Journal of Neuroscience Methods 138 (2004) 81–87

Analysis of cerebral microvascular architecture—application tocortical and subcortical vessels in rat brain

Mei Lu a,∗, Zheng Gang Zhangb, Michael Choppb,c

a Department of Biostatistics and Research Epidemiology, OFP-3E, Henry Ford Health System, One Ford Place, Detroit, MI 48202, USAb Department of Neurology, Henry Ford Health System, 2799 West Grant Boulevard, Detroit, MI 48202, USA

c Department of Physics, Oakland University, Rochester, MI 48309, USA

Received 29 October 2003; received in revised form 2 March 2004; accepted 9 March 2004

Abstract

Cerebral vascular morphology, including vascular diameters, lengths and branch points, was measured using quantitative three-dimensional(3D) imaging software developed in our laboratory, where data were collected from cortical and subcortical regions of a rat brain. To comparemultiple records of the vascular morphological differences between anatomical regions with unpaired data, a simple analysis of variance(ANOVA) model may not be applicable, or may not be valid without considering correlation and unpaired data. In this paper, we formulatea novel outcome to study vascular morphological differences, discuss proper paired test statistics on unpaired data and apply the proposedmethods to study the cerebral vascular morphological differences between cortical and sub-cortical regions in rats (N = 6).

The total vessel surface was analyzed as a composite of diameter, length and branches of vessels. The extended Fisher permutation testprovides the paired test on unpaired data, which is equivalent to a permutation test based on the average of multiple sections; however, thepermutation test may not be exact. The repeated measure analysis of variance using Generalized Estimating Equations (GEE) can be used toconduct the paired test on unpaired data as an alternative, when the permutation test is not exact. These new approaches can be applied in thestudy of treatment effects on changes of vascular morphology under physiological and pathological conditions.© 2004 Elsevier B.V. All rights reserved.

Keywords: Vascular morphology; Brain; Generalized Estimation Equations (GEE); Paired test; Correlated data; Permutation test

1. Introduction

Vascular biology, the structure and characterization,particularly of cerebral microvessel, have profound implica-tions for both health and disease. New vessels, i.e., angio-genesis and vasculogenesis, as well as alterations in vascularstructure are areas of vibrant research effort (Folkman,2001; Yancopoulos et al., 2000). Providing accurate andcomprehensive measurements of vascular morphology is anessential step in the analysis of vascular formation (Drakeet al., 1998; Pettersson et al., 2000). Using novel imageanalysis software in combination with the laser-scanningconfocal microscopy, we quantify cerebral microvascularmorphology in three-dimensions, with the measurement ofbranch points, lengths and diameters of the vessels (Zhanget al., 1999, 2001). In a study of cerebral microvascular

∗ Corresponding author. Tel.:+1-313-874-6413;fax: +1-313-874-6730.

E-mail address: mlu1@hfhs.org (M. Lu).

morphology, vascular characteristics were measured withreplicate tissue samples, called sections, in a region of inter-est (ROI), such as the cortical region and subcortical regionof a rat brain. We hypothesize that the vascular structure inone region differs from the other anatomical region. How-ever, with data collected from multiple sections within eachregion and with two regions involved in each subject, wecannot use a simple statistical approach directly, e.g.,t-testor analysis of variance (ANOVA). In this paper, we: (1)perform an experiment on rats to address the differences inthe cerebral microvascular morphological patterns betweentwo anatomical regions using structural analysis of imagesobtained from laser scanning confocal microscopy; (2) for-mulate a novel microvascular morphological measure thatpermits evaluation of microvascular morphological differ-ences between brain regions; (3) perform statistical analysispaired test data based on unpaired data; and (4) apply thenew measure to analyze the cerebral vascular morphologi-cal difference between the cortical and subcortical tissue inthe rat brain.

0165-0270/$ – see front matter © 2004 Elsevier B.V. All rights reserved.doi:10.1016/j.jneumeth.2004.03.011

82 M. Lu et al. / Journal of Neuroscience Methods 138 (2004) 81–87

2. Experiment: reconstructing the cerebralvessel system

2.1. Brain tissue preparation

Healthy male Wistar rats weighing between 390 and450 mg were administered FITC-dextran (2×106 molecularweight, Sigma, St. Louis, MO; 1 ml of 50 mg/ml) intra-venously. After FITC-dextran was permitted to circulate for1 min, the rats, under deep halothane anesthesia were sac-rificed by decapitation. Brains were rapidly removed andplaced in 4% paraformaldehyde at 4◦C for 48 h. Coronalsections (100�m) were cut on a vibratome. It has been con-firmed that the FITC-dextran fluorescent marker perfusesall microvasculature in the brain (Zhang et al., 1999).

2.2. Three-dimensional image, acquisition and analysis

A vibratome coronal section (100�m) at interaural8.2 mm, bregma 0.8 mm, from each rat was analyzed witha Bio-Rad MRC 1024 laser scanning confocal imagingsystem mounted onto a Zeiss microscope (Bio-Rad, Cam-bridge, MA) (Zhang et al., 1999). Multiple field views wererandomly selected from the parietal cortical and sub-corticalregions with a 40× oil immersion objective size in the260.6�m × 260.6�m × 20�m in x, y and z dimensions.Three-dimensional (3D) quantitative measures of cerebralmicrovessels, perfused by FITC-dextran, were processedusing in-house software (3DVQ). 3DVQ quantitativelymeasures the vessel diameters, lengths and branching points(Zhang et al., 2002). Data illustrated inFig. 1 displays thereconstruction of the cerebral vessel system using the 3Dquantitative image algorithm on one rat, where one fieldreview was selected from the cortical region (1A–1C) andthe sub-cortical region (1D–1F); 1A and 1D are the originalimages, 1B or 1C and 1E or 1F are images after imageacquisition using 3DVQ software.Fig. 2 plots the lengthand diameter of each branch in the cortical region (Fig. 2A)and in the sub-cortical region (Fig. 2B) with data col-lected from the different sections of one rat. The branchesrange from 131 to 210 among the sections in the corti-cal region, and range from 105 to 153 in the sub-corticalregion. We hypothesize that the microvessel morpho-logical in one region differs from the other anatomicalregion.

3. Formulation of the microvascular morphologicalmeasures

In general, the cerebral microvessels are collected ran-domly in multiple sections of the anatomical region withmore than one region involvedin each subject. For simplic-ity, we assume two regions (e.g., cortical and sub-corticalregions of the brain) are involved. SupposeLn

j,mj,land

Dnj,mj,l

are the length and diameter of thelth branch for

l = 1,2, . . . , Lmj in jth region for j = 1,2, . . . , mjthsection for mj = 1,2, . . . ,Mj and nth subject forn = 1,2, . . . , N.

The null hypothesis is that there is no microvascularmorphological difference between the two regions. The alter-native hypothesis is that microvascular morphological struc-ture in one region differs from the other region (a regionaleffect).

To formulate a composite outcome of interest that is ef-ficient and provide an encompassing analysis of microvas-cular morphological structure in a region, we employ thebranch-point,Bj,mj = Lmj (Drake et al., 1998; Zhang et al.,2002), which is an important morphological measure in de-termining newly formed vessels. Vessels can also be classi-fied by their diameters as: capillary (<7.5�m), precapillaryarterioles and postcapillary venules (from 7.5 to 30�m), andsmall arterioles and connecting vessels (>31–50�m) (delZoppo, 1994). However, neither of those two outcomes, i.e.,Bj,mj and the vessel diameter, provides a full assessmentof vessel structure. We therefore, propose novel index, totalvascular surface, as the outcome of interest. The total sur-face has the formula,

Snj,mj= π

Lmj∑l=1

Lnj,mj,l

Dnj,mj,l

(1)

whereLnj,mj,l

and Dnj,mj,l

are the length and diameter ofthe lth branch forl = 1,2, . . . , Lmj in jth region forj =1,2, . . . , mjth section formj = 1,2, . . . ,Mj andnth sub-ject for n = 1,2, . . . , N.

The microvessel surface is a composite outcome of allmajor vessel characteristics (branch, length and diameterof the vessel). The vessel surface provides an index ofvessel morphological structure, which is independent ofblood flow and intake. The surface area gives a quanti-tative measure of vascular morphology and may be usedto measure vessel growth or angiogenesis (newly formedvessels).

In studying microvascular morphology, we are inter-ested in the vessel surface as the primary outcome, thebranch-points and the percentage of diameters<7.5�m asthe secondary outcomes. Note that there is no match amongthe multiple sections selected from one region to the other;e.g., the measurement inSection 1of Region 1 (e.g., thecortex region) does not match the measurement inSection 1of Region 2 (e.g., the sub-cortex region).

4. Statistical considerations

4.1. Paired test on summary data

SupposeZnj,mj

is the outcome of interest, described inSection 3in jth region forj = 1,2, mjth section formj =1,2, . . . ,Mj andnth subject forn = 1,2, . . . , N. We con-sider the mean (average) of the multiple observations at each

M. Lu et al. / Journal of Neuroscience Methods 138 (2004) 81–87 83

Fig. 1. Three-dimensional images of cerebral vessels in the cortical region (A–C) and the sub-cortical region (D–F) from a representative healthy rat.Panels A and D are original composite images of FITC-dextran perfused cerebral microvessels. Panels B, C, E and F were generated by the 3DQVprogram. Green and red colors in panels B and E code for diameters of blood vessels less than 7.5�m and larger than 7.5�m, respectively. Differentcolors in panels C and F represent individual vessels that were not connected to each other. The unit for numbers in all images is�m.

Fig. 2. (A) Length and diameter of cerebral vessel distribution in the cortical region from four sections on one rat. The number of points in the plotsis the number of branches, where variable Location 2 is the section. (B) Length and diameter of cerebral vessel distribution in the sub-cortical regionfrom four sections on one rat. The number of points in the plots is the number of branches, where variable Location 2 is the section.

84 M. Lu et al. / Journal of Neuroscience Methods 138 (2004) 81–87

region per subject, defined as

Zn

j,• = 1

Mj

Mj∑mj=1

Znj,mj

(2)

Let yn• = Zn

1,• − Zn

2,• , the regression modeyn• will be

yn• = µ0 + e (3)

whereµ0 is the coefficient of the microvascular morpholog-ical difference between the regions ande is the error termwith distribution ofN(0,σ), for someσ > 0. Testing the dif-ference between regions is equivalent to testing thatH0µ0 =0 using the pairedt-test, if observations of the differences inmean are normally distributed. A significant regional effectis detected, ift-test statistic̄y•−µ0/(s

√N − 1),> tN−1,0.05

of a one-sided test.If differences are not normal, the nonparametric Wilcoxon

matched-pairs signed-rank test will be considered. The testbased on the summary data may be less efficient comparedto the test that uses the individual data points, because itappears to lack information collected from individual datapoints.

4.2. Paired test on unpaired data using the repeatedanalysis of variance

We consider all possible pairs between two regions (M1×M2) and define the pair difference as

ynk = Zn1,m1

− Zn2,m2

(4)

for m1 = 1,2, . . . ,M1, m2 = 1,2, . . . ,M2 and k =1,2, . . . ,M1×M2(=K). Let vectorY∗

n = (yn1, yn2, . . . , y

nK)

′,for n = 1,2, . . . , N andk = 1,2, . . . , K. The response ofpair difference can be described as a regression model

Y∗n = µ0 + e∗n (5)

whereµ0 is a common intercept among all of the pairs and∼e∗n ∼ N(0, φV), with knownK × K covariance matrixVand unknown scale measureφ.

We could use the repeated measure analysis of varianceto estimate the difference between regions,µ0, as a regionaleffect with the maximum likelihood (ML) estimation, as-suming exchangeable correlations among correlated obser-vations forEq. (4). We could also use the quasi-likelihoodapproach in Generalized Estimation Equations (GEE) withmore causal assumption of the distribution function ofthe vessel difference between the regions,Y∗

n , for n =1,2, . . . , N. A significant regional effect is detected, if theone-sidedP-value for the regional effect<0.05.

Using GEE to estimateµ0, we assume the identity linkfunction and variance structure without attempting to spec-ify the entire distribution of observations ofY∗

n , given thatthe quasi-score function depends on the distributionY∗

n onlythrough its mean,µ0 in Eq. (5), and covariance matrix,V,and assuming that covarianceV is a function ofµ0, which

indicates a weak assumption of the normality on the errorterm, compared to ML estimation. In practice, we often donot know the probability mechanism by which data are gen-erated.Zeger and Liang (1986)showed the robustness ofGEE estimations, even under a weak assumption about theactual covariance structure, by comparing estimates underdifferent covariance structures. One widely accepted prop-erty of the method of GEE is that, if the covariance struc-ture is mis-specified, the estimates of regression parameterswill remain consistent, but may lack efficiency, as previouslydiscussed (McDonald, 1993; Zhao et al., 1992).

Nevertheless, in analyzing microvascular morphology fora regional effect, we encountered abnormality data, espe-cially when the differences were taken between the regions.Although GEE has a weak assumption on the underlying dis-tribution and covariance structure, to increase the efficiencyof the test statistic, the underlying distribution of data shouldbe evaluated to meet, or come close to, the distribution re-quired. The analysis will be conducted using PROC GEN-MOD in SAS (SAS Institution, 1999) assuming an identitylink and exchangeable covariance structure.

4.3. Paired test on unpaired data using permutation test

Fisher (1936) proposed the permutation test for aone-sample test for the paired difference; assuming pairsare random samples from the same population, then all ofthe pairs are equally likely to have positive or negative. Hetherefore determined the significance of the observed sumof the differences with reference to distribution that is ob-tained by taking both signs for each difference with a totalof 2N permutations, ifN subjects are involved. However,for unpaired data, we cannot use Fisher’s permutation testdirectly, because the pairs are no longer random samplesand are restricted by regions. An extended permutation testis proposed based on the subject, but not on pairs, whichincludes the following steps:

1. Chose the observed statistic as the sum of all of the pairdifferences with the expression,

D0 =N∑n=1

K∑k=1

ykk =N∑n=1

M1∑m1=1

M2∑m2=1

(Zn1,m1

− Zn2,m2

)

=N∑n=1

M2

M1∑m1=1

Zn1,m2

−M1

M2∑m2=1

Zn2,m2

=M1M2

N∑n=1

[∑M1m1=1Z

n1,m2

M1−

∑M2m2=1Z

n1,m2

M2

]

=M1M2

N∑n=1

(Zn

1,• − Zn

2,•) = K

N∑n=1

yn• (6)

whereyn• is the average of differences between regionsdefined inSection 4.1.

M. Lu et al. / Journal of Neuroscience Methods 138 (2004) 81–87 85

2. Randomly assign+ or − sign for each subject, wherea negative sign indicatesa region switch. Generate allof the possible pairs and then calculate the test statistic,calledD, with the same calculation inEq. (6).

3. Repeat Step 2 for 2N times or a large number of times,which will be described later ifN is large, each timecalculating the test statistic specified in Step 2, whereNis the number of subjects.

4. Obtain the upper alpha-percentage point of the permuta-tion distribution (e.g.,Dα, whereP(D ≥ Dα) = 0.05)and accept or reject the null hypothesis according towhetherD0 from the original data is larger than valueDα.

A complete permutation test requires 2N permutations,which would be impossible whenN is large. One thousandpermutations are almost certain to give the same results asthe full distribution except in rather borderline cases, withP very close to 0.05 (Manly, 1997).

Note that,Eq. (6) is proportionally equivalent to the dif-ference in averages of multiple sections of each subject,defined inSection 4.1. To our surprise the extended per-mutation test of the difference based on unpaired data isequivalent to the Fisher permutation test based on the sum-mary data by averaging multiple sections in each subject.There may be differences between the pairedt-statistic inEq. (3) and its permutation test. The pairedt-test assumesnormal distribution and determines the significance of theregional effect based on tables of thet-statistic, while thepermutation test determines the significant regional effectbased on permutation distributions.

Cotton (1967)followedFisher (1936)in emphasizing thatdetermining significance by use of conventional statisticaltables (e.g., thet-table for Student’st-test) is of questionablevalidity until it has been shown that the significance obtainedclosely agrees with that given by its corresponding permuta-tion test. These properties make the permutation tests quiteattractive as a validation tool for a conventional test statistic(Lu et al., 2001). Edgington (1995)showed that the permu-tation test is more powerful than thet- or F-test when dataare highly skewed. Nevertheless, we may consider using aparametric test when we have a very small number of obser-vations and assume that the underlying corresponding para-metric test may be relied on. For example, if we only havethree observations with which to test the hypothesis that the

Table 1Distribution of microvascular morphological difference between the cortex and sub-cortical region

Parameter Mean (standard) Median (range) Skewness Kurtosis

Surface 28263.78 (45727.79) 12074.13 (−27755.09, 151953.75) 0.99 1.46No. of branches 67.50 (75.51) 45.50 (−84.00, 270.00) 0.90 0.62Diameter >7.5�m 0.02 (0.09) 0.0 (−0.22,+0.21) 0.16 1.52

R. surface 43.21 (44.39) 34.21 (−72.0, 96.0) −0.60 −0.51R. no. of branches 41.40 (37.69) 41.40 (−72.0, 96.0) −0.91 1.08R. diameter >7.5�m 41.53 (37.54) 41.50 (−71.5, 96.0) −0.91 1.12

mean is zero, the sample space for the permutation test islimited to 23 or 8 permutations. As a result, the smallestprobability ofP(D ≥ Dα) in Step 4 is 1/8 = 0.125. On theother hand, at a significant level of 0.05, a more powerfultest parametric test may be obtained directly from tables oft-statistic, if data are normal.

Manly (1997)agreed withDwass (1957)that the permu-tation test is an exact test, althoughGood (1998)cautionedthat the permutation test is almost, but not quite, an exacttest. A test is said to be an exact test with respect to a hy-pothesis of distribution if the probability of making a type Ierror is exactlyα in any region of hypothesis space (Good,1998). In other words, an exact test (a particular test regard-less of the underlying distribution) cannot overestimate thetype I error. A necessary condition for a one-sample permu-tation test to be an exact test is that observations are symmet-ric (Good, 1998). The GEE approach, defined inSection 4.2,will be considered as an alternative, if the sample size is toosmall or data are not symmetric.

5. Experiment: results

To test that cerebral microvascular morphological struc-ture differs in the cortical tissue from sub-cortical tissuein rats, six healthy rats were employed in this experiment.Cerebral vessel structure measurements were performedin the cortical and sub-cortical regions from four randomsections of each region, respectively. As seen inFig. 2,vessel density is greater in the cortical region than in thesub-cortical region. The other rats had similar patterns. Toexplore the morphological difference between the regions,we chose the vessel surface area, number of branches andpercentage of diameters<7.5�m as outcomes of interest.We considered all possible 16 paired differences inEq. (4)on six rats, defined inEq. (4) with 96 correlated observa-tions in Eq. (5). Spearman Correlation coefficients amongthree outcomes of interest are 0.86 between the vessel sur-face and branch-points, 0.77 between the vessel surface andpercentage of diameters<7.5�m, and 0.97 between thebranch-points and percentage of diameters<7.5�m.

Data were highly skewed with a large variability and a lackof normality (Table 1). We explored various data transfor-mations in an attempt to reduce the variability, but failed toreach normality. Therefore, we considered the signed ranked

86 M. Lu et al. / Journal of Neuroscience Methods 138 (2004) 81–87

Table 2One-sidedP-value for microvascular morphological difference betweenthe cortical and sub-cortical regions

Parameter Method 1 Method 2 Method 3

Surface 0.031 0.018 0.031No. of branches 0.016 0.004 0.016Diameter >7.5�m 0.025 0.019 0.016

Method 1: test based on collapsed data (Wilcoxon matched-pairs signedrank test); Method 2: repeated analysis of variance on signed rank data,Eq. (5); and Method 3: permutation test (on the summary data or onunpaired data of raw data or signed ranked data).

data transformation, defined asy∗ = sign(y) × ranked(y)for all pair differencey. The skewness and Kurtosis wereimproved for the difference in surface (Table 1).

We tested the effect ofµ0 in Eq. (5) based ony∗ usingGEE. The results are presented inTable 2. A significant highdensity of vessel surface was detected in the cortical regioncompared to the sub-cortical region with the median (range)difference of the cortical region from the sub-cortical regionas 12074.13 (−27755.09, 151953.75) andP-value= 0.018using Eq. (5). Furthermore, there were significantly morefrequent branch points and diameters<7.5 in the corticalregion than in the sub-cortical region withP-values= 0.004and 0.019, respectively, based on the sign ranked data andP-values= 0.027 and 0.019 based on the raw data.

Permutation tests were conducted based on the obser-vation test statistic as the sum of all of the pairs. As aresult, there are incidences of higher vessel density in thecortical region compared to the sub-cortical region on thevessel surface (P-value= 0.031), and the branch-points anddiameters<7.5 (P-values= 0.016). Results are the samebased on either the signed ranked data or the raw data.The results are consistent among the proposed analysisapproaches.

6. Discussion and conclusion

In the present study, using the proposed statistical meth-ods, we demonstrate the regional heterogeneity of the cere-bral microvessels, which are consistent with previous studiesby Hudetz (1997)demonstrating that the surface of cere-bral vessels for vascular morphology is as efficient as othervessel structure measurements. The cerebral microvascularsystem is composed of highly irregular patterns of tortu-ous and anastomosing capillaries. Therefore, our proposedmethods may be applied for analyzing vascular morphologyunder physiological and pathological angiogenesis.

The results of test statistics for vascular morphologydifferences between regions are consistent, considering aone-sided test, but we would reach different conclusions if atwo-sided test was considered. Initially, given the data illus-trated inFigs. 1 and 2, we thought that taking the summarydata among multiple sections would reduce the amountof information provided from each individual data point.

However, we have shown, in theory, that the extended per-mutation test of differences on unpaired data is equivalentto the Fisher permutation test of differences based on thesummary data by taking the average of the sections, whichis much easier to compute compared to the extended permu-tation test. The permutation test is not always an exact test,but becomes an exact test with no restrictions for the under-lying distribution if the pairs of differences are symmetric.The GEE approach to conduct a paired test on unpaireddata can be used as an alternative, when data symmetryfails or sample size is too small. Although GEE has weakassumptions on the underlying distribution and covariancestructure, to increase the efficiency of the statistic test, theunderlying distribution of data should be evaluated to meetor come close to the distribution pre-specified in the linkfunction. It has been noted in the literature that, for smallsamples, the GEE approach with sandwich estimator maynot perform well and may lead to inflated type I error forthe Wald chi-square test (Pan and Wall, 2002) which wouldbe a concern for the previous experiment where only sixsubjects are involved. The quasi-likelihood score test usingGEE based on the individual data may improve the preci-sion of the test, but results of GEE need to be interpretedwith caution.

In conclusion, we have developed statistical methods totest for the differences between regions in the same sub-ject with multiple, unequally sized, unpaired observations.We have proven in theory, that the extended permutationtest for the difference between regions based on unpaireddata is equivalent to the test based on the summary data.The GEE analysis approach can handle the test of pairs onunpaired data, specifically when the permutation test is notexact.

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