Bone Adaptation as Level Set Motion

Besler, Bryce A.; Gabel, Leigh; Burt, Lauren A; Forkert, Nils D.; Boyd, Steven K.

doi:10.1007/978-3-030-11166-3_6

Cited by 5 publications

(26 citation statements)

References 26 publications

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“…The concept of bone adaptation as a geometric flow is presented. The work presented here is a significant extension of a conference proceeding [1]. Particularly, an artifact of signed distance transforms of sampled signals has been identified [2] and solved in the case of computed tomography images of biphasic materials [3].…”

Section: Introductionmentioning

confidence: 90%

Bone Adaptation as a Geometric Flow

Besler¹,

Kemp²,

Forkert³

et al. 2021

Preprint

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This paper presents bone adaptation as a geometric flow. The proposed method is based on two assumptions: first, that the bone surface is smooth (not fractal) permitting the definition of a tangent plane and, second, that the interface between marrow and bone tissue phases is orientable. This permits the analysis of bone adaptation using the well-developed mathematics of geometric flows and the numerical techniques of the level set method. Most importantly, topological changes such as holes forming in plates and rods disconnecting can be treated formally and simulated naturally. First, the relationship between biological theories of bone adaptation and the mathematical object describing geometric flow is described. This is termed the adaptation function, F , and is the multi-scale link described by Frost's Utah paradigm between cellular dynamics and bone structure. Second, a model of age-related bone loss termed curvature-based bone adaptation is presented. Using previous literature, it is shown that curvature-based bone adaptation is the limiting continuous equation of simulated bone atrophy, a discrete model of bone aging. Interestingly, the parameters of the model can be defined in such a way that the flow is volume-preserving. This implies that bone health can in principle change in ways that fundamentally cannot be measured by areal or volumetric bone mineral density, requiring structurelevel imaging. Third, a numerical method is described and two in silico experiments are performed demonstrating the nonvolume-preserving and volume-preserving cases. Taken together, recognition of bone adaptation as a geometric flow permits the recruitment of mathematical and numerical developments over the last 50 years to understanding and describing the complex surface of bone.

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Section: Introductionmentioning

confidence: 90%

Bone Adaptation as a Geometric Flow

Besler¹,

Kemp²,

Forkert³

et al. 2021

Preprint

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“…Previously, curvature‐driven bone adaptation was shown to mimic age‐related bone loss in post‐menopausal women implicitly through image processing 12,13 and explicitly through mathematical rules 14 . Thus, we select a speed function with advection and curvature loss 14 :

F = a - italicbκ

where

bold-italicθ = ({}, a, b)

is the vector of model parameters, and

κ

is the mean curvature which can be estimated directly from the embedding image

ϕ

κ = \nabla ∙ ((), \frac{\nabla ϕ}{(||, \nabla ϕ)}) .

For the case of bone adaptation,

a

has units of

μm / year

, and

b

has units of

normalμ m^{2} / year

. These constants determine the speed (or rate) of bone adaptation; they implicitly capture both physiological and mechanical drivers of bone change.…”

Section: Methodsmentioning

confidence: 99%

“…Briefly, osteoclast activity was modeled as a Gaussian filter on binarized images, a threshold was applied which controlled the amount of tissue removed at each iteration, and the activation frequency was modeled by changing the duration of each iteration 12,13 . Alternatively, our lab framed bone adaptation as a surface evolution problem that is solved with level set methods 14 . In this technique, the interface between trabecular bone and marrow was a surface.…”

Section: Introductionmentioning

confidence: 99%

An inverse technique to identify participant‐specific bone adaptation from serial CT measurements

Kemp

Besler

Boyd

2021

Numer Methods Biomed Eng

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Simulated bone adaptation is framed as an interface evolution problem. The interface is extracted from a high‐resolution computed tomography (CT) image of trabecular bone microarchitecture and modified by the level set equation. A model and its parameters determine the bone adaptation rate and thus the bone structure at any future time. This study develops an inverse problem and solver to identify model parameters from multiple high‐resolution CT images of bone within the level set framework. We demonstrate the technique on a model of advection and mean curvature flow, termed curvature‐driven adaptation. The inverse solver uses two CT scans to estimate model parameters, which map the bone surface from one image to the next. The solver was tested with synthetic images of bone changing according to the curvature‐driven model with known model parameters. The algorithm recovered known model parameters excellently (R2 > .99, p < .001). A grid search indicated that the estimated model parameters were insensitive to hyper‐parameter selection for learning rate 1e−5 ≤η≤ 5e−5 and gradient scaling factor 5e−5 ≤γ≤ 5e−4. Finally, we tested the algorithm's sensitivity to salt‐and‐pepper noise of probability P, where .0 ≤P≤ .9. Model parameter accuracy did not change for P < .7, corresponding to Dice coefficients greater than .7. The inverse problem estimates bone adaptation parameters from multiple CT images of changing bone microarchitecture. In the future, this technique could be used to determine participant‐specific bone adaptation parameters in vivo, validate bone adaptation models, and predict bone health.

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“…Second, morphometry can be measured during curve evolution problems where topology can change without explicit splitting and merging techniques Osher and Sethian (1988). This has been the primary feature that made level set methods popular, used extensively in computational fluid dynamics Peng et al (1999); Sussman et al (1994), object segmentation Chan and Vese (2001); Caselles et al (1993); Vese and Chan (2002), and biophysical simulations Besler et al (2018). The one disadvantage is that implicit representations can require large amounts of memory to store and process.…”

Section: Introductionmentioning

confidence: 99%

Local Morphometry of Closed, Implicit Surfaces

Besler,

Kemp,

Michalski

et al. 2021

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Anatomical structures such as the hippocampus, liver, and bones can be analyzed as orientable, closed surfaces. This permits the computation of volume, surface area, mean curvature, Gaussian curvature, and the Euler-Poincaré characteristic as well as comparison of these morphometrics between structures of different topology. The structures are commonly represented implicitly in curve evolution problems as the zero level set of an embedding. Practically, binary images of anatomical structures are embedded using a signed distance transform. However, quantization prevents the accurate computation of curvatures, leading to considerable errors in morphometry. This paper presents a fast, simple embedding procedure for accurate local morphometry as the zero crossing of the Gaussian blurred binary image. The proposed method was validated based on the femur and fourth lumbar vertebrae of 50 clinical computed tomography datasets. The results show that the signed distance transform leads to large quantization errors in the computed local curvature. Global validation of morphometry using regression and Bland-Altman analysis revealed that the coefficient of determination for the average mean curvature is improved from 93.8% with the signed distance transform to 100% with the proposed method. For the surface area, the proportional bias is improved from -5.0% for the signed distance transform to +0.6% for the proposed method. The Euler-Poincaré characteristic is improved from unusable in the signed distance transform to 98% accuracy for the proposed method. The proposed method enables an improved local and global evaluation of curvature for purposes of morphometry on closed, implicit surfaces.

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Bone Adaptation as Level Set Motion

Cited by 5 publications

References 26 publications

Bone Adaptation as a Geometric Flow

Bone Adaptation as a Geometric Flow

An inverse technique to identify participant‐specific bone adaptation from serial CT measurements

Local Morphometry of Closed, Implicit Surfaces

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