Microsphere skimming in the porcine coronary arteries: Implications for flow quantification

Abstract

Particle skimming is a phenomenon where particles suspended in fluid flowing through vessels distribute disproportionately to bulk fluid volume at junctions. Microspheres are considered a gold standard of intraorgan perfusion measurements and are used widely in studies of flow distribution and quantification. It has previously been hypothesised that skimming at arterial junctions is responsible for a systematic overestimation of myocardial perfusion from microspheres at the subendocardium. Our objective is to integrate coronary arterial structure and microsphere distribution, imaged at high resolution, to test the hypothesis of microsphere skimming in a porcine left coronary arterial (LCA) network. A detailed network was reconstructed from cryomicrotome imaging data and a Poiseuille flow model was used to simulate flow. A statistical approach using Clopper–Pearson confidence intervals was applied to determine the prevalence of skimming at bifurcations in the LCA. Results reveal that microsphere skimming is most prevalent at bifurcations in the larger coronary arteries, namely the epicardial and transmural arteries. Bifurcations at which skimming was identified have significantly more asymmetric branching parameters. This finding suggests that when using thin transmural segments to quantify flow from microspheres, a skimming-related deposition bias may result in underestimation of perfusion in the subepicardium, and overestimation in the subendocardium.

Introduction

A range of non-invasive imaging modalities have come into clinical use in recent decades for the assessment of regional cardiac perfusion to diagnose coronary artery disease (CAD) and microvascular disease (CMD). These include single photon emission computed tomography (SPECT), positron emission tomography (PET), and magnetic resonance imaging (MRI) (Salerno and Beller, 2009). Critical to supporting the legitimacy of flow quantification obtained from non-invasive imaging is the independent validation of results via experimental approaches. In this context, validation of perfusion quantification from both PET and MRI has been performed by comparison to microspheres, which are a gold-standard deposition marker for intra-organ perfusion measurements (Prinzen and Bassingthwaighte, 2000).

Previously in the majority of perfusion validation studies, methods have been limited by the imprecise correspondence between in vivo heart segments defined from MR perfusion images and physical segments defined ex vivo upon excision of an animal heart for quantification of MBF from microspheres in standard segments (Christian et al., 2008; Hsu et al., 2012). Langendorff hearts provide experimentalists with control over many aspects of cardiac physiology in an ex vivo setting, and continue to be widely used in cardiac research (Bell et al., 2011; Schuster et al., 2012; Skrzypiec-Spring et al., 2007). Recent development of a MRI-compatible Langendorff system for imaging of isolated, perfused porcine hearts (Schuster et al., 2010) and high-resolution cryomicrotome imaging (Van Horssen et al., 2010) have allowed for greater accuracy in the comparison of perfusion quantified from MR images and microspheres (Schuster et al., 2014). Specifically epifluorescent cryomicrotome imaging has been used to visualise 15 μm diameter fluorescent microspheres, from which 3D microsphere locations within the heart can be determined (Decking et al., 2004; Van Horssen et al., 2010; Krueger et al., 2013). Perfusion computed from microspheres in cryomicrotome images was compared to perfusion quantified with algorithms from dynamic contrast-enhanced (DCE)-MRI in corresponding myocardial tissue segments following an image registration procedure (Schuster et al., 2014). This approach provides flexibility to divide the myocardial tissue in any number of ways for comparison, and offers direct correspondence of cardiac anatomy between cryomicrotome images (containing microspheres) and MR images (the clinical target).

The visualisation of whole-heart coronary arterial networks has also been possible with cryomicrotome imaging (Spaan et al., 2005) following the injection of a vascular cast into the coronary arteries. From this data the detailed 3D coronary geometry can be digitally reconstructed with appropriate algorithms to produce a computational network for analysis (Goyal et al., 2013; Van den Wijngaard et al., 2013). The simultaneous reconstruction of vascular geometry and 3D microsphere locations allows for the distribution of microspheres to be compared directly to metrics derived from the vascular network through which the microspheres have been distributed.

The distribution of microspheres through the coronary circulation is in theory proportional to blood flow, although it is also subject to random variation. To account for this variation, deposition in a given tissue region has previously been approximated using a Poisson distribution (Buckberg et al., 1971). However, studies have revealed a systematic overestimation of flow computed from microspheres in tissue regions of higher perfusion, and with preferential flow to the subendocardium over the subepicardium despite no significant difference in true underlying perfusion (Bassingthwaighte et al., 1990; Utley et al., 1974). Furthermore a distinctly non-random deposition pattern was observed from multiple samples of microspheres sequentially injected into the same organ, where microspheres were found to deposit in single voxels with a significantly higher than expected frequency, in clear contrast to areas of the tissue devoid of microspheres (Decking et al., 2004). These effects have been hypothesised to be due to skimming of microspheres — a tendency for microspheres (as well as other particles) to preferentially enter a daughter vessel receiving a higher flow fraction at a bifurcation. Skimming becomes increasingly pronounced when flow distribution to daughter vessels at a bifurcation becomes more asymmetric, when the diameter of the particles approaches the diameter of the vessel, and at lower particle concentrations (Chien et al., 1985). Numerous in vivo, in vitro and in silico studies have been carried out to investigate determinants of skimming, to which the interested reader is directed (Bugliarello and Hsiao, 1964; Carr and Wickham, 1990; Chien et al., 1985; Fenton et al., 1985; Ofjord and Clausen, 1983; Pries et al., 1989; Schmid-Schönbein and Skalak, 1980). However, to date the effects of skimming on microsphere distribution at the whole-heart level and its implications for perfusion quantification have not been directly investigated.

With the desire to quantify regional intra-organ flow at increasingly high resolutions (Hsu et al., 2012; Krueger et al., 2013; Schuster et al., 2014), the implications of skimming in the coronary arteries need to be understood to circumvent systematic errors in microsphere perfusion measurements (Bassingthwaighte et al., 1990). In this study for the first time microsphere skimming is investigated in the porcine coronary arterial circulation using cryomicrotome image data of network geometry and microsphere distribution in the same heart. Understanding the relation of coronary branching structure and microsphere distribution will provide guidance for regional flow quantification studies using microspheres.

To perform the skimming analysis, the images of the coronary arterial network geometry provided by cryomicrotome imaging were first used to reconstruct a 3D network model. Poiseuille flow solved on the network using a novel terminal flow boundary condition was then used to compare the microspheres distributed throughout the circulation. A statistical approach was tailored to determine if there is a significant prevalence of microsphere skimming at bifurcations in the network. To characterise the bifurcations identified with microsphere skimming, the branching parameters of these bifurcations were compared to non-skimming bifurcations in terms of branching asymmetry parameters related to flow, branching angle and vessel radius.

Materials and methods

Experimental procedure

Data from a Large White Cross Landrace pig was used in this study, and the experiment was conducted with the approval of the UK Home Office in accordance with the UK Animals (Scientific Procedures) Act of 1986 and in compliance with the World Medical Association Declaration of Helsinki regarding the ethical conduct of research involving animals. Further experimental procedures were undertaken in accordance with the Institutional Animal Care and Use Committee (IACUC) of the University of Amsterdam. Briefly, the heart was excised and supported ex vivo in an MRI-compatible Langendorff apparatus for perfusion MR imaging in a non-pumping configuration (Schuster et al., 2012). The right coronary artery (RCA) was occluded and three sets of approximately 100,000 fluorescent 15 μm diameter microspheres (Invitrogen, The Netherlands) were injected into the patent left coronary artery (LCA) for comparison to quantitative perfusion MRI (Schuster et al., 2012). Upon completion of the MRI acquisition, the heart was perfused with adenosine-loaded (100 μg/L) phosphate-buffered saline followed by a fluorescent vascular casting agent, Batson #17 (Polysciences Europe, Eppelheim, Germany), at a pressure of 90 mm Hg. Once the cast had set the heart was suspended in 5% carboxymethylcellulose solution (Alfa Aesar, Karlsruhe, Germany) and frozen. The heart was then serially sectioned from base to apex with a cryomicrotome and imaged with fluorescent optical surface imaging with an in-plane resolution of 50 μm and slice thickness of 64 μm, providing both images of the vascular geometry and microspheres embedded in the heart. For further details of the experimental procedure see previous publications (Van Horssen et al., 2010; Schuster et al., 2012).

Vascular network model reconstruction

The resulting image stack was bilinearly interpolated to produce an isotropic voxel size of 64 μm³ and stack dimensions of 1875 × 1875 × 1280 voxels for vessel network segmentation. A pipeline of algorithms was used to produce a network of 1D cylindrical vessel elements representative of the LCA tree based on a previously reported method (Goyal et al., 2013). Briefly multi-scale vesselness filtering of the image stack (Frangi et al., 1998) was followed by binarisation of the filtered output images separately at large and small vessel scales where manual thresholds were determined based on visual examination of the output stacks. Separate filtering of large and small vessel scales allowed for the removal of non-vessel structures from the image stack, and after combining the binarised outputs of the vesselness filter a fully connected vascular network was retained using connected component analysis (Goyal et al., 2013). Vessel centre-lines were determined using distance-ordered homotopic thinning (Pudney, 1998), and vessel radius was determined at each skeleton voxel using a sphere fitting local thickness algorithm (Hildebrand and Rüegsegger, 1997). Linear cylindrical elements were then fitted to each vessel segment (i.e., between two junctions or a junction and an end-point), where a constant segment radius was approximated by taking the mean radius over each vessel segment. Finally pruning algorithms were applied to remove spurious branches as well as erroneous loop connections within the network. The minimum vessel radius in the network was 64 μm, or one voxel (which is the minimum size that could be reconstructed with reasonable accuracy), and the radius of the LCA stem was 2.8 mm.

Terminal vessel perfusion territories

The tissue mask enclosing the LCA network was manually segmented and an approach proposed in Marxen et al. (2006) was used to divide the myocardial tissue between the terminal vessels of the network. Briefly, each voxel of the myocardial segmentation was assigned to a terminal vessel end-point of the LCA network. This assignment was determined by the terminal vessel, i, to which the Euclidean distance from the voxel, X_i, divided by the terminal segment diameter, D_i, was minimal,

$$\min \left(\frac{X_i}{D_i}\right).$$ (1)

This vessel diameter weighting in Eq. (1) accounts for the expectation that larger diameter terminal vessel segments supply larger tissue volumes. The sum of all voxels assigned to a terminal vessel is referred to as the fed volume of that vessel. The microspheres deposited in each assigned region are also assumed to have passed through the associated terminal vessel. For vessel segments further upstream, the fed volume and microsphere count at downstream vessel segments are progressively summed upward at each junction. Fig. 1 illustrates the myocardial tissue supplied by the LCA divided between the terminal vessels of the reconstructed LCA network.

Fig. 1.

LCA vasculature, microspheres and fed volumes. A 100-slice cross-sectional maximum intensity projection in the base-to-apex direction shows the vasculature (red), microspheres (green) and myocardial segmentation boundary (grey) in the left image. The right image shows an average intensity projection (AIP) in the same slices of the fed volume territories assigned to each of the terminal vessels in the network. Note that the colours of the AIP do not represent any parameter, but rather were chosen to clearly distinguish neighbouring fed volume territories.

Flow derived from the number of fluorescent microspheres passing through a segment, i, is given by

$$Q_f^i=\frac{N_f^i}{N_t}{Q}_t$$ (2)

where $N_f^i$ denotes the number of microspheres passing through segment i, N_t denotes the total number of microspheres injected, and Q_t is the total network inlet flow. Note that while the above fed-volume assignment criterion is not based strictly on a physiologically-derived relationship, it provides an approximation which becomes increasingly accurate for larger (upstream) segments as the exact location of perfusion zone boundaries become relatively less significant.

Flow model

A Poiseuille flow model was used to obtain flow and pressure values throughout the network, an approach that has been used previously for arterial network flow studies (Huo et al., 2009; Hyde et al., 2013; Marxen et al., 2006). Given that the Langendorff-supported ex vivo heart was non-pumping and perfused with a continuous blood flow to the coronary circulation, steady-state Poiseuille flow is considered a suitable model. Briefly, the relation between flow and pressure is given by

$$Q_m=\text{Δ}p\cdot C$$ (3)

where the model flow, Q_m, through a vessel segment is related to the pressure drop across it, Δp, multiplied by the vessel conductance:

$$C=\frac{\pi \cdot r^4}{8\cdot \eta \cdot L}$$ (4)

where L is the segment length, r is the segment radius and η is the dynamic blood viscosity. To account for the Fahraeus–Lindqvist effect of varying viscosity in vessels less than 300 μm in diameter the empirical equation proposed by Pries et al. (1992) was applied,

$${\eta }_{0.45}=220e^{-1.3D}+3.2-2.44e^{-0.06D^{0.645}}$$ (5)

where D is vessel diameter in micrometres and η _0.45 is the relative viscosity for a fixed discharge haematocrit when H_D = 0.45, which is assumed in our model such that η _0.45 in Eq. (5) can be substituted for η in Eq. (4). The following linear system of equations was formed

$$CP=B$$ (6)

where for a network with N nodes, C is a N × N matrix containing the mapped vessel segment conductance values relating nodal pressures in P(N × 1) to nodal flows (or pressures) in the boundary vector B(N × 1). Mass conservation at junctions was imposed with entries of 0 in rows corresponding to internal nodes in B. Inlet or outlet flow values could be set directly in B, or if setting a boundary node pressure in B the corresponding row in C was set to 0, and the diagonal set to 1. P was solved for by inverting C, which in turn allowed for vessel segment flows to be computed using Eq. (3).

Boundary conditions

In order to determine the prevalence of microsphere skimming in the coronary arterial circulation, the flow model needs to be as accurate as possible to provide confidence for a comparison with the distribution of microspheres. An ongoing challenge in the field of truncated arterial network flow modelling is the determination of accurate terminal vessel boundary conditions (BCs). Several approaches have been proposed including deriving terminal resistance to the capillary level from porcine coronary morphometric data (Kassab et al., 1993), using a fitted empirical relation of vessel diameter and pressure from measurements in cat hearts (Chilian et al., 1989; Hyde et al., 2014) and more recently using a relation between supplied tissue volume and vessel resistance to estimate downstream resistance in a rat kidney model (Marxen et al., 2009; Marxen et al., 2006). The present study lends itself to analysis using the volume-scaled resistance BC, which also provides a specimen-specific approach where the volume-resistance relation is derived directly from the vascular network being analysed as detailed below. An additional BC based on the assumption of homogeneous perfusion is also proposed for comparison, the advantages of which are given below and in the Discussion section.

Volume-scaled resistance BC

The volume-scaled resistance BC was introduced by Marxen who proposed a scaling law between vessel conductance and volume order to approximate downstream vessel resistance at each terminal of a network (Marxen, 2004). Briefly, vessel segments are assigned a volume order, U_vol, based on the supplied downstream tissue fed volume, V,

$$U_{vol}={\log }_2\left(\frac{V}{V_0}\right)+1$$ (7)

where V ₀ is the approximate mean volume of tissue supplied by a terminal arteriole of the particular species and organ, i.e., a unit tissue volume containing capillaries supplied by a single terminal arteriole, at which scale a constant perfusion pressure of approximately 25 mm Hg can be assumed. Linear least-squares regression was used to determine a scaling law between volume order, U_vol, and vessel conductance, C, of those vessels in the specimen organ (Marxen et al., 2009). Slope, q, and intercept, L, parameters were estimated from fitting Eq. (8) to the data

$${\log }_2(\text{C})=L+q\cdot U_{vol}.$$ (8)

An additional resistance was added at each terminal vessel end-point based on an extrapolation of this law, where resistance at a given volume order was assumed to be normally distributed with a standard deviation based on the terminal vessels of the existing network, and that asymmetry of the generated daughter vessel volumes was also normally distributed at each generation (for full details see Marxen et al., 2009).

Homogeneous perfusion BC

An outlet BC was also proposed where terminal outlet flows for the Poiseuille simulation were set proportional to the volume fraction of the supplied tissue,

$$Q_k=Q_t\frac{V_k}{V_t}$$ (9)

where Q_k is the outlet flow at terminal vessel k, Q_t is the total flow into the network, V_k is the fed volume of vessel k, and V is the total volume of the LCA tissue segmentation. This BC entails a homogeneous perfusion throughout the myocardial tissue, which while not physiologically realistic at the micro-scale, provides increasingly accurate flows in the higher generations of the network (see the Discussion section for the rationale).

Error metrics

A simple percentage difference between Q_f and Q_m in a vessel segment i is given by

$$E^i=\sqrt{{\left(\frac{Q_f^i-Q_m^i}{Q_m^i}\right)}^2}\times 100.$$ (10)

Vessel segments were also assigned a volume order U ₂ which is given by

$$U_2={\log }_{10}(V).$$ (11)

The range of fed volumes in the LCA network span between 10⁰–10⁵ mm³, and U ₂ spans between 0 and 5. Vessel segments were grouped into 20 bins, w_j, where the U ₂ volume orders of bin w_j are given by

$${\tilde{U}}_2(w_j)-0.125<U_2(w_j)<{\tilde{U}}_2(w_j)+0.125$$ (12)

where Ũ ₂(w_j) is the central value of U ₂ in bin W_j. To visualise the difference between Q_f and Q_m across vessel generations, a root mean squared error (RMSE) is computed for all vessel segments in each fed volume bin w_j. The RMSE is computed from the individual errors, $E_j^i$, of each vessel segment i whose fed volume falls within bin w_j, given by

$$E_j^{rms}=\sqrt{\frac{\sum _i^{n_j}{\left(E_j^i\right)}^2}{}{n}_j}$$ (13)

where n_j is the number of vessel segments in bin j. It is expected that E^rms should decrease in higher volume bins as (i) the number of microspheres passing through vessels increases and hence error due to random variation decreases and (ii) as error in the flow model decreases (for the homogeneous perfusion BC) since larger tissue fed volumes are associated with lower perfusion heterogeneity.

Terminal perfusion perturbation analysis

Physiologically whole-heart perfusion heterogeneity in terms of relative dispersion (RD) has a fractal relationship with tissue volume size (Bassingthwaighte et al., 1989). The aetiology of this heterogeneity has yet to be elucidated, although hypotheses of origins in metabolism differences, electrical activation time and regional cardiac workload have been proposed (Pries and Secomb, 2009). To determine the effect of introducing perfusion heterogeneity at the terminal vessels on the Poiseuille network flow solution when using the homogeneous perfusion BC, a perturbation analysis was performed using a Monte Carlo approach. Keeping terminal fed volumes constant, random perturbations in outlet flow were introduced producing heterogeneity in outlet perfusion, while conserving total flow throughout the network. Specifically flow was perturbed at each terminal vessel, k, according to

$$\begin{array}{ll}{Q}_k^{\prime }=0\hfill & \text{if}{Q}_k+X_k\tilde{Q}\le 0\hfill \\ Q_k^{\prime }=Q_k+X_k\tilde{Q}\hfill & \text{otherwise}\hfill \end{array}$$ (14)

where Q_k′ and Q_k are the perturbed flow and original flow at terminal k respectively, and $\tilde{Q}$ is the median original terminal flow (which is explained further below). X_k is a random variable sampled from a normal distribution with standard deviation σ and a mean of 0. The value σ is chosen to reflect different levels of perfusion heterogeneity in the terminal vessels, and is set to 20%, 40%, 60% and 80% for comparison. The sum of the perturbed flows at each terminal is then corrected in order to match the sum of the original (unperturbed) flows, Q_t,

$${\widehat{Q}}_k^{\prime }=Q_k^{\prime }\left(\frac{Q_t}{Q_t^{\prime }}\right)$$ (15)

where ${\widehat{Q}}_k^{\prime }$ is the corrected, perturbed flow at terminal k, and ${\widehat{Q}}_t^{\prime }$ is the sum of all perturbed flows from Eq. (18). The median terminal vessel flow $\tilde{Q}$ in the network is used since the unperturbed terminal flows Q_k are non-uniform and span several orders of magnitude (as do the terminal fed volumes to which they are proportional). It is established that perfusion heterogeneity decreases with an increase in the size of tissue segments being compared (Bassingthwaighte et al., 1989). This entails that terminal vessels with larger flows (and volumes) should be perturbed by a smaller fraction than those with lower flows (and volumes) in order to produce the appropriate perfusion heterogeneity. Using $\tilde{Q}$ is a simplistic approach for representing this dynamic.

Levels of perfusion heterogeneity at the outlet vessels in terms of σ were set to 20%, 40%, 60% and 80%, where at each level of introduced perfusion heterogeneity a thousand trials were performed. The maximum and minimum RMS errors computed from the 1000 trials in each volume bin w_j are represented by bounds superimposed on the plot of $E_j^{rms}$ versus w_j, denoted by ${\widehat{E}}_j^{rms}$. These bounds indicate a maximal deviation of the flow error from the homogeneous perfusion BC solution for each level of perfusion heterogeneity introduced at the terminal vessels and indicate the effect on the network flow across different volume orders.

Differences in mean perfusion of sub-networks

A further consideration in our analysis is accounting for the observation that perfusion is known to vary in specific regions of the heart, namely the right ventricular (RV) free wall receives a lower mean perfusion than the left ventricular (LV) free wall (Bassingthwaighte et al., 1990), as does the septal wall as demonstrated under stress testing in humans (Muehling et al., 2004). In the reconstructed LCA network, vessel stems were identified with crowns supplying these distinct regions and perfusion was normalised independently for each crown, such that Q_f and Q_m were equal at the stem of each of the subnetworks (see Fig. 2). LCA vessels supplying the RV were few and were removed from the analysis, such that only vessels supplying the LV free wall and the septum were used, as shown in Fig. 2.

Fig. 2.

The LCA sub-networks and their perfused territories. The three highlighted sub-networks in the LCA consist of the LAD (in gold), the LM (in green) and the LCx (in red) arteries. A cross-sectional MIP in the base-to-apex direction (right), indicated by the grey box (left), shows the correspondence between regions in the myocardial segmentation and the perfusion territories of the sub-networks. In the cross-section the LAD artery supplies the septal wall, while the LM and LCx arteries supply the LV free wall. Given that perfusion is typically different in the septal wall compared to the LV free wall, microsphere-derived flow Q_f is computed from Eq. (2) using the values of Q and N_t at the inlet of each sub-network individually to account for this.

Statistical analysis of microsphere skimming

Previous studies into skimming of microspheres and blood cells have typically involved the investigation of flow parameters of isolated junctions (Chien et al., 1985; Dellimore et al., 1983; Fenton et al., 1985; Ofjord and Clausen, 1983; Ofjord, 1981; Schmid-Schönbein and Skalak, 1980). This approach provided the advantage of adjusting flow parameters to assess their importance, where statistical inferences could be made about the distribution of particles at a specific junction after observing a sufficient number of particles passing through to account for random variation. No study to-date however has related microsphere skimming to coronary network architecture on a whole-organ scale. In the present study a detailed LCA network is available for analysis, although only a snapshot of particle distribution is captured — i.e., there is a fixed number of microspheres distributed throughout the LCA network meaning fewer particles have passed through bifurcations further downstream, which in turn limits the confidence of statistical inferences that can be drawn from downstream bifurcations.

To characterise the prevalence of microsphere skimming in the LCA, a method is proposed to analyse microsphere distribution at network bifurcations using binomial distribution Clopper–Pearson confidence intervals. Only a subset of the network bifurcations can be assessed based on the observability of skimming at the bifurcation. We define observability as referring to whether the microsphere distribution to each daughter vessel at a given bifurcation lends itself to a hypothesis test for whether the microsphere distribution differs significantly from the Poiseuille model flow distribution using confidence intervals.

The underlying assumption of microsphere distribution for the purposes of flow quantification is that they distribute in proportion to blood flow. This assumption can also be extended to bifurcations in the network, where a hypothesis test can be performed to determine at which bifurcations the model and microsphere flows deviate significantly, and whether the number of bifurcations that this occurs at is significantly greater than expected. The distribution of microspheres entering a daughter vessel from an upstream bifurcation can be considered as a random variable, X, from a binomial distribution,

$$X\sim B(N,p)$$ (16)

where N is the number of microspheres passing through the upstream bifurcation, and p is the probability of microspheres entering the daughter vessel. Let F* denote the fraction of microspheres passing through the bifurcation entering a daughter vessel, and similarly let Q* denote the fraction of Poiseuille flow. The null hypothesis is that microspheres distribute in proportion to blood flow. Thus let p be given by Q* in Eq. (16), then a hypothesis test can be formulated to identify significant deviations of F* from Q* at bifurcations. Specifically skimming can be characterised as the case where a daughter vessel receiving a high flow fraction (Q* > 0.5) consistently tends to receive an even higher fraction of microspheres (F* > Q*), and as a corollary a daughter vessel receiving a low flow fraction (Q* < 0.5) tends to receive an even lower fraction of microspheres (F* < Q*). Clopper-Pearson confidence intervals, a conservative method for hypothesis testing with binomial distributions, can be used to test the null hypothesis that there is no significant difference between F* and Q* at a bifurcation,

$$H_0:F^{\ast }=Q^{\ast }.$$ (17)

And the alternative hypotheses $H_1^a$ and $H_1^b$,

$$\begin{array}{llll}\hfill & H_1^a:F^{\ast }<Q^{\ast }\hfill & \text{for}\hfill & Q^{\ast }<0.5\hfill \\ \text{or}\text{equivalently}\hfill & H_1^a:F^{\ast }>Q^{\ast }\hfill & \text{for}\hfill & Q^{\ast }>0.5\hfill \end{array}$$ (18)

$$\begin{array}{llll}\hfill & H_1^b:F^{\ast }<Q^{\ast }\hfill & \text{for}\hfill & Q^{\ast }>0.5\hfill \\ \text{or}\text{equivalently}\hfill & H_1^b:F^{\ast }>Q^{\ast }\hfill & \text{for}\hfill & Q^{\ast }<0.5\hfill \end{array}$$ (19)

where the alternative hypothesis $H_1^a$ is a test for classical skimming, and $H_1^b$ acts as a control for the opposite effect to what is expected, i.e., if F* significantly underestimates Q* when Q* > 0.5 (and the converse). Since flow and microsphere distribution fractions sum up to 1 at each bifurcation, the two forms of each alternative hypothesis in Eqs. (18) and (19) will have the same outcome at complimentary daughter vessels branching from each bifurcation.

Coming back to the concept of observability, the choice of confidence level (CL) for the tests above determines the subset of bifurcations which can be used in the analysis, as shown later in Fig. 6. For a given two-tailed confidence interval with a coverage of (1 – α), the alternative hypothesis $H_1^a$ is true $\left(H_1^a=1\right)$ for a daughter vessel if

$$\begin{array}{llll}\hfill & \mathrm{Pr}(X<k)<\alpha /2\hfill & \text{for}\hfill & Q^{\ast }<0.5,F^{\ast }<Q^{\ast }\hfill \\ \text{or}\text{equivalently}\hfill & \mathrm{Pr}(X\ge k)<\alpha /2\hfill & \text{for}\hfill & Q^{\ast }>0.5,F^{\ast }>Q^{\ast }\hfill \end{array}$$ (20)

and similarly for $H_1^b$. The cumulative probability of equal to or fewer than k microspheres entering a daughter vessel, Pr(X ≥ k), is given by the binomial cumulative distribution function (CDF),

$$F(k;N,p)=\text{Pr}(X\le k)={\displaystyle \sum _{i=0}^k\left(\frac{N}{i}\right)p^i{(1-p)}^{N-i}.}$$ (21)

Fig. 6.

Flow fractions and microsphere counts for viable vessels. All daughter vessels of bifurcations characterised in terms of received flow fraction, Q*, and number of microspheres passing through the bifurcation, N. Daughter vessels belonging to bifurcations which are suitable for skimming analysis are marked with crosses, where lower confidence levels (e.g., 60%) include all such daughter vessels from higher confidence levels (e.g., 90%), but non-overlapping viable bifurcations between consecutive confidence levels (e.g., 60% versus 80%) are shown in distinct colours.

The determinants for the observability of a bifurcation are (i) the desired CL (or α/2), (ii) the number of microspheres, N, entering the junction and (iii) the flow fraction to each daughter vessel, Q*. Given a value of α and N, there is a range of values (p _min ≤ Q* ≤ p _max) within which observability is satisfied making a bifurcation amenable for testing with the above hypothesis tests. The bounds of this range are given by the CDF for the most extreme outcomes (i.e., the probability of 0 microspheres or N microspheres entering a daughter vessel),

$$\begin{array}{l}\mathrm{Pr}(X\le 0)=\mathrm{Pr}(X=0)=p_{\min }^0{\left(1-p_{\min }\right)}^N=\alpha /2\hfill \\ \Rightarrow p_{\min }=1-{(\alpha /2)}^{1/N}\hfill \end{array}$$ (22)

$$\begin{array}{l}\mathrm{Pr}(X\ge N)=\mathrm{Pr}(X=N)=p_{\max }^N{\left(1-p_{\max }\right)}^0=\alpha /2\hfill \\ \Rightarrow p_{\max }={(\alpha /2)}^{1/N}.\hfill \end{array}$$ (23)

For a vessel segment in which the received flow fraction lies outside of the range ((p _min ≤ Q* ≤ p _max)), even the most extreme outcome k would not result in an outlier (where $\left(H_1^a=1\right)$), thus skewing the results of the analysis since such a vessel could only be classified as normal. The proportion of outlier bifurcations out of the total number of assessed bifurcations is used to determine if skimming is prevalent. If the distribution of microspheres at all assessed bifurcations were representative of the Poiseuille flow with random variation, then the expected proportion of outliers would be equal to α/2.

The assessed bifurcations are also constrained in terms of vessel generation to determine if there is a major difference in proportion of outliers at different vessel scales (e.g., to compare skimming prevalence in the whole network to just the epicardial arteries). A method for achieving this is to consider bifurcations which have a minimum number of microspheres passing through them, N _min, for a range of values from 5 to 640. Fig. 3 shows the corresponding vasculature retained for analysis at values of N _min between 5 and 320. Additionally the homogeneous perfusion BC becomes increasingly accurate for vessels with larger values of N _min since they supply larger fed volumes.

Fig. 3.

LCA network constrained by N _min. The LCA network constrained to vessels (in red) through which a minimum number of microspheres, N _min, have entered. N _min = 320 corresponds to the epicardial and penetrating transmural arteries, while N _min = 5 contains many of the intramural arterioles.

Finally outlier bifurcations were compared to non-outlier bifurcations in terms of branching asymmetry parameters, which have previously been considered as determinants of skimming. The Mann-Whitney U-test was used to determine if there was a significant difference in branching parameters associated with bifurcations in outlier compared to non-outlier samples since samples were not normally distributed as confirmed with the Kolmogorov-Smirnov Test. Specifically samples included:

$$\begin{array}{l}{\text{S}}_1:H_1^a\text{outliers}\text{where}(Q*<0.5,F*<Q*);\\ {\text{S}}_2:H_1^a\text{outliers}\text{where}(Q*>0.5,F*>Q*);\\ {\text{S}}_3:H_1^b\text{outliers}\text{where}(Q*<0.5,F*>Q*);\\ {\text{S}}_4:H_1^b\text{outliers}\text{where}(Q*>0.5,F*<Q*);\\ {\text{S}}_5:\text{All}\text{non-outlier}\text{bifurcations}.\end{array}$$

The branching parameters which were compared included

$$Q^{\ast }=\frac{Q_1}{Q_1+Q_2}$$ (24)

$${\theta }^{\ast }=\frac{{\theta }_1}{{\theta }_1+{\theta }_2}$$ (25)

$$r_a^{\ast }=\frac{r_1}{r_1+r_2}$$ (26)

$$r_b^{\ast }=\frac{r_1}{r_p}$$ (27)

where subscripts 1 and 2 refer to the current daughter vessel and its complementary daughter vessel at a bifurcation respectively. Parameter θ_i, is the angle between the centrelines of the parent and daughter vessel i. Parameter $r_a^{\ast }$ is the ratio of a daughter vessel’s radius to the summed daughter vessel radii at a bifurcation, and $r_b^{\ast }$ is the daughter-to-parent radius ratio. Values of the asymmetry parameters Q*, θ* and $r_a^{\ast }$ equal to 0.5 indicate symmetric branching with respect to flow, branching angle and daughter radii.

Results

Coronary flow model

Fig. 4 shows the difference between normalised microsphere flow (Q_f/Q_t) and normalised model flow (Q _m/Q _t) in each vessel segment of the network for solutions using (i) the homogeneous perfusion BC and (ii) the volume–resistance scaling law BC. The solution from (i) produces much smaller errors between Q_f and Q_m compared to (ii), where the solution from (ii) contains a large number of vessel segments in which Q_m is several orders of magnitude lower than Q_f particularly in the lower generation segments. The possible reasons for this are explored in the Discussion section, but in the remainder of the Results section the solution from (i) is used.

Fig. 4.

Normalised microsphere versus model flows. Microsphere flow (Q_f) versus Poiseuille flow (Q_m) normalised by inlet flow for each sub-network (Q_t) for the homogeneous BC (left) and volume-scaled resistance BC (right). Each point represents a value in a single vessel segment and the dashed line is unity. There are a large number of vessel segments with a value of Q_m orders of magnitude lower than that of Q_f in the volume-scaled resistance BC flow solution (right), deviating much more from unity than the homogeneous perfusion BC flow solution (left).

Plotting error $E_j^{rms}$ against fed volume reveals several interesting features, shown in Fig. 5. Firstly, the relation between $E_j^{rms}$ and fed volume is monotonically decreasing, which is expected since heterogeneity of perfusion decreases between larger tissue volumes, which are associated with higher generation vessels. Secondly for all levels of introduced random perfusion heterogeneity at the terminal vessels (RD = 20%, 40%, 60% and 80%), the maximum deviation of mean flow errors from the solution with a homogeneous perfusion BC is no greater than approximately 2% in vessels with fed volumes greater than 100 mm³ (0.1 g) as shown in Fig. 5. This shows that the network flow solution produced using the homogeneous perfusion BC is not prone to large errors in terms of vessel flows, particularly in the higher generation vessels where there is also a higher confidence in microsphere distribution given a higher number passing through bifurcations, which is important for the skimming analysis.

Fig. 5.

Flow error versus vessel fed volume generation. The relation of $E_j^{rms}$ and ${\widehat{E}}_j^{rms}$ with fed volume (V_j) on a logarithmic scale (left), with bins (w_j) numbered on the upper x-axis for reference, and the difference $E_j^{rms}-{\widehat{E}}_j^{rms}$ plotted against V_j (right). The red line (left) connects values of $E_j^{rms}$ in each bin, indicating a monotonically decreasing relation (with the exception of bin w ₁₅) with fed volume. The transparent areas enclosed by dashed lines correspond to the bounds of the maximum and minimum errors, ${\widehat{E}}_j^{rms}$, computed from vessel segments in each bin from the 1000 trials of introduced random heterogeneity at the terminal vessel (at σ = 20%, 40%, 60%, and 80%). The right figure illustrates that for vessel segments in bins w ₈ and above, ${\widehat{E}}_j^{rms}$ does not deviate from $E_j^{rms}$ by more than approximately 2% for all levels of introduced perfusion heterogeneity, deviating less at higher vessel generations.

Statistical analysis of microsphere skimming

Confidence levels of 60%, 80%, 90%, 95% and 99% were used for the skimming analysis using the Clopper–Pearson confidence intervals described above. Table 1 shows the number of bifurcations retained for the analysis (from a total of 39,261 bifurcations in the network) after testing for observability with Eqs. (22) and (23). Additionally, the daughter vessels corresponding to these bifurcations are visualised in terms of flow fraction, Q*, and number of microspheres passing through the upstream bifurcation, N, in Fig. 6.

Table 1. Viable bifurcations for skimming analysis at different confidence levels.

Confidence level (%)	α/2 (%)	No. viable bifurcations
60	20	10,713
80	10	7982
90	5	6344
95	2.5	5228
99	0.5	3788

For simplicity, the results of the skimming analysis will be presented for a confidence level of 90%, as results were similar for all confidence levels (indicating that a reduction of sample size from a lower CL to a higher CL did not make a significant difference). When considering all 6344 viable bifurcations at the 90% CL, the test for classical skimming (alternative hypothesis $H_1^a$) produced a proportion of outlier bifurcations of 11.8%, where only 5% (α/2) was expected. The control test for unexpected outliers (alternative hypothesis $H_1^b$) produced a proportion of outlier bifurcations of 8.14%.

To analyse how this changes at higher vessel scales, bifurcations included in the analysis are restricted by a minimum number of microspheres passing through them, N _min, at values between 5 and 640. Fig. 7 shows the proportion of outliers for the two alternative hypotheses $H_1^a$ and $H_1^b$ as a function of N _min, as well as mean parent and daughter vessel radii for bifurcations with a given range of N _min. The proportion of outliers for the classical skimming alternative hypothesis $H_1^a$ increases almost linearly with increasing values of log₂(N _min), reaching 31% of all bifurcations through which at least 320 microspheres pass, while $H_1^b$ is effectively unchanged. The mean radius of the parent vessel of a bifurcation with 320 < N _min < 640 is 0.51 mm, at which scale approximately 30% of bifurcations exhibit classical skimming. For bifurcations with 5 < N _min < 10 the mean parent vessel radius is 0.12 mm with approximately 10% of bifurcations exhibiting classical skimming. The mean daughter vessel radius for the bifurcations with 320 < N _min < 640 is 0.31 mm, and 0.10 mm for bifurcations with 5 < N _min < 10. Fig. 8 shows the vascular network retained for the skimming analysis at N _min = 5 and 320, with vessel segments colour-coded to show the $H_1^a$ outlier vessels. It is evident that many vessel segments along the major epicardial vessels and transmural vessels, which receive higher flow fractions at bifurcations, are also receiving a significantly higher fraction of microspheres than expected (Fig. 8, vessel segments in red). As a corollary, the smaller side branches receiving lower flow fractions at these bifurcations are also receiving a lower than expected fraction of microspheres.

Fig. 7.

Outlier proportion and mean radius versus N _min. (Above) the change in the proportion of $H_1^a$ (solid fitted line) and $H_1^b$ (dashed fitted line) outlier bifurcations with N _min between 5 and 640, at a 90% confidence level (α/2 = 0.05). The proportion of $H_1^a$ outliers increases linearly with N _min while $H_1^b$ remains effectively unchanged, indicating an increase in the prevalence of classical skimming sites at higher vessel generations. Vertical dashed lines are drawn at N _min values of 5 and 320, corresponding to the 3D LCA networks shown with marked skimming sites in Fig. 8. (Below) the mean radius of parent vessels (solid line) and daughter vessels (dashed line) of bifurcations plotted in bins of N _min whose bounds are defined by the points in the above plot.

Fig. 8.

Skimming bifurcations in the LCA network. A long-axis anterior view of the LCA network (left) constrained with N _min values of 5 (top) and 320 (bottom), with a corresponding short-axis cross-section of the vasculature enclosed in the dashed box (right). The bottom panel illustrates how a value of N _min = 320 corresponds to the epicardial vessels, whereas a value of N _min = 5 includes many of the intramural arterioles. The red (high flow fraction) vessel segments correspond to $H_1^a$ outliers where (Q* > 0.5, F* > Q*) (or sample S2), and the blue (low flow fraction) vessel segments correspond to the complementary daughter vessel $H_1^a$ outliers where (Q* < 0.5, F* < Q*) (or sample S1). The green vessel segments consist of non-outlier bifurcations (sample S5) and the $H_1^b$ outliers (samples S3 and S4), the grey transparent vessels were excluded from the analysis, and the infrequent beige segments correspond to trifurcations. Red vessel segments are noticeably more frequent along the large epicardial vessels, and blue segments mainly correspond to the stems of the smaller side branches.

The means and standard deviations (SD) of the branching parameter values for each of the samples S1, S2, S3, S4 and S5 are listed in Table 2 for all bifurcations which satisfied the observability constraint at a CL of 90%. The Mann–Whitney U-test revealed a significant difference (p < 0.01) in the sample median between every pair of samples for the parameters Q*, θ* and $r_a^{\ast }$. Parameters Q* and $r_a^{\ast }$ were significantly lower in sample S1 than any other sample, and higher in sample S2 than any other sample. Parameter θ* was significantly higher in S1 than any other sample, and significantly lower in S2 than any other sample. Parameter $r_b^{\ast }$ was significantly higher for samples S2 than all other samples except S4.

Table 2. Mean ± SD of branching parameters in bifurcation samples.

Parameter	S1	S2	S3	S4	S5
No. vessels	741	741	512	512	10,182
Q*	0.22 ± 0.16	0.78 ± 0.16	0.29 ± 0.14	0.71 ± 0.14	0.50 ± 0.26
θ*	0.67 ± 0.21	0.33 ± 0.21	0.60 ± 0.21	0.40 ± 0.21	0.50 ± 0.24
$r_a^{\ast }$	0.38 ± 0.14	0.62 ± 0.14	0.44 ± 0.10	0.56 ± 0.10	0.50 ± 0.13
$r_b^{\ast }$	0.60 ± 0.29	0.92 ± 0.14	0.74 ± 0.24	0.90 ± 0.15	0.81 ± 0.22

Discussion

The idea of the homogeneous perfusion BC is based on the findings of Bassingthwaighte et al. (1989) who demonstrated a fractal relationship between tissue segment size and perfusion heterogeneity, and Van Beek et al. (1989) who demonstrated a local correlation of perfusion. Specifically Bassingthwaighte et al. showed that the larger the segments of tissue that the LV was divided into, the lower the perfusion heterogeneity observed between them. Van Beek et al. showed that perfusion heterogeneity throughout the myocardium was random except for a weak correlation of perfusion in neighbouring tissue segments. While several network flow modelling studies have been able to reproduce these whole-organ metrics, for example Beard and Bassingthwaighte (2000), Huo et al. (2009), Marxen et al. (2006) and Mittal et al. (2005), these studies were limited by a lack of physiological data for determining suitable terminal vessel boundary conditions and do not necessarily accurately represent local flows. The sole study in which model perfusion predictions were compared to microsphere data demonstrated that there was no correlation between the predictions and data and highlighted the need for further investigation into more accurate boundary conditions (Marxen et al., 2006). The homogeneous perfusion BC essentially sets the mean perfusion of the myocardial tissue at all terminal vessels (distinguishing between the LV septum and free wall), and provides confidence in the model at higher vessel generations since their associated tissue fed volumes are larger and thus perfusion is less heterogeneous, converging towards the prescribed mean perfusion of the myocardium (as illustrated in the left panel of Fig. 4).

The proposed homogeneous perfusion BC yielded a network flow simulation that was able to produce a monotonically decreasing mean error compared to microsphere distribution as a function of vessel volume order (Fig. 5). Additionally, adding random physiological level perfusion heterogeneity at the terminal vessels demonstrated that errors are minimal at higher vessel generations. The homogeneous perfusion BC simulation produced smaller differences compared to microsphere distribution than the volume-scaled resistance BC simulation, most noticeably at lower vessel generations (Fig. 4). Furthermore a physiologically realistic fractal relationship of global perfusion heterogeneity with fed volume was produced from the perturbation analysis (Fig. 9), where from Bassingthwaighte’s experiments using microspheres injected into the coronary arteries of baboon hearts the values of the fitted fractal dimension D ranged between approximately 1.15 and 1.30 (Van Beek et al., 1989). Reproducing this value has previously been used as an indicator of physiological validity for network flow modelling studies (Van Bavel and Spaan, 1992; Huo et al., 2009; Kassab et al., 1997; Smith et al., 2002). The present study demonstrates that the introduction of random perfusion heterogeneity at the network terminal vessels at which perfusion is otherwise considered homogeneous can replicate the same metric, while producing mean segment flow errors which monotonically decrease at higher vessel generations.

Fig. 9.

Displayed with solid lines are the linear least-squares fits to log(RD) of perfusion versus log(V) from vessel segments binned by their fed volume (centred around x-axis positions of data points), with terminal vessel perfusion heterogeneity introduced at levels of 20%, 40%, 60% and 80%. With an increase in terminal heterogeneity level, the fitted fractal dimension D decreases (as shown in the legend) towards a physiological range (Bassingthwaighte et al., 1989). The RD of perfusion computed from the microspheres (FMS) in the same volume bins is higher at all fed volumes than the model, indicated by the red line.

This plot is produced using the ‘model method’ depicted in Fig. 5 in Marxen et al. (2006).

A possible reason for the high error between Q_m and Q_f for the volume-scaled resistance BC is the dependence of the added downstream resistance at each terminal on fed volume. Fed volume is in turn dependent on terminal vessel radius and spatial density as discussed in the Methods section. Such high errors were not observed in a recent kidney model when compared to microspheres (Marxen et al., 2006). Even small errors in terminal vessel radius estimation would have impacted on the allocated terminal fed volumes, which may have compounded with errors owing to the fact that there is transmural variation in vessel volume density in the heart increasing towards the subendocardium (Van Horssen et al., 2014), which is not present in the kidneys. The result was a large number of very low resistance paths in the network, resulting in many more vessels with far lower values of Q_m relative to Q_f, as shown in Fig. 4.

Considering the skimming analysis, our results clearly show that there was a significantly higher than expected proportion of outlier bifurcations exhibiting classical skimming. Moreover, skimming appears to become more prevalent when the network under consideration is constrained to higher generations of the network with evidence of microspheres having preference to remain travelling along the large epicardial coronary arteries instead of entering smaller side branches. Previous studies have identified that flow fraction Q* is the greatest determinant of skimming, while branching angle θ* and daughter-daughter radius ratio $r_a^{\ast }$ do not play a major role, and daughter-to-parent radius ratio $r_b^{\ast }$ affects skimming due to its impact on the shape of dividing streamlines (Chien et al., 1985; Ofjord and Clausen, 1983; Palmer, 1965; Pries et al., 1989; Schmid-Schönbein and Skalak, 1980). Skimming has been shown to increase as particle diameter approaches vessel diameter (Chien et al., 1985; Fenton et al., 1985; Ofjord and Clausen, 1983), and as the concentration of particles decreases (Bugliarello and Hsiao, 1964; Fenton et al., 1985; Pries et al., 1989). Both of these observations are related to the axial streaming (or lateral migration) phenomenon (Segré and Silberberg, 1962; Taylor, 1955) and wall exclusion effect (Ofjord and Clausen, 1983; Ofjord, 1981), resulting in a higher concentration of suspended particles towards the centre of a vessel.

A factor which may contribute to the observed skimming of microspheres in the larger coronary arteries is a change in the fluid separation surface at bifurcations at higher Reynolds numbers. The Reynolds numbers exhibited in the coronary circulation reach approximately 100 in the largest coronary arteries (Kassab et al., 1997). According to the experimental results reported in Carr and Kotha (1995) and numerical results of Enden and Popel (1992), the separation surface is significantly altered at Reynolds numbers above 55 from a concave to a convex shape thus excluding more of the centre of the parent vessel. This dynamics, in turn, would be likely to increase the extent of skimming of RBCs in vessels with high Reynolds number flows, due to the lateral migration of RBCs (Taylor, 1955), and similarly microspheres (Ofjord and Clausen, 1983). Furthermore “inertia enhanced” skimming has been proposed as the origin of varying haematocrit values between different arteries branching from the aortic arch (Mchedlishvili and Varazashvili, 1986), which could also possibly contribute to skimming in the largest coronary arteries.

Another potential factor which could affect skimming is margination, where micro-particles suspended in blood have been observed to migrate towards the vessel wall while leaving a core of RBCs in the centre of the vessel. This phenomenon has been observed for platelets both experimentally (Ofjord and Clausen, 1986; Tangelder et al., 1986) and numerically (Tokarev et al., 2011; Vahidkhah and Bagchi, 2015) as well as for micro- and nano-particles (Gentile et al., 2008; Namdee and Thompson, 2013; Vahidkhah and Bagchi, 2015), all of which however are considerably smaller than the microspheres in this study. Margination has also been observed to affect white blood cells which are closer in size to the 15 μm microspheres, although only at low shear rates in the microcirculation (Fedosov and Gompper, 2014; Fedosov et al., 2012; Goldsmith and Spain, 1984), namely in vessels much smaller than the coronary arteries in this study. Particles which undergo margination are found to have a higher concentration in small side branches compared to RBCs. A study into the deposition of RBCs, WBCs, platelets and microspheres of a range of diameters (0.3 μm–3.5 μm) in the renal circulation revealed that RBCs, WBCs and microspheres were preferentially distributed to the outer cortex, whereas platelets were found in higher concentrations in the deep cortex (Ofjord and Clausen, 1986). It was hypothesised that this was due to margination of platelets and demargination of all the other particles, despite the small size of the smallest microspheres, the cause of which has not yet been clearly identified. Demargination of the microspheres suspended in blood was primarily due to a wall-exclusion effect (as opposed to inertial effects), as demonstrated in earlier tube and slit experiments performed by the same authors (Ofjord and Clausen, 1983; Ofjord, 1981).

Another phenomenon that may be of consequence in this study is the “tubular pinch effect”, where particles migrate to an eccentric equilibrium position with a peak concentration at about 0.6 tube radii from the axis, after travelling some way along the length of a vessel (Segré and Silberberg, 1962). This effect has been characterised for Reynolds numbers lower than 30, above which particle concentrations were more even across the vessel and it took longer to reach an equilibrium position. The tubular pinch effect could possibly explain the observation in this study of a statistically significant proportion of outlier bifurcations exhibiting the opposite effect to skimming – i.e., a disproportionately high fraction of microspheres entering a lower flow side-branch. It is yet unclear how 15 μm microspheres interact with RBCs both with regard to margination as well as the tubular pinch effect, and future experimental and numerical analyses may shed some light on the significance of these phenomena on skimming under different flow regimes in the appropriate range of vessel sizes.

Another factor influencing flow throughout the network is haematocrit which affects viscosity (Eq. (5)) and in turn vessel resistance (Eq. (4)). Since the vessels in our study are greater than or equal to 128 μm in diameter, variability in haematocrit throughout the network should not play a major role. The diameter-dependent empirical model proposed by Pries et al. (1992) was used to provide an approximation for the small variation in viscosity that would be observed in the smaller vessels of the network (<300 μm in diameter). The network Fahraeus Effect (Pries et al., 1986) which addresses the effect of RBC phase separation on haematocrit variability in a network should not have a significant effect in our model since RBC skimming does not occur in vessels greater than 40 μm in diameter (Pries et al., 1989). Thus, haematocrit was assumed to be constant and viscosity was assumed to depend only on vessel diameter.

Branching asymmetry decreases from proximal to distal in vascular networks, in terms of branching angle, daughter radius ratio, fed volume ratio and flow rates (Van Bavel and Spaan, 1992; Kalsho and Kassab, 2004; Marxen, 2004). In the epicardial vessels of the coronary LCA in this study, flow fractions at bifurcations were highly asymmetric, with approximately 30% of bifurcations through which at least 320 microspheres passed having a daughter vessel receiving at least Q* = 0.9. Given that Q* has previously been identified as the primary determinant of skimming, it seems unsurprising that skimming occurs under the conditions observed in the coronary arteries.

There was a significant difference between branching asymmetry parameters of the skimming samples of bifurcations compared to the non-skimming samples. Although branching angle θ* and daughter-daughter radius ratio $r_a^{\ast }$ have previously been shown to have little mechanistic effect on skimming (Fenton et al., 1985; Pries et al., 1989), the observed high asymmetry of both parameters in the proximal vessels are strongly correlated with Q* as shown in Fig. 10. For bifurcations with N _min = 320, a sigmoid function fitted to the relation between Q* and θ* produces an R ² value of 0.81 with higher flow fractions being correlated with lower branching angles, while a sigmoid function fitted to the relation between Q* and $r_a^{\ast }$ produces an R ² value of 0.92, with higher flow fractions being correlated with larger daughter vessels. Thus while $r_a^{\ast }$ and θ* are not direct determinants of skimming, their asymmetry in the proximal circulation reflects the asymmetry of Q*.

Fig. 10.

Sigmoid correlations of branching parameters with flow fraction. Sigmoid functions indicated by the curves are fitted to data from bifurcations with N _min = 320, which provide a strong positive correlation (R ² = 0.92) of $r_a^{\ast }$ and Q* (left), and a strong negative correlation (R ² = 0.81) of θ* and Q* (right). This is intuitive as the conduit epicardial and transmural vessels which receive large flow fractions at bifurcations tend not to decrease in diameter or deviate in direction much from the parent vessel compared to smaller side branches, which supply the more proximally located myocardium.

The identification of microsphere skimming with the Poiseuille flow model suggests that Bassingthwaighte’s hypothesis based on experimental observation of bias in regional microsphere deposition is correct. The most proximal vessels of the coronary arterial circulation consist of large, asymmetrically branching epicardial arteries which transport blood in a base-to-apex direction. These vessels branch into asymmetric transmural arteries which transport blood in an epicardial to endocardial direction, from which subsequent generations of vessel bifurcations become increasingly symmetric. This conclusion has clear implications for the interpretation of results using the microsphere technique. While in sufficiently large tissue regions 15 μm microspheres do not deviate significantly from the molecular marker IDMI (Bassingthwaighte et al., 1990), perfusion quantification from microspheres in small tissue segments particularly in the transmural direction is susceptible to bias introduced by skimming. For validation of quantitative perfusion imaging at high resolutions, with more than three transmural layers for example, this bias will likely result in an overestimation of flow in the subendocardium, and an underestimation of flow in the subepicardium. The next step towards clinical application would be to quantify the level of regional microsphere deposition bias, which may subsequently be used to provide a bias correction for perfusion quantification. More accurate information regarding true regional perfusion would facilitate this, which could be provided by imaging the deposition patterns of the higher fidelity molecular marker IDMI.

Finally, regional cardiac perfusion is well-known to be affected dynamically by contraction of the myocardium (Westerhof and Boer, 2006). This has not been a factor in this study due to the use of a Langendorff heart, which itself is widely used for cardiac research (Bell et al., 2011; Heinzer et al., 2006; Schuster et al., 2012; Schuster et al., 2014; Skrzypiec-Spring et al., 2007). Thus it remains unclear whether the effects of skimming differ greatly in vivo from the current findings. Recent experiments have been performed with in vivo pig hearts for which 3D microsphere deposition and vascular geometry have been acquired, which can be analysed in a procedure similar to that followed in the current study to determine the prevalence of microsphere skimming in vivo.