A study of the effect of non-linearities in the equation of bone remodeling

Zhu, Xinghua; Gong, He; Zhu, Dong; Gao, Bingzhao

doi:10.1016/s0021-9290(02)00028-3

Cited by 51 publications

(50 citation statements)

References 12 publications

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“…The objective function value for this new topology is 0.128177, which is in between the projected Euler method (best) and the standard optimality criteria method (worst). It may also be noted that in other papers where Euler-like methods are applied to this problem, [6,16,28], the solutions presented show separate "legs", as in Figure 6 and in the right hand plot of Figure 11. It may also be notes, in relation to all three topologies generated by different iteration formulas, that the solution is very much determined by the very first iterations in the sequence.…”

Section: Non-convex Problem With Several Equilibrium Pointsmentioning

confidence: 63%

“…We consider a test problem that has been used in several studies directed towards bone remodeling, see [6,16,28]. It seems as if this problem has several local optima with different topologies, but with very similar objective function values.…”

Section: Non-convex Problem With Several Equilibrium Pointsmentioning

confidence: 99%

“…Finally, in Figure 11 we compare the modified optimality criteria method, when the exponent is left exact in equation (28), to the standard optimality criteria algorithm. Again the topology is different and a third local solution in addition to the ones in Figures 6 and 7 is found.…”

Section: Non-convex Problem With Several Equilibrium Pointsmentioning

confidence: 99%

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Dynamical systems and topology optimization

Klarbring

Torstenfelt

2010

Struct Multidisc Optim

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This paper uses a dynamical systems approach for studying the material distribution (density or SIMP) formulation of topology optimization of structures. Such an approach means that an ordinary differential equation, such that the objective function is decreasing along a solution trajectory of this equation, is constructed. For stiffness optimization two differential equations with this property are considered. By simple explicit Euler approximations of these equations, together with projection techniques to satisfy box constraints, we obtain different iteration formulas. One of these formulas turns out to be the classical optimality criteria algorithm, which, thus, is receiving a new interpretation and framework. Based on this finding we suggest extensions of the optimality criteria algorithm.A second important feature of the dynamical systems approach, besides the purely algorithmic one, is that it points at a connection between optimization problems and natural evolution problems such as bone remodeling and damage evolution. This connection has been hinted at previously but, in the opinion of the authors, not been clearly stated since the dynamical systems concept was missing. To give an explicit example of an evolution problem that is in this way connected to an optimization problem, we study a model of bone remodeling.Numerical examples, related to both the algorithmic issue and the issue of natural evolution represented as bone remodeling, are presented.

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Section: Non-convex Problem With Several Equilibrium Pointsmentioning

confidence: 63%

Section: Non-convex Problem With Several Equilibrium Pointsmentioning

confidence: 99%

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Dynamical systems and topology optimization

Klarbring

Torstenfelt

2010

Struct Multidisc Optim

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“…the fabric tensor [17]. Many theories, mostly phenomenological, have been proposed to explain the process of bone remodeling [19][20][21][22][23][24][25][26][27][28]. For example, the apparent density has been used to characterize the internal morphology of the bone and the strain energy density assumed as the stimulus for bone adaptation [19].…”

Section: Introductionmentioning

confidence: 99%

“…These methods, however, are often based on complicated mathematical formulations and/or require extreme computation costs. A high-order nonlinear equation of bone remodeling was utilized in [21] and the influence of nonlinearity on adaptation was studied. Prediction of external shapes and internal density distributions in the proximal femur was also investigated using these non-linear equations [22].…”

Section: Introductionmentioning

confidence: 99%

Using a Truss-Inspired Model with the Uniform Strength Optimization Theory to Predict Spongy Bone Geometry in Proximal Femur

Rouhi¹,

Pishdast²,

Farahmand³

2009

American J. of Applied Sciences

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This paper presents a new naïve approach for simulating bone remodeling process. It is based on the uniform strength theory of optimization and employs a truss-like model for bone. The truss was subjected to external loads including 5 point loads simulating the hip joint contact forces and 3 muscular forces at the attachment sites of the muscles to the bone and the rest are reactions of ligaments. The strain in the links was calculated and the links with high strains were identified. The initial truss is modified by introducing new links wherever the strain exceeds a prescribed or critical value. The critical value was assumed to be equal to an average of the absolute value of strains in the initial model. Each link which undergoes a high strain is replaced by several new links by adding new nodes around it using Delaunay method. Introducing the new links to the truss, which is conducted according to a weighted arithmetic mean formula, will strengthen the structure and reduce the strain within the respective zone. This procedure was repeated for several times. Convergence was achieved when there were no critical links remaining. This method was used to study the 2D shape of proximal femur in the frontal plane and provided results which are in a fairly good agreement with CT image of the human proximal femur.

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Nutrient modulated structural design with application to growth and degradation

2015

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Phenomena such as biological growth and damage evolution can be thought of as time evolving processes, the directions of which are governed by descendent of certain goal functions. Mathematically this means using a dynamical systems approach to optimization. We extend such an approach by introducing a field quantity, representing nutrients or other non‐mechanical stimuli, that modulate growth and damage evolution. The derivation of a generic model is systematic, starting from a Lyaponov‐type descent condition and utilizing a Coleman‐Noll strategy. A numerical algorithm for finding stationary points of the resulting dynamical system is suggested and applied to two model problems where the influence of different levels of nutrient sensitivity are observed. The paper demonstrates the use of a new modeling technique and shows its application in deriving a generic problem of growth and damage evolution.

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A study of the effect of non-linearities in the equation of bone remodeling

Cited by 51 publications

References 12 publications

Dynamical systems and topology optimization

Dynamical systems and topology optimization

Using a Truss-Inspired Model with the Uniform Strength Optimization Theory to Predict Spongy Bone Geometry in Proximal Femur

Nutrient modulated structural design with application to growth and degradation

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