INTRODUCTION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION REFERENCES

PERFUSION IS DEFINED as the amount of blood delivered to a unit mass of tissue per unit time. Average perfusion of an organ can be calculated by dividing the flow through the supplying artery by the mass of the organ. However, local perfusion values within the organ have been found to vary significantly and to show positive correlation in space (1, 2, 17). This means that it is likely that the area surrounding a highly perfused region of the organ is also highly perfused. More importantly, the correlation has been found to be independent of the size (scale) of the measurement volumes, which implies that perfusion heterogeneity is a self-similar or fractal quantity (2, 17). This statement is equivalent to the observation that perfusion heterogeneity as a function of scale can be described by a power law (see METHODS). Differences in local perfusion arise from differences in the resistance of the supplying vascular pathways. Consequently, we can learn something about vascular structure and resistance by measuring the heterogeneity and spatial correlation of perfusion.

Recently, imaging techniques such PET (8) and MRI (4) have been used to acquire data on perfusion heterogeneities. However, without a convincing model that establishes a link between the measured parameters and the underlying vascular structure, a physiologically useful interpretation of these measurements is difficult.

In this article, we present a model that suggests a connection between fractal perfusion heterogeneities and the scaling of vessel resistance as well as branching asymmetry. The model is based on the assumption of fractal vessel resistance, meaning that vessel resistance also obeys a power law as a function of scale similar to the heterogeneity of perfusion. The scale of an artery will be defined here as the volume of tissue that it supplies. This definition of scale also leads to a new approach to vessel ordering that is discussed in detail in METHODS. Branching asymmetry  within the model is defined as the relative tissue volume fraction being fed by the smaller branch of a vessel bifurcation.

Vascular structure in the heart, as depicted in Fig. 1, features highly asymmetric vessel branching and large variations in the distances from the feeding artery to the terminal vessels. The model incorporates these features, and simulations demonstrate that the model is able to reproduce perfusion heterogeneity in the sheep heart as measured by Bassingthwaighte et al. (3) as well as morphological data from the human heart gathered by Zamir (20).

View larger version (116K): [in this window] [in a new window]   Fig. 1.   Rendering of a microCT image of a mouse heart. Vessels and heart chambers are filled with X-ray contrast agent. The branching structure of a coronary vessel is clearly visible. The figure illustrates that vessel branching is highly asymmetric and that large variations in the distances from the feeding artery to the terminal vessels exist.

Certain computer-generated, three-dimensional vascular tree structures have been shown to reproduce perfusion heterogeneities similar to real physiological systems (5, 7, 11). However, the complexity of the generating algorithms may not be essential to account for the perfusion heterogeneity. No concrete morphological parameters have been identified as the cause of fractal perfusion heterogeneity. Parker et al. (11) even observed fractal heterogeneities in a three-dimensional branching model with equal branch flows. However, their model only relies on rules for branching without a mechanism to distribute vessels homogeneously in space.

Alternatively, van Beek et al. (17) proposed a simple fractal flow bifurcation model, which assumes that bifurcating vessels perfuse identical volumes of tissue but carry different amounts of flow. The van Beek model does not include information on the three-dimensional distribution of vessels within the organ. Although the reported experimental data can be fitted with this model, the assumption that both branches of a bifurcation feed the same tissue volume is not physiologically realistic (20).

The proposed model links global properties of vascular structure and multiscale perfusion measurements, which may eventually lead to new interpretations of clinical perfusion data. For example, vascular changes over time or in response to stress or vasodilation in diseases like ischemic heart disease, arteriosclerosis, hypertension, diabetes, and others could be analyzed in terms of the model parameters. Other applications could be the characterization of vascular beds with differing physiology and new ways of categorizing vascular development in mutant animals.

	METHODS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION REFERENCES

Relative dispersion and fractal power laws. The degree of perfusion heterogeneity is commonly quantified by measuring relative dispersion (RD; also called the coefficient of variation), defined as the standard deviation of perfusion measurements in local volumes of defined size divided by the average of these measurements. It has been found that the decrease of RD with increased mass m of the volume elements can be described by a power law function (2, 17)

(1)

Equation 1 is indicative of self-similarity, a basic property of fractals. To understand the concept of self-similarity, let us consider some quantity f: f is considered self-similar if its value measured at an arbitrary scale r differs from the value measured at the scale n · r only by a constant scaling factor k, which is independent of r

(2)

For example, the number of vessels visible in the field of view of a microscope may increase by a factor of k = 3 as one zooms in and magnification is increased by a factor of 2(n = 1/2). The power law for relative dispersion of perfusion (Eq. 1) is consistent with the above definition of self-similarity, with the measured quantity f being the relative dispersion of perfusion RD, m equivalent to the resolution r, and b the scale-independent slope of RD vs. m on a log-log plot resulting in k = n^b. If perfusion is measured at a single scale m (mass of individual volume elements), relative dispersion values at scale n · m can then be obtained by aggregating n neighboring volume elements.

(3)

Defining scale through volume ordering. The observation of fractal perfusion heterogeneities brings up interesting questions: Is the geometry of the underlying vascular structure that distributes blood flow also fractal? And does fractal perfusion heterogeneity arise from fractal vascular geometry? It is important to realize that fractal perfusion heterogeneities are by no means an obvious consequence of fractal vascular geometry and vice versa. To proceed, we need to define what we mean by fractal geometry. Here, fractal vascular geometry refers to the assumption that vessel properties like length, diameter and resistance follow power laws similar to Eq. 1. Vessel length is defined here as the distance between two successive branching points, often referred to as the length of a vessel segment (9).

(4)

(5)

(6)

(7)

View larger version (14K): [in this window] [in a new window]   Fig. 2.   Schematic illustrating the major features of the vascular model. R marks the resistance of a vessel segment, where V is the tissue volume being supplied by the vessel and i is a counting index for equivalent vessels. In this discrete version of the model, all tissue volumes being fed by terminal vessels are of size 1. Subtrees 1 and 2 are feeding the same volume of tissue and represent the same resistance to blood flow. However, subtree 1 will exhibit higher perfusion than subtree 2 because of its higher entrance pressure. To the right of the model tree, supplied tissue volume, volume order [= log2 (supplied volume) + 1] and Strahler order of the vessels ending in the marked bands are tabulated. Pinflow, entrance pressure; Pterminal, exit pressure.

(8)

Proposed model of vascular architecture and flow. To simplify the complicated nature of vascular systems, the following approximations will be used, which have previously been used by others. The vasculature is considered to be a bifurcating tree with a single feeding vessel at entrance pressure P_inflow and a constant exit pressure P_terminal at the capillary level (5, 14, 16). The flow F through a vessel is computed by dividing the pressure difference P between inflow and outflow by the resistance R of the vessel following Poiseuille's law. Branching asymmetry  within the model is defined as  = V₁/(V₁ + V₂), where V₁ and V₂ are the tissue volumes being fed by the child branches of a vessel bifurcation with V₁ < V₂, i.e., 0 <   0.5. The volume supplied by an arbitrary branch of the vessel tree can be computed as the sum of all volumes fed by its individual terminal branches.

q

(9)

(10)

(11)

(12)

View larger version (11K): [in this window] [in a new window]   Fig. 3.   Illustration of the effects of the asymmetry parameter  and the scaling parameter q. VP and VL,R, and RP and RL,R, mark the volume being fed by, and the resistance of, the parent and child vessels, respectively;  determines the asymmetry of the tissue volumes that are being fed by the branches of a bifurcation ( = 0.5 represents total symmetry). The parameter q governs the increase of resistance, which can be interpreted as a decrease in diameter.

View larger version (14K): [in this window] [in a new window]   Fig. 4.   Illustration of the effect of the resistance scaling parameter q. Normalized pressure is plotted as a function of volume order number for a symmetrically branching tree with 21 generations. As q increases, more pressure is lost in vessels of low order.

Model implementation. A flow chart of the computational steps that are involved in the implementation of the proposed model is given in Fig. 5. A typical simulation requires four steps: 1) construction of the vascular tree according to the chosen input parameter set, 2) calculation of flow and pressure distribution within the network, 3) uniform division of outflow volumes (voxelation), and 4) calculation of relative dispersion at different levels of aggregation.

View larger version (26K): [in this window] [in a new window]   Fig. 5.   Flow chart of the algorithm used to implement the presented model. Model parameters are in boldface.

(13)

i  1

i  1

	RESULTS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION REFERENCES

Initially, the independence of the relative dispersion of perfusion from the parameters R₁, P_inflow, and P_terminal was verified (data not presented). The other parameters V₀, V_min, p, , and q have been set to 1,048,576, 2/(1 + ) (see Model implementation), 0.5, 0.1, and 1.0, respectively, unless noted otherwise. These values have been chosen to approximately match experimental data in sheep hearts demonstrating the capacity of the proposed model to generate physiologically reasonable data. Bassingthwaighte et al. (3) have published relative dispersion data on myocardial perfusion in 11 sheep. The average relative dispersion of perfusion as measured with the soluble flow marker 2-iododesmethylimipramine was 33%, ranging from 16% to 47% in individual animals for 254 tissue pieces averaging 217 mg. Aggregation of adjacent pieces resulted in fractal dimensions ranging from 1.07 to 1.30 with a value of 1.16 for the composite data. The number of vessels in the model tree should be sufficient to represent all vessels from the feeding artery to the small arterioles. Kassab et al. (9) have presented data on the number of vessels in pig hearts. Assuming a pig heart weight of ~105 g, they found ~27,900 vessels above 9 µm in diameter per gram of tissue. Assuming that this number includes the terminal arterioles and that vessel density _v scales with animal weight W to the power of 1/4 (19), the relevant number of vessels N in the sheep heart can be estimated to be

In the simulations, we have chosen V₀ to be 2²⁰ = 1,048,576, which results in generating ~2²¹ = 2,097,152 vessels. The average weight of tissue fed by a terminal vessel is ~0.06 mg, equivalent to a volume of ~(400 µm)³. The sample element masses in the experimental data of 217 mg up to 9.6 g are thus approximately equivalent to aggregating 3,600 and 160,000 elements, respectively. Therefore, only relative dispersion values in the simulations from 2,048 to 131,072 aggregated sample elements have been used in an attempt to match the experimental data. Relative dispersion values have not been calculated for sample element volumes >131,072 or <8 sample elements because determination of a standard deviation from only 4 values is unreliable.

Figure 6 shows the logarithm of relative dispersion of perfusion as a function of sample element volume and illustrates the influence of the two major parameters of the model, the asymmetry parameter  and the scaling parameter q. Values of q = 1.0 and  = 0.1 result in a fractal dimension D = 1.15 and a relative dispersion RD of 0.35 at a relative sample volume of 2,048. These values are very close to the average values quoted above for the sheep heart. Both these values for q and  are physiologically reasonable: q = 1.0 is the expected value for nonpulsatile, laminar flow as predicted by West et al. (19), and the asymmetry parameter  = 0.1, which is equivalent to  = 0.51 (q = 1.0 in Eq. 12), falls within the expected range based on data presented by Zamir (20) for distributing vessels in the human heart, which feed large volumes of tissue.

View larger version (20K): [in this window] [in a new window]   Fig. 6.   Log-log plots of the relative dispersion (standard deviation/mean) of perfusion as a function of relative sample element volume (RD plots) for different asymmetry parameters  (A) at q = 1.0 and different scaling parameters q (B) at  = 0.1. RD was computed from 1,048,576 volumes at a relative sample element volume of 1 and from 8 volumes at a relative sample element volume of 131,072. Relative sample element volumes of 2,056 to 131,072 have been fitted with power law functions resulting in the given fractal dimensions D and the correlation coefficients between neighboring volumes C. The fractal dimension D is equivalent to 1 minus the slope of the curves.

It can be noted in Fig. 6A that a variation of  in a range from 0.05 to 0.3 only affects the intercept of the RD curve but not its slope. Of course, this behavior will break down if  is chosen to be very close to 0.5. This is only of theoretical concern, though, because any variance in  will result in a reduction of the mean value of  below 0.5. In contrast, Fig. 6B indicates that a variation of q induces a change in the slope as well as the y-axis intercept. The observation of Bauer et al. (4) that vasodilation changes fractal dimension or the slope of the RD curve could thus be interpreted as a change in the parameter q.

Another interesting finding is that the data in Fig. 6A, if plotted as a function of asymmetry  for different sample element volumes (plot not shown), can be well approximated by an exponential function in the range 0.1 <  < 0.35 for sample element volumes of 32,768 and smaller. Thus the estimation of the asymmetry based on RD data is very easy in this parameter region. The distinct difference in the action of the two parameters  and q allows for a very stable fit of experimental data. The large variations in both the absolute RD values and the fractal dimension of the experimental data (3) indicate that it might be possible to distinguish the underlying vascular morphologies in terms of their asymmetry and resistance scaling.

Although the presence of branching asymmetry is hardly disputable, the scale independence of the asymmetry is only an approximation and seems to change between distributing vessels, which are feeding large volumes of tissue, and delivering vessels, which are supplying small regions (20). In Fig. 7, the effect of a change of  from 0.1 to 0.4 for vessels feeding a volume less than the variable parameter V_-switch is investigated. The  values of 0.1 and 0.4 have been chosen to represent distributing ( = 0.51) and delivering ( = 0.88) vessels, respectively (20). A decrease of the RD values for fed volumes below V_-switch can be observed. The slope of the curve above V_-switch remains unchanged. However, measurements in the rat heart (4) and rat lungs (6) show that perfusion heterogeneities remain fractal down to very small scales. This may indicate that fluctuations in branching asymmetry may be modeled by random fluctuations about a mean value rather than by a distinction between delivering and distributing vessels.

View larger version (21K): [in this window] [in a new window]   Fig. 7.   RD plot with the asymmetry parameter  set to 0.4 for V < V-switch and 0.1 for V > V-switch.

The parameter p of the proposed model determines the probability that the smaller fed volume will be assigned to one of the child vessels. An equally likely assignment to either child (p = 0.5) would represent a very different three-dimensional arrangement of vessels than assigning the smaller fed volume always to the right child (p = 0.0). Relative dispersion curves for different values of p are plotted in Fig. 8. It can be seen that the effect of p on the relative dispersion curves is insignificant. This finding indicates that knowledge of the three-dimensional distribution of vessels, which is ignored in this model, might not be necessary to account accurately for perfusion heterogeneity. The validity of this interpretation remains to be shown by comparison with realistic three-dimensional models.

View larger version (16K): [in this window] [in a new window]   Fig. 8.   RD plot for 3 parameters p, which governs the probability of assigning the larger fed volume to a specific branch of a bifurcation. Only a small change can be noted for totally random (p = 0.5) assignment vs. the totally deterministic (p = 0.0) case.

An important consideration in evaluating the utility of a model is the effect of noise in the parameter space. Small levels of noise should not have a significant influence on model outcome. A true biological vasculature will not exhibit strict self-similarity. The values for  and q for vessels feeding the same volume will fluctuate around a mean value. Simulations were performed to examine changes in the RD curves for noise in the resistance and branching asymmetry of a vessel. The parameter  and the conductance of a vessel C = 1/R were chosen randomly from distributions with means and  = V/R₁. The degree of variability in these parameters was defined by the standard deviation of  and the relative standard deviation _C / of C. A normal distribution with _C/ = 0.3 with C > 0 resulted in a relative standard deviation of the RD values of 8% for nine different seed values at a relative sample element volume (RSEV) of 131,072, which dropped below 2% for RSEV < 4,096. The average over all RSEVs of the absolute deviation of the mean value from the noiseless RD value was 4.5%. To generate a worst case scenario for the variation in , a highly asymmetric distribution () = 0.0253 · ^2.4 for 0.053 <  < 0.288 with  = 0.1 and  = 0.05 was chosen. The relative standard deviation of the RD values was 12% for nine different seed values at a RSEV of 131,072, which dropped below 2% for RSEV < 2,048. The average over all RSEVs of the absolute deviation of the mean value from the noiseless RD value was 3%. Thus a reasonable amount of noise in the simulation resulted in only small changes to the relative dispersion values.

To demonstrate an important property of volume ordering, average volume order of all vessels with the same Strahler order is plotted vs. Strahler order in Fig. 9. For symmetric branching ( = 0.5), volume and Strahler order are equivalent, resulting in a line slope of 1. Obviously, the postulated relationship u_Strahler= log_n(V) + 1 = (U_vol  1)/log₂n + 1 (Eqs. 7 and 8) is not precise for asymmetric branching because Strahler orders are integer valued. Figure 9 shows that, in an average sense, the logarithmic relationship between Strahler order and supplied tissue volume even holds for asymmetric trees as long as the asymmetry is constant at all branch points. However, the line slope increases as asymmetry is increased from 0.5 to 0.1, indicating that n = 2^{(line slope)} also increases. Consequently, it is not possible to infer the volume being fed by a vessel or its absolute scale from its Strahler order without specifying the asymmetry of the vascular system. Thus an analysis based on Strahler order will indicate fractal behavior of systems with constant branching asymmetry, but it is not adequate for fractal analysis of tree structures with changing asymmetry or comparison of fractal dimensions of branching systems with different asymmetry. In contrast, fractal dimensions based on volume ordering are independent of branching asymmetry.

View larger version (21K): [in this window] [in a new window]   Fig. 9.   Plot of average volume order vs. Strahler order for different asymmetry parameters .

	DISCUSSION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION REFERENCES

If perfusion heterogeneity is fractal and exhibits a power law behavior, its scale dependence can be described by only two parameters. However, slope and intercept on a double-logarithmic plot do not reveal the design properties of the underlying vascular structure. The presented model allows a reasonable interpretation of multiscale perfusion heterogeneity data in terms of two morphological parameters  and q describing branching asymmetry and scaling of vessel resistance. It is obvious that two parameters are not sufficient to capture the whole complexity of vascular morphology, but these two parameters contain important physiological concepts that are shown to be of major importance in understanding the remarkable fractal properties of perfusion heterogeneity.

Extensions of the presented model are certainly possible. Examples would be the effect of gravitation on the pressure distribution or a more complicated model of blood flow including pulsatility of flow or the reduction of blood viscosity in small vessels (7, 13). Gravitational influences are expected to be relatively small (7). For pulsatile flow, the anticipated value for s would be 1/2, which would result in q = 5/3 according to West et al. (19). Given that q has been determined to be ~1.0, pulsatility of flow seems to be of minor importance in sheep hearts.

We have argued above that fractal relationships for the length and diameter as a function of scale exist. However, the data gathered by Kassab et al. (9) and vanBavel and Spaan (16) show a large degree of fluctuation of length and diameter measurements. More realistic statistical distributions could be applied in the construction of the vascular tree. It should be noted that a possible correlation of the observed deviations from the expected mean with the distance from the feeding vessel could potentially reduce flow heterogeneities. These correlations have not yet been studied in detail.

Three-dimensional structure could be added to the model, but it is important to realize that every additional parameter in the model will make it more difficult to think about fractal perfusion heterogeneities conceptually. In addition, Glenny and Robertson (7) found that perfusion heterogeneity from a three-dimensional simulation agreed with results based on the aggregation method used by van Beek et al. (17), which is equivalent to the one presented here.

The model currently allows a qualitative understanding of how branching asymmetry and the distribution of vessel resistance influence perfusion heterogeneity. Better data on three-dimensional vascular geometry are still needed to examine the exact meaning of the average parameters  and q as well as to validate the assumption of fractal vascular architecture. Comparison with calculations of relative dispersion of perfusion for a realistic, fully three-dimensional vascular model would also be helpful in this context. To obtain the required data, we are currently studying vascular structure with microcomputed tomography (see Fig. 1). Ideally, one would like to perform a perfusion study with calculation of relative dispersion, extract structural information from a three-dimensional image of the vasculature, and use this to validate the model parameters that have been fitted to the experimental perfusion data. Experiments of this kind will be useful to compare structure and properties of the model tree with those of real vascular trees.

In conclusion, a vascular model based on asymmetric branching and fractal vascular structure has been introduced, which defines branching asymmetry based on the tissue volume that is fed by each vessel. Following the concept of fed tissue volume, a logarithmic volume ordering scheme for vessels is proposed. Fractal dimensions of vascular systems can be compared within this scheme independent of the branching characteristics of the vasculature.

It is shown that the model is able to explain the observed heterogeneity of perfusion and the positive and scale-independent correlation between neighboring perfusion values in organs such as the heart. The two major parameters of the model specify the branching asymmetry of the vasculature and the scaling properties or fractal dimension of vessel resistance. Simulations indicate that branching asymmetry only increases the magnitude of the relative dispersion of perfusion but does not influence its spatial correlation. However, the resistance scaling parameter is related to both the magnitude of relative dispersion and the correlation of perfusion. It is shown that as relative resistance shifts from larger toward smaller vessels, heterogeneity and correlation decrease. The model parameters have been matched to perfusion data in the sheep heart. The results fall within the physiologically expected range. More detailed three-dimensional data on vascular morphology are needed to verify the model assumptions. The model assists in the interpretation of noninvasive perfusion imaging techniques like PET or MRI, allowing the characterization of vascular morphology without acquiring direct information about vascular geometry. This information is very valuable for diagnosis, prognosis, and monitoring therapy response in diseases such as cardiovascular disease, diabetes, arteriosclerosis, hypertension, and cancer.