NASA/TM—2009-215824

Simulating Bone Loss in Microgravity Using

Mathematical Formulations of Bone Remodeling

James A. Pennline

Glenn Research Center, Cleveland, Ohio

November 2009
NASA STI Program . . . in Profile

Since its founding, NASA has been dedicated to the

advancement of aeronautics and space science. The

NASA Scientific and Technical Information (STI)

program plays a key part in helping NASA maintain

this important role.

The NASA STI Program operates under the auspices

of the Agency Chief Information Officer. It collects,

organizes, provides for archiving, and disseminates

NASA’s STI. The NASA STI program provides access

to the NASA Aeronautics and Space Database and

its public interface, the NASA Technical Reports

Server, thus providing one of the largest collections

of aeronautical and space science STI in the world.

Results are published in both non-NASA channels

and by NASA in the NASA STI Report Series, which

includes the following report types:

TECHNICAL PUBLICATION. Reports of

completed research or a major significant phase

of research that present the results of NASA

programs and include extensive data or theoretical

analysis. Includes compilations of significant

scientific and technical data and information

deemed to be of continuing reference value.

NASA counterpart of peer-reviewed formal

professional papers but has less stringent

limitations on manuscript length and extent of

graphic presentations.

• TECHNICAL MEMORANDUM. Scientific

and technical findings that are preliminary or

of specialized interest, e.g., quick release

reports, working papers, and bibliographies that

contain minimal annotation. Does not contain

extensive analysis.

• CONTRACTOR REPORT. Scientific and

technical findings by NASA-sponsored

contractors and grantees.

• CONFERENCE PUBLICATION. Collected

papers from scientific and technical

conferences, symposia, seminars, or other

meetings sponsored or cosponsored by NASA.

• SPECIAL PUBLICATION. Scientific,

technical, or historical information from

NASA programs, projects, and missions, often

concerned with subjects having substantial

public interest.

• TECHNICAL TRANSLATION. Englishlanguage translations of foreign scientific and

technical material pertinent to NASA’s mission.

Specialized services also include creating custom

thesauri, building customized databases, organizing

and publishing research results.

For more information about the NASA STI

program, see the following:

• Access the NASA STI program home page at

http://www.sti.nasa.gov

• E-mail your question via the Internet to help@

sti.nasa.gov

• Fax your question to the NASA STI Help Desk

at 443–757–5803

• Telephone the NASA STI Help Desk at

443–757–5802

• Write to:

NASA Center for AeroSpace Information (CASI)

7115 Standard Drive

Hanover, MD 21076–1320
NASA/TM—2009-215824

Simulating Bone Loss in Microgravity Using

Mathematical Formulations of Bone Remodeling

James A. Pennline

Glenn Research Center, Cleveland, Ohio

National Aeronautics and

Space Administration

Glenn Research Center

Cleveland, Ohio 44135

November 2009
Trade names and trademarks are used in this report for identiÞ cation

only. Their usage does not constitute an ofÞ cial endorsement,

either expressed or implied, by the National Aeronautics and

Space Administration.

Level of Review: This material has been technically reviewed by technical management.

Available from

NASA Center for Aerospace Information National Technical Information Service

7115 Standard Drive 5285 Port Royal Road

Hanover, MD 21076–1320 SpringÞ eld, VA 22161

Available electronically at http://gltrs.grc.nasa.gov
Simulating Bone Loss in Microgravity Using Mathematical

Formulations of Bone Remodeling

James A. Pennline

National Aeronautics and Space Administration

Glenn Research Center

Cleveland, Ohio 44135

Abstract

Most mathematical models of bone remodeling are used to simulate a specific bone disease, by

disrupting the steady state or balance in the normal remodeling process, and to simulate a therapeutic

strategy. In this work, the ability of a mathematical model of bone remodeling to simulate bone loss as a

function of time under the conditions of microgravity is investigated. The model is formed by combining

a previously developed set of biochemical, cellular dynamics, and mechanical stimulus equations in the

literature with two newly proposed equations; one governing the rate of change of the area of cortical

bone tissue in a cross section of a cylindrical section of bone and one governing the rate of change of

calcium in the bone fluid. The mechanical stimulus comes from a simple model of stress due to a

compressive force on a cylindrical section of bone which can be reduced to zero to mimic the effects of

skeletal unloading in microgravity. The complete set of equations formed is a system of first order

ordinary differential equations. The results of selected simulations are displayed and discussed.

Limitations and deficiencies of the model are also discussed as well as suggestions for further research.

1.0 Introduction

Under the conditions of microgravity, astronauts lose bone mass and calcium (Ref. 1). In fact, bone can

atrophy in space at a rate of about 1 percent a month, and some models suggest that a 40 to 60 percent total

loss could eventually be reached (Ref. 13). Data from Buckley (Ref. 1) indicates that the greatest loss occurs

in the lower extremities which experience the higher skeletal loadings during normal activity on earth.

National Aeronautics and Space Administration (NASA) has a specific interest in understanding the

mechanisms for bone loss during space flight for several reasons. Bone recovery has proven problematic.

Based on a recovery model developed from medical data on astronauts who had flown on long duration

missions (4 to 6 months), it could take as much as 9 months to recover 50 percent of the bone loss in some

skeletal sites (Ref. 28). Other health hazards instigated by skeletal bone loss are accumulations of excess

minerals in tissue such as the kidney, and increased risk of fracture. NASA is interested in developing

interventions to sustain and protect the well being of its astronauts. Consequently, understanding bone loss

is one of the research efforts being conducted for development of The Digital Astronaut. The Digital

Astronaut is a an integrated, modular modeling and database system intended to support biomedical

research to help identify and interpret medical and physiological research, to determine the effectiveness of

specific human countermeasures to meet health and performance goals on exploration missions, and to

evaluate medical interventions during mission emergencies, accidents, and illnesses (Ref. 24). Of particular

concern is the effect on bone health from a longer duration mission such as a journey to Mars or the

development of a Moon base. Existing measures of scheduled exercise with specially designed equipment

may not be sufficient for preventing bone loss.

Bone remodeling is the process by which bone is removed and replaced on the same surface site by

highly mediated and coupled action of bone cells. It’s the physiological mechanism for enabling the

normal maintenance of bone in the adult and is achieved by a balance between the processes of bone

formation and bone resorption. These processes can be influenced by skeletal loading, endocrine

regulation, and local biochemical mediators. Bone loss experienced by astronauts is thought to be caused

by a disruption in the resorption-formation balance in favor of resorption, triggered by skeletal unloading.

NASA/TM—2009-215824
Gravity is somehow converted from a mechanical signal to a chemical signal. Although much is known

about the chemical signals, the process by which cells sense mechanical stimuli and then translate the

information to a signal (mechanotransduction) that causes response within the cell or another cell, is not

fully understood (Refs. 13 and 2).

Significant knowledge has been gained in understanding the biomechanical and molecular regulation

of bone remodeling (Ref. 20). Work done in biomechanics has used mathematical formulations to

describe how bone responds to stress and strain under loading. For example, in the study of fracture

mechanics, mechanical and material properties are used to study how bones deform under loads. At the

molecular and cellular level, building mathematical models of cellular dynamics has become an

increasingly active research area in the last decade (Refs. 4, 5, 7, 8, 9, 14, 17, 19, 23, and 25). The cells

that carry out the remodeling process are the osteoclasts that resorb bone by secretion of acid and

proteases, and the osteoblasts that form bone by forming an initial collagen matrix (osteoid) and then

mineralizing the collagen (Ref. 15). Osteocytes, matured osteoblasts encased in calcified bone matrix, are

the cells that have been suggested to be the stimulus sensors or receptors of the stimulus signal. This

suggestion is consistent with histologic and physiologic data (Ref. 2). One of the more descriptive

models, which attempts to accurately detail the biochemical process, is the one by Lemaire et al. (Ref. 7).

That model has also been used by other researchers as a base on which to build an extended model

(Refs. 9, 14, and 17).

The majority of models in the literature lack some elements that prevent them from being used to model

and simulate the effects of microgravity. In particular, there is a lack of a sufficient way to quantify bone

loss or gain. The model in (Ref. 4) describes the change in bone density, as a percent, during a bone

remodeling units cycle as a function of the change in the number of cells. Models in (Refs. 14 and 17) use

similar representations of changes in relative mass. None of the others specify an explicit measure of bone

loss or gain. Models also need to link the effect of the dynamics of loads on bone to the metabolic process.

The model in (Ref. 9) is the first one this author has seen that attempts to combine how bone adapts to both

cellular interactions and mechanical stimulus. It formulates the change in the intracortical radius of a

cylindrical section of cortical bone under a force-induced stimulus that influences the cellular dynamic

effects.

Most of the current cellular models focus on simulating the cause of specific bone diseases that occur

on earth and the result of possible therapeutic strategies. The objective of this work is to investigate the

applicability of some of the cellular dynamic models to NASA’s interest in developing a model to help

simulate the effects of microgravity and to provide quantitative analysis of bone mass and calcium levels

to Digital Astronaut (DA). To do so we describe a model that uses the cellular dynamics from (Ref. 7),

the force induced stimulus function from (Ref. 9), a new equation quantifying bone loss or gain, and an

additional mass balance equation for calcium in bone fluid. In particular, the equation in (Ref. 9) that

governs the changes in the intracortical thickness of a cylindrical section of cortical bone is replaced with

an equation that governs the changes in the area of hard bone tissue in a cross section. The new equation

can be related to the rate of change in the complement of bone porosity as defined in (Refs. 11 and 12).

Section 2 gives a brief description of bone anatomy and the remodeling process. In Section 3, the physical

part of the model is described and in Section 4, the model equations and variables are defined. In Section

5, the results of example simulations are discussed. Conclusions and recommendations for future research

are made in Sections 6.

2.0 Bone Remodeling Process

The skeletal makeup includes cortical bone, the compact bone that forms the hard outside shell that

encloses bone marrow, and trabecular (cancellous) bone, the spongy interior tissue in bone marrow. Trabecular

bone is made up of a network of rod and plate like elements. Cortical bone is organized into Haversian

systems, consisting of concentric layers of bone and a central canal (Haversian canal) which supplies blood.

The boundaries between Haversian systems are referred to as cement lines. Within bone, osteocytes, cells

derived from the bone forming cells, form what is understood to be a signaling network (Ref. 31).

NASA/TM—2009-215824 2
Bone remodeling, the physiological mechanism for maintenance, renewal, and repair in the adult

skeleton, is done threw the replacement of bone in units by the coupled action of bone cells on the same

bone surface. The bone resorbing cells, osteoclasts, remove old or damaged bone. The bone forming cells,

osteoblasts, form an initial collagen matrix and then mineralize the collagen. The replacement unit, often

referred to as a basic multicellular unit (BMU) or bone remodeling unit (BRU), differs between trabecular

bone and cortical bone. In trabecular bone the structural unit is a packet ideally shaped like a shallow

crescent (hemi-osteon) on the surface of a rod or plate like element. In cortical bone the structural unit is a

single Haversian system (Osteon) shaped like a cylinder and is referred to as a cutting cone while forming

(Ref. 32).

The remodeling cycle of a BRU consists of 5 phases, activation, resorption, reversal, formation, and

quiescence. Activation, the conversion of a small area of surface from quiescence to activity with respect

to remodeling, requires recruitment of osteoclasts, a means for them to gain access to bone, and a

mechanism for their attraction and attachment to the surface. Because of variations in age, sex, race, and

metabolic state, and systematic differences between different bones and between different surfaces in the

same bone, activation occurs sometimes at random and sometimes in response to focal events in the bone

structure or biomechanical stimulus. In the normal adult skeleton it occurs about once every 10 sec

(Ref. 15). Following activation, osteoclasts, proliferated from precursor cells, begin to erode a pit referred

to as a Howship’s lacuna in trabecular bone or burrow a tunnel called a cutting cone in cortical bone. (The

pit characteristics were described in the previous paragraph.) The completion of resorption can take from

one to three weeks. Reversal, the time interval between completion of resorption and beginning of

formation at the same location, can last from one to two weeks. Formation by osteoblasts, derived from a

different precursor cell, involves collagen matrix formation and mineralization. It lasts about three

months, but it can take an additional several months for newly formed bone to become mature. Thus, the

complete cycle is on the average about four months (Ref. 15). There are an estimated 10 5 to 106 active

BRUs in the entire body at any one time (Ref. 33).

The processes of remodeling are controlled or influenced by hormones, protein receptors, ligands,

other biochemicals, and mechanical stimulus. A complete description of the entire process of endocrine

regulation, biochemical mediation, and stimulation by skeletal loading involved in bone remodeling is

beyond the scope of this report. We shall briefly discuss the main substances involved in the cellular

process that appear in the model equations of Section 4. The reader can refer to a list of their definitions

in Appendix B. Early in the remodeling cycle, surface bound molecules, referred to as Receptor Activator

for Nuclear Factor icB ligand (RANKL), are expressed on the surface of osteoblasts while the membrane

protein Receptor Activator for Nuclear Factor icB (RANK) is expressed on the surface of preosteoclasts.

NASA/TM—2009-215824
The binding of the receptor RANK with the ligand RANKL causes the derivation of active osteoclasts

from preosteoclasts (Ref. 38). Parathyroid hormone (PTH), secreted by the parathyroid gland, is involved

in promoting bone resorption by binding to osteoblasts and stimulating them to increase expression of

RANKL. Thus PTH indirectly stimulates an increase in osteoclasts. PTH can have opposite effects on

bone mass depending on whether it is released into the plasma continuously or on a pulsatory schedule

(Ref. 5). Osteoprotegerin (OPG) is released by osteoblasts and acts as a decoy receptor for RANKL which

blocks RANK-RANKL binding and inhibits preosteoclasts from maturing into active osteoclasts

(Ref. 38). Transforming Growth Factor Beta (TGF-0) is released by osteoclasts during bone resorption. It

has the ability to stimulate an increase of and activity in the osteoblast population, but it can also inhibit

final maturation into active osteoblasts (Refs. 35 and 36). It’s also thought to induce osteoclast apoptosis

(Ref. 37). Nitric oxide, a gaseous signaling molecule, produced by osteoblasts directly regulates

osteoclastic activity (Ref. 34). Prostaglandidn E2 (PGE2) stimulates osteoblasts to release factors which

stimulate resorption by osteoclasts. The model equations of Section 3 attempt to incorporate the effects of

these substances.

3.0 Model Description

Current models for bone cell dynamics appearing in the literature of the last decade typically consist

of three or more equations. The exact number can vary among models. Usually, there is one governing the

rate of change in the population of the bone forming cells and one governing the rate of change in the

population of the bone resorbing cells. There may or may not be equations for precursor cells and there

may be equations for some or none of the biochemicals. In addition, the mathematical descriptions of the

equations often differ among models. The reason for the difference in the number and form of the

equations is due in part to the scientific assumptions used to create the equations. Some models are based

more on biology (Refs. 5 and 7) while some are based more on experimental evidence (Refs. 4 and 21).

The physical domain in which the cellular dynamics is occurring and upon which the model is based

varies among models. By physical domain, we mean the type, size, and skeletal location of bone used for

numerical values of parameters and initial conditions of the model. For example, some models use a

single BRU (Refs. 4 and 14). Others use an abstract small bone volume (Ref. 5), a section of bone whose

BRUs are under the same external stimulus (Ref. 7), or a cylindrical section of a long bone (Ref. 9). Only

a few of the models have an equation governing the rate of change of bone mass in the physical domain.

In particular, the model of Komorova et al. (Ref. 4) and Moroz et al. (Ref. 14) include a rate equation for

bone mass as a relative change from the initial value in units of percent. The model of Pivonka et al.

(Ref. 17) includes an equation for relative change in bone volume in units of percent. Maldonado et al.

(Ref. 9) use an equation in their model that governs the rate of change in the intracortical radius of a

cylindrical section of cortical bone to quantify the amount of force induced bone gain. None of the other

models referenced in the introduction have an equation governing the amount of bone gained or lost or

the rate of change in the amount of bone. In those models, the gain in the population of active osteoblasts

(loss in the population of active osteoclasts) or the gain in the population of active osteoclasts (loss in the

population of active osteoblasts) is related to bone gain or loss, respectively.

In order to test a model’s ability to simulate bone adaption in the environment of microgravity, the

model must have a suitable quantification of bone mass at skeletal sites, and it must have a relation

describing the bone mass as a function of time or an equation describing the rate of change of bone mass

with time. In addition, the effect of mechanical stimulus and the contribution to bone balance from the

normal skeletal loads on earth must be included.

This motivated the consideration of a model that combines the cellular dynamics model equations of

Lemaire et al. (Ref. 7) with the mechanical stimulus equations in the model of Maldonado et al. (Ref. 9).

The model in Reference 9 actually uses the cellular dynamics equations in Reference 7 coupled with a

simple scalar description of a normal earth load on a segment of bone, but not an equation that governs

bone mass explicitly. Their equation governing intracortical radius appears to have conflicting units

because the rates of resorption and formation use values from the model of Komorova et al. which are in

NASA/TM—2009-215824 4
units that are per cell whereas the rates in the model of Lemaire et al. are in units that are per pM

concentration. The combined set of equations assemble here avoids conflicting units and attempts to

model bone matrix lost or gained within a fixed radius.

According to a study of skeletal data obtained (up to 2007) by the U.S. and Russian space programs,

trabecular bone loss on a percentage basis is greater than cortical bone loss in certain skeletal sites, but on

a mass basis, the greatest loss is from cortical bone (Ref. 29). Therefore, the focus herein will be on

quantifying the change in cortical bone.

Consider a simple model of a cylindrical section of cortical bone of cortical cross sectional area A and

length L subject to a mean load F and stress given by

s  F / A (3.1)

We will formulate the rate of change of bone area in the cortical region similar to the manner first

described by Martin (Ref. 11) and further detailed in Reference 12. Figure 2(a) depicts a cross section

containing secondary osteons or ORUs which are resorbing, refilling, or completed. Pathways of osteons

may be irregular as well as regular and tunnel almost longitudinally but at a slight angle through the

cortex (Ref. 12, p. 84). However it will be assumed that osteons and cutting cones of forming osteons are

perpendicular to the cortical cross section and parallel to the axis of the cylindrical section of cortical

bone. Therefore, within the cortical cross section there are cross sections of osteons. Figure 2(b)

represents the cross section of a forming osteon. Martin defines a quantity called intracortical bone

balance as

Q  QBNF

 QCNR

(3.2)

where QB is the mean rate at which bone is replaced during refilling per osteon in mm2/day, QC is the

mean rate at which bone is resorbed in each osteon during its resorption phase, NF is the number of

osteons or BRUs forming bone per intracortical cross section A, and NR is the number of BRUs resorbing

bone per mm2 cortex. Since resorption begins at the center of a Haversian system and ends at the cement

line, and formation begins at the cement line and ends at the radius of the canal (Fig. 2(b)), these

quantities are defined as



2

 R

2 2

QB 

Rch QC



Rc

(3.3)

TF

TR

where Rc and Rh are the mean radius of an osteon’s cement line and the mean radius of the Haversian

canal, respectively. Quantities TR and TF represent the average time it takes for an osteon to complete its

resorption phase and time to complete refilling phase, respectively. If fa represents the activation

frequency of a BRU, then in a dynamic situation where the normal remodeling balance is perturbed, fa

varies with time so that

t

NF 

t 



T F fa s ds, NR 



t 



T R 

f

a  s ds

If we assume that the activation frequency fa of a BRU is constant (reasonable for bone in a steady state)

then

NF  faTF, NR  faTR

(3.4)

and the times TF and TR will cancel out of Equation (3.2).

NASA/TM—2009-215824
Actually, under constant fa, Equation (3.2) would become a perfectly steady balance ( Q = 0) by

inserting the factor (1–p) in the second term on the right hand side where

p = Rh

2

/ Rc

2

(3.5)

Martin has suggested this alteration of the intracortical balance equation to account for osteons which

eventually begin to overlap one another, which he has shown to limit porosity in sufficiently remodeled

bone (Ref. 26). The value of p has been shown to be the upper limit on porosity when osteons are

randomly located on the cross section (Ref. 26). To further define porosity, we note that the units of Q are

mm2 per day or mm2/A/day and if A,n denotes the area of hard bony matrix, the integral of Q over time is

the amount of change in A,n. Then A –A,n , which we denote as Av , is void area consisting of soft tissue.

Consistent with Martin’s definitions (Refs. 11 and 12) the quantity known as bone volume fraction would

be

Vb  A,n/A (3.6)

NASA/TM—2009-215824 6
and porosity would be

pv  Av/A (3.7)

A proposed equation for allowing changes in the normal bone balance Q or dAm/dt and a proposed

equation for the rate of change of calcium in bone fluid are coupled with the cellular dynamics equations

(Ref. 7) and the mechanical stimulus equations (Ref. 9) to form the model equations.

4.0 Model Equations

The tables in Appendix A provide a definition of all the symbols used in this section. A common

structure can be noted with many of these types of model equations. The rates of change of substances are

constructed as a combination of accumulation terms and degradation terms. By accumulation

(degradation) term we mean a term that increases (decreases) the substance rate of change. The system of

equations here adopts a scheme in which rate coefficients are notated with a plus superscript if they act as

an accumulation constant and a minus superscript if they act as degradation constant. The subscripts

associate them with a specific substance. The current model equations are given by the system of

equations assembled in tabular form below, followed by an explanation for each equation.

Rate of Equation Author(s)

change of

Area of hard dAm



B C

QB NF ( )  QC NR (1  pv)(

dt B0

Co

)

(4.1A) Pennline bone tissue

Osteocyte dO 

 r

sO (B  B0

)  rOc (O  O0) (4.2) concentration dt

(4.3) Maldonado et al. (Ref. 9)

(4.4)

Nitric oxide

concentration

dK"o  r



f r K +I

dt bs – "o K"o "o

PGE2

concentration

dK

 r

pgeAs

 r

("o)(pge)K"o  rpgeKpge  Ipge dt

ge

OPG

concentration

K r

O

 O BR  IO  r(n

o)( o )KNO  rO KO

dt EP

(4.5)

RANKL dKL 

 (k1

/ k2)KO

 (k3

/ k4)K

concentration  rL  IL  r

(a°)(1)KNO  rL L dt N LE C PB 

1

(4.6)

Responding dB 0.05dB

osteoblast  dR

EG BR r

(B)(pge)Kpge dt

E G

(4.7) Lemaire et al. (Ref. 7)

concentration

Active dB 0.05dB

osteoblast 

dt E kBB G

(4.8)

concentration

(4.9)

Active osteoclast dC

— 

 dCER  dAEG C

concentration dt

Calcium

concentration

dCa 

_ r

FB ( B / B0 )  rBF ( C / C0 )  rF  rCF  r

FC  rdCa Ca

dt — (4.10) Pennline

NASA/TM—2009-215824 7
The rate of change of Am (t) we propose is given by Equation (4.1A) where pv is defined in

Equation (3.7). This accounts for osteon overlap using Martin’s factor discussed in Section 3. Note that

it’s modeled as a net gain or loss in cortical bone area (defined in Section 3) as a function of the ratio of

change in cell level to normal cell level for both osteoblasts and osteoclasts. The effects of mechanical

unloading (and loading) and biochemical imbalances are imposed through changes in cell populations

caused by these effects. If an equation were used that is similar to the equation governing the rate of

change of intra-cortical radius in (Ref. 9) it would look like the following

dAm /dt  QBNF (

B

)  QCNR (

C

 )  rm (O)  rm

(Am ) (4.1B)

B0C0

O0

Am 0

where Am replaces intra-cortical radius, QBNF replaces formation rate in radial direction, QCNR replaces

resorption rate in radial direction, andr ; and rm are rate constants. Accumulation due to osteocytes from

the third term is omitted in Equation (4.1A) since one could argue that osteocytes are derived from the

osteoblasts. In comparison to Equations (4.1A), Equation (4.1B) will be shown to give incorrect results

for bone loss.

Since the osteocyte concentration depends on the differentiation of the osteoblast population and the

osteocytes are subject to cell death or apoptosis, an equation for the rate change of osteocyte

concentration is given by Equation (4.2), where rBO and roc are constant rates of differentiation and cell

death, respectively.

Citing references that support their assumptions that osteocytes act as mechanosensors that cause the

release of Nitrix Oxide (NO) and Prostaglandin E2 (PGE2), Maldonado et al. (Ref. 9) define a nonlinear

biochemical stimulus function of osteocyte concentration and stress. The biochemical stimulus function is

given by

sO

(1  exp((rss  rOcO)))

f

bs (s

, O)  (4.11)

where rs and rOc are rate constants. The rates of change of NO and PGE2 are then defined as a linear

function of the biochemical stimulus and given by Equations (4.3) and (4.4) where r o , 



rno , 



r

pge ,

r(no)(pge) , and rpge are rate constants. The quantities Ino and Ipge represent injection inputs of NO and

PGE2, respectively.

Lemaire et al. (Ref. 7), describe a chemical kinetics analysis for RANKL, OPG, the binding complex

of OPG-RANKL, and the RANK-RANKL complex. RANK is held fixed “to reflect the undiminished

availability of osteoclast precursors”. The four differential equations describing the rate of change of

these quantities involving the cytokine RANKL and the receptors RANK and OPG are taken from

(Ref. 6). From those equations Lemaire et al. (Ref. 7) obtain pseudo steady-state expressions for OPG and

RANKL. In fact, if the steady-state expressions for the complexes OPG-RANKL and RANK-RANKL are

substituted into the rate equations for RANKL and OPG, the result is Equations (4.5) and (4.6) where

r

(no)(o) is an accumulation constant, ro and 

r

(no)(l) are degradation constants, rL is the rate of RANKL

production and elimination, k1/k2 is the ratio of rates of OPG-RANKL binding/unbinding, and k3/k4 is the

ratio of the rates of RANK-RANKL binding/unbinding. The constant NL is the maximum number of

RANKL attached on each cell surface, and rO is the minimal rate of production of OPG per cell.

Quantities IO and IL are injection rates of OPG and RANKL, respectively. The symbol EP stands for PTH

receptor occupancy ratio and is a function defined as

NASA/TM—2009-215824


IP (t)  SP

Ep (t)

IP (t)  rp (k6 /k5)

(4.12)

where IP is an injection rate of PTH, SP is the rate of synthesis of systemic PTH, rp is the rate of

elimination of PTH, and k6/k5 is the ratio of rates of PTH unbinding/binding with its receptor.

To understand the formation of cell equations, the authors of (Ref. 7) state that their theory is based

on the assumption that cell “proliferation” (“anti-proliferation”) is proportional (“inversely proportional”)

to receptor occupancy (Ref. 7, p. 297). The equation governing the rate of change of responding

osteoblasts is given by Equation (4.7), where dR is a constant representing differentiation rate of

osteoblast progenitors and EG represents the fraction of occupied TGF-(3 receptors given by

_ C(t)  0.05C

S

EG (t)

C(t)  C

S

(4.13)

The constant C

S

represents the value of C to get half differentiation flux. By citing references that

report that PGE2 promotes bone formation, Maldonado et al. (Ref. 9) added the last term on the right hand

side of Equation (4.7). It increases dBR/dt byr

(B )(pge) Kpge where r



(B )(pge) is a rate constant. Rates of

change for the active osteoblasts and active osteoclasts are then described by Equations (4.8) and (4.9)

where ER = (k3/k4)KL. The constants dB, kB, dC, and dA represent differentiation rate of responding

osteoblasts, rate of elimination of active osteoblasts, differentiation rate of osteoclast precursors, and rate

of osteoclast apoptosis caused by TGF- P, respectively. Note that the NO and PGE2 dynamics is coupled to

the RANK-RANKL-OPG dynamics to influence the osteoclast and osteoblast interactions through

Equations (4.5) to (4.7).

Equation (4.10) is proposed for describing the rate of change of the concentration of calcium in the

bone fluid and is based on a subset of a model by Jaros et al. (Ref. 3) involving the role of bone in

calcium homeostasis. The symbols rFB, rBF, rCF, and rFC represent the rate of flow of calcium from bone

fluid to bone, rate of flow of calcium from bone to bone fluid, rate of flow of calcium from blood

capillary to bone fluid and rate of flow of calcium from bone fluid to blood capillary, respectively. The

constant rF represents the addition of calcium to the bone fluid from bone due to osteocytic osteolysis and

is affected by the concentration of PTH (Ref. 3). The constant rdCa represents a degradation of calcium

due to an increased flow of calcium from bone fluid to blood capillary and is set to balance steady state

for reference values of osteoclasts and osteoblasts. This also accounts for the fact that since an additional

equation governing the concentration of calcium in the blood plasma is missing; the terms involving rCF

and rFC are allowed to remain constant. Note that if the full model included an equation governing PTH

concentration, an additional factor affecting the value of rF would be included. The values for these

constants are given in mmol per hour for the full skeleton. The units were converted to mmol per day and

the values were scaled down. See Table 2.

Equations (4.1) to (4.10) constitute a coupled system of ten 1 st order, ordinary differential equations.

By specifying initial or steady-state values of the variables in Table 1, Equations (4.1) to (4.10) become

an initial value, first-order system.

5.0 Simulation and Results

The quantities QB and QC for the formation and resorbtion rates were computed using human rib

values for the mean Haversian canal and mean cement line radius from Polig and Jee (Ref. 18). An

average value for activation frequency was computed from a range of rib values referenced by Martin

(Ref. 11); while an average value of 20 mm2 for cortical cross-sectional area, A, was chosen, using data

from Abrams et al. (Ref. 39) which reports cortical area in the cross section of human ribs varying from

NASA/TM—2009-215824 9
16.3 to 26.3 mm2 . The primary intent was to choose numerical values from some published data to be

near correct orders of magnitude and not to choose cortical values from a specific long bone. They are not

claimed to be the most accurate currently available. Based on what seemed to be about average normal

values for the stress on long bones estimated using data from Buckley (Ref. 1, p. 16) on ranges of average

values of the load in 1 g at various skeletal sites for a 70 kg body weight, a normal load (value of F) on

earth on the order of 25 N was assumed for the cortical cross sectional area of 20 mm2.

In Equations (4.3) and (4.4), fb, was replaced by (fb,(,, O)–fb,(25/20,O)) (see Eq. 4.11). This was

necessary in order to be consistent with the definition of the initial cell values for the equations from the

model by Lemaire et al. (Ref. 7). Their parameter values of reference were selected to represent normal

bone that is neither gaining nor losing mass, resulting in a system in a state of reference. Therefore, initial

values from the Lemaire model must be assumed to be from bone with a normal physical load. This

requires the combined equations to be in steady state when F = 25. Also, in Equations (4.1) to (4.10), we

replaced the occurrence of each of the variables from Table 1 with the sum of its initial value plus a

change. For example, B(t) was replaced by B(t)+B0, C(t) was replaced by C(t)+C0, and so on. Plots from

simulations will then show the value of changes from the normal or steady state.

The method of solution of the equations was as follows. First, following the approach in (Ref. 9),

Equations (4.3) to (4.6) were solved for their equilibrium expressions by assuming the rates of change are

zero. This simplification was justified in both (Refs. 7 and 9) by stating that the binding reactions and

chemical dynamics are much faster than the rest of the dynamics in the model. The expressions for NO,

PGE2, OPG, and RANKL are then substituted in the remaining equations where they appear. This left an

initial value, first order system of six ODEs, consisting of Equations (4.1), (4.2), (4.7) to (4.10), which

was numerically solved using a 4th order Runge-Kutta algorithm from Matlab. Unless stated otherwise in

a specific simulation, the values used for the initial conditions, parameters, and constants are given in

Tables 2 to 4 and correspond to values in References 7 and 9.

In reviewing the work of Lemaire et al. (Ref. 7), various bone diseases and possible therapies were

simulated using their three cell equations. For example, since postmenopausal osteoporosis induced by

estrogen deficiency is linked to a lack of OPG production (Ref. 27), they ran a simulation by decreasing

rO , the secretion rate of OPG by osteoblasts. They then tested a theoretical therapy by reducing kB, the

rate of elimination of active osteoblasts. In another example, they simulated the catabolic effect of

continuous injection of PTH by running a simulation with IP set to 1000 pM/day.

To check the agreement of the computations from the model Equations (4.1) to (4.10) with those of

Lemaire et al. and the ability of the model to simulate bone disease on earth with no change in the normal

force, a simulation with IP set to 1000 pM/day was performed. Figure 3 shows the results wherein the

normal load of (F(t) = 25) was used. This reduced Equations (4.7) to (4.9) to the same three cell equations

employed in (Ref. 7) to give results for the osteoclasts and osteoblasts curves identical to those displayed

in (Ref. 7, p. 300). The difference here is that simulation results showing the change in the area of hard

bony tissue in the cortical cross section (lower right; Fig. 3) and the change in bone fluid calcium (lower

left; Fig. 3) are displayed, whereas they are not displayed in (Ref. 7).

Note that the curve resulting from the use of (4.1B) shows an increase in bone area during the

continuous injection period which is inconsistent with experimental and clinical observations (Ref. 30),

whereas the response resulting from Equation (4.1A) shows the expected decrease in bone area. Actual

values of the result from Equation (4.1B) were reduced by a factor of 10–1 to enable comparing the

qualitative difference on the same plot.

In reviewing the use of their force-induced bone growth model, Maldonado et al. (Ref. 9) compared

the difference between the effects of bone disease on the intracortical radius with physical activity and the

effects without physical activity. They were able to show that physical activity is beneficial. For example,

in the first test of their model they simulate the effects of an increased load on the intracortical radius.

They ran a 350 day simulation during which the force F was increased on day 50 to mimic physical

activity which increased their assumed average value of F to a value higher than the assumed normal

NASA/TM—2009-215824 10
value over the next 100 days. It was further increased on day 150. On day 250 the force was dropped to a

value lower than the assumed normal value, to mimic a decrease in physical activity over next 100 days.

The results show an increase in the radius between days 50 and 150 and another increase between days

150 and 250. The last one hundred days showed a decrease in the radius to a level slightly below the

equilibrium value. A figure of those results can be seen in Reference 9.

For purpose of this study, a 350 day simulation of the model using Equation (4.1A) was conducted,

where the load (value of F) was first decreased to zero on day 50. This would correspond to unloading the

cylindrical section of bone upon entrance into the environment of microgravity on day 50. Figure 4 shows

the results of such a simulation which also includes the bone area plot obtained using Equation (4.1B).

Once again, the response (change in bone area) resulting from the use of Equation (4.1A) predicts

expected qualitative results in bone loss and gain. There is a continued loss of bone when the load is

decreased to zero on day 50. On day 150 the average load is increased over the next 100 days, as the

cumulative effects of physical activity is accounted for by increasing F to 20, and there is a decrease in

the rate of loss of bone area, as shown. Corresponding to a return to 1 g on day 250, the load is increased

above the normal load and bone loss begins to reverse.

On first glance, the response resulting from the use of Equation (4.1B) also indicates expected

qualitative results in bone loss and gain in the following sense. Bone area in the cross section is lost in the

interval of time beginning with a zero load. Raising the average load to 20 over an interval of time

beginning on day 150 recovers some of the loss. Leaving the environment of a reduced load at day 250

and increasing the average load to 30 with physical activity recovers the lost bone area and then some.

The shape of these curves is similar to the shape of the curves obtained in Reference 9 for bone growth

induced by increasing the average force. However, it is not a good model of bone loss in the environment

of microgravity where bone loss can continue over long periods of time, since the bone curve appears to

level off or almost approach an asymptotic value in the unloaded interval (F = 0) without any

countermeasure.

NASA/TM—2009-215824 11
Thus, the results of the present simulation demonstrate the following. The model Equation (4.1A) for

rate of change of bone, which is more biologically accurate than Equation (4.1B), is able to more

correctly simulate expected bone loss in microgravity which, in this model, means by reducing the normal

value of F (F = 25) assumed on earth.

Biochemical imbalances under the conditions of microgravity may similarly cause disruptions in the

bone remodeling balance in addition to those disruptions explained or modeled strictly by the mechanics

of skeletal unloading. Lemaire et al. (Ref. 7) remark that osteoblast progenitors express TGF-P receptors

so that TGF-P, produced by osteoclasts during bone resorption, leads to differentiation of the progenitors

into responding osteoblasts. Using that knowledge, they associate bone loss, due to senescence on earth,

with a decrease in the production of TGF-P. However, it is not clear what numerical values they modify

to simulate bone loss due to senescence. We associated the consequence of senescence with one of the

possible causes of old age osteoporosis, a decrease in the rate of osteoclast apoptosis due to TGF-P.

Figure 5 shows the result of such a simulation of the model equations (without any skeletal unloading)

using Equation (4.1A) for bone balance. Notice that when the osteoclast death rate due to TGF-P is

reduced from 0.7 to 0.5 in the interval between day 20 and day 80, the loss of bone is similar to the loss in

Figure 4 during the interval between day 50 and day 150. Bone area recovers and returns to the balanced

state when the osteoclast death rate due to TGF-P is returned to its normal value.

It is of interest to examine the results of a simulation of this model that combines unloading with a

reduction in the rate of osteoclast apoptosis due to TGF-P as a possible combination of bone remodeling

imbalances in the environment of microgravity. Figure 6 shows the results of such a simulation using

Equation (4.1A) for bone balance.

NASA/TM—2009-215824 12
Observe that the combined effects of reducing the rate of osteoclast apoptosis and unloading make it

harder to recover bone loss with say physical activity that increases the average load. Increasing the

average value of F above the normal load to 50.0 over the interval between day 150 and 250 still only

showed a decrease in the rate of bone loss whereas without the added effect, bone loss would have been

reversed. Other therapies and countermeasures besides increasing F could also be tested. However, we

defer additional studies until a better model is developed, especially a model with a more robust

description of the mechanical load on bone and mechanotransduction.

Conclusions

The simulations carried out with the simple model in this study show that mathematical formulations

of the bone remodeling function could eventually be used for predicating bone loss in space. It is

important that the model equations have a steady state or semi-steady state accurately perturbed by welldefined metabolic and biomechanical scenarios. However, to have a model with sufficient high fidelity to

be used for application in the development of DA, or any other realistic simulations, much more research

and development is needed. How bone loss or gain is quantified needs to be more accurately linked to the

biological structure and remodeling phase of bone. Theories on the exact biomechanical stimulus for

strain adaptation and mechanotransduction need to be more completely linked to the metabolic process.

To be more specific, consider some of the limitations and flaws of the model in this study.

Although qualitative results look interesting, the modeling is still too abstract to be used for

quantitative purposes, and this is true for other models in the literature. There is not enough dimensional

NASA/TM—2009-215824 13
detail to be able to associate a collection of remodeling units under the same external stimulus with a

specific location and size of bone segment. Our model does not include the complete set of cell types and

biochemicals. For example, the interactions of phosphate ions are missing. The remodeling of trabecular

bone is not addressed. The model here (and most other models) alters the resorption or formation rate

based only on a change in the concentration of cells, and does not include a relation to changes in

resorption depth or direction that Parfitt (Ref. 15, p. 73) associates with certain bone loss conditions.

More research in needed in order to fully understand how the applied loads induce osteogenic response by

various cell types in different regions (Ref. 16). Work is needed in designing experiments to quantify all

model parameters and to insure unit and dimensional consistency, developing a benchmark for meeting

requirements for realistic simulations, and developing validation tests.

References

1. Buckley Jr., J.C., Space Physiology, Oxford University Press, New York, 2006.

2. Cowin, S., The Exact Stimulus for the Strain Adaptation of Bone Tissue is Unknown, Journal of

Biomechanical Science and Engineering, vol. 1, no. 1, pp. 16–28, 2006.

3. Jaros, G.G., Guyton, A.C., and Coleman, T.G. The role of short-term calcium homeostasis: an

analog-digital computer simulation, Ann. Biomed. Eng., 8, pp. 103–141, 1980.

4. Komarova, S.V., Smith, R.J., Dixon, S.J., Sims, S.M., and Wahl, L.M. Mathematical model predicts

a critical role for osteoclast autocrine regulation in the control of bone remodeling, Bone, 33,

pp. 206–215, 2003.

5. Kroll, M.H. Parathyroid hormone temporal effects on bone formation and resorption, B. Math.

Biology, 62, pp. 163–187, 2000.

6. Lauffenburger, D.A., and Linderman, J.J., Receptors: Models for Binding, Trafficking, and Signaling.

Oxford University Press, New York; Toronto, 1996.

7. Lemaire, V., Tobin, F.L., Greller, L.D., Cho, C.R., and Suva, L.J. Modeling interactions between

osteoblasts and osteoclasts activities in bone remodeling, J. Theoretical Biology, 229, pp. 293–309,

2004.

8. Maldonado, S., Borchers, S., Findeisen, R., and Allgower, F., Mathematical Modeling and Analysis

of Force Induced Bone Growth, Proceedings of the 28th IEEE, EMBS Annual International

Conference, New York City, USA, Aug. 30 to Sept. 3, 2006.

9. Maldonado, S., Findeisen, R., and Allgower, F., Phenomenological Mathematical Modeling and

Analysis of Force-Induced Bone Growth and Adaptation, Proceedings of the FOSBE 2007, Stuttgart,

Germany, Sept. 9–12, 2007.

10. Manolagas, S.C. Choreography from the tomb: an emerging role of dying osteocytes in the

purposeful and perhaps not so purposeful targeting of bone remodeling, Osteovision, 3(1), pp. 5–14,

2006.

11. Martin, R.B., The Usefulness of Mathematical Models for Bone Remodeling, Yearbook of Physical

Anthropology 28, pp. 227–236, 1985.

12. Martin, R.B., Burr, D.B., and Sharkey, N.A., Skeletal Tissue Mechanics, Springer-Verlag, New York,

1998.

13. Miller, K., Gravity Hurts (So Good). (2001). Science@NASA, August 2, 2001.

<http://science.nasa.gov/headlines/y2001/ast02sug_1.htm>.

14. Moroz, A., Crane, M.C., Smith, G., and Wimpenny, D.I., Phenomenological model of bone

remodeling cycle containing osteocyte regulation loop, Biosystems 84, pp. 183–190, 2006.

15. Parfitt, A.M., Bone Remodeling: Relationship to Amount and Structure of Bone, and the

Pathogenesis and Prevention of Fractures, Osteoporosis: Etiology, Diagnosis, and Management,

edited by H. Lawrence Riggs and L. Joseph Melton, III., Raven Press, New York, 1988.

16. Pearson, O.M. and Lieberman, D.E., The Aging of Wolff’ s “Law”: Ontogeny and Responses to

Mechanical Loading in Cortical Bone, Yearbook of Physical Anthropology, 47, pp. 63–99, 2004.

NASA/TM—2009-215824 14
17. Pivonka, P., Simak, J., Smith, D.W., Gardiner, B.S., Dunstan, C.R., Sims, N.A., Martin, T.J., and

Mundy G.R., Model Structure and Control of Bone Remodeling: A Theoretical Study, Bone 43,

pp. 249–263, 2008.

18. Polig, E., and Jee, W.S.S., A Model of Osteon Closure in Cortical Bone, Calcified Tissue

International, 47, pp. 261–269 1990.

19. Rattanakul, C.R., Lenbury, Y., Krishnamara, N., and Wolkind, D.J. Modeling of bone formation and

resorption mediated by parathyroid hormone: resonse to estrogen/PTH therapy, BioSystems, 70,

pp. 55–72, 2003.

20. Robling, A.G., Castillo, A.B., and Turner, C.H. Biomechanical and molecular regulation of bone

remodeling, Annu. Rev. Biomed. Eng., 8, pp. 455–498, 2006.

21. Ruimerman, R., van Reitbergen, B., Hilbers P., and Huiskes, R. A 3-dimensional computer model to

simulate trabecular bone metabolism, Biorheology, 40, 315–320, 2003.

22. Talmage, R.V. and Myer Jr., R.A. Physiological role of parathyroid hormone, In Handbook of

Physiology, vol. III Endocrinology, G.D. Aurbach (Ed.) American Physiological Society, Washington

D.C., pp. 343–351, 1976.

23. Wang Z., Song, W., Li, Y., Dhar, P., Tsuchiya, M., and Mondry, A., Quantitative study of calcium

homeostasis maintenance through systemic modeling, FOSBE 2005, Santa Barbara, California, USA,

August 2005.

24. White, R.J. and McPhee, J.C., The Digital Astronaut: An integrated modeling and database system for

space biomedical research and operations, 60, pp. 273–280, 2007.

25. Wimpenny, D.I., and Moroz, A., On allosteric control model of bone turnover cycle containing

osteocyte regulation loop, Biosystems 90, pp. 295–308, 2007.

26. Martin, R.B., Overlap feedback: An autonomous mechanism for controlling porosity during haversian

bone remodeling, Proceedings, Fourth Annual New England Bioengineering Conference, New

Haven, CT, pp. 41–44, 1976.

27. Hofbauer, L.C., Khosla, S., et al., Estrogen stimulates gene expression and protein production of

osteoprotegerin in human osteoblastic cells, Endocrinology 140 (9), 4367–4370, 1999.

28. Sibonga, J.D., and et al., Recovery of spaceflight-induced bone loss: Bone mineral density after longduration missions as fitted with an exponential function, Bone 41, 973–978, 2007.

29. LeBlanc, A.D., et al., Skeletal responses to space flight and the bed rest analog: A review,

J Musculoskelet Neuronal Interact 7 (1), 33–47, 2007.

30. Watson, P.H., Fraher, L.J., et al., enhanced osteoblast development after continuous infusion of

hPTH(1-84) in the rat, Bone 24 (2), 89–94, 1999.

31. Bone Structure and Function, ASBMR Bone Curriculum, Susan Ott website

<http://depts.washington.edu/bonebio/ASBMRed/structure.html>

32. Bronner, F. and Worrell, R., Orthopaedics: principles of basic and clinical science, CRC Press LLC,

1999.

33. Langton, C.M. et al., Dynamic stochastic simulation of cancellous bone resorption, Bone, vol. 22,

no. 4, 1998.

34. Evans D.M., Ralston S.H., Nitric oxide and bone, J. Bone Miner Res. Mar; 11(3):300-5. 1996.

35. Bonewald, L.F. and Dallas, S.L., Role of active and latent TGF- 0 in bone formation, J. Cell.

Biochem. 55(3), 350–357, 1994.

36. Alliston, T. and Choy, L., et al., TGF- 0-induced repression of CBFA1 by Dmad3 decreases cbfal and

osteocalcin expression and inhibits osteoblast differentiation, Embo J. 20 (9), 2254–2272, 2001.

37. Roodman, G.D., Cell biology of the osteoclast, Exp. Hematol, 27 (8), 1299–1241, 1999.

38. Hofbauer, L.C., Khosla, S., et al., The roles of osteoprotegerin and osteoprotegerin ligand in the

paracrine regulation of bone resorption, J. Bone Mineral Res. 15 (1), 2–12, 2000.

39. Abrams, E., Mohr, M. et al., Cross-Sectional Geometry of Human Ribs, Legacy Clinical Research &

Technology Center Report, Portland, OR., 2000.

NASA/TM—2009-215824 15

Appendix A.—Tables

TABLE 1.—NOMENCLATURE

Symbol Definition Unit

A.(t) Area of bone mm2

B(t) Concentration of osteoblasts pM

BR (t) Concentration of responding osteoblasts pM

C(t) Concentration of osteoclasts pM

O(t) Concentration of osteocytes pM

Ca(t) Calcium concentration mmol

KL (t) Concentration of RANKL pM

Kno (t) Concentration of nitric oxide pM

KO (t) Concentration of osteoprotegerin (OPG) pM

K'ge (t) Concentration of prostaglandin E 2

 (PGE2) pM

IL (t) Injection rate of RANKL pM/day

Ino (t) Injection rate of nitric oxide pM/day

IO (t) Injection rate of OPG pM/day

IP (t) Injection rate of PTH pM/day

I'ge (t) Injection rate of PGE2 pM day–1

TABLE 2.—INITIAL VALUES

Symbol Value Definition

A.

0

18 mm2Initial value bone area

B0

0.0007282 pM Initial value active osteoblasts

BR0

0.0007734 pM Initial value responding osteoblasts

C0

0.0009127 pM Initial value active osteoclasts

C 0.0073 pM Initial value osteocytes

Ca0 0.696 mmol Initial value calcium (Bone fluid)

NASA/TM—2009-215824 17
TABLE 3.—PARAMETER VALUES

Symbol Value Definition

QBNF 0.173 x 10–3 mm2 day–1 Rate of formation of bone area by osteoblasts

QCNR

0.156x 1 0

–3 mm2

 day–1 Rate of resorption of bone area by osteoclasts

r

+

m

1.0 mm2 day–1 Rate of increase of bone area due to osteocytes

r

-

m

1.0 day–1 Rate of degradation of bone area

r

+

O

0.1 day–1 Osteocyte production rate from differentiation of osteoblasts

r

-

Oc

1 day–1 Osteocyte degradation rate

rs 1 mm2

/N Stress influence rate

rOc 1.0 pM–1 Osteocyte influence rate

r

+

no

2.0x 104mm2day/N Rate of release of NO

r

-

no

1.0x103 day–1 NO elimination rate

r

+

pge

100 mm2day/N Rate of release of PGE2

 due to biochemical stimulus

r

(

+

no)(pge)

1.0 day–1 or 10.0 day–1 PGE2

 rate increased by NO

r

pge

1.0x102 day–1 PGE2

 elimination rate

rO

2x 105 pmol day–1/pmol cells Minimal rate of production of OPG per cell

r

+ 10 day–1 OPG rate increase by NO

r

6

0.35 day–1 Rate of elimination of OPG

rL

103

 pM day–1 Rate of RANKL production and elimination

r

100 day–1 RANKL rate decreased by NO

r

p

86 day–1 Rate of elimination of PTH

r

+

(B)(pge)

1.0x 10–4 day–1

BR rate increased by PGE2

NASA/TM—2009-215824 18
TABLE 4.—PARAMETER VALUES

Symbol Value Definition

C

S 5× 10–

 pM Value of C to get half differentiation flux

dA 0.7 day–1 Rate of osteoclast apoptosis caused by TGF R

dB 0.7 day–1 Differentiation rate of responding osteoblasts

dC

2.1 × 10–3 pM day–1 Differentiation rate of osteoclast precursors

dR 7× 10–4 pM day–1 Differentiation rate of osteoblast progenitors

K 10 pM Fixed concentration of RANK

k1 10-2 pM–1 day–1 Rate of OPG-RANKL unbinding

k2

10 day–1 Rate of OPG-RANKL unbinding

k3 5.8 × 10–4 pM–1 day- Rate of RANK-RANKL binding

k4 1.7× 10–2 day-1 Rate of RANK-RANKL unbinding

k5 0.02 pM–1 day–1 Rate of PTH binding with its receptor

k6 3 day–1 Rate of PTH unbinding

NL

3 × 106

 pmol/pmol cells Maximum number of RANKL attached on each cell surface

kB 0.189 day–1 Rate of elimination of active osteoblasts

rFB 0.072 mmol day–1 Rate of flow of calcium from bone fluid to bone

rBF 0.0696 mmol day–1 Rate of flow of calcium from bone to bone fluid

rCF

1.2696 mmol day–1 Rate of flow of calcium from blood capillary to bone fluid

rFC 1.2 mmol day–1 Rate of flow of calcium from bone fluid to blood capillary

rF 0.0024 mmol day-1 Rate of accumulation of calcium in bone fluid due to osteocytic osteolysis

rdCa 0.1 day–1 Calcium degradation rate

NASA/TM—2009-215824 19

Appendix B.—Substance Definitions

Unless specifically referenced, definitions were taken from Wikipedia, The Free Encyclopedia,

http://en.wikipedia.org/wiki/.

1. Receptor.—A protein molecule, embedded in either the plasma membrane or cytoplasm of a cell, to

which a mobile signaling molecule may attach.

2. Ligand.—A molecule that binds to a receptor.

3. Receptor Activator of Nuclear Factor x B (RANK).—Membrane protein which is expressed on the

surface of osteoclasts and is involved in the activation of osteoclasts upon ligand binding.

4. Receptor Activator for Nuclear Factor x B Ligand (RANKL).—A surface bound molecule expressed

on osteoblasts that stimulates osteoclast differentiation from preosteoclasts through binding with

RANK.

5. Parathyroid Hormone (PTH).—A hormone secreted by parathyroid gland that enhances release of

calcium from bone and indirectly stimulates osteoclasts by binding to osteoblasts which in turn

stimulates osteoblasts to increase expression of RANKL.

6. Osteoprotegerin (OPG).—A decoy receptor for RANKL that prevents osteoclast maturation by

blocking the RANKL-RANK interaction.

7. Transforming Growth Factor beta (TGF-β).—Controls cellular differentiation and other functions in

most cells and induces apoptosis in numerous cell types. TGF- P is released by osteoclasts during bone

resorption. It has the ability stimulate addition in and activity in the preosteoblast population but it

can also inhibit final maturation into active osteoblasts (Refs. 35 and 36). It’s also known to induce

osteoclast apoptosis (Ref. 37).

8. Nitric Oxide (NO).—A gaseous signaling molecule produced by osteoblasts which directly regulates

osteoclastic activity (Ref. 34).

9. Prostaglandin E2 (PGE2).—A substance that stimulates osteoblasts to release factors which stimulate

bone resorption by osteoclasts.

NASA/TM—2009-215824 21
REPORT DOCUMENTATION PAGE Form Approved

OMB No. 0704-0188

The public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the

data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this

burden, to Department of Defense, Washington Headquarters Services, Directorate for Information Operations and Reports (0704-0188), 1215 Jefferson Davis Highway, Suite 1204, Arlington, VA 22202-4302.

Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to any penalty for failing to comply with a collection of information if it does not display a currently valid OMB

control number.

PLEASE DO NOT RETURN YOUR FORM TO THE ABOVE ADDRESS.

1. REPORT DATE (DD-MM-YYYY) 2. REPORT TYPE 3. DATES COVERED (From - To)

01-11-2009 Technical Memorandum

4. TITLE AND SUBTITLE 5a. CONTRACT NUMBER

Simulating Bone Loss in Microgravity Using Mathematical Formulations of Bone

Remodeling 5b. GRANT NUMBER

5c. PROGRAM ELEMENT NUMBER

6. AUTHOR(S) 5d. PROJECT NUMBER

Pennline, James, A.

5e. TASK NUMBER

5f. WORK UNIT NUMBER

WBS 516724.02.02.08.01

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) 8. PERFORMING ORGANIZATION

National Aeronautics and Space Administration REPORT NUMBER

John H. Glenn Research Center at Lewis Field E-17086

Cleveland, Ohio 44135-3191

9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSORING/MONITOR'S

National Aeronautics and Space Administration ACRONYM(S)

Washington, DC 20546-0001 NASA

11. SPONSORING/MONITORING

REPORT NUMBER

NASA/TM-2009-215824

12. DISTRIBUTION/AVAILABILITY STATEMENT

Unclassified-Unlimited

Subject Categories: 52, 59, and 64

Available electronically at http://gltrs.grc.nasa.gov

This publication is available from the NASA Center for AeroSpace Information, 443-757-5802

13. SUPPLEMENTARY NOTES

14. ABSTRACT

Most mathematical models of bone remodeling are used to simulate a specific bone disease, by disrupting the steady state or balance in the

normal remodeling process, and to simulate a therapeutic strategy. In this work, the ability of a mathematical model of bone remodeling to

simulate bone loss as a function of time under the conditions of microgravity is investigated. The model is formed by combining a

previously developed set of biochemical, cellular dynamics, and mechanical stimulus equations in the literature with two newly proposed

equations; one governing the rate of change of the area of cortical bone tissue in a cross section of a cylindrical section of bone and one

governing the rate of change of calcium in the bone fluid. The mechanical stimulus comes from a simple model of stress due to a

compressive force on a cylindrical section of bone which can be reduced to zero to mimic the effects of skeletal unloading in microgravity.

The complete set of equations formed is a system of first order ordinary differential equations. The results of selected simulations are

displayed and discussed. Limitations and deficiencies of the model are also discussed as well as suggestions for further research.

15. SUBJECT TERMS

Bone remodeling; Mathematical model; Microgravity; Bone demineralization; Differential equations; Simulation

16. SECURITY CLASSIFICATION OF: 17. LIMITATION OF 18. NUMBER 19a. NAME OF RESPONSIBLE PERSON

ABSTRACT OF

PAGES

STI Help Desk (email:help@sti.nasa.gov)

a. REPORT b. ABSTRACT c. THIS 19b. TELEPHONE NUMBER (include area code)

U U PAGE UU 25 443-757-5802

U

Standard Form 298 (Rev. 8-98)

Prescribed by ANSI Std. Z39-18