Dynamic Changes in Microvascular Flow Conductivity and Perfusion After Myocardial Infarction Shown by Image‐Based Modeling

Background Microcirculation is a decisive factor in tissue reperfusion inadequacy following myocardial infarction (MI). Nonetheless, experimental assessment of blood flow in microcirculation remains a bottleneck. We sought to model blood flow properties in coronary microcirculation at different time points after MI and to compare them with healthy conditions to obtain insights into alterations in cardiac tissue perfusion. Methods and Results We developed an image‐based modeling framework that permitted feeding a continuum flow model with anatomical data previously obtained from the pig coronary microvasculature to calculate physiologically meaningful permeability tensors. The tensors encompassed the microvascular conductivity and were also used to estimate the arteriole–venule drop in pressure and myocardial blood flow. Our results indicate that the tensors increased in a bimodal pattern at infarcted areas on days 1 and 7 after MI while a nonphysiological decrease in arteriole–venule drop in pressure was observed; contrary, the tensors and the arteriole–venule drop in pressure on day 3 after MI, and in remote areas, were closer to values for healthy tissue. Myocardial blood flow calculated using the condition‐dependent arteriole–venule drop in pressure decreased in infarcted areas. Last, we simulated specific modes of vascular remodeling, such as vasodilation, vasoconstriction, or pruning, and quantified their distinct impact on microvascular conductivity. Conclusions Our study unravels time‐ and region‐dependent alterations of tissue perfusion related to the structural changes occurring in the coronary microvasculature due to MI. It also paves the way for conducting simulations in new therapeutic interventions in MI and for image‐based microvascular modeling by applying continuum flow models in other biomedical scenarios.

shlould be noted that periodicity is necessary in order to ignore secular terms.
Furthermore, blood flow of the microvessels is considered incompressible and dominated by viscous forces (Stokes flow). Taking these assumptions into account and the fact that volume averaged tissue-scale blood velocity equals the surface mean blood velocity to leading order, 2 the homogenization technique of Shipley and Chapman 3 is applied and the volume averaged form of Darcy's law is deduced to: where K(I) is the permeability tensor of the image volume I and given the 3D nature of I it is of size [3 × 3]. i.e. equation 1 is written analytically as: where ∇p i stands for the pressure gradient applied on the tissue along direction i, while u j in 3D is the surface mean blood velocity component along direction j and it Data S1.
is thus given by: where S j is the area of the outflow surface at direction j and Q j represents the sum of the flow rates of the capillaries on that surface.
By combining equations 2 and 3, the elements of the permeability tensor can be calculated by: where M j is the number of capillaries on surface j.

Derivation of myocardial blood flow equation
Perfusion can be calculated by integrating the velocity over the cross-sectional area (A) of the material through which flows the fluid, i.e. in this work, the myocardium. If the velocity is parallel to the axis of one of the main directions of flow, then by using the corresponding diagonal element of the permeability tensor (k ii , i = 1, 2, 3) which encompasses the microvascular conductivity of the microvasculature feeding the myocardium, perfusion can be calculated by: where ∆p stands for the AV pressure drop in mmHg over a microvascular path of length l given in µm. It should be noted that 0.133 × 10 −3 is a conversion factor for pressure units from mmHg to kgµm −1 s −2 , while the cross-sectional area over which the velocity is integrated is expressed in µm.
Myocardial blood flow can be expressed: where ρ represents myocardial density (ρ), while 60 and 100 are conversion factors from minutes to seconds, and 1g to 100g respectively.
Taking into account the aforementioned relationships, i.e. the relation between perfusion and the permeability tensors (eq. 5), and the relation between perfusion and MBF (eq. 6), 4 the latter can be calculated using the permeability tensors by:

Incorporation of phase separation effect into the calculation of the permeability tensors
We incorporated an iterative procedure in the calculation of the tensors for each sub-network as show in Supplemental Fig. 2.
In brief, a constant hematocrit (0.4) was initially assumed in order to calculate the permeability tensor along the x-direction. The pressure solution that resulted in the calculation of the permeability tensor elements k 11 , k 12 , k 13 was used as an input to an iterative publicly available algorithm 5 for the calculation of varying hematocrit when separation effect on junction points is taken into account. 6 It is worth mentioning that the equations regarding apparent viscosity of this implementation, although were derived from data for the rat mesentery, are expressed for human blood parameters. 7 In order to account for differences in blood parameters between different species, the diameters of the vessels should be scaled using the cubic root of the ratio of the volume of red blood cells for humans and the species under investigation. Therefore, here, the diameters of the vessels were scaled by 8 Once the solution converges or reaches the maximum number of permitted iterations (100), the permeability tensors are calculated with the new varying values of the hematocrit.
Once the new permeability tensors differ less than 2 × 10 −17 mm 3 s kg −1 in all elements from the last iteration, the procedure is considered as converged and the calculated tensor is the final one. As for some volumes did not converge, an additional limit of a maximum of 300 iterations was set.

Calculation of cut-off size of Representative Volume Element (REV)
To define the cut-off side length along the x and y directions below which calculation of permeability tensors could lead to errors, we considered the capillary bed of each volume as the meso-scale which is homogeneous inside smaller areas of itself. More precisely, the volume was divided into smaller microscopic units with a size of 256 × 256× Nz, 512 × 512× Nz, 1024 × 1024× Nz voxels, where Nz stands for the size of the original image I along the z-axis (in voxels). We applied the developed framework at each unit to calculate permeability tensors for each one of them. We concluded that the permeability tensor for volumes below 512 voxels were frequently overestimated, with the larger element of the permeability tensor being an order higher than values reported in literature 4 ( Supplementary Fig. 6). Therefore, in our subsequent analysis, we excluded sub-networks whose size was smaller than this cut-off limit. It should be noted that the voxel size of the majority of our images is [0.379, 0.379, 1.007] µm. Thus, the calculated cut-off limit of 512 voxels along the x and y directions corresponds to a physical size of 194 µm. After inspection of the results for all images, the cut-off size for excluding connected components from subsequent analysis was given a small margin and was set equal to 170 µm in order to avoid excluding connected components with size very close to 194 µm for which the tensors were correctly calculated. In sum, the permeability tensor of a 3D image was calculated as the weighted sum of the connected components whose side length along x-y axis was larger than 170 µm prior to application of the mirroring procedure.

Calculation of scaling parameter
We performed correlation and regression analysis and discovered a statistically significant linear relation of the permeability tensor to the ratio of sub-network vascular volume (V (cc)) to the vascular volume in the corresponding volume of the image (V init (cc)) prior to any post-processing having taken place. More precisely, the radii and length of all vascular segments, prior and after repetitive elimination of the blind-ends until the microvascular network under investigation is fully connected and has no blind-ends, had already been calculated. Therefore, we used them to calculate the two volumes, i.e. V (cc) and V init (cc), by considering each microvessel as a tube of constant radius. Using the basal condition, we calculated the Kendall's Tau correlation coefficient to study whether there is any association between the permeability tensor and the ratio of volume of the sub-network prior and after the elimination of the blind ends, i.e. V (cc) V init (cc) . The coefficient was equal to 0.47 (p-value= 0.008) and 0.66 (p-value = 8 × 10 −5 ) for k 11 and k 22 respectively denoting a dependency of the permeability tensor on the ratio. We applied a scaling factor equal to the reverse of the ratio ( V (cc) V init (cc) ) to the permeability tensors in order to adjust them on the basis of this dependency. The scaling factor was incorporated by multiplying the weights w ij (cc) by it. By applying linear regression analysis before and after application of the scaling factor, we noticed that the permeability tensors were no longer linearly dependent on V (cc) V init (cc) (y ∼ β 0 , β 0 = 4 × 10 3 (p-value= 9.49 × 10 −28 )).  Figure S1. Mirroring of the original image colour-coded with light green (7). 2D slices along x,y and z directions of the 3D resulting mirrored image that con-tains 8 copies of the original image and periodic boundaries on opposites faces.  Figure S2. Overview of the approach for incorporating the phase sepa-ration effect in the calculation of permeability tensors.  Figure S3. Summary of the changes in microvascular conductivity, AV pressure drop and MBF after MI. Dashed lines correspond to remote areas, while solid to infarcted. The scheme highlights the bimodal changes obtained for the per-meability tensors which increase at day 1 and day 7 after MI with partial restora-tion at day 3 (closer to basal). MBF calculated assuming physiological AV pressure drop independently of tissue condition, i.e. 19mmHg, follows the bimodal increase of the tensors at infarcted areas (blueish green line). On the contrary, when condition-dependent AV pressure drop is used, MBF remains reduced at infarcted areas (pur-ple line).  Figure S4. Impact of vascular remodeling strategies on the microvas-cular conductivity obtained by simulations using images of basal conditions. The pseudocolored map represents the impact, i.e. percentage of change (%) of k 11 , on the permeability tensors of the simulations for distinct vascular remodeling strategies described on the left. Every column corresponds to an image with the last column representing the mean among all images for the specific tissue condition.  Figure S5. Impact of vascular remodeling strategies on the microvascu-lar conductivity obtained by simulations using images on day 3 post-MI. The pseudocolored map represents the impact, i.e. percentage of change (%) of k 11 , of the simulations for distinct vascular remodeling strategies described on the left. a Change in k 11 when vascular remodeling is simulated using images from remote areas on day 3 post-MI. b Change in k 11 when vascular remodeling is simulated for images from infarcted areas on day 3 post-MI. Every column corresponds to an image with the last column representing the mean among all images for the specific tissue condition.  Figure S6. Dependency of the permeability tensor on Representative Volume Element. Three different sizes were investigated: 256 × 256× Nz, 512 × 512× Nz, 1024 × 1024× Nz voxels, with Nz standing for the size of the image along z-axis. The original image volume was therefore decomposed into 16, 4, and one volume(s)/unit(s) respectively depending on the unit size used. Tensors for the sub-volumes under investigation were calculated by applying the proposed approach for the calculation of permeability tensors from anatomical data on each sub-volume. The asterisk size is proportional to the volume of the units normalized by a volume size of 0.0074mm 3 corresponding to 1024 × 1024 × 50 voxels of size [0.379, 0.379, 1007]µm. The different asterisk sizes are due to the fact that the resulting connected network inside the unit might be smaller than the initial unit to which the image was decomposed. For each time point, the plots on the left show the maximum element of the permeability tensor in relation to volume size of the subnetworks of all images available in our dataset. On the right, there are plots for every image (one line per image) after fusing the tensors of the units in which it was decomposed. The permeability tensor here is given as the median of the different subvolumes for simplicity reasons.