Impact of Effective Detector Pixel and CT Voxel Size on Accurate Estimation of Blood Volume in Opacified Microvasculature

Abstract

Rationale and Objectives:

The purpose of this study was to determine the impact of effective detector-pixel-size and image voxel size on the accurate estimation of microvessel density (ratio of microvascular lumen volume/tissue volume) in an excised porcine myocardium specimen using microcomputed tomography (CT), and the ability of whole-body energy-integrating-detector (EID) CT and photon-counting-detector (PCD) CT to measure microvessel density in the same ex vivo specimen.

Materials and Methods:

Porcine myocardial tissue in which the microvessels contained radio-opaque material was scanned using a micro-CT scanner and data were generated with a range of detector pixel sizes and image voxel sizes from 20 to 260 microns, to determine the impact of these parameters on the accuracy of microvessel density estimates. The same specimen was scanned in a whole-body EID CT and PCD CT system and images reconstructed with 600 and 250 micron slice thicknesses, respectively. Fraction of tissue volume that is filled with opacified microvessels was determined by first subtracting the mean background attenuation value from all voxels, and then by summing the remaining attenuation.

Results:

Microvessel density data were normalized to the value measured at 20 μm voxel size, which was considered reference truth for this study. For emulated micro-CT voxels up to 260 μm, the microvessel density was underestimated by at most 11 %. For whole-body EID CT and PCD CT, microvessel density was underestimated by 9.5% and overestimated by 0.1 %, respectively.

Conclusion:

Our data indicate that microvessel density can be accurately calculated from the larger detector pixels used in clinical CT scanners by measuring the increase of CT attenuation caused by these opacified microvessels.

INTRODUCTION

One of the earliest signs of atherosclerosis is an increased density in the microvascular network of an arterial wall, the vasa vasorum, which occurs prior to the development of vascular dysfunction and plaque formation (1–5). While evidence of the relationship between vasa vasorum density and atherosclerosis in knock-out mice, rabbits, and pigs is strong, the evidence in humans is weak. This is almost certainly due to inability to measure the microvascular density of vasa vasorum in a longitudinal, minimally invasive, clinical setting. The goal of this current study, therefore is to examine how well the blood fraction of tissue can be estimated with CT images, as a surrogate measure for vasa vasorum density, even though clinical CT images cannot spatially resolve these microvessels. Our method is intended to allow the needed measurements to be made noninvasively in a clinical setting, which will help to answer these questions in humans.

Though different from the vasa vasorum, the microvasculature density is also of interest in the myocardium, as it is the primary determinant of tissue perfusion. However, despite their physiologic importance, individual microvessels, ranging from approximately 8 μm to several hundred microns (6), cannot be resolved using conventional clinical imaging modalities (7). Thus, because they cannot be resolved using noninvasive imaging techniques, studies of microvessels have been limited to postmortem studies using microcomputed tomography (micro-CT) (8). This study explored the possibility that clinical computed tomography (CT) might be able to indirectly estimate micro vascular density in tissues, such as the wall of arteries or the myocardium.

In a single cross-sectional micro-CT image, microvascular density can be expressed as the number of vessel cross-sections per unit area of tissue (n/mm²). The fraction of tissue volume that is filled with intravascular blood can be expressed as the summed cross-sectional lumen area per area of tissue, which results in a dimensionless parameter (Σlumen area/Σtissue area). We hypothesize that if this is done for all contiguous cross-sectional images, the ratio of microvascular lumen volume/tissue volume can be derived. This volume density can be obtained by histological or contrast-enhanced micro-CT imaging methods, which can spatially resolve the microvessels and thereby allow computation of the true cumulative luminal volume per tissue volume. In contrast to micro-CT, whole-body CT has effective detector pixel dimensions approximately equal to or larger than the largest microvessel lumen diameters; it therefore cannot resolve the individual vessels and hence cannot be used to count them. In spite of its limited resolution, we hypothesize that whole-body CT can be used to estimate microvascular density by measuring the increase of CT image attenuation caused by these opacified microvessels (see Appendix A for the theoretical basis for this hypothesis).

The purpose of this study was to examine the accuracy with which the increased X-ray attenuation of tissue during the presence of intravascular contrast agent can be used to quantify microvascular density, and to determine how the accuracy depends on CT detector pixel size and CT image voxel size. Additionally, we sought to determine the accuracy with which microvessel density can be derived from CT images acquired using whole-body, energy-integrating-detector (EID) CT and photon-counting-detector (PCD) CT. To accomplish this, we used a micro-CT system to generate scan data with a range of detector pixel sizes and CT image voxel sizes to explore the impact of these parameters on the accuracy of microvessel density estimates in porcine myocardium in which the microvessels contained radio-opaque material. After scanning the same specimen with the whole-body CT systems, we compared the microvessel density from EID CT (600 μm image thickness) and PCD CT (250 μm image thickness) with those from micro-CT (20–260 μm image thickness).

MATERIALS AND METHODS

Animal Specimen Preparation

This protocol was approved by the Institutional Animal Care and Use Committee of the Mayo Clinic College of Medicine and Science. A 30 kg female pig was used, which was anesthetized by induction with a 2.2 mg/kg mixture of telazol/ketamine/xylazine and maintained with 1.5%–3% isoflurane. Prior to euthanizing the animal with an intravenous injection of 100 mg/kg pentobarbital, 10,000 international units of Heparin were injected intravenously to prevent blood coagulation.

The heart was removed and the left main coronary artery cannulated so that the blood could be flushed out with heparinized saline and the artery injected with a silicon-based polymer doped with lead (Microfil, Flow Tech, Carver, MA) at 100 mmHg infusion pressure. Once the particle-free polymer emerged from the coronary sinus, the cannula was clamped and the heart was left over night in a refrigerator. The next day, a transmural tissue sample (approximately 2 cm long and 1 cm × 1 cm in cross-section) was cut out from the free left ventricular wall. The specimen was then immersed in 10% formalin for several hours, patted dry and placed in a thin-wall plastic cylinder (18 mm diameter) and immersed in low-melting-point paraffin to prevent drying and movement of the specimen during the subsequent micro-CT and CT scanning. The volume of tissue cut from the myocardium was selected to include only microvessels, with the largest vessel in the specimen being below 0.5-mm-luminal diameter. This relatively large specimen provided sufficient statistical power, since the specimen contained thousands of microvascular branch segments, whose lumen diameters were considerably smaller and of similar size as the vasa vasorum.

Micro-CT Imaging

The myocardial tissue sample was scanned on a micro-CT scanner (North Star Imaging, Rogers, MN). This scanner has a fixed microfocus X-ray source (Hamamatsu Photonics K.K, Hamamatsu, Japan), a flat panel X-ray detector (Varian Medical Systems, Palo Alto, CA), a high precision translation stage (Newport Corporation, Irvine, CA), and a high precision rotation stage (Mitsubishi Electronic, Tokyo, Japan). The flexible geometry of the scanner provides selectable voxel sizes (5 μm-127 μm), different focal spot sizes (small 7 μm, medium 20 μm, and large 50 μm), and different tube potential settings (40 kV–150 kV). The flat panel detector features a 1024 by 1024 array of 127 micrometer square pixels. In a micro-CT system, the effective detector pixel size can be adjusted by changing the geometric magnification of the X-ray source and detector. The following formula is used to define the geometric magnification for a given set up: M = SDD/SOD, where SDD is source to detector distance and SOD is source to object distance.

We adjusted the scanner geometric magnification to 6.25X, so that the effective detector pixel size (at axis of rotation) was 20 μm. The exposure settings were 40 kV and 250 μA. The specimen was mounted on the computer controlled precision stages and rotated 360° about the vertical axis during the scan. After each projection acquisition, the detector’s pixel signals were transferred to the associated computer, via a 16-bit analog-to-digital converter, and the specimen was rotated by a small angular increment (0.24°) before acquiring the next projection. This process was repeated until data were acquired for a full 360°. The projections were reconstructed by the software provided with the micro-CT system using a modified Feldkamp tomographic reconstruction algorithm (9) to generate a 3D stack of CT images encompassing the specimen. The CT image stack had a cumulative volume of 10⁹ (20 μm)³ voxels.

The projection data were subsequendy used to determine the impact of the effective detector-pixel-size and reconstructed voxel size. For this purpose, the effective detector-pixel-size in the projections was increased using a pixel binning procedure (grouping detector elements in different configurations, e.g., 2 × 2 or 4 × 4). Using this technique, we generated effective detector-pixel-sizes (at the isocenter) from 20 × 20 μm² to 260 × 260 μm², in step sizes of 20 μm per detector pixel side. The dimension of each side of the reconstructed cubic image voxels was matched to the effective pixel dimension (at isocenter) for the projection data used. Thus, the dimensions of the effective detector pixels and of the cubic image voxels were identical for micro-CT data.

Whole-Body CT Imaging

After micro-CT, the myocardial tissue sample was scanned on a whole-body research PCD CT scanner (SOMATOM CounT, Siemens Healthcare GmbH, Forchheim, Germany). The system is a modified second generation dual-source scanner (Somatom Definition Flash, Siemens Healthcare GmbH) equipped with two X-ray tubes and two corresponding detectors, mounted onto the rotating gantry with an angular offset of approximately 90°. One tube/detector pair used an EID array and had a 50 cm field-of-view (FOV). The second tube/detector pair was equipped with a PCD array and had a 27.5 cm FOV (10). The specimen (housed in the same plastic cylinder) was scanned with the PCD system at 140 kV and 44 mAs using a multidetector-row collimation of 48 × 0.25 mm =12 mm. Images were reconstructed with a 250-μm-thickness, a 65 mm × 65 mm FOV, a 1024 × 1024 image matrix and a D50 reconstruction kernel, yielding a 63 × 63 × 250 μm³ CT voxel size.

In addition, the same myocardial tissue sample was scanned on a clinical dual-source CT system (SOMATOM Definition Flash, Siemens Healthcare GmbH, Forchheim, Germany) equipped with two conventional EIDs. Data acquisition was performed with a tube potential of 140 kV and 52 mAs, matching the volume CT dose index of the PCD acquisition (because of the narrower collimation of the PCD CT system, the CTDIvol per mAs is larger, thus a smaller mAs value was required on the PCD to yield the same dose). The clinical EID CT system has a detector size of 0.6× 0.6 mm² (in plane) at isocenter, and a longitudinal collimation of 32 × 0.6 mm = 19.2 mm. The acquired data were then reconstructed into images with a 65 mm × 65 mm FOV using a 1024 × 1024 matrix size and a weighted filtered back projection algorithm with a quantitative D50 kernel. An image slice thickness of 600 μm was used, the smallest thickness possible on this system, yielding a voxel size of 63 × 63 × 600 μm³.

CT Image Analysis

For micro-CT, the average X-ray attenuation of the specimen’s soft tissue (myocardium) was about one-tenth of the attenuation of the lead-based polymer within the microvascular lumens. This large difference in attenuation facilitated easy automated discrimination of the opacified microvessels from the surrounding muscle tissue, which we denote as background. However, for the whole-body CT data, this is not possible due to the lower resolution. So the whole-body data sets were scaled to the same values of micro-CT. The scaling was done using the values of air, paraffin, and Microfil (from the largest vessel lumen). Those values were used to find a curve that scaled whole-body CT values to the equivalent micro-CT values. Then, the threshold-based region in the micro-CT was matched with the same region in the whole-body CT using 3D registration.

To confirm our results for the background value, we also obtained a measure of the myocardial attenuation before contrast was injected using another available specimen. A piece of unenhanced pig myocardium was scanned at different effective pixel sizes and results confirmed that the background attenuation for both micro-CT and whole-body CT before the contrast injection was very similar to what we used in our “vessel-free” region of interest method. For example, for 20-μm-resolution, the background value for tissue using the reported method was 0.540 cm⁻¹ (−18 HU) and using second contrast-free specimen was 0.515 cm⁻¹ (−63 HU); both of these low attenuation values (i.e., negative CT numbers) are readily separate from the attenuation of the contrast-enhanced vessels.

To determine the density of the microvessels, we subtracted the mean attenuation value of the unenhanced background from all of the voxels within a given volume of interest (VOI). As a result, the voxels containing contrast-filled vessels retained positive-nonzero attenuation values, while the myocardial background attenuation values within the VOI were reduced to a mean of 0. In order to make accurate comparisons between the evaluated effective detector pixel size/cubic voxel size conditions, the attenuation values of the voxels within the selected VOI’s were normalized by the number of voxels in the VOI. In this way, we ensured that the contribution of each voxel’s attenuation value to the sum of attenuation values over the VOI was consistent as effective detector pixel and reconstructed image voxel sizes were varied. The normalized attenuation values for voxels within the VOI were summed, which resulted in the total attenuation due to the opacified microvessels. Normalizing to the volume of the VOI yielded the micro vessel density in attenuation units. A measurement of the attenuation of the lead-based polymer over a VOI containing only the contrast agent, if available, would allow normalization of the attenuation units (total attenuation × micro vessel volume /VOI volume) to yield micro vessel density (unidess fraction).

Similarly, the microvessel density was obtained for the whole-body CT image data by subtracting the mean background attenuation value measured in a region of interest with minimal vessels (in this case guided by the micro-CT data) from all attenuation values, resulting in a mean background value of zero. The attenuation values in the VOI were then summed and divided by the volume of the VOI.

RESULTS

Image noise in the micro CT images, where dose was not a limitation, was 0.4% of the mean contrast-filled vessels attenuation value, whereas in the EID and PCD data, where typical technique settings were used, it was 2% of the mean contrast-filled vessels attenuation.

The impact that the micro-CT effective detector-pixel-size and image voxel size had on spatial resolution is visually demonstrated in upper row of Figure 1. Blurring of the vessels increased as the size of the effective detector-pixel-size and voxel size increased, as expected. Similar cross-sections of tissue using PCD and EID CT scans are shown in the second row of the figure, with slice thicknesses of 250 μm and 600 μm, respectively. The largest lumen diameters in this tissue specimen were about 500 μm. While vessels of this size would be countable at lower resolutions, many of the smaller diameter vessels are not countable due to partial volume averaging of the attenuation of the small vessels with the background tissue attenuation in the larger voxels.

Figure 1.

Upper Panel: Images reconstructed at increasing voxel sizes, which were set to be the same dimension as the effective detector-pixel-sizes in the micro-CT projection data. Lower Panel: Whole-body CT images scanned and reconstructed with PCD CT and EID CT systems. CT, computed tomography; EID, energy-integrating-detector; PCD, photon-counting-detector.

Although the individual small vessels are not resolvable in the 260 μm voxel images, the attenuation from the opacified vessels is still reflected in the attenuation value of the larger voxels. This is demonstrated in Figure 2, where the fraction of tissue volume that is filled with opacified microvessels, derived from summing all attenuation above background in the micro-CT data, is shown as a function of effective detector-pixel-size, which was equal to the voxel size. The data were normalized to the value measured for the 20 μm voxel size, which we considered as reference truth for this study because these data were measured with the highest spatial resolution (Previous work has confirmed the accuracy of micro CT for measuring micro vessel density compared to histology (11)). At 260 μm, the microvessel density was underestimated by 11% compared to the 20-μm CT voxel size micro-CT data.

Figure 2.

Plot showing the impact of effective detector-pixel-size and voxel size on the accuracy of estimating microvessel density for the same volume of interest using micro-CT data (triangle markers). The micro-CT cubic voxels have an image thickness equal to the dimension of the side of an in-plane pixel. Thus, the volume of a voxel with isotropic 20 μm resolution is (20 μm). CT, computed tomography.

The microvessel density derived data from EID CT and PCD CT, as a fraction of the value measured using micro-CT (20 μm)³ voxels, are shown in Table 1. The vessel density is underestimated by 9.5% for the whole-body EID CT and overestimated by 0.1% for the whole-body PCD CT relative to the 20 μm CT voxel size micro-CT data.

TABLE 1.

Microvessel Density (Normalized to That at 20 μm) for Conventional EID CT (63 × 63 × 600 μm Voxel Size) and a Research PCD CT (63 × 63 × 250 μm Voxel Size)

Voxel Dimension	Whole-Body CT
63 × 63 × 600 μm (EID CT)	0.905
63 × 63 × 250 μm (PCD CT)	1.001

CT, computed tomography; EID, energy-integrating-detector; PCD, photon-counting-detector.

DISCUSSION

Our data indicate that accurate estimates of the volume density of microvessels within tissue can be obtained from the larger detector pixels used in clinical CT scanners for an image thickness of 250 μm and 600 μm, which are available on the evaluated whole-body, research PCD CT and clinical EID CT. While smaller voxels can be created for a given detector-pixel-size by use of a smaller FOV and/or a larger image reconstruction matrix, this does not improve the limiting spatial resolution of the system, which is determined by the focal spot and detector pixel sizes. Smaller voxel dimensions can help, however, in more precise delineation of regions of interest.

While smaller effective detector pixels will increase the inherent spatial resolution of a CT system (up to the point where the focal spot size blurring dominates), realization of this resolution with use of sharp reconstruction kernels and small voxel sizes dramatically increases image noise. For the evaluated specimen, the measurements of mean CT number used to estimate microvessel density could be made over the relatively large VOI of the myocardial tissue specimen. Spatial variability in this value that extends beyond the size of the region of interest used for measurement, which may exist due to regional disease, should be observable.

For assessment of microvessel density in smaller tissue samples, for example for the assessment of vasa vasorum density in arterial walls, which are 1–2 mm thick, the noise associated with the mean CT numbers over such a small region of interest will pose a considerable challenge, hkely requiring use of noise reduction techniques. Such noise reduction techniques must maintain accurate attenuation values and must not cause spatial blurring. The impact of CT image noise due to X-ray dose limitations, the small size of the region of interest under consideration, and the patchy distribution of microvasculature, a feature of particular importance in vasa vasorum distribution in arterial walls, remain to be determined.

Limitations of this work include the measurement of microvessel density on only two whole-body CT systems, a PCD CT using 250 μm detector-pixel-sizes (at isocenter) and an EID CT using a 600 μm detector-pixel-size (at isocenter). Also, the lead-based polymer used to enhance the microvessels is considerably more enhancing than the iodine contrast agents used in vivo, which for a cardiac CT exam is in the range of approximately 300–400 HU. The attenuation of the resultant specimen using whole-body CT over one of the larger vessel lumens (1400 HU at 140 kV) is much brighter than would be achievable clinically. Furthermore, the initial appearance of the bolus of contrast agent opacifies the vascular lumens, but after a few heart cycles a fraction of the intraluminal contrast diffuses into the extravascular space. Thus, in a clinical study, it needs to be established if the delayed opacification of the extravascular space augments or detracts from the index of vascular blood volume. If this were to be the case, we could use the method introduced by Daghini et al. (12), which showed that the extravascular component can be separated from the intravascular component. Motion may also be a problem for in vivo studies. However, current state of the art CT scanners have effective temporal resolution of less than 70 ms, which is fast enough to capture stop-motion images of the myocardium or vessels in diastole.

Our ultimate goal is to quantify the density of vasa vasorum in arterial walls. Prior to attempting that more challenging task, it was necessary to demonstrate in a larger specimen that microvascular density could be accurately measured with large detector pixels, even though the individual microvessels are not resolvable. Thus, we evaluated the volume density of the microvessels in a large piece of myocardium so that we could avoid the confounding issue of partial volume averaging due to CT image voxels straddling the interface between the inner vessel wall and the lumen or between the outer vessel wall and adjacent extravascular tissue. Future work involving measurement of vasa vasorum within an arterial wall will be needed to demonstrate the ability of whole-body CT to measure microvessel density accurately in such small regions of interest (arterial wall thicknesses are approximately 1 mm).

In summary, however, the reported work demonstrates that the volume density of microvasculature that is smaller than the resolution limits of a whole-body CT system can be measured by use of mean attenuation in the volume of interest. This finding opens new possibilities for in vivo assessment of these fundamentally important components of the cardiovascular system.

Abbreviations

CT

computed tomography

EID

energy-integrating-detector

FOV

field-of-view

Micro-CT

microcomputed tomography

PCD

photon-counting-detector

VOI

volume of interest

APPENDIX A. ASSESSMENT OF MICROVESSEL DENSITY USING MEAN CT ATTENUATION MEASUREMENTS

Abbreviations

Contrast attenuation in micro vessels: {C,mv}

Tissue attenuation in myocardium: {T,m}

Myocardium: m

Pure contrast agent: contrast

Water: water

Tissue: T

Micro vessels: mv

Notations for Physical Units

• Linear attenuation coefficient of a tissue/material x is μ_x

• Mass attenuation coefficient of a tissue/material x is ${\left(\frac{\mu }{\rho }\right)}_{\text{x}}$

• Density is ρ, mean density is $\overline{\rho }$

• Volume is V

• Mass is M

• CT number of a specific material (say contrast in microvessels) is given by:

$${\text{CT}}_{\{\text{C},\text{mv}\}}=\left(\frac{{\mu }_{\{\text{C},\text{mv}\}}-{\mu }_{\text{water}}}{{\mu }_{\text{water}}}\right)\times 1000$$

Note: μ in this appendix refers to linear X-ray attenuation.

Baseline Correction for Contrast Signal in Myocardium

The mass of contrast in the myocardium is given by:

$${\text{M}}_{\{\text{C},\text{m}\}}={\overline{\rho }}_{\{\text{C},\text{m}\}}\cdot {\text{V}}_{\text{m}}$$ (1)

Since myocardium constitutes of tissue and mv, we can rewrite the above equation with respect to the mv component alone as:

$${\text{M}}_{\{\text{C},\text{m}\}}={\rho }_{\{\text{C},\text{mv}\}}\cdot {\text{V}}_{\text{mv}}$$ (2)

Note in Equation 2 above, the value of ρ is the density of contrast in individual microvessels, while the density measured in the myocardium shown in Equation 1 is a mean density.

The mean density of contrast in the myocardium can be related to the density of contrast in a region of pure contrast as shown below:

$${\overline{\rho }}_{\{\text{C},\text{m}\}}\cdot {\text{V}}_{\text{m}}={\rho }_{\{\text{C},\text{contrast}\}}\cdot {\text{V}}_{\text{mv}}$$ (3)

This assumes that the density of contrast in the microvessels is the same as the density of contrast in a region of pure contrast. That is, the microvessels will have the same density of contrast as in the larger arteries that supply blood to the microvessels. The diffusion of contrast into the myocardial tissue is not considered here.

The fraction of myocardium density to pure contrast density can then be obtained from Equation 3:

$$\text{F}=\frac{{\text{V}}_{\text{mv}}}{{\text{V}}_{\text{m}}}=\frac{{\overline{\rho }}_{\{\text{C},\text{m}\}}}{{\rho }_{\{\text{C},\text{contrast}\}}}$$ (4)

The CT number measured in the myocardium using a region or volume of interest can be written as:

$${\text{CT}}_{\text{m}}=\left(\frac{{(\mu /\rho )}_{\text{C}}\cdot {\overline{\rho }}_{\{\text{C},\text{m}\}}+{(\mu /\rho )}_{\text{T}}\cdot {\rho }_{\{\text{T},\text{m}\}}-{\mu }_{\text{water}}}{{\mu }_{\text{water}}}\right)\times 1000$$ (5)

Equation 5 can be rewritten as:

$$\begin{gathered} {\text{CT}}_{\text{m}}=\left(\frac{{(\mu /\rho )}_{\text{C}}\cdot {\overline{\rho }}_{\{\text{C},\text{m}\}}}{{\mu }_{\text{water}}}\right)\times 1000 \\ +\left(\frac{{(\mu /\rho )}_{\text{T}}\cdot {\rho }_{\{\text{T},\text{m}\}}-{\mu }_{\text{water}}}{{\mu }_{\text{water}}}\right)\times 1000 \end{gathered}$$ (6)

$${\text{CT}}_{\text{m}}=\left(\frac{{(\mu /\rho )}_{\text{C}}\cdot {\overline{\rho }}_{\{\text{C},\text{m}\}}}{{\mu }_{\text{water}}}\right)\times 1000+{\text{CT}}_{\text{T}}$$ (7)

Equation 7 can be further rewritten as:

$$\left(\frac{{(\mu /\rho )}_{\text{C}}\cdot {\overline{\rho }}_{\{\text{C},\text{m}\}}}{{\mu }_{\text{water}}}\right)\times 1000={\text{CT}}_{\text{m}}-{\text{CT}}_{\text{T}}$$ (8)

Further rearranging Equation 8 w.r.t. mean contrast density in myocardium, we get:

$${\overline{\rho }}_{\{\text{C},\text{m}\}}=\frac{{\text{CT}}_{\text{m}}-{\text{CT}}_{\text{T}}}{{(\mu /\rho )}_{\text{C}}\times 1000}\times {\mu }_{\text{water}}$$ (9)

Similarly, the CT number of contrast can be written in terms of its attenuation and density components as:

$${\text{CT}}_{\text{contrast}}=\left(\frac{{(\mu /\rho )}_{\text{C}}\cdot {\rho }_{\{\text{C},\text{contrast}\}}-{\mu }_{\text{water}}}{{\mu }_{\text{water}}}\right)\times 1000$$ (10)

$${\rho }_{\{\text{C},\text{contrast}\}}=\frac{{\text{CT}}_{\text{contrast}}+1000}{{(\mu /\rho )}_{\text{C}}\times 1000}\times {\mu }_{\text{water}}$$ (11)

Combining Equations 9 and 11, we get:

$$\text{F}=\frac{{\overline{\rho }}_{\{\text{C},\text{m}\}}}{{\rho }_{\{\text{C},\text{contrast}\}}}=\frac{{\text{CT}}_{\text{m}}-{\text{CT}}_{\text{T}}}{{\text{CT}}_{\text{contrast}}+1000}$$ (12)

Note CT_T corresponds to the attenuation from myocardial tissue without any contrast signal and CT_contrast corresponds to the attenuation from the pure contrast material.

From Equation 12, we can conclude that the absolute density of the microvessels in the myocardium at any location is represented by the fraction of contrast enhancement in the myocardium to the total attenuation in pure contrast.

Knowledge of the attenuation of the pure contrast material was not used in the described work, as all data were normalized to the microvascular density at 20 0μm.