Nilima Nigam

Mathematical Modeling of Spatio-Temporal Dynamics of a Single Bone Multicellular Unit†

During bone remodeling, bone‐resorbing osteoclasts and bone‐forming osteoblasts are organized in bone multicellular units (BMUs), which travel at a rate of 20–40 μm/d for 6–12 mo, maintaining a cylindrical structure. However, the interplay of local BMU geometry with biochemical regulation is poorly understood. We developed a mathematical model of BMU describing changes in time and space of the concentrations of proresorptive cytokine RANKL and its inhibitor osteoprotegerin (OPG), in osteoclast and osteoblast numbers, and in bone mass. We assumed that osteocytes surrounding a microfracture produce RANKL, which attracted osteoclasts. OPG and RANKL were produced by osteoblasts and diffused through bone, RANKL was eliminated by binding to OPG and RANK. Osteoblasts were coupled to osteoclasts through paracrine factors. The evolution of the BMU arising from this model was studied using numerical simulations. Our model recapitulated the spatio‐temporal dynamics observed in vivo in a cross‐section of bone. In response to a RANKL field, osteoclasts moved as a well‐confined cutting cone. The coupling of osteoclasts to osteoblasts allowed for sufficient recruitment of osteoblasts to the resorbed surfaces. The RANKL field was the highest at the microfracture in front of the BMU, whereas the OPG field peaked at the back of the BMU, resulting in the formation of a RANKL/OPG gradient, which strongly affected the rate of BMU progression and its size. Thus, the spatial organization of a BMU provides important constraints on the roles of RANKL and OPG as well as possibly other regulators in determining the outcome of remodeling in the BMU.

Geometric integration on spheres and some interesting applications

Geometric integration theory can be employed when numerically solving ODEs or PDEs with constraints. In this paper, we present several one-step algorithms of various orders for ODEs on a collection of spheres. To demonstrate the versatility of these algorithms, we present representative calculations for reduced free rigid body motion (a conservative ODE) and a discretization of micromagnetics (a dissipative PDE). We emphasize the role of isotropy in geometric integration and link numerical integration schemes to modern differential geometry through the use of partial connection forms; this theoretical framework generalizes moving frames and connections on principal bundles to manifolds with nonfree actions.

A Predictive Model of Rectal Tumor Response to Preoperative Radiotherapy Using Classification and Regression Tree Methods

Purpose: The ability to predict rectal tumor response to preoperative radiotherapy before treatment would significantly impact patient selection. In this study, classification and regression tree (CART) methods were used to model tumor response to preoperative conformal high-dose rate brachytherapy by assessing the predictive value of vascular endothelial growth factor (VEGF), Bcl-2, p21, p53, and APAF-1. Experimental Design: Immunohistochemistry was used to detect VEGF, Bcl-2, p21, p53, and APAF-1 from 62 pretreatment rectal tumor biopsies. Scores were assigned as percentages of positive tumor cell staining and were used in CART analysis to identify the proteins that best predicted response to radiotherapy. Ten-fold cross-validation was used to prevent overfitting and multiple cross-validation experiments were run to estimate the prediction error. Results: Postoperative pathologic evaluation of the irradiated tumor bed revealed 43 responsive tumors [20 with complete response (T0) and 23 with partial response] and 19 nonresponsive tumors. The optimal tree resulting from CART analysis had five terminal nodes with a misclassification rate of 18%. Of the five proteins selected for their predictive value, VEGF and Bcl-2 contributed most to the classification of responsive and nonresponsive tumors. All 10 tumors with no VEGF were completely responsive (T0) to radiotherapy; 85% of those with VEGF and negative for Bcl-2 were responsive to therapy. Conclusions: VEGF and Bcl-2 status in pretreatment rectal tumor biopsies may be predictive of response to preoperative high-dose rate brachytherapy.

How large can the first eigenvalue be on a surface of genus two

AbstractSharp upper bounds for the first eigenvalue of the Laplacian on asurface of a fixed area are known only in genera zero and one. Weinvestigate the genus two case and conjecture that the first eigenvalue ismaximized on a singular surface which is realized as a double branchedcovering over a sphere. The six ramification points are chosen in sucha way that this surface has a complex structure of the Bolza surface.We prove that our conjecture follows from a lower bound on the firsteigenvalue of a certain mixed Dirichlet-Neumann boundary value problemon a half-disk. The latter can be studied numerically, and we presentconclusive evidence supporting the conjecture. Keywords: Laplacian, first eigenvalue, surface of genus two, mixed boundaryvalue problem. ∗ Department of Mathematics and Statistics, McGill University, 805 Sherbrooke Str. West,Montreal, QC H3A 2K6, Canada; e-mail jakobson@math.mcgill.ca † Department of Mathematics, Heriot-Watt University, United Kingdom; e-mailM.Levitin@ma.hw.ac.uk

The effect of intramuscular fat on skeletal muscle mechanics: implications for the elderly and obese

Skeletal muscle accumulates intramuscular fat through age and obesity. Muscle quality, a measure of muscle strength per unit size, decreases in these conditions. It is not clear how fat influences this loss in performance. Changes to structural parameters (e.g. fibre pennation and connective tissue properties) affect the muscle quality. This study investigated the mechanisms that lead to deterioration in muscle performance due to changes in intramuscular fat, pennation and aponeurosis stiffness. A finite-element model of the human gastrocnemius was developed as a fibre-reinforced composite biomaterial containing contractile fibres within the base material. The base-material properties were modified to include intramuscular fat in five different ways. All these models with fat generated lower fibre stress and muscle quality than their lean counterparts. This effect is due to the higher stiffness of the tissue in the fatty models. The fibre deformations influence their interactions with the aponeuroses, and these change with fatty inclusions. Muscles with more compliant aponeuroses generated lower forces. The muscle quality was further reduced for muscles with lower pennation. This study shows that whole-muscle force is dependent on its base-material properties and changes to the base material due to fatty inclusions result in reductions to force and muscle quality.

The Cellular Dynamics of Bone Remodeling: a Mathematical Model

The mechanical properties of vertebrate bone are largely determined by a process which involves the complex interplay of three different cell types. This process is called bone remodeling and occurs asynchronously at multiple sites in the mature skeleton. The cells involved are bone resorbing osteoclasts, bone matrix producing osteoblasts, and mechanosensing osteocytes. These cells communicate with each other by means of autocrine and paracrine signaling factors and operate in complex entities, the so-called bone multicellular units (BMUs). To investigate the BMU dynamics in silico, we develop a novel mathematical model resulting in a system of nonlinear partial differential equations (PDEs) with time delays. The model describes the osteoblast and osteoclast populations together with the dynamics of the key messenger molecule RANKL and its decoy receptor OPG. Scaling theory is used to address parameter sensitivity and predict the emergence of pathological remodeling regimes. The model is studied numerical...

Complex dynamics of osteoclast formation and death in long-term cultures.

Background Osteoclasts, cells responsible for bone resorption, contribute to the development of degenerative, metabolic and neoplastic bone diseases, which are often characterized by persistent changes in bone microenvironment. We aimed to investigate the dynamics of osteoclast formation and death in cultures that considerably exceeded the length of standard protocol and to design a mathematical model describing osteoclastogenesis. Methodology/Principal Findings RAW 264.7 monocytic cells fuse to form multinucleated osteoclasts upon treatment with pro-resorptive cytokine RANKL. We have found that in long-term experiments (15–26 days), the dynamics of changes in osteoclast numbers was remarkably complex and qualitatively variable in different experiments. Whereas 19 of 46 experiments exhibited single peak of osteoclast formation, in 27 experiments we observed development of successive waves of osteoclast formation and death. Periodic changes in osteoclast numbers were confirmed in long-term cultures of mouse bone marrow cells treated with M-CSF and RANKL. Because the dynamics of changes in osteoclast numbers was found to be largely independent of monocytes, a two-species model of ordinary differential equations describing the changes in osteoclasts and monocytes was ineffective in recapitulating the oscillations in osteoclast numbers. Following experimental observation that medium collected from mature osteoclasts inhibited osteoclastogenesis in fresh cultures, we introduced a third variable, factor f, to describe osteoclast-derived inhibitor. This model allowed us to simulate the oscillatory changes in osteoclasts, which were coupled to oscillatory changes in the factor f, whereas monocytes changed exponentially. Importantly, to achieve the experimentally observed oscillations with increasing amplitude, we also had to assume that osteoclast presence stimulates osteoclast formation. Conclusions/Significance This study identifies the critical role for osteoclast autocrine regulation in controlling long-term dynamic of osteoclast formation and death and describes the complementary roles for negative and positive feedback mediators in determining the sharp dynamics of activation and inactivation of osteoclasts.

Error analysis of an enhanced DtN-FE method for exterior scattering problems

In this work we analyze the convergence of the high-order Enhanced DtN-FEM algorithm, described in our previous work (Nicholls and Nigam, J. Comput. Phys. 194:278–303, 2004), for solving exterior acoustic scattering problems in

Regionalizing muscle activity causes changes to the magnitude and direction of the force from whole muscles—a modeling study

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