A Stochastic Model for the Self-Similar Heterogeneity of Regional Organ Blood Flow

A stochastic model for the self-similar heterogeneity of regional organ blood flow

Wayne S. Kendal†

Department of Radiation Oncology, Ottawa Regional Cancer Centre, 503 Smyth, Ottawa, Ontario K1H 1C4, Canada

Edited by Herman Chernoff, Harvard University, Cambridge, MA, and approved November 21, 2000 (received for review July 25, 2000) The theory of exponential dispersion models was applied to Exponential Dispersion Models construct a stochastic model for heterogeneities in regional organ A brief introduction to the relevant aspects of exponential blood flow as inferred from the deposition of labeled micro- dispersion models that closely follows the work of Jørgensen (10, spheres. The requirements that the dispersion model be additive 11) will be presented here. These models serve as error distri- (or reproductive), scale invariant, and represent a compound Pois- butions for generalized linear models (12), and they provide a stan- means to study a large variety of non-normal data. The term ؍ son distribution, implied that the relative dispersion (RD dard deviation͞mean) of blood flow should exhibit self-similar exponential dispersion model reflects, in part, the exponential scaling in macroscopic tissue samples of masses m and mref such form of the generalized linear model. One class of such models 1؊D -RD(mref). (m͞mref) , where D was a constant. Under may be expressed by the canonical equation using the interre ؍ (that RD(m these circumstances this empirical relationship was a consequence lated measures ␯␭, of a compound Poisson-gamma distribution that represented macroscopic blood flow. The model also predicted that blood flow, at ␭ ␪͑ ␧ ͒ ϭ ͵ ͕␪ ⅐ Ϫ ␭␬͑␪͖͒ ⅐ ␯␭͑ ͒ the microcirculatory level, should also be heterogeneous but obey P , Z A exp z dz , a gamma distribution—a prediction supported by observation. A

exponential dispersion model ͉ fractals which describes the distribution P␭,␪ corresponding to the random variable Z defined upon the measurable sets A; ␪ is the ␪⅐ canonical parameter, ␬(␪) ϭ 1͞␭ log ͐ e z⅐v␭(dz) is the egional organ blood flow exhibits a significant degree of cumulant function, ␭ the index parameter, and z the canonical Rspatial heterogeneity when measured by using labeled mi- ␭ ␪ ␪ statistic. P , , for a range of values of , represents a family of crospheres or by other means (1, 2). This heterogeneity is distributions ED*(␪,␭) completely determined by the parame- apparently not random (there exist correlations between neigh- ters (␪,␭) and the cumulant function. This family has the property boring regions of an organ) and over a certain range the relative that the distribution of the sum of independent random vari- ϭ ϩ ϩ ϳ ␪ ␭ dispersion of blood flow reveals a self-similar scaling with respect ables, Zϩ Z1 ... Zn with Zi ED*( , i) corresponding to the size of the region (3–5). Bassingthwaighte and colleagues to fixed ␪ and various values of ␭, belongs to the family of ␪ ϳ ␪ ␭ ϩ ϩ ␭ (1, 6), based on their observations from the deposition of labeled distributions with the same , Zϩ ED*( , 1 ... n). microspheres, have inferred that the relative dispersion RD(m) These distributions, ED*(␪,␭), are called additive. An additive of blood flow (ml͞min⅐g) within tissue pieces of mass m will scale exponential dispersion model will be employed below to describe relative to that from pieces of reference mass mref according to the quantity of radioactivity deposited by blood flow within the equation, individual tissue fragments of equal mass. ␬ ␪ Ϫ Returning now to the cumulant function, ( ), this may be m 1 D ͑ ͒ ϭ ͑ ͒ ⅐ ͩ ͪ used to construct what is known as a cumulant generating RD m RD mref . [1] mref function K*(s) for the additive exponential dispersion model corresponding to the random variable Z: In their work, the constant D has been identified as a spatial ⅐ fractal dimension. K*͑s͒ ϵ logE͑es Z͒ ϭ ␭͕␬͑␪ ϩ s͒ Ϫ ␬͑␪͖͒ . [2] This empirical scaling has been related to the self-similar branching of vascular trees (3), and to the demands placed by Here, s is a variable used to construct the generating function, local tissue metabolism (4). As the regions under observation are and E represents the expectation operator. The cumulant gen-

scaled down to the level of the microcirculation, this relationship erating function determines the distribution function of Z;itis APPLIED presumably would require some modification. We know that the particularly useful when the distribution function is not easily MATHEMATICS expressed in closed form (as is the case with the blood flow capillary network can contribute to the heterogeneity of capil- ϭ lary perfusion through a complex and variable pattern of redis- model). In addition, the first two derivatives of K*(s)ats 0 provide the mean and variance of Z. And, K*(s) can be used to tributed blood flow (7). Blood flow, as indexed by erythrocyte construct new random variables from simpler variables, a prop- velocity within individual capillaries, also appears quite heter- erty exploited in the blood flow model. ogeneous; the related velocity distribution has been character- The function ␶(␪) ϭ ␬Ј(␪) gives the relationship between the ized by a gamma distribution (8, 9). How this microcirculatory canonical parameter ␪ and the mean, ␮ ϭ ␬Ј(␪), is called the heterogeneity might relate to the observed macroscopic heter- mean value mapping. Using this mapping we can define the ogeneity will be considered here. variance function, V(␮) ϵ ␶Ј {␶Ϫ1(␮)}, where ␶Ϫ1(␮) denotes In this article, the theory of exponential dispersion models the inverse function rather than the reciprocal. This function is (10) will be employed to provide a stochastic description for the designed to isolate how the variance behaves with respect to the macroscopic and microscopic heterogeneities in regional organ blood flow. The scaling relationship (Eq. 1) will be shown to be a direct consequence of an exponential dispersion model based This paper was submitted directly (Track II) to the PNAS ofﬁce. on a scale invariant compound Poisson-gamma distribution. A †E-mail: [email protected]. hypothesis, relating this Poisson-gamma distribution to possible Article published online before print: Proc. Natl. Acad. Sci. USA, 10.1073͞pnas.021347898. microcirculatory dynamics, will be presented. Article and publication date are at www.pnas.org͞cgi͞doi͞10.1073͞pnas.021347898

PNAS ͉ January 30, 2001 ͉ vol. 98 ͉ no. 3 ͉ 837–841 Downloaded by guest on October 2, 2021 Table 1. Summary of Tweedie exponential dispersion models (10) Distributions p Domain on R Support ␣ Domain on R ␪ Domain ␮ Domain

Extreme stable p Ͻ 0R1Ͻ ␣ Ͻ 2R0 Rϩ Normal p ϭ 0R ␣ ϭ 2RR

(do not exist) 0 Ͻ p Ͻ 1 nil ␣ Ͼ 2R0 Rϩ Poisson p ϭ 1N0 ␣ ϭϪϱ RRϩ Compound Poisson 1 Ͻ p Ͻ 2R0 ␣ Ͻ 0RϪ Rϩ Gamma p ϭ 2Rϩ ␣ ϭ 0RϪ Rϩ 1 Positive stable 2 Ͻ p Ͻ 3Rϩ 0 Ͻ ␣ Ͻ ⁄2 (Ϫϱ,0] Rϩ 1 Inverse Gaussian p ϭ 3Rϩ ␣ ϭ ⁄2 (Ϫϱ,0] Rϩ 1 Positive stable p Ͼ 3Rϩ ⁄2 Ͻ ␣ Ͻ 1(Ϫϱ,0] Rϩ

R ϵ real numbers, R0 ϵ positive real numbers with zero, Rϩ ϵ positive real numbers, RϪ ϵ negative real numbers, N0 ϵ positive integers with zero.

␣ mean. The mean and variance of an additive random variable can ␣ Ϫ 1 ␪ be expressed by using these quantities as E(Z) ϭ ␭␮ and ␬͑␪͒ ϭ ͩ ͪ , [3] ␣ ␣ Ϫ 1 var(Z) ϭ ␭V(␮). In the model for deposited radioactivity given below, both Z and E(Z) will be expressed in units of counts, where the parameter ␣ ϭ (p Ϫ 2)͞(p Ϫ 1). In this case, the 2 whereas var(Z) will be in units of counts . cumulant generating function for the additive form of the At this point, a second class of exponential dispersion models distribution, K*(s) is (10, 14): should be introduced, with random variable Y ϭ Z͞␭ ϳ ␣ ␮ ␴2 ␴2 ϭ ͞␭ ␮ ␴2 s ED( , ), where 1 . These models, ED( , ), are ͑ ͒ ϭ ␭␬͑␪͒ͭͩ ϩ ͪ Ϫ ͮ termed reproductive exponential dispersion models, and they K* s 1 ␪ 1 . [4] are characterized by a convolution property: For n independent ϳ ␮ ␴2͞ random variables Yi ED( , wi), where the weighting By referring to the properties of cumulant generating functions, ϭ ¥n factors wi are summed thus, w iϭ1 wi, we have under the Eq. 4 can be shown to represent a compound Poisson distribution ͞ ¥ n ϳ weighted averaging of the variables 1 w iϭ1 wiYi generated by a gamma distribution (10, 14, 15). This is essentially ED(␮,␴2͞w). This weighted average of independent random the distribution of the sum of N independent identically distrib- variables, corresponding to fixed ␮ and ␴2 and various values of uted random variables with a gamma distribution where N is a ␮ ␴2 wi, belongs to the family of distributions with the same and . random variable with a Poisson distribution. Eq. 4, with the use A reproductive exponential dispersion model will be employed of Eq. 3, yields for this Poisson-gamma distribution a variance to ͞ ␣Ϫ in the discussion to describe blood flow when expressed in the mean power function: var(Z) ϭ a ⅐ E(Z)p, where a ϭ ␭1 ( 1) physiologic units, ml͞min⅐g. is a proportionality constant. Another property that certain exponential dispersion models In general, the Tweedie exponent p characterizes the related may possess is called scale invariance. For a reproductive expo- distribution (10). Table 1 gives the different distributions that nential dispersion model ED(␮,␴2) we can require that for any correspond to the values of p along with the domains of support positive constant c, c ⅐ ED(␮,␴2) ϭ ED(c␮,c2Ϫp ␴2), where p is for the associated random variable, and the distributional pa- a real-valued and unitless constant. Under this scale transfor- rameters a, ␪, and ␮. In the case of regional blood flow, a mation, the new random variable Yˆ ϭ cY belongs to the same compound Poisson-gamma distribution is employed with 1 Ͻ family of distributions with fixed ␮ and ␴2, but different values p Ͻ 2 and the random variables being supported as positive real of c. Under such transformation the variance function obeys the values with 0. The value of p Ͼ 1, indeed, implies a degree of equation, V(c␮) ϭ g(c) ⅐ V(␮), for an appropriate function g(c). clustering of blood flow within certain regions of a given organ. Scale invariance implies that g(c) ϭ V(c) and V(␮) ϭ ␮p. Scale From a practical standpoint, the basic mathematical tools invariant exponential dispersion models have been termed needed to construct a blood flow model would include the Tweedie models, in recognition of M. C. K. Tweedie’s contri- properties of additivity and scale invariance, and the resultant bution (13) in this area, and p has been called the Tweedie variance to mean power function. With these tools it is possible exponent (10). to construct a model characterized by Bassingthwaighte’s equa- Two differential equations follow from the properties of tion (Eq. 1). Because of its nonlinearity, the Poisson-gamma exponential dispersion models. The first equation, distribution is more difficult to work with. The reader is referred to Jorgensen’s book (10) for details regarding exponential Ϫ Ѩ␶ 1͑␮͒ 1 dispersion models. ϭ , Ѩ␮ V͑␮͒ Regional Blood Flow Described by an Exponential Dispersion gives the relationship between the mean value mapping ␶(␪) and Model the variance function; the second equation, Here, a stochastic model for regional organ blood flow will be presented that yields Bassingthwaighte’s relationship (Eq. 1). Ѩ␬͑␪͒ ϭ ␶͑␪͒ One of the most common methods for measuring regional blood Ѩ␪ , flow employs radiolabeled microspheres (16). Here, microspheres 10–15 ␮m in diameter are injected intravascularly and expresses how the mean value mapping relates to the cumulant then become entrapped in the capillaries [diameters 2.5–9 ␮m function ␬(␪). These two equations may be solved under specific (17)] of the target organ (16). The injected animals are killed, the conditions to obtain the cumulant function ␬(␪), and thus specify target organ is removed and sectioned into equally sized pieces different exponential dispersion models. of tissue, and the amount of radioactivity deposited in each piece In the scale invariant case where V(␮) ϭ ␮p and p Þ 1or2 is measured. The amount of deposited radioactivity within each the cumulant function takes the form (10, 14), piece is assumed proportional to the relative amount of blood

838 ͉ www.pnas.org Kendal Downloaded by guest on October 2, 2021 flow (ml͞min). Blood flow through larger conduit vessels would A relationship resembling Bassingthwaighte’s equation (Eq. not be associated with any significant deposition, and thus would 1), therefore, can be derived to describe the regional heteroge- be excluded by this approach. In this initial formulation, blood neity in the macroscopic deposition of labeled microspheres flow will be considered in the context of engineering units within an organ. This relationship can be viewed as a conse- (ml͞min) so that it can be treated as an additive quantity. A quence of an additive and scale invariant exponential dispersion generalization to include physiologic units (ml͞min⅐g) will be model based on a compound Poisson-gamma distribution. How provided in the discussion to follow. In strict terms, the stochastic appropriate this dispersion model is to the description of blood model to be derived here describes the heterogeneity in the flow (in both engineering terms and physiologic terms) and how deposited radioactivity within organs. Since the deposited ac- it might relate to the microcirculation will be discussed in the tivity has been accepted in the context of experimental blood sections below, but first we will examine the implications of this flow measurements (16), it should similarly be considered a model at the level of the capillary circulation. reasonable surrogate for blood flow in the theoretical model. To proceed with the model, consider an organ that can be Implications for the Microcirculation subdivided into n pieces of equal mass. The deposited radioac- An exponential dispersion model thus seems reasonable to tivity within these pieces will be modeled using an additive describe the macroscopic heterogeneities in the regional depo- exponential dispersion model, and the deposited radioactivity sition of radioactivity as a result of organ blood flow. To within the individual pieces (measured in terms of the number understand how this model might relate to the microcirculation of radioactive counts per piece) will be represented by the we need to first review how the deposition of microspheres independent random variables Z1,...,Zn. Therefore, it should relates to blood flow, and to how blood flow is defined. The sizes be possible to sum the activity contained within adjacent organ of the radiolabeled microspheres are chosen so that virtually all pieces to reconstitute larger pieces. Thus, in theory, one may injected microspheres become entrapped at restrictive sites consider the regional distribution of deposited radioactivity within the microcirculation. The probability that a restrictive site within a series of reconstituted organ pieces, where the recon- might entrap a microsphere presumably depends upon the blood structed pieces can be scaled upwards in size. For simplicity, the flow through the site relative to the flow through all other development of this model was restricted to the case where the restrictive sites within the catchment volume (1). In the context volumes were scaled radially, through one degree of freedom. of an individual restrictive site, blood flow would be best ͞ Thus, the organ tissue was assumed isotropic with respect to all expressed in normal engineering units such as ml min. It is this of its properties, including the heterogeneity of deposited flow through the restrictive site that presumably relates to the radioactivity. chance that a microsphere might be carried to and entrapped If the model were required to be scale invariant, the variance within the site. of the deposited radioactivity within the reconstituted pieces To interpret the exponential dispersion model in the context would relate to the mean radioactivity within the pieces accord- of the capillary entrapment of microspheres requires a number ing to a power function, of assumptions: (i) Within each equally sized sample of tissue, there exists a random number (i.e., Poisson-distributed) of var͑Z͒ ϭ a ⅐ E͑Z͒p . [5] restrictive sites; (ii) the blood flow (ml͞min), at the level of the restrictive sites, obeys a gamma distribution; (iii) the probability The variances for deposited radioactivity as measured for two of entrapment is directly proportional to the blood flow through different sizes of reconstituted tissue pieces would then be the potential traps; and (iv) the blood flow between the indi- related by the scaling relationship, vidual traps is uncorrelated. (Uncorrelated blood flow between traps does not imply that the blood flow between adjacent pieces E ͑Z͒ p ͑ ͒ ϭ ͩ 1 ͪ of tissue is also uncorrelated—this will be discussed below.) var1 Z ͑ ͒ var2(Z) . E2 Z Granted these assumptions, a compound Poisson-gamma distribution may be constructed, in accordance with the cumulant Some further manipulation gives a relationship resembling generating function of Eq. 4. Scale invariance and additivity Bassingthwaighte’s: represented additional properties that were imposed for reason ͑ ͞ ͒ Ϫ of macroscopic requirements to yield Eq. 1. ͱvar ͑Z͒ E ͑Z͒ p 2 1 ͱvar ͑Z͒ 1 ϭ ͩ 1 ͪ 2 A relationship resembling Bassingthwaighte’s equation (Eq. 1) ͑ ͒ ͑ ͒ ͑ ͒ E1 Z E2 Z E2 Z can therefore be derived, granted certain assumptions regarding both blood flow at the microscopic level and the macroscopic ͑ ͒ ͑ p͞2͒ Ϫ 1 E1 Z ϭ ϭ ͩ ͪ deposition of radioactivity consequent to this blood flow. The APPLIED RD1 ͑ ͒ RD2 . E2 Z biophysical justification for these assumptions, and the validity of MATHEMATICS the derived equations, will be discussed below. Here the exponent p corresponds to Bassingthwaighte’s dimension D such that p ϭ 4 Ϫ 2D, and the ratio of means corresponds Discussion to that of the masses. A number of assumptions were made that warrant closer exam- The distribution function for the model can now be specified. ination. First, there was the use of an exponential dispersion If the deposited radioactivity in the ith tissue piece Zi is model to describe the deposition of labeled microspheres. Ex- represented by an additive and scale invariant exponential ponential dispersion models have been particularly useful in dispersion model, and if the exponent p is restricted such that 1 Ͻ describing overdispersed distributions, and they allow the de- p Ͻ 2, then the compound Poisson-gamma distribution in the pendence of the variance upon the mean to be closely examined form given by Eq. 4 would be used to describe the distribution (10). Because the observed range for D (1–5) specified an of radioactivity among the tissue pieces. overdispersed distribution and because Bassingthwaighte’s The restricted range of the exponent p implies that Bassingth- equation (Eq. 1) implied a particular relationship between the waighte’s spatial dimension D will also be restricted: 1 Ͻ D Ͻ variance and the mean, an exponential dispersion model seemed 1.5. This restriction was consistent with the usual range observed appropriate. Moreover, the use of an exponential dispersion for D (18). Other values for p (that would be consistent with Eq. model allowed for the imposition of additivity and scale invari- 1) were possible, but would have required the formulation of ance, two properties that were useful in characterizing the different dispersion models (10, 14). macroscopic distribution of deposited radioactivity.

Kendal PNAS ͉ January 30, 2001 ͉ vol. 98 ͉ no. 3 ͉ 839 Downloaded by guest on October 2, 2021 The assumption of additivity was reasonable, because the total tionship seen for the relative dispersion of blood flow. This deposition of microspheres within a number of tissue samples relationship is apparent for blood flow defined either in engi- should represent the sum from individual samples. However, this neering terms (ml͞min) or physiologic terms (ml͞min⅐g). Both assumption did limit the description of blood flow to that of models are closely related and they rest upon the same biophys- engineering terms (ml͞min), and it was not immediately obvious ical assumptions. that a model so derived would generalize to blood flow measured Another assumption that requires justification was that the in physiologic units (ml͞min⅐g). Scale invariance was justified on blood flow at the different restrictive sites should be uncorre- the basis of the scale invariance inherent to Bassingthwaighte’s lated. If significant correlations were to exist between the blood empirical relationship (Eq. 1), and that arising from work with flow of neighboring capillaries this could prevent the construc- artificial networks (19). tion of a compound distribution. Nevertheless, light microscopic In this model, the deposition of microspheres was considered studies of the microcirculation done by Ellis et al. (7) revealed a an indicator of macroscopic blood flow. Blood flow though large highly complex and uncorrelated movement of blood cells within conduit vessels is not associated with significant deposition of adjacent vessels. In addition, Tyml (20) measured the coefficient microspheres, nor physiologically relevant materials, and thus of variation of red cell velocity in skeletal muscle capillaries and was excluded from consideration. found this to range from 49% to 60%, reflecting a considerable The physiologic definition of blood flow (ml͞min⅐g) can be heterogeneity in the spatial distribution of the microcirculatory problematic, particularly if one wishes to consider regional organ blood flow. And finally, Groom et al. (21) observed a significant blood flow within tissue pieces of nonuniform mass. Because of temporal heterogeneity in capillary blood flow within skeletal the normalization with respect to mass, a flow measurement is muscle, and they proposed that ‘‘even if arteriolar flow were specific to the whole piece; one cannot apportion out regional homogeneous, there would still be heterogeneity of capillary contributions of the (mass-normalized) flow from various seg- flow, because of the structure of the capillary network.’’ On the ments of the piece. Similarly, to reconstitute the blood flow for basis of these observations, the presumption that any such a collection of tissue pieces requires that the flow (ml͞min⅐g) of correlations could be considered negligible therefore seemed each piece be weighted by its individual mass, the weighted flows reasonable. summed, and then the sum divided by the total mass to obtain The assumption of uncorrelated flow between neighboring ͞ ⅐ the reconstituted flow (ml min g). The additive exponential capillary traps did not mean that the flows between adjacent dispersion model, presented above, did not account for blood macroscopic tissue pieces would also be uncorrelated. Macro- flow measured in this manner. scopic correlations in regional blood flow have been well doc- Nevertheless, a closely related exponential dispersion model umented (22). Moreover, such correlations are in fact implied by can be applied to physiologic blood flow measurements. Let the Poisson-gamma distribution through the power variance the independent random variables Y1,..., Yn, describe the function. ͞ ⅐ blood flow (ml min g) in n pieces of tissue that make up an The distinction here is that uncorrelated flow at the level of organ, each with corresponding masses w1,..., wn.Wecan the microcirculation does not imply a lack of correlation within ϭ ͞ estimate the blood flow in the reconstituted organ Yϩ 1 w macroscopic flows. In the exponential dispersion model, the ¥n iϭ1 wiYi. With a nonadditive variable like Y, a reproductive gamma-distributed deposition of microspheres at the level of exponential dispersion models is appropriate. The reproduc- multiple capillaries level was necessarily uncorrelated so that the tive model can also be made scale invariant, and thus can distributions could be summed to yield the Poisson-gamma possess a variance to mean power function. Scale invariant distribution. The total deposited activity within macroscopic reproductive models are also Tweedie models, and they have tissue fragments would involve the summed contributions from a direct correspondence with their additive counterparts. The multiple capillaries, and it would be these macroscopic deposits reproductive form of the cumulant generating function K(s), that would exhibit near-neighbor correlations by virtue of the that corresponds to Eq. 2 is (10): imposed scale invariance. Correlations such as these have been well studied for the one-dimensional case in the context of fractal K͑s͒ ϭ ␭͕␬͑␪ ϩ s͞␭͒ Ϫ ␬͑␪͖͒ stochastic processes (23). The imposition of scale invariance and the requirement that The assumption that the chance of entrapment should be p Þ 1 or 2 gives an equation analogous to Eq. 4 (10): directly proportional to the blood flow through the restrictive sites in any tissue sample seemed reasonable. This assumption ␣ s has been implicitly used and verified in macroscopic studies (2). ͑ ͒ ϭ ␭␬͑␪͒ͭͩ ϩ ͪ Ϫ ͮ K s 1 ␪␭ 1 . [6] As for the assumption that the number of entrapment sites should be distributed randomly within each tissue piece, accord- In short, regional blood flow in physiologic terms (ml͞min⅐g) can ing to a Poisson distribution, this too seemed reasonable as a first be represented by a scale invariant reproductive exponential approximation. Indeed, the anatomic placement of microvessels dispersion model that is characterized by a power function seems to obey a Poisson distribution (24), and presumably the relationship between the variance and the mean, var(Y) ϭ placement of restrictive sites would follow similar statistics. E(Y)p͞␭, resembling that seen with the additive model pre- We are left with the assumption that the blood flow through sented above (Eq. 5). Bassingthwaighte’s relationship can be the sites of entrapment should be gamma-distributed. Capil- derived from this exponential dispersion model for blood flow lary blood flow has been noted to obey a gamma distribution expressed in physiologic units (remembering that E(Y) ϭ (8, 9). Since the entrapment sites are likely associated with E(Z)͞␭). The probability distribution that arises from this capillaries, and the gamma distribution is additive, it seemed reproductive model (Eq. 6) is also a compound Poisson-gamma reasonable that the flow through the restrictive sites should distribution, albeit parameterized differently. also be gamma-distributed. Again, at this point it is important The different parameterizations for the reproductive model to emphasize the distinction between the flow observed be- should be emphasized. The mean and variance of the reproduc- tween tissue fragments at the macroscopic level and that tive random variable Y are, respectively, E(Y) ϭ ␮ and var(Y) ϭ observed within capillaries. Bassingthwaighte has observed ␴2V(␮). Despite the different parameterizations used in the macroscopic flow distributions that are fairly symmetrical (1), additive and reproductive models for blood flow, it is the and not as skewed as seen with those measured within underlying scale invariance that yields the power function rela- individual capillaries by microscopic techniques (8, 9). As

840 ͉ www.pnas.org Kendal Downloaded by guest on October 2, 2021 noted above, and according to the proposed model, these dispersion model provided a framework from which Bassingth- phenomena are described by two different distributions. waighte’s empirical relationship for blood flow (Eq. 1) was a Some of these assumptions, made at the microcirculatory direct consequence of the scale invariance. Although the model level, may not be strictly accurate. However, in the context of our was itself stochastic, it nevertheless possessed a nonrandom incomplete understanding of microcirculatory hemodynamics, symmetry—scale invariance. This symmetry underlies these assumptions seemed reasonable. The scale invariance Bassingthwaighte’s empirical relationship for blood flow (Eq. 1), inherent to Bassingthwaighte’s relationship (Eq. 1) may reflect and it relates to the correlations, which appear between neigh- vascular structure; it may also reflect long-range and continuing boring organ regions (22). demands of local tissue metabolism. These two processes, more- In conclusion, two stochastic models, based upon scale invari- over, may be related. Other interpretations for the exponential ant Poisson-gamma distributions, have been derived to describe dispersion model at the level of the microcirculation, neverthe- regional heterogeneities in organ blood flow. The additive model less, are also conceivable. was applied to blood flow described in engineering units As previously mentioned, Bassingthwaighte’s relationship (ml͞min); the reproductive model to blood flow in physiologic (Eq. 1) has been attributed to the branching structure of the units (ml͞min⅐g). Both models implied power-law scaling of the vasculature (3), and to the demands of local tissue metabolism relative dispersion of their respective blood flows attributable to (4). The stochastic model presented here provided a kinematic scale invariance. These models were explained on the basis of description for the heterogeneities in blood flow, but it did not gamma-distributed blood flow through random restrictive sites specify the biophysical origins of scale invariance or additivity. within the microcirculation; the scale invariance was attributed Presumably, these properties could be reflective of vascular to the structure of the vascular tree. The theory of exponential structure. Similarly, the reasons why microcirculatory flow might dispersion models, used to derive these models, provides a be gamma-distributed remain unclear. A better understanding of practical means to examine the effect of scale invariance upon microcirculatory dynamics and the dynamics of blood flow in a stochastic models, and thus potentially may provide descriptions dichotomous vasculature would be required to pursue these for other random processes with fractal symmetries. concerns. This article presents an application of exponential dispersion I would like to acknowledge the support provided by the Beattie Library models, in particular Tweedie models (10, 13), to describe the and the Department of Radiation Oncology, both of the Ottawa self-similar pattern of heterogeneous organ blood flow. The Regional Cancer Centre.

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