1. Topic 6: Bone Growth and Remodeling
Bone continually remodels
growth, reinforcement, resorption
depends on stress and strain
There is an optimal range of stress for maximum strength
understressed or overstressed bone can weaken
stresses on fractured bone affect healing
stress-dependent remodeling affects surgical implant and prosthesis design, e.g. fracture fixation plates, surgical screws, artificial joints
1978: radiographic evidence of bone resorption seen in 70% of total hip replacement patients
14 January 1999
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2. Stress-Dependent Remodeling
Osteoclasts - cells responsible for resorption
Osteoblasts - cells responsible for growth (hypertrophy)
compressive stress stimulates formation of new bone and is important for fracture healing
loss of normal stress  loss of calcium and reduced bone density
Time scales:
remodeling - months/years
fastest remodeling is due to change in mineral content
healing - weeks
growth/maturation - years
14 January 1999
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3. Types of Bone Remodeling
Two types of remodeling in bone:
1. surface (external) remodeling
change in bone shape and dimensions
deposition on to or resorption of bone material from inner or outer surfaces
2. internal remodeling
change in:
bulk density
trabecular size
orientation
osteon size, etc.
14 January 1999
This fits comfortably in one lecture (80 mins). 5 mins left for a feew more nice pictures but probably enough numbers and facts already. This should be a whole lecture supplemented only with more nice pictures aas they become available.

4. Functional Adaptation
Principal of Functional Adaptation, Roux (1895):
"the ability of organs (and cells, tissues and organisms) to adapt their capacity to function in response to altered demands by practice”
Functional adaptation in bone is remodeling of structure, geometry and mechanical properties in response to altered loading
Related to the engineering concept of optimal design
14 January 1999
This fits comfortably in one lecture (80 mins). 5 mins left for a feew more nice pictures but probably enough numbers and facts already. This should be a whole lecture supplemented only with more nice pictures aas they become available.

5. Some Optimal Design Principles
Theory of Uniform Strength — attempts to produce the same maximum normal stress (brittle material) or shear stress (ductile material) throughout the body for a specific loading
Theory of Trajectorial Architecture — concentrates material in the paths of force transmission, such as principal stress lines, e.g. fiber reinforcing of composite (kevlar-mylar) sails
Principle of Maximum-Minimum Design — maximize strength for minimum weight or cost
These theories have been verified in many cases
14 January 1999
This fits comfortably in one lecture (80 mins). 5 mins left for a feew more nice pictures but probably enough numbers and facts already. This should be a whole lecture supplemented only with more nice pictures aas they become available.

6. Stress Adaptation of Trabecular Bone
G.H. von Meyer’s trabecular bone architecture in human femur (1867)
Principal stress trajectories of Culmann’s crane
14 January 1999
This fits comfortably in one lecture (80 mins). 5 mins left for a few more nice pictures but probably enough numbers and facts already. This should be a whole lecture supplemented only with more nice pictures as they become available.

7. Remodeling of Trabecular Bone: Wolff's Law
Wolff (1872): when loads are changed by trauma or change in activity, functional remodeling reorients bone trabeculae so they align with the new principal stress axes
Wolff never actually proved this
Wolff's “law of bone transformation” (1884): “there is a perfect mathematical correspondence between the structure of cancellous bone in proximal femur and Culmann’s trajectories”
Culmann’s trajectories and other of Wolff’s assertions were suspect, but photoelastic studies (Pauwels,1954) confirmed Wolff's law
14 January 1999
This fits comfortably in one lecture (80 mins). 5 mins left for a feew more nice pictures but probably enough numbers and facts already. This should be a whole lecture supplemented only with more nice pictures aas they become available.

8. Cowin's Mathematical Statement of Wolff's Law
In remodeling equilibrium:
T•H = H•T
The stress tensor T and the fabric tensor H commute (they have the same principal axes)
The principal axes of H are parallel to the trabecular trajectories
14 January 1999
This fits comfortably in one lecture (80 mins). 5 mins left for a feew more nice pictures but probably enough numbers and facts already. This should be a whole lecture supplemented only with more nice pictures aas they become available.

9. Theories of Stress-Adaptive Bone Remodeling
Two main analytical models of bone adaptation to stress:
1. The theory of surface bone remodeling
2. The theory of internal bone remodeling
14 January 1999
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10. Remodeling Theories: Assumptions
1. Bone is a linearly elastic porous solid (matrix, mineral, cells) perfused with fluid (interstitial and blood)
2. Fluid phase can be converted to solid and vice- versa by cells. Mass transfer to or from the bone matrix is slow compared with the time constants for dynamic mechanical effects
3. Inertial and other dynamic mechanical effects are neglected
4. Reaction rates of solid-fluid conversion at a point depend on strain, stress or strain energy
5. Internal and external remodeling differ by the site of reactions and how mass is added/removed
14 January 1999
This fits comfortably in one lecture (80 mins). 5 mins left for a feew more nice pictures but probably enough numbers and facts already. This should be a whole lecture supplemented only with more nice pictures aas they become available.

11. Theory of Surface Remodeling
Solid-fluid conversion occurs only on boundary surfaces
Mass is only added/removed to/from bone solid by changes in external shape of bone
Bone interior bulk density is unchanged
The rate of surface deposition/resorption at a point is a function of surface strain, stress or strain energy at that point
14 January 1999
This fits comfortably in one lecture (80 mins). 5 mins left for a feew more nice pictures but probably enough numbers and facts already. This should be a whole lecture supplemented only with more nice pictures aas they become available.

12. Theory of Internal Remodeling
Solid-fluid conversion occurs everywhere throughout porous bone
Mass is added/removed by altering bulk density through changes in bone porosity, mineral content, etc.
Exterior bone dimensions are constant
Rate of change of density at a point is a function of strain, stress or strain energy at that point
14 January 1999
This fits comfortably in one lecture (80 mins). 5 mins left for a feew more nice pictures but probably enough numbers and facts already. This should be a whole lecture supplemented only with more nice pictures aas they become available.

13. Example: Diaphysial Surface Remodelingb

14. Example: Diaphysial Surface Remodeling

15. Diaphysial Surface Remodeling: Solution
8 steady-state solutions for a and b at remodeling equilibrium
solutions depend on P, E and remodeling rate constants, Rp and Re, for given initial conditions, a0, b0 and T0zz

16. Bone growth and remodeling: Summary of key points
Historical principles
Wolff’s Law
Functional adaptation (stress-adaptive remodeling)
Types of bone remodeling
internal remodeling
changes of bone density (and hence strength and stiffness)
changes of trabecular architecture
external remodeling
changes of bone geometry
Remodeling laws
strain-energy dependent density remodeling (Carter)
strain-dependent surface remodeling (Cowin)
stress-dependent “fabric tensor” remodeling (Cowin)