From www.bloodjournal.org by guest on June 14, 2017. For personal use only.
Morphogenesis of the Bone Marrow: Fractal Structures and
Diffusion-Limited Growth
By Faramarz Naeim, Farhad Moatamed, and Muhammad Sahimi
All three methods, in all
Bone marrow (BM) provides a particular spatial organization clear area, and (C) cell numbers.
cases, showed that thespatial structure of the BMis fractal.
that allows interaction between its variouscomponents.
The average values of the fractal dimensions (D‘) were 1.7
Characterization ofthe spatial patterns in the BM and underf 0.08, 1.64 f 0.1, and 1.69 f 0.04 for categories A, B, and
standing the mechanismsthat give riseto them may play a
role in better understandingof the BM pathologic processes. C, respectively. The overall valueof D, for the cellularity in
0% to 60% was about 1.67 k 0.09. Fractal
the range of 4
Morphometric analyses were performed in BM biopsy samdimensions of 1.6 to 1.7 represent configurations
that correples from 30 patients (16 men and 14 women) with an averspond to two-dimensional diffusion limited aggregation
46 years, rangingfrom 17 to 77 years.The biopsies
age age of
structures, suggesting that the structural configuration of
were obtained during the course of patient care to rule out
the diffusion of regulaBM involvementin a variety of hematologic disorders before hematopoietic cells is dependent on
tory cytokines in the BM.
or aftertherapy.Threedifferent,
but structurally interre0 1996 by The American Society of Hematology.
lated, parameters were measured: (A) cellular area, (B) nu-
B
ONE MARROW (BM) provides the microenvironment
for hematopoietic progenitor cells, extremely labile
cells that continuously go through self-replication and/or differentiation. Such cells are supported by a variety of factors,
including (1) a seemingly random network of irregularly
branching vascular structures, (2) stromal cells that are distributed unevenly throughout the BM, and (3) a diverse extracellular matrix that is laid down in various directions. The
BM structure provides a particular spatial organization that
allows interactions between its various components.’ These
interactions are coordinated by complex regulatory mechanisms, leading to patterns and average productivity that are
known to us as normal range.
However, as with any other biologic systems, the spatial
patterns in the BM are very complex. Characterizing such
patterns and investigating the mechanisms that give rise to
them may be the key to understanding the pathologic processes of the BM. In the past several years, several biologic
systems with complex spatial patterns have been analyzed
and have been shown to possess a fractal and self-similar
str~cture,2.~
ie, a structure that repeats itself at various length
scales. For example, Smith et a14and Caserta et als suggested
that the shapes of the dendritic branching of neurons growing
in culture and in the cat retina are fractal. Family et a16
provided the first quantitative analysis and characterization
of the geometry of blood vessels in human retina and showed
thatthey form fractal patterns. Mainster’ analyzed retinal
arterial and venous patterns and concluded that they possess
a fractal structure. Bifurcation patterns of vessels in the cat
brain and in the human retina were investigated by Matsuo
et a18 and were shown to be fractal and self-similar. BenJacob et a19 have proposed an interesting model for the
growth of bacterial colonies in culture and have shown that,
under certain circumstances, bacterial colonies display a
fractal structure. We have undertaken a systematic investigation of the spatial patterns in the BM. In this report, we
present compelling evidence that such patterns are fractal
and propose a mechanism that gives rise to them.
MATERIALS AND METHODS
BM biopsy samples from the iliac crest of 30 patients were studied. Sixteen patients were men and 14 were women, with an average
age of46 years, ranging from 17 to 77 years. The BM biopsy samples
were retrieved from the laboratory tissue archival storage (1988
Blood, Vol 87, No 12 (June 15). 1996: pp 5027-5031
through 1995) at the UCLA Medical Center. The biopsies were
performed during the course of patient care to rule out BM involvementin a variety of diseases such as anemia, thrombocytopenia,
leukemia, multiple myeloma, and metastatic tumor before or after
therapy. Specimens, measuring 1 to 2 cm in length, had been prepared by routine procedures of fixation, decalcification, dehydration,
and paraffin-embedding;by cutting sections with a thickness of about
3 to 5 pm; and by staining with hematoxylin and eosin.
The stained sections were evaluated under conventional transmission light microscopy for structural studies and detection ofBM
lesions. All biopsy sections included in this study depicted normal
morphology, with no evidence of any structural abnormality or disease process. Simultaneous bilateral (left and right iliac crest) biopsy
specimens were available in 3 patients, and 2 patients had more than
one BM biopsy taken in a time period ranging from 1 to 2 months.
For the morphometric analyses, a Nikon (Melville, NY) Microphot-SA microscope, equipped with plan-apochromat lenses, was
used. A Sony (Montevale, NJ) 3 CCD (3 chips) high-resolution color
video camera model DXC-960MD was attached to the microscope
for delivering images to a Matrox (Dorval, Quebec, Canada) imagegrabber board model MPV AT, installed in the computer. An Intel
100-MHZ Pentium-based computer (Gateway 2000, North Sioux,
SD) equipped with a rewriteable magneto-optic storage device, operating under MS-DOS version 6.22 and Microsoft (Redman, WA)
Windows version 3.1 environments, was used. The computer was
also equipped with another Matrox graphics-board, capable of displaying images with a resolution of 1,024 X 768 pixels and 24 bits
of color depth. Adobe Photoshop and Novell Presentation graphics
software programs (Novell Corp, Orem, UT) were used to arrange
and print the images on a Mitsubishi color video copy processor
From the Hematopathology Laboratories, Department of Pathology and Laboratory Medicine, UCLASchool of Medicine, Los
Angeles, CA; the Immunology/Molecular Pathology Laboratories,
Department of Pathology and Laboratory Medicine, Wadsworth VA
Medical Center, Los Angeles, CA; and the Department of Chemical
Engineering, University of Southern California, Los Angeles, CA.
Submitted October 2, 1995; accepted February 12, 1996.
Supported in part by the Department of Veterans Affairs.
Address reprint requests to Faramarz Naeim, MD, Department
of Pathology and Laboratory Medicine, UCLA Medical Center, Los
Angeles, CA 90024.
The publication costs of this article were defrayed in part by page
charge payment. This article must therefore be hereby marked
“advertisement” in accordance with 18 U.S.C. section I734 solely to
indicate this fact.
0 1996 by The American Society of Hematology.
0006-4971/96/8712-0039$3.00/0
5027
From www.bloodjournal.org by guest on June 14, 2017. For personal use only.
Fig l. A representative measurement of
the cellular areas(S.) is performed on one ofthe microscopic
images (using l o x objective lens) from patient no.
19. The largeyellow areas representfat cells outside
the measurement area. In the circular region,the artifactual cracks (red areas) are excluded and only
the
surface area of the fat cells (blue areas) are measured. S. is determinedby subtracting the fat surface
area from the entire circular region.
Fig 2. (A) A representative measurement of the
nuclear area (S.) is shown on thesame image as in
Figs l and 28. In the large circular region, the selected nuclei (blue areas) are measured. The pink
dots within the large circle are the excluded pixels
based on the size criteria.Two (largestand smallest)
of the eight circular regions are
shown in this photograph. (B)A hematoxylin & eosin-stained photomicrograph from patient no. l 9 showing the BM structure. Two of the four circular regions (largest and
smallest) are overlaid on the photomicrograph. For
analysis, the confined regions within each of the
for the manual counteight circles were used except
ing in which only four of the concentric circleswere
Used.
From www.bloodjournal.org by guest on June 14, 2017. For personal use only.
5029
BONE MARROW MORPHOGENESIS
in 10 specimens, the images of 40 randomly selected microscopic
fields (4 for each specimen) were printed, and the total cell numbers
were counted manually in 4 concentric circular regions with increasing radii (Fig 2B).
Thus, three different, but structurally interrelated, parameters were
used for the analysis of the patterns and determination of their fractal
dimension in the BM sections, organized in three categories as follows: (A) cellular area (the total area of the BM minus the fat), S,;
(B) the area occupied by nuclei of BM cells (nuclear area), S.; and
(C) the number of the BM cells, N.
If X denotes S,,S., or N, and if the spatial pattern of the BM
components is fractal with a fractal dimension Df, then for a circle
of radius r, one must have X ,Df, and thus Dr is the slope of the
straight line that one obtains when log X is plotted versus log r. The
radius of the innermost circle was r = 15.6 pm and that of the
outermost was r = 173 pm, more than one order of magnitude larger
than the innermost circle.
model CP21OU (Mitsubishi Electronics, Inc, Cypress, CA). Image1 software from Universal Imaging Corp (West Chester, PA) was
the main program used to perform the necessary morphometric analyses. The software feaNres included acquiring color images directly
from the microscope through the video camera and storing them at
a resolution of 512 X 480 pixels with a color depth of 24 bits.
Images were calibrated in micrometers for the final system magnification using the 1OX objective lens.
Duplicate sections were prepared for each biopsy specimen. Four
different microscopic fields were randomly selected in each BM
specimen. A cellular cluster was consistently placed at the center of
each image. A total of 38 BM specimens and 148 microscopic
images were evaluated. In each image, 8 centered concentric circular
regions were analyzed. Each region had a surface area twice as large
as the next immediate inner circle. In each region, total cellular area
(the entire region minus the corresponding fat area; Fig 1) and all
the surface areas of the nuclei (Fig 2A) were measured. In addition,
-
Table 1. Cellularity and Fractal Dimensionsof Three Structural Categories in the BM Biopsy Sections From30 Patients:
Average of Four Randomly Selected Microscopic Fields
% Cellularity
SexlAge
Patient No.
l-L
2-R
2
3
4
5
6
7
8
9
10
11
12
13
14
15-L
15-R
16
17
18
19
20-L
20-R
21
22
23 # l
#2
#3
24 # l
#2
F135
M159
MI38
MI52
MI50
M121
Mi63
W40
Mn7
M137
F133
M144
F154
MI37
F/48
MD6
F148
MI17
F160
F142
F165
F154
W37
M/20
#3
25
26
27
28
29
30
Average
W54
F146
Mi75
Mi53
F151
F137
Cellular Area
Nuclear Area
Cell Number
Average
SD
DI
SD
DI
SD
73.50
74.75
49.00
60.50
62.00
45.50
45.75
54.75
68.75
42.75
53.25
51.75
44.00
75.25
62.75
56.50
63.50
55.50
65.00
61.OO
53.00
40.75
52.75
51.25
57.00
59.75
78.25
57.75
55.00
55.50
35.00
45.50
56.75
52.75
51.75
44.25
55.25
10.50
3.00
16.75
6.00
2.80
8.80
4.50
4.50
3.80
6.50
8.40
2.00
9.00
12.70
7.25
7.20
5.50
1.70
6.30
1.70
2.20
2.30
10.00
3.50
6.00
3.00
2.00
4.00
8.50
5.00
4.00
5.00
2.00
5.00
1.30
2.40
9.00
1.82
1.86
1.59
1.75
1.74
1.60
1.61
1.71
1.81
1.59
1.69
1.66
1.56
1.88
1.75
1.71
1.76
1.72
1.77
1.76
1.69
1.58
1.67
1.67
1.70
1.74
1.88
1.74
1.69
1.71
1.54
1.68
1.69
1.66
1.67
1.60
1.71
0.07
0.03
0.16
0.05
0.03
0.07
0.09
0.03
0.03
0.10
0.10
0.03
0.09
0.03
0.04
0.08
0.04
0.05
0.07
0.02
0.03
0.05
0.10
0.01
0.08
0.03
0.02
0.03
0.09
0.03
0.06
0.08
0.09
0.05
0.03
0.05
0.07
1.66
1.81
1.47
1.65
1.75
1.48
1.61
1.65
1 .87
1.55
1.71
1.60
1.45
1.86
1.64
1.67
1.77
1.71
1.72
1.72
1.56
1.61
1.54
1.55
1.65
1.60
1.72
1.73
1.81
1.62
1.59
1.58
1.56
1.52
1.44
1.59
1.63
0.37
0.07
0.1 4
0.14
0.05
0.18
0.07
0.10
0.12
0.12
0.19
0.10
0.07
0.04
0.16
0.10
0.04
0.1 1
0.03
0.18
0.10
0.20
0.15
0.02
0.18
0.08
0.1 1
0.06
0.12
0.16
0.08
0.05
0.08
0.12
0.1 1
0.1 1
0.13
55.89
9.96
1.70
0.08
1.64
0.1 1
Abbreviations: L, left iliac crest; R, right iliac crest.
D,
SD
1.73
1.70
1.66
1.75
1.65
1.66
1.71
1.72
1.65
1.71
0.07
0.06
0.06
0.09
0.05
0.18
0.04
0.03
0.12
0.02
1.69
0.04
From www.bloodjournal.org by guest on June 14, 2017. For personal use only.
5030
NAEIM,MOATAMED.AND
Df= 1.70
A
Cornlation: r = .W
4.8
Y
/'
,
1
4.4
4
4
3.6
1
3.2
1
1.2
1.4
1.6
1.8
2
2.2
2.4
Logr
D/= f.65
Cornlation: r = .W
B
4.4
i
l
',
I
Y
41
3.6
3.2
2.8
/'
,/
SAHlMl
The resulting typical plots are shown in Fig 3. All the three
methods indicate that, in all cases studied, the spatial structure of the cellular pattern is fractal. The mean values of DI
were 1.72 0.08, 1.64 ? 0.1, and 1.69 2 0.04 for categories
A, B, and C, respectively. Thus, the overall value of Dt, for
the cellularity in the range 40% to 60% is estimated to be
Df = 1.67 2 0.09.
The mean cellularity of the 148 microscopic fields was
56% 2 10%.To test the existence of any possible deviation
from the estimated D,., we also analyzed the data for the
five
largest concentric circles and estimated the
fractal dimensions. The estimated values were consistent with the above
results, indicating that no significant deviation from our results should be expected if using larger concentric circles
were possible.
Table I indicates that there may be some correlation between the BM cellularity and Df. For example, in normocellular marrows, ie, those with cellularity in the range 40% to
60%, the fractal dimension isintherangegivenabove,
whereas the apparent fractal dimension for the hypercellular
BMs, ie, those with cellularity greater than 60%, seems to
be larger than our overall estimates. Moreover, in patients
1, 15, and 20 in Table
with bilateral biopsies (patients no.
l ) and in those with recurrent biopsies (patients no. 23 and
24) depicting more cellular marrow, there seem to be higher
apparent fractal dimensions. Finally, in hypocellular BMs,
ie, those with low (<40%) cellularity, the fractal dimension
is lower than the mean estimate.
DISCUSSION
I
I
1
1.2
C
1.6
1.4
1.8
22.4
2.2
Df = 1.69
Correlation:r = .W
3.2
2.8
2.4
2
1.6
l
_I'
d
,
1.2 /,
1.2
_1
1.4
1.6
1.8
2
2.2
2.4
Logr
Fig 3. Typical plots of the logarithm of the radius (r) versus the
logarithm of the cellular area (Se),the nuclear area En), and the cell
number (N).The data are obtainad from imageanalysisof a randomly
selected microscopic field from the BM sections of patients no. 19
(A), 20 (B), and 7 (C).
RESULTS
The average and standard deviationof the BM cellularity
and thecorresponding fractal dimensions in each BM
biopsy
section for categories A, B, and C are presented in Table 1.
Fractal dimensions of approximately 1.6 to I .7 may representtwo-dimensionalfractalstructures
that are formed as
the result of adiffusion-limitedaggregation
(DLA)process.'o,'' In this process, one starts with a seed particle located at the center of an empty system. Diffusing particles
are then released into the system, far from the seed and one
at a time, and are allowed to move(diffuse) until they arrive
at the vicinity of the existing particles, at which
time they
stop their diffusion and adhere to the system. If this process
i s repeated for a large numberof particles, a fractal aggregate
is formed whose fractal dimension (in two dimensions) is
D, = 1.6to 1.7. In thisstudy, thin BM biopsysections,
analyzed under a transmission light microscopy, represent a
two-dimensional system; a fractal system in which the overall value of fractal dimension, for the cellularity in the range
40% to 60%, is estimated to be 1.67 -+ 0.09, consistent with
a DLA-like process. Proliferation and differentiation of the
hematopoietic cells in the BM is regulated by complex interacting processes that send stimulatory or inhibitory signals
from the environment to the cells. Cytokines, one
of the
major carriers of the signals, diffuse in the BM and communicate with the cells. Numerous cytokines involved in the
regulation of hematopoiesis have been identified and structurally characterized."." Most of theseregulatorshavea
broadnetwork of respondingtarget cells and overlapping
pathways by signaling through mechanisms that are fundamentally similar to the diffusion-limited process disclosed
here. The effect of a cytokine on a cell at a particular stage
of differentiation depends on itslocal concentration, itsdura-
From www.bloodjournal.org by guest on June 14, 2017. For personal use only.
BONE MARROW MORPHOGENESIS
tion of exposure, and its interaction with other regulatory
cytokines and on the topographical status of the cell in its
en~ironment.""~Many of these features reflect the rate of
the cytokine diffusion in the BM. In this study, the demonstration of BM fractal structure and fractal dimensions compatible with a DLA-like process strongly suggests that the
spatial organization of the hematopoietic cells in the BM is
dependent on the diffusion of the regulatory cytokines. If a
DLA-like process is the dominant mechanism for the formation of the spatial patterns of the BM in healthy subjects,
then such patterns in the three-dimensional space of the BM
have a fractal dimension Df 2.45. Such a structure is very
complex and branched."
The existence of a DLA-like process in the BM may
explain some of the pathologic processes. For example,
structural abnormalities of the BM, such as myelofibrosis,
can disturb diffusion of the regulatory cytokines and consequently affect production and distribution of the hematopoieticcells.Alternatively,thelack
of production or
reduction of the cytokines may affect their rate of diffusion,leading to BM hypocellularity with lowerfractal
dimensions than that of the DLA-like processes (Table 1).
The autoregulatory mechanisms that are present inthe
malignant neoplasms, such as leukemias, appear to be inconsistent with the DLA-like processproposed here. They
lead to the expansion of the tumor cells, marked BM hypercellularity, and loss of the fractal structure in the involved areas. We suggest that the fractalstructure and
spatial organization of the BM may change in pathologic
conditions and that this alteration may be in part the result
of an ineffectiveDLA-like process. Therefore,fractal
studies of BM lesions may provide additional help in diagnosis and/or prognosis of certain hematologic disorders.
Work in this direction is in progress. In addition, showing
that the pattern of cell distribution in BM is fractal provides a new approach for modeling self-renewal and dif-
5031
ferentiation of the hematopoietic cells. This has already
been shown in our recent report.I5
REFERENCES
I . Naeim F: Topobiology in hematopoiesis. Hematol Pathol
9: 107, I995
2. West BJ, Deering W: Fractal physiology for physicists: Levy
statistics. Phys Rep 246: 1, 1994
3. Vicsek T: Fractal Growth Phenomena. Singapore, World Scientific, 1989
4. Smith TA Jr, Marks WB, Lange GD, Shriff WH Jr. Neale EA:
A fractal analysis of cell images. J Neurosci Methods 27:173, 1988
5 . Caserta F, Stanley HE, Elder WD, Doccord G, Hansman RE,
Nittman J: Physical mechanisms underlying neurtie outgrowth. Physio1Rev Lett 64:95, 1990
6. Family F, Masters BR, Platt DH: Fractal pattern formation in
human retinal vessels. Physica D 38:98, 1989
7. Mainster MA: The fractal properties of retinal vessels: embryological and clinical implications. Eye 4:235, 1990
8. Matsuo T, Okeda R, Takohashi M, Funata M: Forma 5:19,
1990
9. Ben-Jacob E, Schochet 0, Tenenbaum A, Cohen I, Czirok
A, Vicsek T: Generic modelling of cooperative growth patterns in
bacterial colonies. Nature 368:46, 1994
10. Witten TA Jr, Sander LM: Diffusion-limited aggregation, a
kinetic critical phenomenon. Physiol Lett 47:1400, 1981
11. Meakin P: The growth of fractal aggregates and their fractal
measures, in Domb C, Lebowitz JL (eds): Phase Transitions and
Critical Phenomena, v01 12. London, UK, Academic, 1988, p 335
12. Kaushansky K, Karplus PA: Hematopoietic growth factors:
Understanding functional diversity and structural terms. Blood
82:229, 1993
13. Moore MAS: Clinical implications of positive and negative
hematopoietic stem cell regulators. Blood 78:1, 1991
14. Mayani H, Guilbert U,Clark SC, Janowska-Wieczorek A:
Inhibition of hematopoiesis in normal human long-term marrow cultures treated with recombinant human macrophage colony stimulating factor. Blood 78:651, 1991
15. Sahimi M, Mehrabi AR, Naeim F A discrete stochastic model
for self-renewal and differentiation of progenitor cell. Physiol Rev
Lett (manuscript submitted)
From www.bloodjournal.org by guest on June 14, 2017. For personal use only.
1996 87: 5027-5031
Morphogenesis of the bone marrow: fractal structures and diffusionlimited growth
F Naeim, F Moatamed and M Sahimi
Updated information and services can be found at:
http://www.bloodjournal.org/content/87/12/5027.full.html
Articles on similar topics can be found in the following Blood collections
Information about reproducing this article in parts or in its entirety may be found online at:
http://www.bloodjournal.org/site/misc/rights.xhtml#repub_requests
Information about ordering reprints may be found online at:
http://www.bloodjournal.org/site/misc/rights.xhtml#reprints
Information about subscriptions and ASH membership may be found online at:
http://www.bloodjournal.org/site/subscriptions/index.xhtml
Blood (print ISSN 0006-4971, online ISSN 1528-0020), is published weekly by the American
Society of Hematology, 2021 L St, NW, Suite 900, Washington DC 20036.
Copyright 2011 by The American Society of Hematology; all rights reserved.