A Patient-Specific Anisotropic Diffusion Model for Brain Tumour Spread

Gliomas are primary brain tumours arising from the glial cells of the nervous system. The diffuse nature of spread, coupled with proximity to critical brain structures, makes treatment a challenge. Pathological analysis confirms that the extent of glioma spread exceeds the extent of the grossly visible mass, seen on conventional magnetic resonance imaging (MRI) scans. Gliomas show faster spread along white matter tracts than in grey matter, leading to irregular patterns of spread. We propose a mathematical model based on Diffusion Tensor Imaging, a new MRI imaging technique that offers a methodology to delineate the major white matter tracts in the brain. We apply the anisotropic diffusion model of Painter and Hillen (J Thoer Biol 323:25–39, 2013) to data from 10 patients with gliomas. Moreover, we compare the anisotropic model to the state-of-the-art Proliferation–Infiltration (PI) model of Swanson et al. (Cell Prolif 33:317–329, 2000). We find that the anisotropic model offers a slight improvement over the standard PI model. For tumours with low anisotropy, the predictions of the two models are virtually identical, but for patients whose tumours show higher anisotropy, the results differ. We also suggest using the data from the contralateral hemisphere to further improve the model fit. Finally, we discuss the potential use of this model in clinical treatment planning.     

A Mathematical Model for the Effects of HER2 Overexpression on Cell Proliferation in Breast Cancer

We present a mathematical model to study the effects of HER2 over-expression on cell proliferation in breast cancer. The model illustrates the proliferative behavior of cells as a function of HER2 and EGFR receptors numbers, and the growth factor EGF. This mathematical model comprises kinetic equations describing the cell surface binding of EGF growth factor to EGFR and HER2 receptors, coupled to a model for the dependence of cell proliferation rate on growth factor receptors binding. The simulation results from this model predict: (1) a growth advantage associated with excess HER2 receptors; (2) that HER2-over-expression is an insufficient parameter to predict the proliferation response of cancer cells to epidermal growth factors; and (3) the EGFR receptor expression level in HER2-over-expressing cells plays a key role in mediating the proliferation response to receptor-ligand signaling. This mathematical model also elucidates the interaction and roles of other model parameters in determining cell proliferation rate of HER2-over-expressing cells.     

Mathematical Model for Spatial Segregation of the Rho-Family GTPases Based on Inhibitory Crosstalk

Cdc42, Rac, and Rho are small GTPases known to play a central role in signal transduction to the actin cytoskeleton. These proteins regulate cell motility, by affecting nucleation, uncapping, and depolymerization of actin filaments, and acto-myosin contractility. Studies of crosstalk and mutual feedbacks in these three proteins have led to a number of proposals for their interaction. At the same time, observations of the spatio-temporal dynamics of Rho-family proteins give evidence of spatial polarization and mutual exclusion between Cdc42/Rac and Rho. In this paper, we formulate a mathematical model to account for such observations, based on the known underlying biology of these proteins. We first investigate which of the crosstalk schemes proposed in the literature is consistent with observed dynamics, and then derive a simple model that can correctly describe these dynamics (assuming crosstalk is mediated via Rho GEFs). We show that cooperativity is an essential ingredient in the interactions of the proteins. The co-occurrence of a stable rest state with the possibility of fast spatial segregation can be related to bistability in a set of underlying ODEs in which the inactive forms of these proteins are fixed at a constant level. We show that the fast diffusion of the inactive forms is essential for stabilizing the transition fronts in the PDE formulation of the model, leading to robust spatial polarization, rather than traveling waves.     

A Mathematical Model for the Regulation of Tumor Dormancy Based on Enzyme Kinetics

In this paper we present a two-compartment model for tumor dormancy based on an idea of Zetter [1998, Ann. Rev. Med. 49, 407–422] to wit: The vascularization of a secondary (daughter) tumor can be suppressed by an inhibitor originating from a larger primary (mother) tumor. We apply this idea at the avascular level to develop a model for the remote suppression of secondary avascular tumors via the secretion of primary avascular tumor inhibitors. The model gives good agreement with the observations of [De Giorgi et al., 2003, Derm. Surgery 29, 664–667]. These authors reported on the emergence of a polypoid melanoma at a site remote from a primary polypoid melanoma after excision of the latter. The authors observed no recurrence of the melanoma at the primary site, but did observe secondary tumors at secondary sites 5–7 cm from the primary site within a period of 1 month after the excision of the primary site. We attempt to provide a reasonable biochemical/cell biological model for this phenomenon. We show that when the tumors are sufficiently remote, the primary tumor will not influence the secondary tumor while, if they are too close together, the primary tumor can effectively prevent the growth of the secondary tumor, even after it is removed. It should be possible to use the model as the basis for a testable hypothesis.     

Chemically Based Mathematical Model for Development of Cerebral Cortical Folding Patterns

The mechanism for cortical folding pattern formation is not fully understood. Current models represent scenarios that describe pattern formation through local interactions, and one recent model is the intermediate progenitor model. The intermediate progenitor (IP) model describes a local chemically driven scenario, where an increase in intermediate progenitor cells in the subventricular zone correlates to gyral formation. Here we present a mathematical model that uses features of the IP model and further captures global characteristics of cortical pattern formation. A prolate spheroidal surface is used to approximate the ventricular zone. Prolate spheroidal harmonics are applied to a Turing reaction-diffusion system, providing a chemically based framework for cortical folding. Our model reveals a direct correlation between pattern formation and the size and shape of the lateral ventricle. Additionally, placement and directionality of sulci and the relationship between domain scaling and cortical pattern elaboration are explained. The significance of this model is that it elucidates the consistency of cortical patterns among individuals within a species and addresses inter-species variability based on global characteristics and provides a critical piece to the puzzle of cortical pattern formation.     

DeepCKI:一个基于变分图自编码器预测细胞-细胞因子相互作用的生物信息学模型

细胞因子(cytokine)是一类由免疫细胞和某些非免疫细胞合成和分泌的信号分子,在免疫系统中通过结合相应受体调节细胞生长、分化和调控免疫应答。目前研究多侧重于通过实验方法检测细胞因子和受体的相互作用来研究细胞间的通讯网络,但存在实验周期长、设备要求高和成本高等不足。因此,有必要通过计算方法来加快对细胞-细胞因子相互作用(cell-cytokine interactions, CKI)的系统研究。本文提出一种基于变分图自编码器(variational graph auto-encoder, VGAE)预测细胞-细胞因子相互作用的深度学习模型——DeepCKI。该模型可有效融合蛋白质相互作用网络和不同类型的蛋白质特征,充分挖掘网络拓扑结构和节点属性中的有效信息,实现对细胞-细胞因子相互作用的高效预测。与变分自编码和深度神经网络方法相比,采用图结构设计的DeepCKI表现出了最优的预测性能。DeepCKI模型对4种不同类型细胞-细胞因子相互作用的ROC曲线下面积均高于0.8,模型具有一定的鲁棒性和有效性。预测打分排名前100的细胞-细胞因子相互作用中,有36对已被最新发表文献验证,表明该模...     

Mathematical Model of Growth Factor Driven Haptotaxis and Proliferation in a Tissue Engineering Scaffold

Motivated by experimental work (Miller et al. in Biomaterials 27(10):2213–2221, 2006, 32(11):2775–2785, 2011) we investigate the effect of growth factor driven haptotaxis and proliferation in a perfusion tissue engineering bioreactor, in which nutrient-rich culture medium is perfused through a 2D porous scaffold impregnated with growth factor and seeded with cells. We model these processes on the timescale of cell proliferation, which typically is of the order of days. While a quantitative representation of these phenomena requires more experimental data than is yet available, qualitative agreement with preliminary experimental studies (Miller et al. in Biomaterials 27(10):2213–2221, 2006) is obtained, and appears promising. The ultimate goal of such modeling is to ascertain initial conditions (growth factor distribution, initial cell seeding, etc.) that will lead to a final desired outcome.     

Stress-Based Plaque Vulnerability Index and Assessment for Carotid Atherosclerotic Plaques Using Patient-Specific Vessel Material Properties

Cardiovascular diseases are closely linked to atherosclerotic plaque development and rupture. Assessment of plaque vulnerability is of fundamental significance to cardiovascular research and disease diagnosis, prevention, treatment and management. Magnetic resonance image (MRI) data of carotid atherosclerotic plaques from 8 patients (5 male, 3 female; age: 62-83, mean=71) were acquired at the University of Washington (UW), Seattle by the Vascular Imaging Laboratory (VIL) with written informed consent obtained. Patient-specific vessel material properties were quantified using Cine MRI data for modeling use. 3D thin-layer models were used to obtain plaque stress and strain for plaque assessment. A stress-based plaque vulnerability index (SPVI) was proposed to combine mechanical analysis, plaque morphology and composition for more complete carotid plaque vulnerability assessment. The five intervals (unit: kPa) [0, 46.8), [46.8, 80), [80, 92), [92, 103), and [103, +∞) from in vivo material models were used for SPVI values of 0, 1, 2, 3 and 4, respectively. The optimized agreement rate was 85.19%. The use of patient-specific material properties in plaque models could potentially improve the accuracy of model stress/strain calculations. SPVI has the potential to improve the current image-based screening and plaque vulnerability assessment schemes.     

Composite Subsidence Vulnerability Assessment Based on an Index Model and Index Decomposition Method

The threat of damage to buildings and other infrastructures resulting from land subsidence associated with groundwater pumping in urbanized areas is an ongoing problem requiring assessment. An important goal of subsidence vulnerability assessment is to construct a composite subsidence vulnerability index (SVI) that is represented by a set of indicators that focuses on four different thematic factors: physical, social, economic, and environmental vulnerability. These indicators are evaluated on the basis of indicator selection principles and then weighted by their contribution rate to the overall index. The weights reflect different measures assigned to the township-specific conditions. A complete and composite subsidence vulnerability assessment is developed in which future vulnerability management decision-making processes can be readily made. The vulnerability assessment includes not only the construction of the SVI, which involves selecting, assigning value to, weighting, and aggregating the vulnerability indicators, but also the presentation of the SVI decomposition. Research results demonstrate that a composite subsidence vulnerability assessment method can be made by first constructing and then decomposition-presenting the overall SVI. This allows for the relative comparison of subsidence vulnerability and the identification of the main vulnerable indicators; thus providing subsidence risk, which represents an important step toward vulnerability management of water resources.     

基于均匀设计的主成分分析-支持向量机模型及其在几丁质酶最适pH建模中的应用

采用主成分分析法对样本数据集进行预处理,将得到的新样本数据集输入支持向量机,籍均匀设计,构建了几丁质酶氨基酸组成和最适pH的数学模型。当惩罚系数C为10,epsilon值为0.7,Gamma值为0.5,模型对pH值拟合的平均绝对百分比误差为3.76%,同时具有良好的预测效果,预测的平均绝对误差为0.42个pH单位。该方法比用BP神经网络方法效果更佳。     

A Biophysically Based Mathematical Model for the Kinetics of Mitochondrial Na-Ca Antiporter

Sodium-calcium antiporter is the primary efflux pathway for Ca²⁺ in respiring mitochondria, and hence plays an important role in mitochondrial Ca²⁺ homeostasis. Although experimental data on the kinetics of Na⁺-Ca²⁺ antiporter are available, the structure and composition of its functional unit and kinetic mechanisms associated with the Na⁺-Ca²⁺ exchange (including the stoichiometry) remains unclear. To gain a quantitative understanding of mitochondrial Ca²⁺ homeostasis, a biophysical model of Na⁺-Ca²⁺ antiporter is introduced that is thermodynamically balanced and satisfactorily describes a number of independent data sets under a variety of experimental conditions. The model is based on a multistate catalytic binding mechanism for carrier-mediated facilitated transport and Eyring's free energy barrier theory for interconversion and electrodiffusion. The model predicts the activating effect of membrane potential on the antiporter function for a 3Na⁺:1Ca²⁺ electrogenic exchange as well as the inhibitory effects of both high and low pH seen experimentally. The model is useful for further development of mechanistic integrated models of mitochondrial Ca²⁺ handling and bioenergetics to understand the mechanisms by which Ca²⁺ plays a role in mitochondrial signaling pathways and energy metabolism.     

A Biophysically Based Mathematical Model for the Kinetics of Mitochondrial Calcium Uniporter

Ca²⁺ transport through mitochondrial Ca²⁺ uniporter is the primary Ca²⁺ uptake mechanism in respiring mitochondria. Thus, the uniporter plays a key role in regulating mitochondrial Ca²⁺. Despite the importance of mitochondrial Ca²⁺ to metabolic regulation and mitochondrial function, and to cell physiology and pathophysiology, the structure and composition of the uniporter functional unit and kinetic mechanisms associated with Ca²⁺ transport into mitochondria are still not well understood. In this study, based on available experimental data on the kinetics of Ca²⁺ transport via the uniporter, a mechanistic kinetic model of the uniporter is introduced. The model is thermodynamically balanced and satisfactorily describes a large number of independent data sets in the literature on initial or pseudo-steady-state influx rates of Ca²⁺ via the uniporter measured under a wide range of experimental conditions. The model is derived assuming a multi-state catalytic binding and Eyring's free-energy barrier theory-based transformation mechanisms associated with the carrier-mediated facilitated transport and electrodiffusion. The model is a great improvement over the previous theoretical models of mitochondrial Ca²⁺ uniporter in the literature in that it is thermodynamically balanced and matches a large number of independently published data sets on mitochondrial Ca²⁺ uptake. This theoretical model will be critical in developing mechanistic, integrated models of mitochondrial Ca²⁺ handling and bioenergetics which can be helpful in understanding the mechanisms by which Ca²⁺ plays a role in mediating signaling pathways and modulating mitochondrial energy metabolism.     

Multi-Objective Ant Colony Optimization Based on the Physarum-Inspired Mathematical Model for Bi-Objective Traveling Salesman Problems

Bi-objective Traveling Salesman Problem (bTSP) is an important field in the operations research, its solutions can be widely applied in the real world. Many researches of Multi-objective Ant Colony Optimization (MOACOs) have been proposed to solve bTSPs. However, most of MOACOs suffer premature convergence. This paper proposes an optimization strategy for MOACOs by optimizing the initialization of pheromone matrix with the prior knowledge of Physarum-inspired Mathematical Model (PMM). PMM can find the shortest route between two nodes based on the positive feedback mechanism. The optimized algorithms, named as iPM-MOACOs, can enhance the pheromone in the short paths and promote the search ability of ants. A series of experiments are conducted and experimental results show that the proposed strategy can achieve a better compromise solution than the original MOACOs for solving bTSPs.     

A Mathematical Model for Extrachromosomal Heredity

During embryogeny the components of the cytoplasm of a zygote are subject to certain irregularities. This has been investigated with the aid of some probability models and distribution functions. For mitochondria, which are the most important cell components with respect to extra-chromosomal heredity, a four-phase model with computer programs was developed. The complete arithmetical results total 15,000 pages from a fast printer. For the jobs a CPU-time of 500 hours, using a SIEMENS 4004 system, was required.     

A Mathematical Model for Bacterial Chemotaxis

A differential equation describing the chemotactic migration of a bacterial population in a fixed exponential gradient of attractant has been integrated using the appropriate boundary conditions. The solution predicts an initial bacterial accumulation at the concentration “knee” with the final distribution of bacteria approaching a time-independent state. Specific additional experiments to obtain further data for a rigorous test of the theory are suggested.     

一个人体运动的数学模型

本文采用L-E法和铰分解法研究了人体运动的数学模型问题,给出了一种人体运动的规范建模方法。该方法适用于多种拓扑结构形式的人体运动（如空翻、步行、滑行等）,所得方程具有适合于计算机程式求解的特点。     

A New Conceptual Model of Nitrogen Saturation Based on Experimental Nitrogen Addition to an Oak Forest

The dominant conceptual model of nitrogen (N) saturation in forests predicts the temporal patterns of key N cycling indicators as an initially N-limited forest is progressively enriched in N. We present the results from a long-term N addition experiment in an oak forest in southeastern New York State, USA, which do not conform to the predictions of the conceptual model in several ways. In contrast to the predictions of the conceptual model, the foliar N concentrations in the N-treated stands of our study increased to about 20% above the levels in the control stands and then remained essentially constant, and nitrogen leaching from the treated stands increased almost immediately after the start of the experiment, prior to the onset of elevated nitrification. Concentrations of N in soil solution of the N-treated stands peaked at over 150-fold greater than the concentrations in the control stands. There were no significant changes in potential net N mineralization. Tree mortality increased in the treated stands, but the tree mortality did not appear to be the primary cause of the excess nitrate leaching. Based on these results and those of other recent studies, we present a new conceptual model of the N saturation process focused on the mass balance of N rather than the temporal dynamics of N cycling indicators. The mass balance is characterized by inputs of N from atmospheric deposition and fertilization, internal sinks in the vegetation and soils, and outputs to leaching and gaseous losses. The key points of the conceptual model are (1) added N can flow simultaneously to all sinks and losses in the system, (2) the fate of the added N and the temporal patterns of flow of N depend on the strength of the sinks and the factors that control them, and (3) the movement of N to the various sinks determines how N saturation is manifested in the ecosystem. We distinguish capacity N saturation, in which the sinks in the vegetation and soil are zero or negative, from kinetic N saturation, in which the sinks are positive but lower than the N input rate. The sink strengths in the vegetation and soil have two components, one due to carbon (C) accumulation in the system and the other due to change in the stoichiometry (C:N ratio) of the pool. Further work quantifying the magnitudes and controlling factors for the N sinks will allow better prediction of the dynamics of N saturation in different types of forested ecosystems.     

害虫生物防治的数学模型

文章讨论一类捕食者（天敌）具脉冲放养与食饵（害虫）具阶段结构时滞的捕食-食饵模型,得到了害虫灭绝周期解全局吸引的充分条件和害虫的密度可以控制在经济危害水平E（EIL）之下的脉冲存放周期．为现实的害虫管理提供一定的理论依据．