© 2003 Hindawi Publishing Corporation

Journal of Biomedicine and Biotechnology • 2003:3 (2003) 170–193 • PII. S1110724303209165 • http://jbb.hindawi.comREVIEW ARTICLE

Practical Modeling Concepts for Connective TissueStem Cell and Progenitor Compartment Kinetics

George F. Muschler,1,2∗ Ronald J. Midura,2 and Chizu Nakamoto2

1Department of Orthopeadic Surgery (A-41), The Cleveland Clinic Foundation, 9500 Euclid Avenue, Cleveland, OH 44195, USA2Department of Biomedical Engineering (ND-20) and The Orthopeadic Research Centre,

The Cleveland Clinic Foundation, 9500 Euclid Avenue, Cleveland, OH 44195, USA

Received 13 September 2002; revised 15 January 2003; accepted 23 January 2003

Stem cell activation and development is central to skeletal development, maintenance, and repair, as it is for all tissues. However, anintegrated model of stem cell proliferation, differentiation, and transit between functional compartments has yet to evolve. In thispaper, the authors review current concepts in stem cell biology and progenitor cell growth and differentiation kinetics in the contextof bone formation. A cell-based modeling strategy is developed and offered as a tool for conceptual and quantitative explorationof the key kinetic variables and possible organizational hierarchies in bone tissue development and remodeling, as well as in tissueengineering strategies for bone repair.

THE PARADIGM OF STEM CELLS ANDPROGENITOR CELLS

Stem cells and progenitors are essentially present inall normal tissues [1, 2, 3, 4, 5, 6, 7]. “Stem cells” are de-fined, in general, as resting cells (not actively proliferat-ing) that are present in small numbers in normal tissues.They share one important feature: the capacity for “asym-metric” cell division and “self-renewal” [8, 9]. In this pro-cess, a stem cell is activated by some signal or event toleave its normal resting state and to divide. However, theresult of this cell division provides two daughter cells thatare not identical. One daughter cell proliferates symmet-rically, often for many cell divisions, to produce an abun-dance of progeny referred to as progenitors. These pro-genitors subsequently differentiate to form a mature tis-sue. In contrast, the second daughter cell returns to theoriginal resting state of the mother cell until a new activat-ing signal or event occurs. It retains a stem cell phenotypeand all of the capabilities of the original mother cell in aprocess referred to as “self-renewal.” This process is criti-cally important to the preservation of the stem cell com-partment. If both daughter cells were to become progeni-tors, then the pool of stem cells would be progressively de-pleted with each activation event. Such an outcome wouldrapidly deplete the stem cell population that is necessaryto support ongoing tissue remodeling and repair requiredfor long-term health.

During embryonic development, cells of the innermass of the blastocyst retain the capability to regeneratean entire individual, and are therefore “totipotent” intheir differentiation potential. However, convention hasheld that as the progeny of these totipotent stem cells

become dispersed throughout the organism and localizedwithin specific tissues or organs, the stem cells in eachof these tissues become progressively determined andconfined transiently or permanently within definedstem cell compartments or niches. Stem cell populationsinitially become committed as “pleuripotent” stem cellsconfined to selected groups of tissues within a devel-oping embryo (endoderm, ectoderm, or mesoderm).As development proceeds, some stem cell populationsmay remain “multipotent,” capable of differentiationalong one of several cell lineages (eg, cell populationsin the neural tube, neural crest cells, hemangioblasts,and the mesenchymal mass of fetal limb buds). Otherstem cell populations become intrinsically limited to thegeneration of only one mature cell type (eg, intestinalendothelium or skin keratinocytes). Such monopotent orunipotent stem cells were considered to be “committed,”“restricted,” or “determined” as a result of irreversiblechanges in the cell nucleus.

The transient pleuripotent and multipotent stem cellpopulations of embryonic and fetal life have appeared todisappear in postnatal life, leaving behind populations ofmore restricted adult stem cells that support virtually ev-ery organ system (eg, skin, intestinal mucosa, liver, vascu-lar endothelium, the central nervous system, hematopoi-etic stem cells in bone marrow, and connective tissue ormesencymal stem cells) [1, 2, 3, 4, 5, 6, 7]. These adultstem cell populations are of central importance in adulthealth and in all settings requiring tissue repair, remod-eling, or regeneration. In fact, the health of a given tis-sue might even be defined by the state and kinetics ofthe supporting infrastructure of stem cells and progeni-tors.

2003:3 (2003) Connective Tissue Stem Cell Kinetics 171

The progressive restriction of stem cells from embry-onic stem cells, to fetal stem cells to adult stem cells, toterminally differentiating cells, and to eventual cell deathcan be seen as a series of progressive transitions as cellsmove from one defined population to another. Cell pro-liferation is an integral part of this process. The period ofresidence or life-span of cells within each compartmentor the process of transition or “transit” between compart-ments is often associated with cycles of cell division, andproliferation is not always followed by terminal differen-tiation. The dramatic and rapid expansion of tissue massand stem cell diversity that is associated with embryonicdevelopment, as well as fetal and postnatal growth, alsorequires expansion of stem cell populations as much asdiversification. Therefore, stem cells must not be limitedto only asymmetric cell division. Stem cell expansion re-quires “symmetric” stem cell division or “self-expansion,”as discussed below.

Challenges to the traditional stem cell paradigm

In recent years, a number of lines of observation havechallenged some of the traditional views of lineage restric-tion among adult stem cell pools. For example, cloning bytransfer of somatic nuclei into activated oocytes providesevidence that the apparent restriction of somatic cells maybe regulated by factors that are extrinsic to the nucleus[10]. However, the mechanism of reversibility induced bythe cytoplasmic environment of an activated oocyte maynot be relevant to events in normal stem cell physiology.

Of more physiologic relevance, are a number of ob-servations that suggest that adult mammals may retainone or more stem cell populations which either retain themultipotent or pleuripotent biological potential or canbe induced to exhibit such potential. Several reports havedemonstrated that, following transplantation of marrow-derived cells or enriched marrow-derived hematopoi-etic stem cells (HSC), donor-derived cells can be de-tected contributing to hepatocyte and biliary epithelium[11, 12, 13], cardiac myocytes [14, 15], skeletal myocytes[16, 17], lung, intestinal, and skin epithelium [18], andneuroectoderm [19, 20, 21, 22]. Others have shown thatneural-derived [23, 24] or muscle-derived [25, 26] cellscan contribute to hematopoiesis. Similarly, several reportshave demonstrated that culture-expanded bone marrow-derived cells [27] may contribute to a broad range of tis-sues [28, 29], such as skeletal muscle [29, 30], cardiacmuscle [31, 32], liver [12, 33, 34], intestinal mucosa [35],lung [36], vascular endothelial cells [37, 38, 39, 40], boneand cartilage [41, 42], glial and neuronal tissues [43, 44],and other sites [45]. Multipotentiality has also been re-ported in cells derived from muscle [46, 47] and fat [7].

Most recently, a multipotent adult progenitor cell(MAPC) has been proposed [18], based on evidence thatsome cells from adult marrow can be expanded for over80 population doublings, and if transplanted into a blas-tocyst will contribute to the tissue of all three germ lay-ers. Furthermore, these authors report that cells expanded

in this way can be infused into a mouse host and can befound to engraft and contribute to blood, bone marrow,spleen as well as epithelium in lung, liver, and intestine.

These results all suggest that adult mammals may re-tain one or more populations of adult progenitor cellsthat retain the intrinsic biological potential to generateprogeny which can potentially differentiate into many en-dodermally, mesodermally, and/or ectodermally derivedmature tissues. However, several possible mechanismscould contribute to these observations. One possibility isthat a small number of intrinsically multipotential cellsmay persist in marrow and other tissue niches. These cellsmight be quiescent in adults or may function upstream ofmore easily identified stem cell pools. If so, they mightbe present in very low abundance and function with avery low turnover rate and still feed into or supplementmore restricted downstream adult stem cell populations.It is also possible that the apparent restriction of most, ifnot all, stem cells in marrow and in other tissues, may beimposed by factors extrinsic to the stem cell. The stem cellniche and milieu within each organ system may define thephenotype(s) expressed by the local stem cell pool, as aresult of the unique signaling environment from the localmatrix, cytokines, and cell-cell interactions in each tissue.Changing the niche of a stem cell pool, by transplanta-tion into a new niche or exposing a stem cell to uniquetissue culture conditions, may unmask a broader intrin-sic biological potential. Finally, it is possible that the ob-served properties of a stem cell pool might be changedthrough selective pressures that exist during prolonged invitro culture or during radical procedures such as trans-plantation.

Regardless of the mechanism(s) that are at work, theserecent observations are of tremendous interest for thosewho would seek to develop stem cell therapy strategies us-ing adult cells [48]. All of the diverse stem cell populationsthat reside in or can be derived from adult tissues have po-tential value in therapeutic efforts to regenerate, preserve,or repair tissues. This fact also presents a challenge to thestem cell field to define practical strategies for character-izing and modeling the kinetics of stem cell function andvarious stem cell populations during normal tissue for-mation and remodeling, as well as in settings of repair.These models must include means of accommodating ad-ditional and as yet uncharacterized pools of stem cells aswell as more fluid relationships between stem cell poolsthan have previously been recognized.

Recognizing this need, this paper presents the ratio-nale for and development of a practical model systemrelevant to investigation of the kinetics of the stem cellpopulations contributing to the formation and remod-eling of bone tissue. The conceptual starting point forthis discussion is the relatively traditional vision of thelife cycle of a stem cell and its progeny; it is illustratedin Figure 1. In this model, functionality of stem cell andits progeny is regulated by five primary events or behav-iors: activation, proliferation, migration, differentiation,and survival (or death). Once introduced, the model is

172 George F. Muschler et al 2003:3 (2003)

then further developed to incorporate strategies that ac-commodate concepts of multiple stem cell pools or tran-sit populations and the relationships between these stemcells and transit cell pools.

Bone formation and the connective tissuestem cell system

Background and terminology

In the 1960s, Burwell showed that the bone forma-tion induced by implantation of cancellous bone graftswas derived from primitive osteogenic cells in bone mar-row [49, 50, 51]. Friedenstein et al [52] showed that newbone was formed by proliferative fibroblast-like marrowcells and that the number of these proliferative cells couldbe assayed by counting the number of fibroblastic colonyforming units (CFU-Fs) in vitro. It was later shown that,at least some of these colony forming cells are multipotentand can differentiate into bone, cartilage, fibrous tissue,fat, or muscle [41, 42]. Several reviews nicely summarizethe many contributions in this field [42, 53, 54, 55, 56, 57].

Many names have been used to describe the colonyforming cells found in bone marrow, periosteum, or tra-becular bone, in addition to CFU-Fs. These terms includemechanocytes, bone marrow stromal cells, and mesenchy-mal stem cells, although the precise definition and bio-logic capabilities ascribed by these terms are not entirelysynonymous. A large subset of the colony forming popu-lation has been suggested to be resident in tissue in a qui-escent (Go) state in vivo, supporting the concept that thesecells may have stem cell-like function and self-renewal po-tential [58].

We have previously proposed and provided the ratio-nale for the term connective tissue progenitors (CTPs) forthe heterogeneous population of proliferative cells thatcan be harvested from bone marrow and other tissues,and can be shown to differentiate into one or more con-nective tissue phenotypes [59]. (See Figure 2.) We use theterm CTP throughout the following discussion. This termrecognizes that these tissue-derived cells are not a pure oruniform population, and may be derived from more thanone pool of stem cells and progenitors in native tissues.These cells may include true resting multipotent stem cellsthat become activated after harvest and are capable ofself-renewal. However, colonies may also be formed bycells that are already proliferating in vivo, that lack self-renewal capabilities and may exhibit intrinsic commit-ment to various stages of diverse lineages [53, 57, 60].This diversity can be a source of frustration for those look-ing for homogeneous purified populations of cells. How-ever, this diversity can also be a source of valuable in-formation which can be dissected experimentally usingin vitro CFU assays to understand variation in intrinsicproperties, the prevalence and kinetics of various con-nective tissue stem cell populations, and how these pop-ulations change with aging, gender, disease states, phar-macologic intervention, and tissue engineering strategies[59, 61, 62, 63].

Adult connective tissue progenitor populations

Multipotent CTPs are resident in many musculoskele-tal locations. The osteogenic and chondrogenic potentialof periosteum, as recognized long ago [64], is derivedfrom cells resident in the outer cambial layer of perios-teum [65, 66, 67]. Multipotent CTPs are present on thesurface of bone trabeculae, in peritrabecular soft tissues,within haversian canals of cortical bone, and in the bonemarrow space, including bone marrow harvested by aspi-ration [62]. Recently, CTPs have also been demonstratedto be resident in adipose tissue [7, 68] and muscle [46].

A potentially unifying concept to explain the presenceof CTPs in fat, muscle, and other tissues is the presencein each of vascular pericytes. The pericyte, a unique cellfound outside the basement membrane of small bloodvessels, is present in all vascularized tissues. Several inves-tigators have found that pericytes isolated from many tis-sues can be induced to differentiate into various connec-tive tissue phenotypes [69, 70], suggesting that pericytesmay represent a widely distributed population of mul-tipotent CTPs. In bone marrow, pericytes may give riseto the Westen-Bainton cells, fibroblast-like marrow stro-mal cells associated with the outer surface of marrow si-nusoids expressing alkaline phosphatase [71]. Bianco etal [72, 73, 74] have suggested that pericytes and Westen-Bainton cells are part of an integrated system of stemand progenitor cells in bone marrow. They argue thatthese two cell types contribute to the formation of thefibroblastic stromal network in marrow that supportshematopoiesis and to the formation and remodeling ofmarrow fat as well as of cortical and trabecular bone(Figure 3). However, the pericyte alone does not accountfor all the progenitors outside bone. Satellite cells har-vested by digestion of isolated skeletal muscle fibers canundergo connective tissue differentiation independent ofthe pericyte population [6, 75].

The widespread distribution of multipotent CTPs isparticularly relevant to the field of orthopedic tissue engi-neering. It provides many potential sources of stem cellsand progenitors that can be harvested, selected, concen-trated, and manipulated or “engineered” to improve clin-ical outcomes. This system also provides many potentialbiologic targets for specialized matrix materials and lo-cally or systemically active pharmaceuticals, hormones,growth factors, and cytokines. Moreover, it offers a fer-tile system in which to explore possible intrinsic differ-ences between CTPs in these disparate and distinct stemcell/progenitor cell compartments, as well as the uniquefeatures of the stem cell niches within each compartment.

THE MATRIX-BASED MODEL FOR BONE TISSUEFORMATION AND REMODELING

A remarkable set of histologic observations andthe application of innovative and painstaking methodsof quantitative histomorphometry pioneered by Parfittand Frost have provided a robust understanding of the

2003:3 (2003) Connective Tissue Stem Cell Kinetics 173

Activation

Self-renewal

Proliferation

Migration

Differentiation

Apoptosis/death

Figure 1. Six stages in the stem cell life cycle. In the simplest scenario, the life cycle of cells in a stem cell system involves at least sixstages. This begins with the cycle of stem cell activation and cell division producing a progenitor cell and self-renewal of the stem cell,but is continuous with the process of proliferation of progenitors, migration, differentiation, and eventual apoptosis or cell death.

Bone

CartilageFat

Stromalcell

Fibroustissue

NeuronsGlia

Muscle

LiverCardiacmuscle

Cbfa1/Osf-2/Runx2Osf-1Osx

ADD1PPAR γ

Sox 9Sox 5Sox 6

Myo DMyf 5MRF4

Myogenin

Figure 2. Differentiation pathways for connective tissue progenitors. This diagram illustrates the potential differentiation pathwaysavailable to connective tissue progenitor cells. Some of the transcription factors that regulate these pathways are shown in associatedboxes.

functional and dynamic parameters associated with boneformation and remodeling at the tissue level [76, 77, 78,79, 80]. In adult bone remodeling, these processes of boneformation and bone resorption generally take place in thecontext of the basic multicellular unit (BMU) describedby Frost [81]. A conceptual illustration of one BMU is pre-sented in Figure 3.

In an average BMU, a group of 6–10 osteoclasts movesforward resorbing bone at a linear rate of approximately20–40 µm per day. This group of osteoclasts constitutesa “cutting cone” and will continue to erode bone for aperiod of as long as 100 days. The deepest point in theeroded surface marks the trailing edge of the osteoclastfront, and is usually about 200 µm behind the first osteo-clast.

The wave of bone resorption is followed immedi-ately by a wave of bone formation that is mediated byosteoblasts. Osteoblasts are rapidly added to the newly

eroded bone surface very near the trailing edge of the os-teoclast front at a rate that is sufficient to cover the surfaceof the newly eroded bone. Osteoblasts begin secreting ma-trix within a day, and matrix synthesis increases over sev-eral days to a maximum rate of approximately 1.5 µm perday over an area of approximately 150 µm2 per osteoblast,resulting in a maximal rate of synthesis of approximately225 µm3 per day per osteoblast. The wave of osteogene-sis fills in the defect created by the osteoclasts, a depthof about 40–60 µm, over a period of about 50 days. Thetotal matrix synthesis per osteoblast is therefore approxi-mately 6000–9000 µm3, or 3–5 times its cell volume. Areasof increased osteoblast density (smaller surface area perosteoblast) are associated with proportionately increasedlinear rates of matrix synthesis [82]. In a fully active BMU,approximately 2000 active osteoblasts will be assembled asa functional unit behind the bone resorption front, trail-ing over a distance of 1600–2000 µm.

174 George F. Muschler et al 2003:3 (2003)

Figure 3. Schematic diagram of the osteoblastic stem cell sys-tem. This conceptual drawing illustrates the primary candidatepopulations of stem cells and transit cells thought to be asso-ciated with bone formation and remodeling. Vascular pericytes(green), Westen-Bainton cells (orange), type I or pre-osteoblasts(pink), secretory osteoblasts (maroon), osteocytes (brown), lin-ing cells (purple), and adipocytes (yellow). Vascular pericytesmay give rise to the Westen-Bainton cells. Pericytes and Westen-Bainton cells may contribute to the formation of pre-osteoblastsand also adipocytes. New osteoblast are added in the region im-mediately behind the advancing front of osteoclastic resorption.Secretory osteoblasts produce new bone matrix until they be-come quiescent on the surface of bone as a lining cells (purple)or become embedded in the matrix as osteocytes (brown), ordie via apoptosis. Osteoclast formation is also illustrated. A frac-tion of the monocytes population in systemic circulation (blue)will become resident in the bone marrow space. Osteoclasts areformed by fusion of monocytes resident in bone marrow to formmultinucleated functional units. The nuclei in active osteoclastscontinue to be turned over as a result of nuclear loss and on-going fusion events with new marrow-derived monocytes [103].The black arrow indicates the direction of bone resorption bythe osteoclastic front, followed by bone formation.

During the process of bone formation, some os-teoblasts become embedded in the newly synthesized ma-trix as osteocytes. These osteocytes reside within cavi-ties known as lacunae and interconnect with one anotherthrough multiple processes extending through an inter-connected plexus of channels called canaliculi. Some os-teoblasts also undergo apoptosis. The osteoblasts remain-ing on the bone surface gradually cease making new ma-trix and spread out as lining cells to occupy a surfacearea of approximately 2500 µm2, or about 16 times thearea of a secretory osteoblast. Therefore, the mean life-span of an osteoblast is slightly less than the 50 daysneeded to complete the wave of bone formation, proba-bly about 40 days. However, published estimates of meanosteoblast life-span vary widely from 10 to 100 days[59, 83, 84].

In contrast, both the lining cells and the osteocytescreated in this process appear to survive for a long period,

usually until the next wave of bone remodeling movesthrough the region. The mean life-span of osteocytes hasbeen estimated to be 15 years in cancellous bone (∼ 5000days) [85] and 25 years in cortical bone (∼ 9000 days)[86, 87], with a range of 3 to 30 years (1000–10,000 days).However, osteocytes can be lost before remodeling occurs,leaving lacunae empty through processes of osteonecrosisor micropetrosis [88].

In general, after skeletal maturity, bone formationdoes not occur without the prior stimulus of bone resorp-tion. Therefore, the gain or loss of bone in a region oftissue is dependent primarily on the balance of resorp-tion and formation in the individual BMUs. However,new bone formation can occur without bone resorptionon periosteal surfaces and occasionally within nonbonesites in soft tissues (heterotopic bone formation) and evenin perivascular tissues (Gorski JP and Midura RJ, unpub-lished data).

The number of BMUs within any region of bone tis-sue will depend on the rate of activation events in that tis-sue volume, and varies widely from region to region, boneto bone, and individual to individual, but can be esti-mated using techniques of fluorochrome double labeling,described first by Frost [78]. The number of osteoblastsper unit volume of bone will also vary widely, but can beestimated based on measurements of the surface to vol-ume ratio in the region of interest, the area fraction of thesurface that is occupied by osteoblasts, and the mean areaof an osteoblast. Similarly, the number of osteocytes perunit volume can be estimated based on the volume frac-tion of bone in a region and the mean osteocyte density inthe region.

Data with respect to osteocyte density (Do) is availablewith increasing precision. An age-related decline in osteo-cyte viability was observed long ago [87], as well as thepossibility that many canaliculi and some empty lacunaemay become filled with mineralized debris (micropetro-sis) [87]. A study of iliac cancellous bone samples takenfrom skeletally healthy white women (age 20–73), foundno evidence of age-related changes in osteocyte densitynear the surface of bone (< 25 µm), while an age-relateddecline was seen in deeper regions in bone [89].

The osteocyte density (Do) is reported to be greater incancellous bone (0.000047 osteocytes/µm3) [90] than incortical bone (0.000026 osteocytes/µm3) [91]. Estimatesof osteocyte density as low as 0.000013 osteocytes/µm3

have been reported in the human iliac crest [92] and it hasbeen suggested that osteocyte density may be increased inthe setting of osteoporosis [93, 94]. However, as a first ap-proximation, based on a mean bone volume of 11 to 25%in cancellous bone, one can estimate the number of osteo-cytes in one cubic centimeter of cancellous bone to be inthe range of 5 to 10 million.

A concise review of these concepts and mechanismsinvolved in the regulation of cellular activity in boneremodeling has been provided recently by Manolagas[84].

2003:3 (2003) Connective Tissue Stem Cell Kinetics 175

A THEORETICAL CELL-BASED MODEL FOR BONETISSUE FORMATION AND REMODELING

Available knowledge of the likely stem cell and pro-genitor cell populations in bone and the biologic path-ways that are available to progenitors of osteoblasts andosteoclasts provides a useful structure in which to ex-plore the biologic events using a cell-based approach, fo-cusing on the key variables in stem cell kinetics associ-ated with bone remodeling. A cell-based approach hasbeen the starting point for many other stem cell systemswhere the volume of matrix and matrix turnover does notdominate organ function. However, application of thesestrategies in the skin, intestinal mucosa, and the carti-laginous growth plate in long bones has been facilitatedby the relative ease of histologic assessment in these sys-tems, the limited number of cell types in these tissues, andthe close physical proximity of the stem cell compartmentand maturing cells in these systems [95, 96, 97, 98, 99,100].

A cell-based mathematical model system requires afunctional understanding of the stem cell and progen-itor cell populations in the system in order to gener-ate a mathematical strategy that has fidelity to the ob-jective hierarchy and kinetic parameters governing thecell populations involved. The cellular heterogeneity andthree-dimensional complexity of bone have hinderedthis kind of investigation in bone. However, the ex-tensive literature in bone morphology and the estab-lished matrix-based model for the kinetics of bone ma-trix turnover, provide a robust set of data and parametersthat facilitate this approach, despite its apparent complex-ity.

Exploration using a cell-based strategy is necessary inorder to provide insight into the kinetics of stem cell andprogenitor cell populations upstream and downstream ofosteoblasts and osteoclasts. This approach will also becritical as a platform for rational analysis of the CTP pop-ulation present in bone and other tissues, for understand-ing the response of CTPs tissue in injury and repair, andin the rational design of strategies to use CTPs therapeu-tically to augment or induce the repair and regenerationof bone and other tissues.

In a previous publication, we introduced a cell-basedmodeling strategy for bone formation, and applied thismodel to explore the likely range of some kinetic pa-rameters in the setting of bone grafting and bone re-modeling [59]. In this paper, we expand upon this ap-proach to further explore key variables in bone forma-tion and remodeling, and particularly the relationship be-tween stem cell pools in bone and the transit of cellsfrom one pool to another. Although the discussion be-low is particularly focused on bone, it is expected thatsimilar concepts will apply to the regeneration or re-pair of any tissue that might be derived from connec-tive tissue stem cells and CTPs, and may also find ap-plication in modeling of stem cell pools in other organsystems.

Tissue formation and remodelingIn any region of tissue regeneration, repair, or remod-

eling, the quantity of new tissue formed (T) will representthe difference between the amount of tissue formed (F)and the amount of tissue simultaneously lost due to re-sorption or removal (R):

T = F − R. (1)

This balance between formation and resorption ofbone in an idealized system of tissue remodeling is a cen-tral theme in the pathogenesis of age-related bone lossand postmenopausal osteoporosis. Similarly, the balanceof tissue formation or the rate of tissue formation (or loss)can be represented as difference between the rate of tissueformation and the rate of tissue loss:

dT

dt= dF

dt− dR

dt. (2)

Under steady state conditions, the rate of formation willequal the rate of resorption and there will be no netchange in the amount of tissue present over time (dT/dt =0). Furthermore, the total amount of any given tissuewithin an organism at any time will equal the integral ofdT/dt over the life of the organism (0 to t ), beginning atthe fertilized egg.

Bone tissue formationAll bone formation occurs as the result of work

performed by active mature osteoblasts. Taking thisparadigm, the rate of bone tissue formation within a giventissue volume (dFb/dt) can be represented as the productof the total number of active osteoblasts in that volume(Nob) and the mean rate of bone tissue formation per cell(dσob/dt), in units of volume (both bone matrix plus thevolume of osteocytes and canaliculi) per unit time:

dFbdt

= Nob dσobdt

. (3)

Under steady state conditions, the number of activeosteoblasts in a region (Nob) of tissue will be determinedby the rate at which osteoblasts are formed in this region(Nob/dt) and the mean life-span of an osteoblast (lob) inthat region, where

Nob = Nobdt

lob. (4)

Furthermore, under steady state conditions, the meanamount of bone matrix produced by single osteoblast dur-ing its lifetime (σob) can be expressed as the product of themean rate of bone formation per osteoblast and the meanlife-span of an active osteoblast:

σob = dσobdt

lob. (5)

This strategy has practical utility. Since histomorpho-metric measurements can be used to directly measure therate of bone formation (using double labeling techniques)

176 George F. Muschler et al 2003:3 (2003)

and to estimate the amount of bone matrix formed perosteoblast [101], allowing the mean life-span of an os-teoblast to be calculated [76, 77, 78, 79, 80, 84, 102]. Withthe addition of reliable means of directly measuring thetotal number of osteoblasts within a tissue region, the rateof formation of new osteoblasts in the region (Nob/dt)could be readily estimated.

Tissue removal

The same approach can be taken to describe the re-moval of bone tissue, mediated by osteoclastic bone re-sorption. The rate of bone resorption can be expressed as

dR

dt= Noc dσoc

dt, (6)

where Noc is the number of active osteoclasts and dσoc/dtis the mean rate of bone resorption for an osteoclast, andthe number of active osteoclasts is determined by the re-lationship:

Noc = Nocdt

loc, (7)

where Noc/dt is the rate of formation of osteoclasts and locis the mean life-span of an osteoclast.

Overall balance of bone tissue formation

Substituting values from (3), (4), (6), and (7) into (2)provides a representation of the overall rate of bone tissuemass in any tissue volume as

dT

dt=(Nobdt

lob

)dσobdt

−(Nocdt

loc

)dσocdt

. (8)

Based on this model, the balance of bone tissue for-mation is dependent on only the rate of formation of os-teoblasts and osteoclasts, the life-span of osteblasts andosteoclasts, and the rate of bone formation or resorptionof bone tissue per osteoblast or osteoclast, respectively.This strategy provides the opportunity to explore the pos-sible range of variation in these parameters, and to definethose parameters that are likely to be most important, orthat exert greatest effects on variation in bone formationand removal.

Many variables will influence the life-span of thesecells (lx) and the rate or efficiency with which they formor remove matrix (σx). However, these variables are func-tions of mature cells, therefore they are outside of the fo-cus of this paper, which is stem cell kinetics. Therefore,the remainder of this discussion will focus on dissect-ing the variables related to the rate of formation of os-teoblasts (Nob/dt), and the rate of formation of osteoclasts(Noc/dt).

The rate of formation of new osteoblasts

In a previous publication, we proposed and developedthe rationale for a mathematical model to describe therate of osteoblast formation (Nob/dt) under steady state

conditions [59]. This relationship is expressed as

Nobdt

= Nshε2µPob. (9)

For the purposes of this paper, it is desirable to usenomenclature that clearly distinguishes between connec-tive tissue stem cells (CTS) and the HSC that give rise toosteoclasts. Therefore, we offer a modified nomenclature,as follows:

Nobdt

= NCTShCTSεCTS2µCTSPob, (10)

where

(i) both Ns and NCTS = the number of cells in the stemcell pool upstream of the osteoblast,

(ii) h = the frequency of stem cell activation events inthe defined tissue volume,

(iii) ε = the efficiency of stem cell activation events inthe defined tissue volume,

(iv) µ = the effective number of symmetric mitotic cy-cles between the time of stem cell activation andthe time of differentiation into mature secretory os-teoblasts.

The factor Pob represents the cumulative probabilitythat the progeny of an initial stem cell activation would re-tain an osteoblastic phenotype during the period of clonalexpansion:

Pob = ρob 1ρob 2ρob 3 · · · ρob x, (11)

where ρob x = the probability after each mitotic cycle “x”that the progeny created will retain osteoblastic potential.The application of this equation to the progeny and com-mitment events of one hypothetical stem cell is shown inFigure 4.

It is useful to note that under ideal circumstances, ρobwill be equal to 1.0 for each sequential symmetrical mito-sis and therefore Pob will equal 1.0. However, this prob-ability will be reduced by the fraction of progeny afterany mitotic event that undergo apoptosis, or the fractionof cells that irreversibly commit to an alternative maturephenotype (eg, an adipocyte). The contribution of any in-cremental increase in the number of symmetric mitoses(µ) to additional osteoblast formation depends on main-taining a value for ρob that is greater than 0.5. Similarly,few osteoblasts will be formed unless the overall value ofPob remains substantially greater than 2−µ.

It is also useful to note that product of εh can also berepresented as the activation rate (AR) or the inverse ofthe mean cycle time of the stem cell population (tCTS) be-ing modeled. Therefore,

εCTShCTS = ARCTS = 1tCTS

. (12)

2003:3 (2003) Connective Tissue Stem Cell Kinetics 177

Osteoblasts

2

4

6

12

18

36

36

ρap5 = 0.25ρob5 = 0.75

ρap3 = 0.25ρob3 = 0.75

Nob = Nshε2µ(ρob1ρob2ρob3 · · · ρobx)Nob = 1 × 1 × 26 × (1 × 1 × 0.75 × 1 × 0.75 × 1) = 64 × (0.5625) = 36

Figure 4. Competing differentiation options and events. The effect of alternative commitment events on the number of matureosteoblasts produced after the activation of one connective tissue stem cell is shown. In this example, the probability of retaining anosteoblastic phenotype (ρob) was 1 after all mitotic events, with the exception of the third and fifth symmetric mitosis. After the thirdmitosis, 25% of the cells undergo apoptosis (ρob 3 = 0.75). As a result, 16 fewer cells are produced. After the fifth mitosis, 25% of thecells commit to adipocytic differentiation (ρob 5 = 0.75). Twelve adipocytes are the result. These events leave an overall probability ofosteoblastic differentiation (Pob) among the possible progeny of 0.5625 (0.75 × 0.75). The result is an approximately 44% reductionin the total number of osteoblasts produced, from 64 to 36. Even if these progenitors continue to proliferate for many more celldivisions before differentiating without further commitment to nonosteoblastic phenotypes, these two events related to the third andfifth mitosis effectively limit the number of mature osteoblasts to only 56% of the theoretical maximum.

The rate of formation of new osteoclasts

An analogous strategy can be applied to modeling thesteady state rate of formation of osteoclastic progenitors.However, this requires a two-step approach. The precur-sor cells that form the osteoclast are derived from theHSC, but also transit through the compartment of cir-culating monocytes before fusing to contribute nuclei tomature multinucleated osteoclasts [103]. By analogy to(10), the systemic rate of formation of mature monocytes(dsNm/dt) can be described as

dsNmdt

= sNHSC ARHSC 2µHSCPm, (13)

where

(i) sNHSC = the total number of HSC available for ac-tivation, systemically,

(ii) ARHSC = the mean HSC activation rate,(iii) µHSC = the effective number of symmetric mitotic

cycles between the time of HSC activation and thetime of differentiation into mature monocytes,

(iv) Pm = the cumulative probability that the progeny ofthe initial stem cell activation will retain monocytephenotype during the period of clonal expansion.

Similarly, the total number of systemic monocytes(sNm) available to contribute to osteoclast formation bycell fusion can be defined by the product of the rate of

monocyte formation (dsNm/dt) and the mean life-span ofa monocyte (lm):

sNm = dsNmdt

lm. (14)

However, only a fraction of the monocytes that arepresent systemically will contribute to osteoclast forma-tion. To accommodate this variable, the probability thatany individual systemic monocyte will be activated tocontribute to osteoclast formation within a defined regionof bone in a defined unit of time can be assigned a value,Pm→oc/dt. Furthermore, since multiple monocytes mustcontribute to form a mature multinucleated osteoclast (amean of∼ 8 cells) [103], a factor of n must be introduced,equal to the mean number of monocytes contributing to amature osteoclast. Using these two additional factors, therate of formation of osteoclasts at steady state in a definedregion of bone (dNoc/dt) can be expressed as

dNocdt

=[sNHSC ARHSC 2µHSCPmlmPm→oc/dt

]n

. (15)

Combined expression for bone tissue formationand remodeling

Substituting factors for the rate of formation of os-teoblasts and osteoclasts from (10) and (15) into (8) pro-vides the following combined expression for the overallbalance of bone tissue formation within a given region of

178 George F. Muschler et al 2003:3 (2003)

bone:

dT

dt= [NCTS ARCTS 2µCTSPob]lob dσob

dt

−[sNHSC ARHSC 2µHSCPmlmPm→oc/dt

]n

locdσocdt

.

(16)

Limitations of the model

All models are inevitably based on simplifying as-sumptions, which may not (and likely are not) univer-sally true. Several of these assumptions require explo-ration. Some assumptions require further refinement asdiscussed below.

One of the assumptions in the model, as presentedthus far, is that the stem cell activation and new osteoblastformation both take place within the same region of in-terest or observation. In contrast, the activation of theHSC need not to occur within the region where the ma-ture progeny are active. The validity of this assumptionfor bone is dependent upon the relative size of the re-gion of observation or sampling and the mean distancebetween the site of connective tissue stem cell activationand the site of mature osteoblast function. If this dis-tance is on the order of 100–5000 µm, then data avail-able from conventional histomorphometry is likely to of-fer wide enough sampling. However, there is a possibilitythat this distance between the initiating stem cell nicheand the site of osteoblast function might be much largerthan the field of sampling. For example, it has been sug-gested that, like osteoclasts, some or all of the precursorsof mature osteoblasts may migrate for relatively long dis-tances [104, 105, 106] or even circulate in blood as an os-teoblastic transit cell population (OT) [107]. If this is thecase, it would be necessary to accommodate a systemicallydistributed osteoblastic transit cell population. Taking thisstrategy, the expression for overall bone tissue formationwithin a region of tissue could be written as

dT

dt= [sNCTS ARCTS 2µCTSPotlotPot→ob/dt]lob dσob

dt

−[sNHSC ARHSC 2µHSCPmlmPm→oc/dt

]n

locdσocdt

,

(17)

where

(i) Pot = the systemic cumulative probability that theprogeny of an activated connective tissue stem cellwould become an osteoblastic transit cell,

(ii) lot = the mean life-span of an osteoblastic transitcell,

(iii) Pot→ob/dt = the mean probability that any individ-ual osteoblastic transit cell will become an active os-teoblast within the region of interest per unit time.

Another limitation in generalizing this strategy is thefact that bone formation in different locations and set-tings may be derived from different stem cell populations

having different intrinsic capabilities and pathways. Tra-becular bone remodeling, cortical haversian remodeling,periostial new bone formation, myositis ossificans, ossi-fication of a fracture callus, endochondral ossification ofprimary and secondary ossification centers, ossificationof an advancing growth plate, and ossification within anatherosclerostic plaque may each rely on the activation ofa different pool or pools of connective tissue stem cellshaving different intrinsic attributes and extrinsic modu-lating factors. Each stem cell pool may have intrinsicallydifferent activating signals, different thresholds for acti-vation, and different activation rates. Each pool may giverise to progeny that have intrinsically different patterns ofproliferation or/and probabilities of differentiation alongan osteoblastic pathway. Furthermore, each pool of stemcells will also be exposed to a different set of extrinsic in-fluences (ie, biochemical, cytokine, matrix, and mechani-cal environment) that is imposed by each tissue and loca-tion or each stem cell niche. These differing sets of intrin-sic and extrinsic attributes would combine to create differ-ences in mean activation frequency (h) and efficiency (ε),cycle time (t), and activation rate (AR) for each stem cellpopulation and setting, as well as differences in numberof symmetrical mitoses in the clonal expansion phase (µ)and the cumulative probability that an osteoblastic phe-notype would be preserved at the completion of clonal ex-pansion (Pob). Recognizing this limitation calls attentionto the fact that settings in which this strategy is appliedmust be carefully defined. Parameters determined in onesetting may not be generalizable in another (eg, trabecularversus cortical remodeling).

The model, as described above, has at least three othermajor limitations. One limitation, and perhaps the great-est, is that this model assumes that the pathway leadingto osteoblast development is associated with a single stemcell activation event and a single stem cell population. Infact, as discussed above, there is abundant evidence tosuggest that bone formation in trabecular bone and likelyother settings is associated with transit of cells throughmore than one cell phenotype or transit cell compart-ment. These transit steps likely involve a series of activa-tion events. Recognizing this hierarchy of osteoblastic celldevelopment, the model is expanded below to accommo-date multiple transit cell populations.

A second limitation is that this model does not con-sider the fate of the osteoblast population after they con-tribute to the population of active secretory osteoblasts.The transit of these cells into the downstream populationsof osteocytes and trabecular and osteonal lining cells andultimate cell death also has important implications in theprocess of bone formation and skeletal health, and shouldbe included in a cell-based modeling approach.

Finally, the model does not address the issue of stemcell renewal and expansion, which is clearly a criticalvariable in the development, regeneration, and long-termhealth of the connective tissue stem cell system.

The remainder of this paper will attempt to addressthese three issues: upstream transit cell populations, the

2003:3 (2003) Connective Tissue Stem Cell Kinetics 179

downstream fate of osteoblasts, and stem cell renewal andexpansion.

THE TRANSIT CELL PARADIGM

The concept of transit cell populations has been ap-plied to several models of stem cell kinetics, particularlyin the stem cell systems in dermal epithelium and in smallintestinal mucosa [98, 99, 108, 109, 110, 111, 112]. Tran-sit populations have generally been defined as cell pop-ulations or stages of differentiation that are intermediatebetween stem cells and mature cells. Transit cell popula-tions can be defined as compartments of either proliferat-ing cells or nonproliferating cells. It is generally assumedthat the cells in each compartment are intrinsically differ-ent from the cells in another compartment, and the cellsin all transit compartments tend to progress irreversiblytoward the mature phenotype.

The concept of “proliferating transit populations” isused most commonly. A proliferating transit population isgenerally envisioned to have the capacity for proliferation,and some capacity for self-renewal or self-maintenance,reducing the demand required for further activation of anupstream stem cell compartment. However, if there is anyongoing contribution from an upstream compartment,stable regulation of cell numbers requires that the rate ofself-renewal in a proliferating transit population must beless than 100% [111]. When a proliferating transit popu-lation exists, it provides a means of cellular expansion. Aproliferating transit cell population is also inevitably as-sociated with a physical migration of cells away from thesite of the upstream stem cell, since new cells must moveaway or be pushed away from the site of cell division ascell expansion occurs.

A highly simplified model involving three proliferat-ing transit cell populations (T2, T3, and T4) in a con-tiguous linear array feeding a population of mature cells(M/T5) is illustrated in Figure 5. For simplicity, this is amodel composed entirely of asymmetric cell division. Thestem cell (S/T1) divides to renew itself and to produce aT2 cell. The T2 cell divides as a transiently self-renewingcell with a cycle time (t2) and life-span (l2) for a numberof cycles (µ2) giving rise to a number of T3 transit cells(also equal to µ2) before its death. The T2 cell that dies isthen replaced by a new T2 cell generated by a subsequentdivision of the upstream stem cell. The T3 population oftransit cells feeds the T4 population in the same way. Foreach Tx compartment, µx = lx/tx. Ultimately, the T4 pop-ulation gives rise to only cells that mature without divid-ing (M/T5). These mature cells live out their functionallife-span (l5) and die. The table within Figure 5 illustrateshypothetical values for tx, lx, µx, and the resulting numberof cells in each transit compartment (Nx) at steady state.

Figure 5 illustrates several features of the transit cellparadigm. First, the change in any one parameter will havesecondary effects on the number of cells in each compart-ment (Nx), which is determined by the product of the rate

of cells entering that compartment (dNx/dt) and the life-span of cells within that compartment (lx):

Nx = dNxdt

lx. (18)

It is also possible to define a velocity of cells leaving eachcompartment in this linear model (Vx), where

Vx = 1tx= ARx . (19)

Figure 5 and the associated table also illustrate thenumber of cell divisions that the stem cells in each com-partment will be burdened with over the life of a hy-pothetical individual (25,000 days, ∼ 68.5 years). Thisdemonstrates the principle value of proliferating transitpopulations, which is the protection of the original stemcell from the burden and genetic risk associated with di-rect generation of each mature cell. In the case of the sys-tem illustrated in Figure 5, in the absence of any transitpopulations, the T1 stem cell would have needed to divide25,000 times, rather than 25 times, to generate the samenumber of mature cells over the life of the individual.

The transit cell model above is based on contiguousunidimensional single-file cell to cell displacement. Theseconditions are appropriate to models in the skin and inthe intestinal lining cell systems. A similar system mightalso be relevant to modeling the progression of cell com-partments in the active growth plate, in articular cartilage,in the setting of periosteal new bone formation.

In the case of organizationally complex and heteroge-neous tissues, such as bone, a contiguous physical chain ofcells beginning at the stem cell is not applicable. Given therequirement in bone for episodic formation of new sites ofbone tissue formation in response to local tissue signals inmarrow or near the bone surface, it would appear that thetransit cell pools upstream of the osteoblast must includeone or more migratory transit populations that provide amechanism of physical migration and homing of progen-itor cells from the (as yet uncharacterized) upstream stemcell niche to a site near where they will activated to leavethe transit compartment and further differentiate.

Any system involving one or more transit populationsalso requires some means of regulating of the total num-ber of transit cells in each compartment. This regulationcould be mediated through modulation of the AR of theupstream stem cell or rate of entry of upstream transitcells. However, feedback regulation in this setting wouldneed to occur over significant and potentially impracticaldistances. As a result, regulation of the size of the localtransit population (ie, a function of the rate of entry, pro-liferation, and residence time of cells within each com-partment) is more likely to be mediated by the effect oflocal signals on the activation/migration AR, proliferationkinetics (µ), differentiation (P), or life-span (lx).

Transit populations can also serve to distribute theprogeny of stem cells beyond the limited domain of theupstream stem cell niche. This may occur by migration of

180 George F. Muschler et al 2003:3 (2003)

T1

T2

T3

T4

T5

tx lx µx Nx Vx1000 days 25,000 days 25 1 .001100 days 1000 days 10 1 .0120 days 200 days 10 2 .11 day 10 days 10 1 1∞ 10 days 0 10 10

tx = cycle time

lx = life-span

µx = total mitoses in lifetime = lx/tx

Nx = number of cells = Nx−1lx/tx−1

Vx = velocity of cells leaving this compartment (cells/day) (Nx/tx)

Figure 5. Conceptual model of transit cell amplification. This schematic diagram and associated table illustrate several central con-cepts in the proliferating transit cell paradigm. A stem cell (T1) and three proliferating transit cell populations (T2, T3, and T4) aremodeled at steady state in a contiguous linear array supporting a population of mature cells (M/T5). All cell division is modeled asasymmetric events associated with renewal of the founding cell. The table illustrates hypothetical values for the cycle time (tx), life-span for cells in each compartment (lx), the effective mean number of cell divisions in each compartment (µx), the resulting numberof cells resident in each transit compartment (Nx), and a velocity or rate at which cells leave each compartment (Vx). In this model,the originating stem cell survives throughout the life of the individual (l1 = 25,000 days ∼ 68 years), cycling as a slow rate of one celldivision every 1000 days (t1 = 1000 days). During the life of the stem cell, it divides a total of 25 times (µ1 = 25). The velocity of cellsleaving the stem cell compartment and entering the T2 compartment is 1 cell per 1000 days, or V1 = 0.001 cells/day. Cells in the T2compartment function similarly to feed the T3 compartment, and so on. Cells in downstream populations (T2, T3, and T4) dividemore rapidly than cells in upstream compartments. In contrast to the originating stem cells, the cells in downstream compartmentsalso have limited self-renewal capacity, resulting in decreasing functional life-span for cells in each compartment. Note that, in theabsence of any transit populations, the upstream stem cell would need to divide 25,000 times to generate the same number of maturecells over the life of the individual.

transit cells through tissue or by transport within systemiccirculation, as is the case with the transit monocyte popu-lation that contributes to osteoclast formation. A broadlydistributed migratory transit population, having the po-tential for proliferation, also provides advantages in thesetting of tissue injury and repair. Locally resident tran-sit cells are better positioned to respond to changes in lo-cal tissue conditions and signaling events, and potentiallyavoid the inevitable delay that would result if tissue repairwas to require the activation, proliferation, and migrationof cells from a remote upstream stem cell niche.

Transit populations upstream of theosteoclast compartment

A diagram of transit cell compartments upstreamof the osteoclast is relatively simple to illustrate con-ceptually. (See Figure 6.) The diagram begins with thesmall population of adult pluripotent hematopoietic stemcells that have long-term repopulating potential (HSC-LT) [113, 114]. These are activated to divide and theirnonstem cell progeny undergo symmetric clonal expan-sion, passing through a series of downstream “prolifer-

ating transit populations.” These downstream transit cellpools include a small population of cells that have lim-ited self-renewal capacity, resulting in short-term repop-ulating activity (HSC-ST) but still give rise to multipo-tent progeny. Further downstream are a population ofcommon myeloid progenitors followed by granulocytemacrophage precursors (CFU-GM) and finally commit-ted macrophage forming progenitors (CFU-M).

Cells in the CFU-M compartment in marrow exitfrom the marrow space and enter into a “nonproliferatingtransit population,” as circulating monocytes, with someprobability (PM). Monocytes are then distributed system-ically in circulation, making them accessible to local acti-vation signals for a period of time (lM). These local sig-nals can result in their subsequent activation to move intoother cell compartments, including tissue monocytes andmacrophages. The third transit compartment in the os-teoclast lineage is the tissue monocyte that has left circu-lation to reside in the bone marrow or osteonal compart-ment of bone. The fourth and final transit compartmentis the osteoclast compartment, where monocyte-derivednuclei fuse to transiently contribute to the osteoclast pop-ulation. The osteoclast population persists throughout the

2003:3 (2003) Connective Tissue Stem Cell Kinetics 181

NHSC -LT

NHSC -ST

NCMP

NGMP

NM

NBTM

NOC

dNM/dt

CLP

MEP

Granulocytes

lM

Figure 6. Transit cell populations in osteoclast formation. Theconceptual hierarchy of transit cell populations upstream of theosteoclast is illustrated. Triangles indicate phases of clonal ex-pansion arising from proliferating transit populations. Black ar-rows indicate transit events in which cells move from one com-partment to another associated with changes in their intrinsicbiological properties. Grey arrows indicate departure of cellsfrom the upstream compartment to other cell compartments.Green boxes illustrate the conceptual size of each cell popula-tion, where the width of the box represents the rate at which cellsare added to or leave each compartment at steady state, and theheight of each box represents the mean life-span or residencetime of cells within each compartment. (Abbreviations: HSC-LT, long-term repopulating hematopoietic stem cell; HSC-ST,short-term repopulating hematopoietic stem cell; CMP, com-mon myeloid progenitor; CLP, common lymphoid progenitor;MEP, megakaryocyte erythroid progenitor; GMP, myelomono-cytic progenitor; M, monocyte; BMT, bone tissue monocyte;OC, osteoclast nuclei [114].)

life of the cutting cone of a BMU. However, this pop-ulation is continually fed by the addition of new nucleithrough new fusion events, balancing the simultaneousturnover of other nuclei. The transit time for nuclei in theosteoclast compartment has been estimated to be approx-imately 12.5 days [115].

Each of these transit compartments is associated withand defined by an overall AR, mean number of effec-tive mitoses (µ), mean life-span (l), and probability oftransit to the downstream population (P). The processof monocytes nuclei contributing as a transit population

to osteoclast generation is rather unique. In terms of ab-solute cell number, transition from monocyte to osteo-clast represents a reverse amplification event, requiringseveral monocytes (∼ eight) to make one osteoclast (ie,µ ∼ −3).

Using this model concept, access to quantitative infor-mation about the number of cells (nuclei) in each com-partment and the mean life-span of cells within each com-partment can be used to gain significant insight into thepossible range of kinetic parameters governing the transitprocesses leading to osteoclast development.

Transit populations downstream of theosteoblast compartment

Using the transit cell paradigm described above, it isalso possible to begin to build a model system of transitcell compartments that contribute upstream to osteoblastformation, and to model the downstream transit cell com-partments that contribute to the removal of osteoblasts.

Much more is known about the downstream transitcompartments, as illustrated in Figure 7, than about com-partments that are upstream of the osteoblast. Removalof osteoblasts from a region (ie, the transit of cells outof the osteoblast compartment) occurs through three pri-mary pathways or transit events: formation of an osteo-cyte, formation of a lining cell, and cell death via apopto-sis. These variables are absent from the model developedabove, because the model was based on the variables in-fluencing the rate of bone tissue formation and removal,and the contribution of osteoblasts to bone matrix vol-ume ends when they transit out of the osteoblast compart-ment. However, the transit of osteoblasts into the down-stream populations of osteocytes and lining cells, whilenot a determinant of the rate at which new bone tissue isformed, is a critical variable determining the density anddistribution of osteocytes and lining cells, and thereforethe histologic features, biologic environment, and long-term health of the newly formed bone tissue. The distri-bution of these cells in bone tissue is likely to have signifi-cant effects on the function and maintenance of the newlyformed bone and on the initiation and propagation of fu-ture cycles of bone resorption and bone formation in thattissue volume.

The mean probability that any given osteoblast willfollow one of these pathways can be represented as

ρo + ρl + ρap = 1, (20)

where

(i) ρo = the probability of forming an osteocyte,(ii) ρl = the probability of forming a lining cell,

(iii) ρap = the probability of apoptosis.Based on this concept, and substituting the expression

for the rate of osteoblast formation (Nob/dt) into (10), the

182 George F. Muschler et al 2003:3 (2003)

dNOb/dt

NOb

NO NLApoptosis

lOb ∼ 0.1 years

ρo = .20ρA = .75ρ1 = .05

lO ∼ lL ∼ 15–25 years

Figure 7. Transit cell populations downstream of the osteoblast. The three differentiation pathways available to osteoblasts and the twotransit populations downstream of the osteoblast are illustrated, using the same illustration strategy described in Figure 6. Probabilityvalues that would be common in cortical osteonal bone remodeling are illustrated.

rate of formation of new osteocytes (No/dt) can be ex-pressed as

Nodt= Nob

dtρo = NCTS ARCTS 2µCTSPobρo. (21)

Assuming steady state conditions, then the total num-ber of osteocytes in a region (No) will expressed as

No = Nobdt

ρolo = NCTS ARCTS 2µCTSPobρolo, (22)

where, lo = the mean life-span of an osteocyte.Similarly, at steady state, the relative number of osteo-

cytes and osteoblasts in a given region of bone can be ex-pressed as

NoNob

= lo ρolob

. (23)

Based on rough estimates of these values (ρo ∼ 0.2,lo ∼ 20 years, lob ∼ 0.1 years) [85, 86], the mean ratioof osteocytes to active osteoblasts should be in the rangeof 40 to 1. However, this is expected to vary significantlybetween sites. The difference in remodeling rate betweencortical and cancellous bone results in a generally longerlife-span of osteocytes in cortical bone than in trabecularbone. Similarly, ρo will change significantly with the ge-ometry of the site of bone formation, as discussed below.

The rate of formation of new osteoblasts (Nob/dt) canalso be investigated beginning with data available fromhistomorphometric measurements. At steady state, thisrate will be equal to the rate at which osteoblasts transitout of the osteoblast compartment (rNob/dt). The rate ofremoval will be related to the total number of osteoblasts

in the region of interest (Nob) and life-span of the os-teoblast, based upon the relationship derived from (4):

rNobdt

= Nobdt

= Noblob

= NCTS ARCTS 2µ

CTSPobρololob

.(24)

Since the rate of removal of osteoblasts (rNob/dt) canalso be expressed as a sum of the rate of the three path-ways, or

rNobdt

= Nodt

+Nldt

+Napdt

, (25)

where,

No/dt = the rate of formation of new osteocytesNl/dt = the rate of formation of new lining cellsNap/dt = the rate of osteoblast loss due to apopto-sis,

the relative velocity of the three rates is determined bythe relative probability that a mature osteoblast will fol-low each of the pathways (ρo, ρl, and ρap).

Finally, the density of osteocytes within the newlyformed bone matrix (Do) will be determined by the rateof formation of new osteocytes (No/dt), the number ofactive osteoblasts (Nob), and the rate of formation of newbone matrix per osteoblast (dσob/dt), according to the re-lationship

Do = No/dtNob

· dσobdt

. (26)

2003:3 (2003) Connective Tissue Stem Cell Kinetics 183

Substituting terms for No/dt and Nob from (21) and (4),respectively, provides that

Do =[NCTS ARCTS 2µCTSPob

]

× ρo[NCTS ARCTS 2µCTSPob

]

× lob dσobdt

or Do = ρolob

dσobdt

.

(27)

It is interesting to note that, osteocyte (lacunar) den-sity has been reported to be higher in females than inmales [116], and higher in osteoporosis subjects than inage-matched normal subjects [94]. These findings wouldsuggest that the pathomechanics of osteoporosis may beassociated with a decreased rate of matrix synthesis per os-teoblast (dσob/dt), a decrease in osteoblast life-span (lob),and/or an increase in the probability of osteocyte forma-tion (ρo).

It is also interesting to note that the anatomic site orgeometry of the BMU will have an profound influenceover the likely fate of an osteoblast with respect to theprobability of apoptosis or differentiation as a lining cell[79]. Figures 8a through 8c and data presented in Table 1illustrate the predicted range of variation in the probabil-ity factors regulating the fate of osteoblasts with geometryof the site (ie, the contour and thickness of the new boneformed) and with osteocyte density (eg, cortical versustrabecular cancellous bone). Increasing matrix thicknessand increased osteocyte density are associated with an in-creased probability (ρo) of osteocyte formation, and a de-crease in the allowable probability of apoptosis (ρA). Sim-ilarly, the transition from concave surfaces (such as the in-terior of an osteon in Figure 8a) to formation of new boneon a flat surface (such as a periosteal surface or trabecu-lar plate as shown in Figure 8b) or to formation of newbone on a convex surface (such as the cylindrical sectionof a trabecular strut illustrated in Figure 8c) is associatedwith increasing demands and probability of transit to lin-ing cell and osteocyte population, and a decreasing allow-able range of apoptosis. These changes are also associatedwith an increase in the mean volume of new bone synthe-sis required per starting osteoblast (σob). The need for os-teoblast retention as osteocytes and lining cells effectivelylimits the maximal thickness of new bone matrix produc-tion with each remodeling cycle, particularly on convexsurfaces.

Transit populations upstream of theosteoblast compartment

Direct objective information upon which to build aconceptual model for transit cell compartments upstreamof the osteoblast is much more difficult. Regardless of this,there is a significant volume of data and observation thatcan be assembled in an attempt to strategically dissectquestions related to the likely size, hierarchy, and kineticsof transit populations upstream of the osteoblast.

(a) (b) (c)

Figure 8. The effect of surface geometry on the fate of 128 os-teoblasts. A cell-based model is shown representing bone forma-tion on three different surface geometries: (a) a concave segmentof a 200 µm diameter osteon in cortical bone, (b) a flat surfacein cortical or cancellous bone, and (c) the convex surface of a100 µm diameter trabecula in cancellous bone. In each case, aset of 128 osteoblasts is shown at t = 0 as ∼ 12.5 µ cuboidalcells covering an appropriate segment of the bone surface. Be-low, the same surface and the new bone formed by this set of128 osteoblasts are shown. In each case, some fraction of cellsmust become embedded in the matrix as osteocytes to maintainan appropriate osteocyte density and some osteoblasts must beretained as lining cells covering the remaining surface. Cells thatare not required as osteocytes or lining cells are presumed to belost through apoptosis. The probability of osteoblast transit intothe osteocyte or lining cell population, and the probability ofapoptosis are dependent on surface geometry, the density of os-teocytes in the matrix, and the thickness of the new bone that isformed at the site.

As discussed above, Bianco et al [72, 73] have pre-sented histologic observations to support the concept thatosteoblasts in bone may be derived from a populationof fibroblastic cells in bone marrow known as Westin-Bainton cells. Other evidence indicates that cells derivedfrom the perivascular compartment (vascular pericytes)have the capacity to contribute to the osteoblast compart-ment [69, 70]. There is also recent evidence suggests thatosteogenic cells may also transit through peripheral blood[107, 117]. As a result, it is necessary for any model ofosteogenic transit populations to include not only an up-stream stem cell niche, but also possible transit compart-ments of circulating cells, vascular pericytes, and Westin-Bainton cells.

Detailed histologic analysis of BMUs in cortical boneand radioactive labeling studies has also suggested thepresence of another small compartment of proliferatingcells that is located very close to the junction of the osteo-clasts in the cutting cone of the BMU and the region whereall new osteoblasts are incorporated. Radionucleotide la-beling is seen within this population of cells early afterinjection, suggesting a high proliferation rate. Further-more, by 1–1.5 days after labeling, radiolabel remains ev-ident in the type I population and is also seen in the new

184 George F. Muschler et al 2003:3 (2003)

Table 1. The effect of osteocyte density and surface geometry on osteoblast fate and function. This table provides a quantitativeassessment of the end result in each geometric configuration illustrated in Figures 8a, 8b, and 8c. Each geometry calculations arebased on formation of 40 µm or 60 µm thick volume new bone. The table illustrates the fate of the initial set of 128 osteoblasts,identifying the number of osteocytes (No) and lining cells (NL) that are required and the probability of an initial osteoblast form anosteocyte (ρo), a lining cell (ρl), or to undergo apoptisis (ρA). Calculations for cortical and trabecular cancellous bone differ based onpublished values for osteocyte density in cortical and trabecular bone. Note that the required probability for osteocyte formation (ρo)increases dramatically in these examples from 0.13 to 0.69 as the surface geometry changes from concave to convex, as the osteocytedensity changes from cortical to cancellous bone, and as the thickness of new bone increases.

Nob

No

NL

NA

Po

PL

PA

Do (osteocytes/µm3)

σob (µm3)

Total matrix (µm3)

Matrix thickness (µm)

128

17

5

106

0.13

0.04

0.83

0.000026

5152

659400

40

Corticoid

128

23

4

101

0.18

0.03

0.79

0.000026

6991

894900

60

Corticoid

128

21

8

99

0.16

0.06

0.78

0.000026

6,250

800,000

40

Corticoid

128

31

8

89

0.24

0.06

0.70

0.000026

9,375

1,200,000

60

Corticoid

128

38

8

82

0.30

0.06

0.64

0.000047

6,250

800,000

40

Cancellous

128

56

8

64

0.44

0.06

0.50

0.000047

9,375

1,200,000

60

Cancellous

128

51

14

63

0.40

0.11

0.49

0.000047

8,517

1,090,000

40

Cancellous

128

88

17

23

0.69

0.13

0.18

0.000047

8,517

1,090,000

60

Cancellous

Osteon Flat Convex

Table 2. Maximum matrix thickness of new bone formation on a concave, flat, or convex surface. This table illustrates the maximumtheoretical thickness of new bone formation for the 128 cells illustrated in each geometric configuration shown in Figures 8a, 8b, and8c. For flat and convex surfaces, the limit occurs when osteoblasts become osteocytes and lining cells and no osteoblast undergoesapoptosis (ρA = 0). In contrast, in the concave configuration of an osteon, the thickness is limited by the maximum diameter of acylinder that can be occupied by a single row of 128 osteoblasts (∼ 250 µm). In this case, approximately half of the initial osteoblastsmust still undergo apoptosis if the observed osteocyte density is to be maintained at or near the normal osteocyte density in corticalbone.

Nob

No

NL

NA

Po

PL

PA

Do (osteocytes/µm3)

σob (µm3)

Total matrix (µm3)

Matrix thickness (µm)

128

61

1.5

64

0.48

0.01

0.51

0.000026

18,408

2,356,192

250

Corticoid

128

120

8

0

0.94

0.06

0.00

0.000026

36,058

4,615,385

231

Corticoid

128

120

8

0

0.94

0.06

0.00

0.000047

19,947

2,553,191

128

Cancellous

128

109

19

0

0.85

0.15

0.00

0.000047

18,099

2,316,692

70

Cancellous

Osteon Flat Convex

2003:3 (2003) Connective Tissue Stem Cell Kinetics 185

dNOb/dt

1ObNOb

NPre-Ob

NWB

NP

NC

NS

?

?

??

Adipocytes

Figure 9. Transit cell populations upstream of osteoblasts. Theputative transit cell populations that are upstream of the os-teoblast are shown, using the same illustration strategy describedin Figures 6 and 7. The upstream originating stem cell, the peri-cyte, and the pre-osteoblast (type I osteoblast) are all presumedto be proliferating transit populations. Precise features of thisdiagram (the magnitude of expansion, the life-span, the rateof transit, and even pathways of transit between these potentialpopulations) must be considered highly speculative, though it isconsistent with available data and prevailing theory. Regardlessof this speculation, it is useful to compare this diagram to thatillustrating the events that occur downstream of the osteoblastshown in Figure 7. This comparison illustrates the very smallrate of cell division and small rate of transit that must be ex-pected in transit populations upstream of the osteoblast, rela-tive to downstream events. Similarly, it also illustrates that oneor more of these upstream populations (eg, the Westin-Baintoncell) might be present in comparable numbers to the active os-teoblast population, if the life-span of cells in these transit com-partments significantly greater that was the life-span of the se-cretory osteoblast. (Abbreviations: S, upstream stem cell; C, cir-culating stem cell; P, vascular pericytes; WB, Westen-BaintonCells; Pre-Ob, Pre-osteoblast (Type I osteoblast); Ob, mature se-cretory osteoblast.)

osteoblastic cells that are added to the advancing frontof osteoblasts. This unique population of cells, about 8cells per BMU, has been referred to as type I osteoblasts[102, 118]. Based on these observations, we interpret thispopulation of cells to represent a small proliferating tran-sit population that is immediately upstream of the secre-tory osteoblast which has some self-renewal capacity, sim-ilar to the transit populations illustrated in Figure 5.

Many possible models could be proposed to linkingthese compartments into a hierarchy. However, in the ab-sence of compelling data to the contrary, the simplestpossible model involving all of these compartments is amodel of linear progression of cells through these com-partments in an order that is based on physical proxim-ity to the bone forming surface analogous to the previousmodel for osteoclast formation, as illustrated in Figure 9.In this model, osteogenic cells may be envisioned to tran-sit through a stem cell and progenitor cell system with upto five compartments upstream of the mature secretory

osteoblast. This model begins with a true initiating stemcell population (NS or T1), followed by a circulating tran-sit cells (NC or T2), the pericyte compartment (NP or T3),the Westin-Bainton compartment (NWB or T4), Type I os-teoblasts (NobI or T5), and secretory osteoblasts (Nob orT6).

With the exception of the type I osteoblast compart-ment, there is no objective data to demonstrate the tran-sit of cells between these compartments, nor prolifera-tion within any one compartment, including the relativelyabundant population of Westin-Bainton cells. As a re-sult, the transit of cells between these compartments andthe stem cell kinetics associated with these compartments(proliferating or non-proliferating transit) under normalremodeling conditions is entirely speculative. It seems rea-sonable to assume with some confidence that the origi-nating stem cell compartment represents a proliferatingtransit population, though the kinetics of this compart-ment is entirely unknown. Similarly, given the capacityof pericytes to be cultured in vitro to produce a prolif-erating population of osteogenic cells, it seems reason-able to expect that the pericyte compartment also repre-sents a proliferating transit population, as illustrated inFigure 9. Also represented in Figure 9 is the seemingly rea-sonable assumption that the life-span of a pericyte and ofa Westin-Bainton cell is quite long in comparison to thelife-span in the compartments containing the circulatingosteoblastic progenitor, type I osteoblast, or osteoblaststhemselves, though the actual life-span of cells in thesecompartments is not known. The number of symmetricmitoses in the stem cell compartment or pericyte com-partment is also entirely speculative. The same is true forthe probability that the cells leaving each of these com-partments will transit to the next, though for graphic pur-poses Figure 9 arbitrarily illustrates a probability (P) oftransit to the next compartment of ∼ 0.5.

Despite these limitations, this organizational hierar-chy provides a starting point from which to explore andtest assumptions regarding the relative size of these com-partments, the presence or absence of transit events be-tween these compartments, and the rate and kinetics thatmay be associated with these events in order to supportbone remodeling and/or in settings of injury or repair.This is illustrated below.

In accord with the previous discussion, for each com-partment, the rate at which cells are added to a compart-ment (dNx/dt) is also equal to the rate at which cells movefrom the upstream compartment to the next, and can berepresented as

dNx

dt= Nx−1 ARx−1 2µx−1Px, (28)

where

(i) Nx−1 = number of cells in the upstream (x−1) pop-ulation,

(ii) ARx−1 = mean activation rate of cells in the up-stream (x − 1) population,

186 George F. Muschler et al 2003:3 (2003)

(iii) µx−1 = mean number of symmetric mitoses in theupstream (x − 1) population,

(iv) Px = the probability of cells generated in the up-stream (x − 1) population to become a transit cellin the x population.

Based on this relationship, with the exception of theinitiating stem cell population (T1), the total number ofcells in each compartment (Nx) can be represented as

Nx = dNxdt

lx = Nx−1 ARx−1 2µx−1Pxlx, (29)

where lx = mean life-span of a cell in the x population.Note. If the upstream compartment is composed of

only nonproliferating transit cells (ie, µ = 0), the term ARis equal to the rate at which cells in this compartment areactivated to leave their current state and transit to anothercompartment.

Based upon the relationship described in (29), the ra-tio of the number of cells observed in one compartmentand the number of cells observed in an adjacent compart-ment becomes a tool in the assessment of the kinetics be-tween two adjacent compartments, since

Nx−1Nx

= 1(ARx−1 2µx−1Pxlx

) . (30)

The logical starting point to begin to evaluate the utilityof this theoretical relationship is in an exploration of thekinetic interface between the T5 population of type I os-teoblasts and the T6 compartment of active secretory os-teoblasts, where at least some objective data exists. Thisdata suggests that the ratio of type I osteoblasts to se-cretory osteoblasts is approximately 1 : 125 (∼ 8 type Iosteoblasts to 2000 secretory osteoblasts in a fully activecutting cone) [102, 118] and that this population prolifer-ates relatively rapidly, allowing cells to exit this compart-ment within about 24–36 hours. Beyond this information,the remainder of a first-order analysis must be based on aset of assumptions. One set of possible parameters that isuseful for a first-order exploration includes the followingthree assumptions: (1) all type I osteoblasts become os-teoblasts (PobI = 1), (2) the activation rate of the cells inthe type I compartment is approximately one cell divisionper day (ARobI = 1, ie, a cycle time of 24 hours), and (3)the total number of asymmetric mitoses per cell duringresidence within the type I osteoblast compartment is inthe range of 13 (lobI ∼ 13 days).

If these assumptions are correct, then two other pa-rameters follow. First, based on substitution into (30),the effective number of symmetric mitotic events (µobI)predicted among the progeny of an activated type I os-teoblast would slightly be greater than 3, resulting in ap-proximately 10 new osteoblasts for each activation event.Second, the rate at which new type I osteoblasts wouldneed to be added from the upstream compartment wouldbe approximately 0.6 cells per day (dNobI/dt = NobI/lobI =8/13).

While these assumptions and the calculated kineticparameters resulting from this example are internally con-sistent and within the range of predicted biological feasi-bility, this example must not be over interpreted. Currentmarkers for stem cell and progenitor cell populations andhistomorphometric methods for counting cells and mea-suring proliferation rates in vivo, have not yet providedthe means of reliably testing the validity and utility of thisapproach. Without these data, the model is primarily use-ful as a conceptual tool for interpretation of increasinglyrich and quantitative histologic and histomorphometricdata.

Application of cell-based modeling to the clinicaland experimental settings

While the quantitative data to support the full applica-tion of a cell-based modeling strategy is not yet available,an increasing number of publications are providing datathat will allow these kinds of analysis. For example, in thesetting of estrogen deficiency, there is recent evidence ofa decreased life-span among both osteoblasts and osteo-cytes [119]. Systemic exposure to corticosteroids also hasthese effects [120]. These findings are consistent with theobserved contraction of both of these transit cell compart-ments in both settings.

Cell-based modeling may also be useful in interpre-tation of the apparent accumulation of Westin-Bainton-like alkaline phosphatase-positive cells within the areas ofintramedullary fibrosis that is observed in hyperparathy-roidism [121]. In this context, an increase in size of aWestin-Bainton-like compartment would be attributed toincreased activation (ARP) or proliferation (µP) in the up-stream compartment (eg, pericytes), an increased proba-bility that cells from the upstream compartment will en-ter the WB compartment (PWB), or an increase in life-span of cells within the WB compartment (lWB). An in-crease in life-span could in turn be mediated by a de-crease in the rate with which cells in the WB popula-tion are activated to transit downstream compartments(ARWB). The accumulation of intramedullary fibrous tis-sue in the setting of fibrous dysplasia, a condition re-sulting from constitutive activation of a Gα S-protein,similar to the pathway activated by tonic PTH stimu-lation, may also be interpreted in this way [106, 122,123].

Cell-based modeling may also be instructive in the de-sign and selection of experimental strategies. For exam-ple, the model predicts that the magnitude or velocityof transit events into the type I osteoblast pool does notneed to be very large in order to support ongoing boneremodeling activity. This would imply a low basal acti-vation rate in the upstream compartment. Activation inthe Westin-Bainton population (ARWB) could be partic-ularly rare in light of the relative abundance of Westin-Bainton cells in bone marrow. As a result, observationof these transit events during normal bone remodelingwould be highly unlikely. The setting of intramedullary

2003:3 (2003) Connective Tissue Stem Cell Kinetics 187

α 2α µ

Figure 10. Three options for stem cell division. Three possibleoutcomes of stem cell activation and cell division are illustrated:an “α” division represents the classic asymmetric cell divisionwith renewal of the mother cell and generation of a daughterthat enters a downstream transit population. A “2α” cell divi-sion generates two identical stem cells, increasing the numberof total stem cells by one. Finally, a “µ” division generates twocells that enter a downstream transit population, depleting thenumber of stem cells by one. The balance between “2α” and “µ”events determines whether a given stem cell pool will increase ordecrease in number.

trauma or fracture healing might be expected to be verydifferent, however, since the Westin-Bainton population(and/or the pericyte population) would appear to be theosteogenic transit compartments that are most effectivelypositioned to respond in the rapid regional mobilizationof the bone healing response that is required in these set-tings.

Finally, as it is further developed, cell-based modelingwill offer many new capabilities for interpretation and in-vestigation of in vivo phenomenon in the setting of em-bryonic development, tissue remodeling, disease states,responses to targeted drug and cell therapies, the tissuelevel effects of targeted mutations and knockouts. For ex-ample, if a pharmacologic agent, disease, mutation, orknockout was found to be associated with a significant in-crease in the number and prevalence of type I osteoblasts,then targeted assessment of the ARobI, lobI, µobI, Pob, orlob should reveal the underlying kinetic process that is af-fected and responsible for this change.

Ultimately, the utility and validity of cell-based mod-els must be tested against experimental data involvingdirect measurement of the number of cells in the rele-vant compartment and their associated kinetic parame-ters. This will require significant improvement in the cellspecific markers and methods that are currently available.

STEM CELL AND TRANSIT CELL SELF-RENEWALAND SELF-EXPANSION

Cell-based modeling also requires strategies which de-scribe the origin, expansion, and maintenance of the stemcell and transit cell populations throughout the life of anindividual, via the biologically essential process referred tocommonly as self-renewal. An extensive literature is avail-able on this subject [99, 111], which cannot be reviewedhere. However, some exploration of these concepts is ap-propriate to the development of the current model.

When a stem cell is activated, at any stage of devel-opment, there are conceptually three possible outcomes,as illustrated in Figure 10. The classic mechanism for self-

renewal is “asymmetric division” producing one daughtercell that is identical to the mother cell and one daugh-ter cell that is intrinsically different than the mother celland goes on to proliferate and mature. For the purpose ofthis discussion, and to remain consistent with a previouspublication, we referred to this as an “α” division [59].This mechanism by itself would be sufficient to maintainthe stem cell population in adults. However, at least twoother options are possible. Two stem cells that are iden-tical to the mother could also be produced (a “2α” celldivision in Figure 10). Alternatively, two daughters couldresult which are both different than the mother cell, whichwe have referred to as a “µ” division. As a result, in anypopulation of stem cells, any individual activation eventwould have some probability of each of these three path-ways (pα, p2α, and pµ, respectively), where

pα + p2α + pµ = 1. (31)Each of these pathways must be possible. In adult life,

when maintenance of stem cell populations would seem tobe the goal, one might expect that the α division would bestrongly favored, and that the frequency of the other twopathways would be small (pα � p2α, pµ). However, thissituation is not required for stem cell maintenance. A stemcell population can be maintained even if the probabilityof α division is low, provided that the probability of 2αand µ divisions were equal over time (p2α = pµ).

The mechanism by which each stem cell populationmaintains functional self-renewal is therefore uncertain,and may vary greatly depending upon the population.What is definite, however, is that the probabilities of thesedifferent stem cell division pathways shift significantlyduring growth and developm