Mathematical Descriptions of
Bone Remodeling Dynamics
in Myeloma Bone Disease
Bruce P. Ayati
Department of Mathematics and Program in Applied and
Computational Mathematics
University of Iowa
Presented at the 13th International Myeloma Workshop
Carrousel du Louvre, Paris, France
May 3, 2011
Conflicts of Interest
I have no conflicts of interest.
Value of a Mathematical
Model
Basic science:
Link mechanisms to phenomena
Generate hypotheses
Rapid initial screening of drugs and therapies
Predictive models facilitate patient specific medicine
Level of Abstraction
Picasso's Bulls
Nature of Abstraction
Abstraction in a mathematical
model:
Emphasize important
relationships
Represent things by what they
do more than what they are
Georges Braque, Woman
with a Guitar, 1913. Musée
National d'Art Moderne
NonSpatial Model
Our representation of normal bone remodeling
dynamics is taken from Komarova et al. 2003
Power laws in the equations for cell types
implicitly capture the cell signaling.
NonSpatial Model
A model with explicit compartments for the signaling
molecules is given by Pivonka et al. 2008
An approach of this type could form the basis of a
more mechanistic model
NonSpatial Model
Tumor Cel s
Osteoblast
precursors
Osteoclast
precursors
g
g
11
g
22
21
g12
Osteoblasts
Osteoclasts
Trabecular Bone
Model Equations
We modify the Komarova power laws to
incorporate the impact of myeloma cells on bone
remodeling
We add an equation for growth of myeloma cells
which can include the effects of a proteasome
inhibitor:
direct antimyeloma effects and direct
stimulation of osteoblast differentiation
Model Equations
Ayati, Edwards, Webb, Wikswo, Biology Direct 2010
Model Results
The model was found to reflect accurately the
basics of myeloma bone disease
tumor burden is decreased and bone
volume or markers of bone formation are
increased in response to proteasome
inhibitor
Spatial model
Current work: embedding the models for
local interactions into a spatial model
that reflects what are seen in sections of
bone marrow biopsy
2D levelset formulation with normal bone
remodeling is complete (Graham, Ayati,
Ramakrishnan, Martin, submitted)
Need refinement and parameterization of the
local dynamics and extension to 3D
Spatial model
(proof of concept)
We use a level set to define regions of bone
and marrow.
The interface moves according to the local
dynamics of the interacting cell types (no
myeloma case), which are also simulated.
A circular geometry is chosen for illustrative
purposes and simplicity.
Spatial model
(proof of concept)
0.1
0.1
0.05
0.05
Snapshots during
0
0
remodeling of a
0
0.05
0.1
0
0.05
0.1
t=0
t=0 day
da s
y
circular section of
t=0 days
t=20 days
trabecular bone at
0.1
0.1
three remodeling
sites.
0.05
0.05
0
0
0
0.05
0.1
0
0.05
0.1
t=30 days
t=150 days
Spatial model
(proof of concept)
In addition to the bone/marrow interface,
we compute the densities of osteoclasts
and osteoblasts (and in other simulations
the explicit concentrations of RANKL and
OPG concentrations).
The local Ob/Oc interactions are what
actually drive the bone/marrow interface
dynamics in our simulation.
Spatial model
(proof of concept)
0.1
0.1
350
350
300
300
0.05
0.05
250
250
Snapshots during
0
0
0
0.05
0.1
0
0.05
0.1
remodeling of
osteoblast densities
t=0 days
t=20 days
at three remodeling
0.1
0.1
sites.
350
350
300
300
0.05
0.05
250
250
0
0
0
0.05
0.1
0
0.05
0.1
t=30 days
t=150 days
Spatial Model
(Extension)
The level set method is
designed for much more
complicated geometries.
Framework extends naturally
to these geometries at a
range of spatial scales
Figure courtesy of C. Edwards
Thanks
My graduate student Jason Graham
Claire Edwards, Glenn Webb and John Wikswo at
Vanderbilt University
The Vanderbilt Integrative Cancer Biology Center
(VICBC)
Jim Martin and Prem Ramakrishnan at University of
Iowa Orthopaedics and Rehabilitation
Régis Bataille and the organizers of the 13th IMW for
their kind invitation to speak