A Mathematical Model of Intermittent Androgen Suppression for Prostate Cancer

For several decades, androgen suppression has been the principal modality for treatment of advanced prostate cancer. Although the androgen deprivation is initially effective, most patients experience a relapse within several years due to the proliferation of so-called androgen-independent tumor cells. Bruchovsky et al. suggested in animal models that intermittent androgen suppression (IAS) can prolong the time to relapse when compared with continuous androgen suppression (CAS). Therefore, IAS has been expected to enhance clinical efficacy in conjunction with reduction in adverse effects and improvement in quality of life of patients during off-treatment periods. This paper presents a mathematical model that describes the growth of a prostate tumor under IAS therapy based on monitoring of the serum prostate-specific antigen (PSA). By treating the cancer tumor as a mixed assembly of androgen-dependent and androgen-independent cells, we investigate the difference between CAS and IAS with respect to factors affecting an androgen-independent relapse. Numerical and bifurcation analyses show how the tumor growth and the relapse time are influenced by the net growth rate of the androgen-independent cells, a protocol of the IAS therapy, and the mutation rate from androgen-dependent cells to androgen-independent ones.

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A robust finite difference scheme for strongly coupled systems of singularly perturbed convection‐diffusion equations

This paper is devoted to developing an Il'in‐Allen‐Southwell (IAS) parameter‐uniform difference scheme on uniform meshes for solving strongly coupled systems of singularly perturbed convection‐diffusion equations whose solutions may display boundary and/or interior layers, where strong coupling means that the solution components in the system are coupled together mainly through their first derivatives. By decomposing the coefficient matrix of convection term into the Jordan canonical form, we first construct an IAS scheme for 1D systems and then extend the scheme to 2D systems by employing an alternating direction technique. The robustness of the developed IAS scheme is illustrated through a series of numerical examples, including the magnetohydrodynamic duct flow problem with a high Hartmann number. Numerical evidence indicates that the IAS scheme appears to be formally second‐order accurate in the sense that it is second‐order convergent when the perturbation parameter ϵ is not too small and when ϵ is sufficiently small, the scheme is first‐order convergent in the discrete maximum norm uniformly in ϵ.     

Solitary solutions to an androgen-deprivation prostate cancer treatment model

A model of androgen deprivation treatment for prostate cancer is considered in this paper. Bright/dark solitary solutions to the model are constructed using inverse balancing and generalized differential operator techniques. It is shown that solitary solutions correspond to biomedically relevant sets of model parameters. Dynamical properties of solitary solutions are analyzed in the phase plane. It is demonstrated that such solutions closely reflect the real-world phenomena observed during androgen deprivation treatment. Computational experiments are used to illustrate these effects.     

MIP Models and BB Strategies in Brachytherapy Treatment Optimization

Brachytherapy (brachy being derived from a Greek word meaning short) is the treatment by means of radioactive sources that are placed at short distances from the target cells. This form of therapy is becoming common in the treatment of early stage prostate cancer, the most common cancer and the second leading cause of cancer deaths among American males. We consider the use of mixed-integer programming (MIP) models and branch-and-bound (BB) methods to optimize the placement within the prostate of the radioactive seeds used in this procedure. Several different optimization models are considered along with a number of branchand- bound strategies. With appropriate combinations of modelling and solution strategies, nearoptimal seed placements can be generated for each two-dimensional ultrasound section of the prostate in less than five minutes on a 333-MHz workstation. The original three-dimensional problem can then be solved by considering an appropriately interrelated sequence of these two-dimensional problems.     

Using eigenstructure of the Hessian to reduce the dimension of the intensity modulated radiation therapy optimization problem

Optimization is of vital importance when performing intensity modulated radiation therapy to treat cancer tumors. The optimization problem is typically large-scale with a nonlinear objective function and bounds on the variables, and we solve it using a quasi-Newton sequential quadratic programming method. This study investigates the effect on the optimal solution, and hence treatment outcome, when solving an approximate optimization problem of lower dimension. Through a spectral decompostion, eigenvectors and eigenvalues of an approximation to the Hessian are computed. An approximate optimization problem of reduced dimension is formulated by introducing eigenvector weights as optimization parameters, where only eigenvectors corresponding to large eigenvalues are included. The approach is evaluated on a clinical prostate case. Compared to bixel weight optimization, eigenvector weight optimization with few parameters results in faster initial decline in the objective function, but with inferior final solution. Another approach, which combines eigenvector weights and bixel weights as variables, gives lower final objective values than what bixel weight optimization does. However, this advantage comes at the expense of the pre-computational time for the spectral decomposition. A preliminary version of this paper was presented at the AAPM 46th annual meeting, held July 25–29, 2004 in Pittsburgh, PA.     

Statistics of a parametric oscillator in a thermal reservoir

A parametric Bose oscillator evolving from a state in the form of a signal-noise superposition is considered. The region of values of its frequency, the temperature, the optical pumping, and the time for which the occupation numbers satisfy sub-Poisson statistics is found. It is noted that the sub-Poisson effect is delayed by thermal fluctuations in the long-wavelength region and suppressed by quantum fluctuations in the short-wavelength region, the suppression taking the form of a frequency threshold condition at low temperatures.Joint Institute for Nuclear Research, Dubna. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 92, No. 1, pp. 158–165, July, 1992.     

A Mathematical Model of Prostate Tumor Growth Under Hormone Therapy with Mutation Inhibitor

This paper extends Jackson’s model describing the growth of a prostate tumor with hormone therapy to a new one with hypothetical mutation inhibitors. The new model not only considers the mutation by which androgen-dependent (AD) tumor cells mutate into androgen-independent (AI) ones but also introduces inhibition which is assumed to change the mutation rate. The tumor consists of two types of cells (AD and AI) whose proliferation and apoptosis rates are functions of androgen concentration. The mathematical model represents a free-boundary problem for a nonlinear system of parabolic equations, which describe the evolution of the populations of the above two types of tumor cells. The tumor surface is a free boundary, whose velocity is equal to the cell’s velocity there. Global existence and uniqueness of solutions of this model is proved. Furthermore, explicit formulae of tumor volume at any time t are found in androgen-deprived environment under the assumption of radial symmetry, and therefore the dynamics of tumor growth under androgen-deprived therapy could be predicted by these formulae. Qualitative analysis and numerical simulation show that controlling the mutation may improve the effect of hormone therapy or delay a tumor relapse.     

支持向量机在临床疾病诊断中的应用

基于支持向量机的最优二分类方法,以癌症诊断为例,构建了疾病诊断的支持向量机模型.对50例非癌症患者和100例癌症患者的腺苷三磷酸酶(ATP酶)和琥珀酸脱氢酶(SDH酶)活性两项指标分组进行训练和仿真诊断,检测样本的诊断正确率为98.03%,故可以用支持向量机建立临床疾病诊断系统.     

Heuristics for integrated optimization of catheter positioning and dwell time distribution in prostate HDR brachytherapy

High dose-rate (HDR) brachytherapy is one kind of radiotherapy used to treat different forms of cancer, such as prostate cancer. When this treatment is used for prostate cancer, a radioactive source is moved through catheters implanted into the prostate. For each patient, a treatment plan is constructed. This plan determines for example catheter positions and dwell time distribution, that is, where to the radioactive source should stop and for how long.     

Comparing the efficiency of Wolbachia driven Aedes mosquito suppression strategies

Wolbachia is an endosymbiotic bacterium which manipulates host reproduction by cytoplasmic incompatibility, and restrains the transmission of dengue virus in Aedes mosquitoes. A novel strategy for dengue control involves releasing Wolbachia infected males into nature to suppress wild Aedes mosquito population. We develop a model of delay differential equations, integrating larval density-dependent competition and diapausing eggs, to compare the efficiency of different suppression strategies. The global asymptotical stability of the complete suppression state identifies the releasing amount threshold ascertaining suppression. Based on the experimental data for Aedes albopictus population in Guangzhou, our simulations show that the mosquito density in the highest peak season can be reduced by more than $95\%$ when the number of released males is above the releasing threshold. The best time to initiate the suppression is in early March, lasting until the end of June, followed by the parallel releasing policy from July to November. However, the egg bank has neglectable effects on the control of dengue vector in Guangzhou.     

Dynamic optimal strategy for monitoring disease recurrence

Surveillance to detect cancer recurrence is an important part of care for cancer survivors.In this paper we discuss the design of optimal strategies for early detection of disease recurrence based on each patient’s distinct biomarker trajectory and periodically updated risk estimated in the setting of a prospective cohort study.We adopt a latent class joint model which considers a longitudinal biomarker process and an event process jointly,to address heterogeneity of patients and disease,to discover distinct biomarker trajectory patterns,to classify patients into different risk groups,and to predict the risk of disease recurrence.The model is used to develop a monitoring strategy that dynamically modifies the monitoring intervals according to patients’ current risk derived from periodically updated biomarker measurements and other indicators of disease spread.The optimal biomarker assessment time is derived using a utility function.We develop an algorithm to apply the proposed strategy to monitoring of new patients after initial treatment.We illustrate the models and the derivation of the optimal strategy using simulated data from monitoring prostate cancer recurrence over a 5-year period.     

Indexing Distributions of Data: Preservice Teachers' Notions of Representativeness

The purpose of the study was to identify strategies used by preservice elementary teachers to index distributions of data. A secondary purpose of the study was to investigate whether representational form influenced the type of representative values constructed. Two hundred eighty‐three preservice teachers were administered a selection of tasks requiring them to construct representative values for a variety of distributional shapes. Participants were asked to provide justifications for their chosen representative measures. Data were analyzed from two perspectives: nature and adequacy of representative measure. Results indicated that when presented with skewed data sets, over half of the participants constructed poorly representative values. Symmetric data sets were best indexed with 67% of participants constructing highly representative values. Representational form played a significant role in determining representative values. Data presented graphically were more likely to be represented using measures of variability than were those presented nongraphically. Conversely, measures of central tendency were more frequently used to represent nongraphical data. The information arising from this study indicates that many preservice teachers construct poorly representative values to index data sets and utilize identical measures of representativeness regardless of distributional shape. The findings of the study highlight the need for exploratory data activities at the preservice level, call for provision of experiences affording insights into construction of representative values, and thus, have implications for the design of preservice mathematics education curricula.     

Fluid models for many-server Markovian queues in a changing environment

Motivated by service systems with time-varying customer arrivals, we consider a fluid model as a macroscopic approximation for many-server Markovian queues alternating between underloaded and overloaded intervals. Our main result is a refinement of the piecewise stationary approximation (PSA) for the stationary distribution of the fluid model. The form of the refined approximation suggests simple metrics for assessing the accuracy of PSA for underloaded and overloaded intervals respectively.     

公路货运平台延迟赔付服务定价策略研究

考虑延迟赔付率、车主延迟风险、货主物流时效关注度的影响下,研究两种延迟赔付定价策略的选择问题,通过逆向求解法获取最优均衡。发现:(1)车主网络外部性大于货主两倍的情况,平台始终选择C策略。反之,若车主延迟风险低于某临界值、货主对物流时效的关注度高于某临界值,应选F策略;否则选C策略。(2)无论F策略还是C策略,平台收益均随车主延迟风险的降低、货主对时效性关注度的提升而提升,不受赔付率影响。(3)若选F策略,则不对货主补贴。若选C策略,车主的网络外部性大于货主的情况下始终实行补贴;反之,只有当车主延迟风险、货主物流时效的关注度满足条件时进行补贴。最后,用数值仿真结论进行验证。     

Finite element analysis of evaporative cutting with a moving high energy pulsed laser

A computational model for simulation of pulsed laser-cutting process has been developed using a finite element method. An unsteady heat transfer model is considered that deals with the material-cutting process using a Gaussian wave laser beam in a pulsed mode. An iterative scheme is used to handle the geometric nonlinearity due to the melting region. The convergence study with mesh refinements and time steps first identifies optimal mesh and time step for the present analyses. Numerical analyses are carried out on the amount of material removal and groove smoothness with laser power (LP) and number of pulses (NPS) while other laser cutting parameters are fixed. The results show that there exist threshold values in number of pulses and laser power in order to achieve two predetermined conditions: (1) amount of material removal and (2) smoothness of groove shape. These values form an envelope called threshold curve that separate the acceptable region from unacceptable one for quality pulsed laser cutting. The effect of velocity also leads to another threshold curve which is determined from both number of pulses and velocity. Finally, the convergence of results in error domain is shown oscillating due to geometric nonlinearity.     

A three-phase algorithm for the cell suppression problem in two-dimensional statistical tables

To obtain full cooperation from respondents, statistical offices must guarantee that confidential data will not be disclosed when their reports are published. For tabular data, cell suppression is one of the preferred techniques to control statistical disclosure. When suppressing only confidential values does not guarantee the desired data protection, it is also necessary to suppress the values in some non-confidential cells. The problem of finding an optimal set of complementary suppressions—the cell suppression problem (CSP)—is NP-hard. We present a three-phase algorithm for the CSP based on a binary relaxation derived from row and column protection conditions. To enforce violated single cell conditions, integer cuts are added to the CSP relaxation. The numerical results obtained in 1410 instances with up to more than 250?000 cells, which were generated to reproduce two classes of real-world data, indicate that the algorithm is quite effective for both classes of instances and that it outperforms state-of-the-art algorithms for one of them.     

按时滞转化的阶段结构SIS传染病模型

对一类按时滞转化的具有两个阶段结构的SIS传染病模型进行了分析,得到了传染病最终消除和成为地方病的阈值.即当传染率小于该阀值时,传染病最终消除;反之,此种传染病将成为地方病.     

单次重复试验的三水平析因设计的分析

由于试验材料、费用和时间等条件的限制,仅有单次重复试验的三水平析因设计经常要应用在农业、工业和医学临床试验等领域。例如,在医学临床试验中,为找到影响治疗关节炎效果的重要因子和最佳治疗方案需要考虑2个三水平的因子:A(药物治疗)和B(运动治疗),由于只能找到9位病情相似的病人进行试验,故只能实施仅有单次重复试验的三水平析因设计3~2。不幸的是,交互作用A×B也可能存在,这样就没有剩余自由度用于估计误差的方差,从而通常的方差分析方法不再能用于数据分析。针对上述问题,本文提出了三个基于均方误差的检验统计量用于分析单次重复试验的三水平析因设计。通过实例表明用这些方法不仅能检验所考虑因子的主效应,而且还能同时检验交互效应。相应检验所用的一些常用临界值提供在附录中。并且,还通过大量的模拟研究对所提出的三个检验方法进行了比较。结果显示,T_~((3))检验在三个检验方法中具有最大的功效。     

Nanoscale dendritic organoplatinum antitumor drugs

Organoplatinum compounds exhibit broad antitumor activity. Cisplatin [cis- dichlorodiammine platinum (II)] is the most prominent of this family and is currently the most prescribed cancer drug. It is widely used for the treatment of human testicular, ovarian, bladder, head and neck carcinomas. However, it is an extremely toxic compound and side effects such as nephrotoxicity and myelotoxicity are major drawbacks for its use in clinical applications. A way to combat the side effects associated with the administration of platinum drugs would be to generate a prodrug conjugate that, when present in the extracellular fluid, would slowly release active platinum species at a level below the threshold for side effects. Traditionally, linear hydrophilic polymers have been examined for potential application as carrier platforms for these drugs. The advent of dendrimers, which are highly branched macromolecules with precisely controlled nanoscale size, shape and end-group functionality, has provided an excellent opportunity to design and formulate novel multivalent time-release drug systems. For example, treatment of a generation 4.5 (carboxylate surface) poly(amidoamine) [PAMAM] dendrimer with diaquo(1,2-diaminocyclohexane) platinum(II) generates a nanoparticle with forty platinum moieties attached at the surface. This polymer-drug conjugate displays a very good release profile for the platinum species. Under physiological conditions about 80% of the platinum units are released over the first 24 hours. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)