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 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.

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 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.

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.

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.

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.

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.

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.

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.

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 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.

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.

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.

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.

Example: Diaphysial Surface Remodeling b

Example: Diaphysial Surface Remodeling

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

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)