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The fractal spatial distribution of pancreatic islets in three dimensions: a self-avoiding growth
model
View the table of contents for this issue, or go to the journal homepage for more
2013 Phys. Biol. 10 036009
(http://iopscience.iop.org/1478-3975/10/3/036009)
Home Search Collections Journals About Contact us My IOPscience
The fractal spatial distribution ofpancreatic islets in three dimensions: aself-avoiding growth modelJunghyo Jo1,5, Andreas Hornblad2, German Kilimnik3, Manami Hara3,Ulf Ahlgren2 and Vipul Periwal1,4
1 Laboratory of Biological Modeling, National Institute of Diabetes and Digestive and Kidney Diseases,National Institutes of Health, Bethesda, MD, USA2 Umea Center for Molecular Medicine, Umea University, Umea, Sweden3 Department of Medicine, The University of Chicago, Chicago, IL, USA
Received 4 December 2012Accepted for publication 8 April 2013Published 29 April 2013Online at stacks.iop.org/PhysBio/10/036009
AbstractThe islets of Langerhans, responsible for controlling blood glucose levels, are dispersed withinthe pancreas. A universal power law governing the fractal spatial distribution of islets intwo-dimensional pancreatic sections has been reported. However, the fractal geometry in theactual three-dimensional pancreas volume, and the developmental process that gives rise tosuch a self-similar structure, has not been investigated. Here, we examined thethree-dimensional spatial distribution of islets in intact mouse pancreata using opticalprojection tomography and found a power law with a fractal dimension of 2.1. Furthermore,based on two-dimensional pancreatic sections of human autopsies, we found that thedistribution of human islets also follows a universal power law with a fractal dimension of 1.5in adult pancreata, which agrees with the value previously reported in smaller mammalianpancreas sections. Finally, we developed a self-avoiding growth model for the development ofthe islet distribution and found that the fractal nature of the spatial islet distribution may beassociated with the self-avoidance in the branching process of vascularization in the pancreas.
1. Introduction
Improving glucose homeostasis by regenerating the insulin-secretion capability in diabetics is a major medical priority.Insulin is secreted by beta cells in the endocrine pancreas. Anormal beta-cell function depends on the collective behaviorof endocrine cells organized in the islets of Langerhans. Thus,apart from the crucial question of increasing beta-cell mass,the process of formation and function of the islet distributionin a normal pancreas is of topical interest in the light of rapidadvances in bioengineering.
The fractal geometry of self-similar structures is observedin many organs such as the lung, kidney, brain and vasculature[1–5]. Such structures may originate from developmental
4 Author to whom any correspondence should be addressed.5 Present address: Asia Pacific Center for Theoretical Physics, Pohang, Korea.
processes using basic building rules repeatedly, as confirmedin lung formation [6]. A defining property of fractals is thescale independence of self-similar patterns. The space fillingproperty is measured by the fractal dimension, a quantitythat can take non-integer values. The methods for measuringfractal dimensions from biological images have been reviewed[7]. The fractal dimensions, D2, measured usually from two-dimensional (2D) organ images are 1.8 in the bronchial tree [1],1.6 in the renal arterial tree [2], 1.5 in the cerebral cortex [3],1.3 in the pial vasculature [4] and 1.7 in the retinal vasculature[5]. In addition to organ structures, the spatial distributionsof islets on pancreas sections have been reported to follow afractal geometry with D2 = 1.5 in guinea pigs [8] as well asin dogs, pigs, goats and cows [9].
These fractal dimensions were computed from 2D imagesof organs. Recent advances in imaging methods, however,have provided three-dimensional (3D) information, e.g., the
3D fractal dimension of the lung, D3 = 2.3–2.5 [10]. Inparticular, optical projection tomography (OPT) has provided3D spatial distributions of islets in intact mouse pancreata[11]. In addition to such advances in imaging methods, theavailability of computational resources has enabled practicaladvances in automated image analysis [12]. Applying these tohuman tissues has allowed the examination of the geometryof islet distributions in the entire human pancreata, althoughthe data obtained from large human organs are still limitedto 2D sections [13]. In this study, we examine the spatialorganization of islets in 3D mouse pancreata and 2D humanpancreatic sections using data obtained with these advances.
It has been speculated that the fractal geometry of thespatial islet distribution is associated with vascularization inthe pancreas [8]. Here we explicitly examine this hypothesiswith a self-avoiding growth model and show that the spatialfractal distribution of islets may result from self-avoidance inthe branching process of vascularization.
2. Self-avoiding growth model
The model starts with a single node (islet) at a 3D position,y1. At the next growth step, the node can generate (branchinto) a new node at y2 = y1 + �y, where �y is a random 3Dvector with |�y| < l. Here l is a distance scale obtainable fromthe mean distance between the nearest islets in experiments.In the following step, every node including the mother anddaughter nodes can generate new nodes. In the growth process,we introduce self-avoidance to inhibit close overlaps betweennodes. A standard mechanism for the directionality of growthprocesses is the diffusion of special molecules such as chemo-attractants and chemo-repellents, e.g., in axon guidance [14].Using the idea of chemo-repulsion, we assume that every nodeat yi produces a chemo-repellent factor, which diffuses outfrom the existing nodes and finally degrades. Given N nodes,the potential (concentration) of the chemo-repellents, �(x, t),at position x at time t can be described by
∂t� =N∑
i=1
δ(x − yi) + D∂2x � − d�, (1)
where D and d are diffusion and degradation constants. Sincethe diffusion and degradation are expected to occur muchfaster than the growth process, we consider a stationary stateof the above equation and obtain a spatial gradient of thechemo-repellents. We can solve this differential equation bythe Fourier transform:
�(k) = 1
Dk2 + d
N∑
i=1
e−ık·yi . (2)
Finally, the inverse Fourier transform of �(k) gives
�(x) =N∑
i=1
λ
|x − yi| e−|x−yi|/λ, (3)
where λ = √D/d is an effective distance of the chemo-
repulsion from the existing nodes. In this study, we set λ = lto match the effective distance as the mean minimal distancebetween islets. It is of interest that �(x) has the form of the
Yukawa potential in particle physics. Therefore, a new nodecan be generated in a unit volume around a point, x, near theexisting ith node (|x − yi| < λ) with the probability density
P(x) = A�−m(x), (4)
which is assumed to be proportional to the inverse ofthe chemo-repellent gradient from the existing nodes. Theparameter m determines how strongly the chemo-repulsionaffects the growth process with a corresponding normalizationconstant, A. When no additional nodes exist near the ith node,the normalization constant is approximated as A−1(m) ≈∫ 1
0 rmemrdr. However, depending on the exponent m, a fewnodes exist together within the effective distance λ fromthe ith node. For example, there are ten and five nodeson average within the effective distance for m = 0.5 and3, respectively. In practice, it is not necessary to computethe exact normalization constant A, which requires a heavycomputation at every addition of new nodes. Instead, we use apractical normalization constant,A′(<A), that equally reducesP(x) at every position, x. We have checked that the growthprocess is independent of the specific value of A′ if it is smallenough (e.g. A′ = 0.01A), while it must also be large enoughnot to reject every trial of node addition.
The practical algorithm for adding a new node to theexisting N nodes is as follows.
1. Calculate the chemo-repellent gradient �(x) inequation (3), generated by the existing N nodes.
2. Randomly select one node (e.g. the ith node) among theN nodes.
3. Randomly pick a coordinate near the ith node, x = yi+δy,where yi is the coordinate of the ith node and δy is arandom 3D vector with |δy| < l.
4. Accept the addition of the new node with a probability,P(x), in equation (4).
5. Iterate these steps to add more nodes.
3. Results
3.1. Fractal spatial distribution of pancreatic islets
OPT imaging captured the 3D spatial distribution of theislets scattered in the exocrine tissue of the mouse pancreas(figure 1(A)). Based on the centroid coordinate of everyislet, we calculated the average number 〈n〉 of neighboringislets within a distance r. This allowed us to determine thecorrelation dimension D [15], a type of fractal dimension,from the relation 〈n〉 ∝ rD. For example, when points arehomogeneously distributed in a line, surface and volume, theirfractal dimensions become 1, 2 and 3, respectively, as expectedin the Euclidean space. The logarithmic plot of 〈n〉 versus rshowed the expected power-law behavior in 3D (figure 1(B)),first observed in the 2D pancreatic sections in guinea pigs withthe fractal dimension D2S = 1.5 [8]. Independent of pancreaticlobes (splenic, duodenal and gastric lobes), the 3D fractaldimension was D3 = 2.1 (table 1). The box-counting methodgave a similar 3D fractal dimension, 2.03 ± 0.24 (P = 0.3).
The spatial islet distribution information in the pancreasis usually extracted from 2D pancreatic sections, especially
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100
101
102
103
104
102 103 104
⟨ n⟩
r (μm)
(A) (B )
Figure 1. Spatial distribution of islets in the three-dimensional (3D) pancreas. (A) Optical projection tomography-based iso-surfacereconstruction of a splenic lobe from a C57BL/6 pancreas at eight weeks. The islets are reconstructed based on the signal frominsulin-specific antibody staining (red) whereas the exocrine parenchyma is reconstructed based on the signal from endogenous tissuefluorescence (gray). The scale bar is 1 mm. (B) Average number 〈n〉 of islets within a distance r. The linear relation in their log–log plotrepresents a power law, and its slope (2.1) corresponds to the fractal dimension of the spatial islet distribution. Note that the finite size ofpancreata gives a cut-off distance above which power-law behavior disappears. Therefore, we excluded data points at large separations ifthey changed the first decimal place of the fractal dimension (slope), the confidence interval in this study.
Table 1. Fractal dimensions of the spatial islet distributions in mouse and human pancreata. The fractal dimensions of islet distributions inthree-dimensional volume (D3) and two-dimensional projections (D2P) and sections (D2S) of the pancreas were calculated based on isletcentroid coordinates. Ntotal represents the total number of islets in the pancreas volume, while Nsection represents the number of islets on thepancreas section. Mean ± s.d. (n samples).
a Pancreata from female mice were isolated, stained for insulin and subjected to optical projection tomography as describedin our previous study [24]. The 3D islet centroid coordinates were extracted using the Imaris software. For the estimation ofthe 2D projection and section fractal dimensions, the projection plane was selected by rotating the pancreas in 3D tomaximize the averaging distance between islets on the projected plane, and the section plane along with the given projectionplane was selected to include the maximal number of islets within a section depth of 100 μm.b Pancreata were excised from MIP-GFP mice in which beta cells are genetically tagged with green fluorescent protein underthe control of the mouse insulin 1 promoter [36]. Fixed and cleared pancreata were then placed between a slide glass andcover slip. To capture entire islets including those that are not in perfect focus, an epifluorescent configuration with theuncertainty of Z depth was used with a confocal microscope. The islet centroid coordinates were extracted using the ImageJsoftware. Detailed methods are described in our previous study [37].c Automated image analysis was used to examine centroid coordinates of islets on pancreatic sections from human autopsies[13].d The fractal dimensions of projection D2P showed no significant difference between C57BL/6 and MIP-GFP mice.e The fractal dimensions of sections D2S showed a significant difference between C57BL/6 mice and humans (P < 0.01)
for large animals including humans. To examine the methoddependence, we compared islet distributions in the 3D, 2Dprojections and 2D sections of the full 3D data (figure 2(A)).Note that random sections of the non-symmetric 3D structurein figure 1(A) could result in large variations in the fractaldimension measurement. Therefore, we consistently chosethe unique 2D projection plane and section that maximize
the average distances between islets and the number of isletswithin the section, respectively, by rotating the 3D pancreas.Although all three methods gave power-law behaviors inthe spatial distribution, they had different fractal dimensions(figure 2(B) and table 1). The 3D fractal dimension has beenextrapolated as D3 = D2S + 1 by adding a unit co-dimensionbased on the fractal dimension D2S measured on the 2D
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101
102
103
104
102 103 104
⟨n⟩
r (μm)
3 dimensionprojectionsection
(A) (B )
Figure 2. Fractal dimensions depending on organ preparations. (A) Spatial distribution of islets in a splenic lobe of the 3D pancreas (black),the two-dimensional (2D) pancreatic section (section, red) and with uncertain Z values (projection, blue) in an eight-week C57BL/6 mouse.(B) Average number 〈n〉 of islets within a distance r for the three preparations: slopes, fractal dimensions, in the log–log plots are 2.1 (3D),1.6 (projection) and 1.3 (section).
pancreatic section [8]. Our direct observation demonstratedthat D3 was different from D2S + 1 (P < 0.01), suggestingthat islet distributions are not symmetric in space. This wasobvious in the 3D OPT image (figure 1(A)).
We also examined islet distributions in intact mousepancreas with uncertainty in Z depth, which corresponds tothe 2D projection. The fractal dimension did not change withage (table 1), which is consistent with the age independencefound in guinea pigs [9].
Furthermore, the recent availability of human tissuesallowed us to examine the spatial islet distribution in humanautopsies [13]. We analyzed the fractal dimension withpancreatic sections from adult pancreata. They also followedthe power law with the fractal dimension D2S = 1.53 ± 0.07(table 1), which is very close to 1.5, as reported for othersmaller mammals [9]. Note that a relatively smaller animal,mouse, showed a significantly smaller fractal dimension,D2S = 1.3 (table 1). In our model, this corresponds to a smallervalue of m.
3.2. The emergence of fractal geometry
Pancreatic islets are co-localized with pancreatic ducts [16]and blood vessels [17] in the pancreas. Thus, it has beenspeculated that the fractal geometry of spatial islet distributionsis associated with the vascular structure of pancreatic ducts[8, 18]. In pancreatic development, the primitive endodermalepithelium differentiates into the pancreatic duct and theendocrine and exocrine cells [19]. Note that the clustersof endocrine cells finally form islets. Blood vessels play acritical role for organ development by providing not onlyoxygen and nutrients, but also inductive signals for endocrinecell differentiation [20]. On the other hand, endocrinecells produce the vascular endothelial growth factor, whichattracts endothelial cells of blood vessels. Therefore, theorganogenesis of endocrine pancreas results from the mutualinteraction between the endocrine cells and blood vessels.Surprisingly, however, Magenheim et al have recently reportedthat blood vessels reduce the branching and differentiation of
epithelial cells [21]. The dual effect of inducing and inhibitingendocrine cell differentiation by blood vessels may lead to aself-avoiding formation of islets in the pancreas.
Here, we developed an organ growth model consideringself-avoidance. The model focused on islet formation withoutintroducing the epithelium and blood vessels explicitly. Thusislets, represented as nodes in space, were assumed to producea chemo-repellent potential, �, to implicitly incorporate theself-avoiding effect resulting from the mutual interactionbetween the islets and blood vessels. In the model, newislets (nodes) were generated from the existing nodes witha probability P ∝ �−m. The parameter m controls the degreeof self-avoidance. For example, a vanishing m amounts toneglecting self-avoidance, while a large m limits the growthto locations furthest from the existing nodes. Figure 3 showsexamples of the self-avoiding growth with various values ofm. With a negligible self-avoidance (m = 0.5; figure 3(A)),nodes were densely distributed with a high fractal dimension(D3 = 2.74 ± 0.04). On the other hand, with a relativelystrong self-avoidance (m = 3; figure 3(C)), nodes weresparsely distributed with a small fractal dimension (D3 =1.88 ± 0.05). Finally, the self-avoiding growth model withm = 1.5 could produce a node distribution pattern resemblingthe islet distribution in the pancreas (figure 3(B)). The fractaldimensions of the 3D, 2D projections and 2D sections of themodel were consistent with the fractal dimensions of isletdistributions, when the node number is truncated to 2000 asthe total islet number in adult mice (tables 1 and 2). The self-avoiding growth model could generate node distributions withdistinct fractal dimensions (figure 3(D) and table 2). Notethat during the growth process with a fixed m, the fractaldimension generally increased until the node number becamelarge enough to make finite size effects negligible (table 2).
4. Discussion
A universal power law in the spatial distributions of pancreaticislets has been reported in guinea pigs, dogs, pigs, goats andcows with the fractal dimension D2S = 1.5, measured in 2D
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101
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102 103 104
⟨ n⟩
r (μm)
(A) (B )
(C ) (D)
m = 0.5 m = 1.5
m = 3.0
m = 0.5m = 1.5m = 3.0
N=100N=1000N=2000
Figure 3. Self-avoiding growth model. In the growth process, new nodes (points) are added with a lower probability proximal to existingnodes. Suppose that existing nodes produce a chemo-repellent potential, �(x) (see the text for details). Then, the probability for generatinga new node at a position x is inversely proportional to the potential, P(x) ∝ �−m(x). The exponent m determines how strongly thechemo-repulsion affects the growth process: (A) negligible (m = 0.5), (B) medium (m = 1.5) and (C) large (m = 3.0) effects ofself-avoidance. The evolution of the growth process is represented by color dots: 100 (red), 1000 (black) and 2000 (gray) nodes added. Axisarrows at every left-bottom represent the same scale for the three models. In addition, the right panel for every model displays a skeletalview where neighboring dots are connected by gray lines. (D) Average number 〈n〉 of islets within a distance r for the three models: slopes,fractal dimensions, in the log–log plots are 2.5 (m = 0.5), 2.1 (m = 1.5) and 1.9 (m = 3.0).
Table 2. Fractal dimensions in the self-avoiding growth model. Thedegree of self-avoidance produced different fractal structures. Alarger m produces stronger self-avoidance. The model had finite sizeeffects on the fractal dimension. As the model with m = 1.5 growsfrom 1000 to 5000 nodes, its fractal dimensions of the 3D, 2Dprojections and sections approach equilibrium. Mean ± s.d.(n = 10).
pancreatic sections. In this study, we demonstrated that humansalso followed the universal power law with the same fractaldimension. Furthermore, with the advanced imaging method,OPT, we found the fractal dimension in the 3D pancreas tobe D3 = 2.1 in mice. Note that the null hypothesis, D3 = 2(surface in the Euclidean space), was rejected (P < 0.001).
The pancreas has distinct developmental origins of thedorsal (splenic) and ventral (duodenal) pancreas, which later
fuse [22]. The gastric lobe is formed by a perpendiculargrowth from the dorsal pancreas a few days after its formation[23]. Our recent study has shown regional differences in thepancreas that the gastric lobe has a higher relative number ofislets (i.e. islets mm−3 of pancreatic tissue) than the splenicand duodenal lobes [24]. However, we found indistinguishablefractal dimensions of spatial islet distributions in all three lobesof the pancreas, although the higher islet density in the gastriclobe was reflected as a slightly higher fractal dimension thatwas not statistically significant. This suggests that the threeregions may be governed by the same developmental rule.
The fractal dimension in the 3D pancreas has beenextrapolated as D3 = 2.5 by adding a unit co-dimension to the2D section fractal dimension D2S = 1.5, under the assumptionof symmetric islet distributions in space [8]. Because thediffusion-limited aggregation (DLA) in 3D generates a tree-like structure with the fractal dimension D3 = 2.4 [25], it hasbeen proposed as the underlying mechanism of the spatialdistribution of pancreatic islets [18]. However, our directmeasurement of the 3D fractal dimension, D3 = 2.1, wasclearly different from D3 = 2.4 of the DLA. Furthermore,endocrine cells are continuously nucleated from the sourceof the endodermal epithelium during pancreatic development.Therefore, the organ expansion is fundamentally differentfrom the DLA in which particles scattered in a large volumeaggregate to the center of a cluster [26]. The self-avoidinggrowth model, however, captured the organ expansion process.
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The DLA-like branching pattern has also been described bya neurite growth model attracted (not avoided) by chemicalgradients [27]. Our model also explained the symmetrybreaking, D3 �= D2S + 1, because the self-avoiding growthprefers to grow in directions as orthogonal as possible tothe locations of the proximal existing nodes. When theself-avoiding effect becomes negligible (smaller m), spatialsymmetry is restored.
The model with m = 1.5 could reproduce similar 3D,2D projections and 2D section fractal dimensions of thespatial islet distributions in mice, considering the uncertaintyof measurement. In particular, the uncertainty in the 2Dsection is considerable because the fractal dimension of the2D section depends greatly on the position and orientation ofthe section plane in the pancreas volume. In general, smalleranimals had smaller fractal dimensions (D2S) in the spatialislet distributions: mice (1.31 ± 0.07), rats (1.42 ± 0.04) [9]and humans (1.53 ± 0.07). To reproduce the spatial isletdistributions in larger animals including humans, the samemodel with a smaller m (weaker self-avoiding effect) couldrecreate a larger fractal dimension in the larger pancreas withmore islets. This growth algorithm can be generally applicableto generate self-similar structures such as the vasculature withflexible fractal dimensions.
The inter-species differences of lung airway structureshave been carefully examined by considering a multiplicityof scales [28]. Outer scales and, in more generality, thequestion of multi-fractal scaling would indeed be an interestingexplanation for the islet data in different species. However, ourtheoretical attempts were limited by the amount of data we hadavailable. The species difference can be simply understood interms of self-avoidance, if we accept that different species canhave different strengths of self-avoidance in islet development.Our data are limited to just center coordinates of islets, unlikethe lung data with specific branch lengths and diameters.Nevertheless, multi-fractal scaling might lead to a modelwithout self-avoidance that might apply to all species.
Node generation and self-avoidance in thephenomenological model could reflect the dual roles ofblood vessels for inducing and inhibiting endocrine celldifferentiation. Nevertheless, this model gives little insightinto the specific mechanisms of self-avoidance. In addition,the model did not generate a conserved fractal dimensionduring its evolution. Ignoring the measurement uncertainty,the fractal dimension does not change with age in guineapigs [9] and also in the mice shown in this study. Duringthe evolution of the model, however, the fractal dimensionslightly increased until the node number became large enoughto overcome finite size effects.
The self-avoidance of islet formation may play a role forproducing islets with optimal sizes. Endocrine cells tend toaggregate in embryonic development [29] and even in vitroculture [30]. The continuous generation of proximal cells,therefore, could lead to a gigantic cluster of cells. Constraintson endocrine cell differentiation by blood vessels [21] mayinhibit the formation of large aggregates of cells and help togenerate optimal sizes of islets evenly distributed throughoutthe pancreas. The functional importance of islet sizes has
been emphasized [31–35]. As progress is made in both tissuescaffold engineering and the induction of desired lineagecharacteristics in induced pluripotent stem cells, the geometryof how functional replacement organs might be best engineeredbecomes important. As such, our growth model provides asimple characterization of how to grow finite samples withthe desired geometric properties for the optimal function ofpancreatic islets.
Acknowledgments
The work was supported by funding from the KempeFoundations, Umea University, and the Intramural ResearchProgram of the National Institutes of Health, National Instituteof Diabetes and Digestive and Kidney Diseases.
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