Fig. 5). The resistance of each collateral configuration was calculated via Poiseuille's Law, given by equation (2): where µ is the viscosity, L is length of the collateral, n is the number of collaterals in the configuration and r is the radius of the collateral.

3D resistance.
To calculate the resistance of the 3D model, we first generated vessel centerlines via the Vascular Modeling Tool Kit (VMTK; http://www.vmtk.org/). Each point in the centerline was identified as a branch segment if a perpendicular cross-section at that point did not intersect with any other centerline point. If the cross-section intersected more than one centerline point, then it was labeled as a junction region. This separated the centerline into junctions and branch segments between junctions. After labeling every point, we determined the parent (upstream) branch segment and child (downstream) branch segments for each junction region. We then calculated the resistance for each branch segment based on the pressure difference from the most proximal point to the most distal point and the flow within that segment from the simulation. Finally, the overall 3D resistance was calculated starting from the most distal branches using a recursive method to add the segment resistances in parallel or in series based on the connectivity.
Diameter-defined Strahler ordering. We utilized the diameter-defined Strahler ordering system to compare morphometric and hemodynamic quantities at similar positions in the coronary tree between the neonate and adult. This system has been used in previous morphometric studies to classify branch segments into orders that describe the hierarchical nature of a vascular tree 42,43,65 . Using the same labels for branch segments and junction regions as in the 3D resistance calculations, we determined the initial Strahler ordering by setting the most distal segments to order 1 and working backwards up the coronary tree to the aorta. Parent segment orders were set to either equal the greater child order if the two children orders were different or incremented by one if the two child orders were the same. As neither 3D model of the mouse vasculature included all arteries down to the capillary level (only 5 distinct orders here versus 11 in other studies 43 ), we translated all the orders by a constant such that the order of the most proximal segment of the coronaries was 12 and the aorta was order 13 to ensure consistency with previous studies. Segments were then reorganized based on their diameter to ensure that unbalanced branching (that is, a very small vessel branching from a large one) was properly accounted for.
To do this, we iteratively moved segments to higher or lower orders such that every segment within an order was within 1 s.d. of that order's mean diameter. With the final diameter-defined Strahler ordering, we compared quantities such as diameter, length, flow and pressure between the same orders of the neonate and the adult.
User study of diameter variability. To measure the human source of uncertainty in vessel diameter, we asked 5 users to segment the same 16 locations along 3 different vessels. These segmentations were sampled from both proximal (large) and distal (small) vessels. The users had extensive knowledge of vascular anatomy and were given the same instructions of how to segment the vessels using SimVascular. The coefficient of variation was measured for each segmentation and averaged to use for sensitivity analysis.
0D simulation and sensitivity analysis. We extracted radius, length, and connectivity information from centerlines generated through VMTK from the neonate and adult coronary tree ('3D resistance') to create an input file for 0D simulation. Any modification of the radius for sensitivity analysis was directly scaled in the input file of the simulation. We ran 0D simulations using the svzerodsolver code in the SimVascular software package as previously described 45 .
Each arterial segment was modeled as a resistor with a resistance value following Poiseuille's law. The resistors were connected in parallel at each junction depending on the connectivity of coronary network. The input and output boundary conditions were the same as 3D simulations: a pulsatile waveform at the inlet, an RCR boundary condition at the aortic outlet and coronary LPN at coronary outlets. Simulations ran for ten cardiac cycles with timesteps of 0.001 s taking approximately 20 s of computational time.
Semiautomated artery tracing. Subsequential images were imported into the National Institutes of Health (NIH) Fiji ImageJ 1.53 software as stacks files, and these stacks were then converted into 8-bit and the resolution was reduced to one-fourth of the original. Using an ImageJ plug-in, Simple Neurite Tracer, the branch structures of the LCx were able to be drawn by placing seed points along the length of α-SMA + vessels 32 . Once every α-SMA + artery in the LCx branch was completely accounted for within the trace, isolation of the traces was performed by the Fill Out option within the plug-in. The resulting image stack was used as a 3D outline of the arterial structure as the foundation for further modeling and analysis. After discontinuation of Simple Neurite Tracer, the updated version SNT was used in similar manner as above 31 . Quantitative metrics from tracing were similar between four independent researchers.
Tortuosity index quantification. Tortuosity index of collaterals was defined according to equation (3): where L is the total length along the centerline of the collateral and L 0 is the distance between the start and end of the collateral.
3D rendering. The non-traced image stack was overlaid with the filled LCx stack using the Add Channel option in Imaris 9.5.0 software (Bitplane). Pixel dimensions were updated from the non-reduced 16-bit image metadata. The Filament Object Tracer module was used to generate an Imaris customizable 3D LCx branch model. Branch tips and length were measured by automatically generated data under Number of Terminal Points, and Total Length fields, respectively. Branch levels were obtained from the Filaments Branch Hierarchy field. Surface objects in Imaris were used for quantifying the sample heart volumes. Myocardium volume was calculated by creating surface objects surrounding the entire sample surface and objects encompassing the lumen of the ventricles. The volumes of the ventricles were then subtracted from the entire heart volume to result in the myocardium tissue volume.
Mouse left coronary artery ligations. Neonatal LCA ligations were performed as previously described 7 with minimal modifications. P2 neonates were cooled on ice for 6 min to induce hypothermic circulatory arrest and placed in a supine position followed by disinfecting with iodine and ethanol. Dissection was carried through the pectoralis major and minor muscles, and the thoracic cavity was entered via the fourth intercostal space. The LCA was identified and ligated with a double knot using an 8-0 nylon suture, leaving the LCx intact. The chest muscle and skin were then closed (independently) with interrupted 7-0 prolene sutures. The neonate was then allowed to recover on a warm plate at 37 °C and, when conscious, were returned to its mother's care.
Adult mice ligations were performed as previously described 66 . Adult mice were subjected to permanent coronary artery ligation, under anesthesia using initially 1.5-4% isoflurane chamber for induction. The chest cavity was opened, and a 7-0 silk suture was placed around the left coronary artery, with occlusion verified by blanching of the underlying myocardium. The chest was then sutured closed. Following surgery, buprenorphine (0.1 mg per kg body weight) was used as an analgesic.
In vivo CD31 labeling. Four days after MI, neonates were cooled on ice for 6 min to induce hypothermic circulatory arrest and placed in a supine position. Mice received an intravenous 10 µl retro-orbital injection 67 of rat anti-mouse CD31 Alexa Fluor 647 in sterile 1× PBS (1:5 dilution; BioLegend, 102516). Neonates were then allowed to recover on a warm plate at 37 °C and, when conscious, were returned to their mother's care for 30 min before euthanasia.
Immunohistochemistry and confocal microscopy. Neonatal or adult hearts were fixed in 4% PFA for 1 h at 4 °C, and then rinsed 3× with PBS, blocked in 5% NDS, 0.5% Triton X-100 in PBS for 1 h at room temperature, and then incubated with Cy3-conjugated αSMA (1:300 dilution; Sigma, C6198) in 0.5% Triton X-100 in PBS overnight at 4 °C. Next, the hearts were rinsed 3× in 0.5% Triton X-100 in PBS and stored in Fluoromount G (SouthernBiotech, 0100-01) overnight at RT. Tissue was imaged using an inverted Zeiss LSM-700 confocal microscope at ×5 objective. Digital images were captured with Zeiss Zen 2.3 software and measured using ImageJ.
Human hearts. Under protocols approved by the Institutional Review Board, human fetal hearts were obtained for developmental analysis 68 . Gestational age was determined by standard dating criteria by last menstrual period and ultrasound 69 . Tissue was processed within 1 h following procedure. Tissue was extensively rinsed with cold, sterile PBS while placed on ice, followed by incubation in sterile 4% PFA for 4 h at 4 °C before further iDISCO processing. Pregnancies complicated by multiple gestations and known fetal or chromosomal anomalies were excluded.
Human adult samples were acquired from the Stanford Catheterization Angiography Laboratory. All individuals displayed symptoms of chronic angina and were scheduled to receive conventional coronary angiography, which was performed according to local clinical standards. Collateral number and size were confirmed by an experienced cardiologist. All participants were consented under an approved Institutional Review Board protocol at Stanford University.
Statistical analysis. Graphs represent mean values obtained from multiple experiments and error bars represent standard deviation. A two-tailed unpaired Student's t-test was used to compare groups within an experiment and the level of significance was assigned to statistics in accordance with their P values (*0.05, **0.01, *<0.001 and ****<0.0001). All graphs were generated using GraphPad Prism 9.1 software. Error bars represent ± standard deviation.
Reporting summary. Further information on research design is available in the Nature Research Reporting Summary linked to this article.

Code availability
The code used for flow simulations can be found on the SimVascular GitHub (https://simvascular.github.io/). Custom code for Strahler ordering and volume perfusion using publicly available modules, such as the vascular toolkit, can be found on GitHub (https://github.com/StanfordCBCL/Collateral). Fig. 3 | Investigation into parameters related to collateral artery placement. a,b, Location of collateral bridges in relationship to infarcted tissue. (a) Schematic of categorization shown in b. b, Pie chart showing distribution of collateral location (n = 87 collaterals, n = 6 hearts). Distribution by collateral configuration type (n = 47 collaterals, n = 3 hearts). c, Distances between the aorta and collateral bridges (n = 47 collaterals, n = 3 hearts). d-f, Septal artery evaluation. (d) Distances between most distal RCA and SpA branch tips to the aorta. (e) Representative image of 3D rendered coronary artery branch tips and distances. (f) Quantification of distances in e between most distal branch tips of RCA and SpA to LCA. Right (RCA), left (LCA), and septal (SpA) coronary arteries. d-f, n = 3 hearts. Error bars are mean ± st dev: ****, p≤0.0001 by two-sided Student's t-test.