A Matrix Iteration for Finding Drazin Inverse with Ninth-Order Convergence

The aim of this paper is twofold. First, a matrix iteration for finding approximate inverses of nonsingular square matrices is constructed. Second, how the new method could be applied for computing the Drazin inverse is discussed. It is theoretically proven that the contributed method possesses the convergence rate nine. Numerical studies are brought forward to support the analytical parts

SPECIFIC INTERNAL ENERGY OF RELATIVISTIC RANKINE-HUGONIOT EQUATIONS

Abstract – The stress energy tensor and the mean velocity vector of a simple gas are expressed in terms of the Maxwell-Boltzman distribution function. The rest density 0 � , pressure, �, and internal energy per unit rest mass � are defined in terms of invariants formed from these tensor 0 quantities. It is shown that � cannot be an arbitrary function of �and � but must satisfy a 0 certain inequality. Thus � � ( 1

Four-Point Optimal Sixteenth-Order Iterative Method for Solving Nonlinear Equations

We present an iterative method for solving nonlinear equations. The proposed iterative method has optimal order of convergence sixteen in the sense of Kung-Traub conjecture (Kung and Traub, 1974); it means that the iterative scheme uses five functional evaluations to achieve 16(=25-1) order of convergence. The proposed iterative method utilizes one derivative and four function evaluations. Numerical experiments are made to demonstrate the convergence and validation of the iterative method

A Family of Iterative Methods for Solving Systems of Nonlinear Equations Having Unknown Multiplicity

The singularity of Jacobian happens when we are looking for a root, with multiplicity greater than one, of a system of nonlinear equations. The purpose of this article is two-fold. Firstly, we will present a modification of an existing method that computes roots with known multiplicities. Secondly, will propose the generalization of a family of methods for solving nonlinear equations with unknown multiplicities, to the system of nonlinear equations. The inclusion of a nonzero multi-variable auxiliary function is the key idea. Different choices of the auxiliary function give different families of the iterative method to find roots with unknown multiplicities. Few illustrative numerical experiments and a critical discussion end the paper

A family of iterative methods for solving systems of nonlinear equations having unknown multiplicity

The singularity of Jacobian happens when we are looking for a root, with multiplicity greater than one, of a system of nonlinear equations. The purpose of this article is two-fold. Firstly, we will present a modification of an existing method that computes roots with known multiplicities. Secondly, will propose the generalization of a family of methods for solving nonlinear equations with unknown multiplicities, to the system of nonlinear equations. The inclusion of a nonzero multi-variable auxiliary function is the key idea. Different choices of the auxiliary function give different families of the iterative method to find roots with unknown multiplicities. Few illustrative numerical experiments and a critical discussion end the paper

Predictive Modeling of Drug Response in Non-Hodgkin's Lymphoma.

We combine mathematical modeling with experiments in living mice to quantify the relative roles of intrinsic cellular vs. tissue-scale physiological contributors to chemotherapy drug resistance, which are difficult to understand solely through experimentation. Experiments in cell culture and in mice with drug-sensitive (Eµ-myc/Arf-/-) and drug-resistant (Eµ-myc/p53-/-) lymphoma cell lines were conducted to calibrate and validate a mechanistic mathematical model. Inputs to inform the model include tumor drug transport characteristics, such as blood volume fraction, average geometric mean blood vessel radius, drug diffusion penetration distance, and drug response in cell culture. Model results show that the drug response in mice, represented by the fraction of dead tumor volume, can be reliably predicted from these inputs. Hence, a proof-of-principle for predictive quantification of lymphoma drug therapy was established based on both cellular and tissue-scale physiological contributions. We further demonstrate that, if the in vitro cytotoxic response of a specific cancer cell line under chemotherapy is known, the model is then able to predict the treatment efficacy in vivo. Lastly, tissue blood volume fraction was determined to be the most sensitive model parameter and a primary contributor to drug resistance

Theory and Experimental Validation of a Spatio-temporal Model of Chemotherapy Transport to Enhance Tumor Cell Kill

<div><p>It has been hypothesized that continuously releasing drug molecules into the tumor over an extended period of time may significantly improve the chemotherapeutic efficacy by overcoming physical transport limitations of conventional bolus drug treatment. In this paper, we present a generalized space- and time-dependent mathematical model of drug transport and drug-cell interactions to quantitatively formulate this hypothesis. Model parameters describe: perfusion and tissue architecture (blood volume fraction and blood vessel radius); diffusion penetration distance of drug (i.e., a function of tissue compactness and drug uptake rates by tumor cells); and cell death rates (as function of history of drug uptake). We performed preliminary testing and validation of the mathematical model using <i>in vivo</i> experiments with different drug delivery methods on a breast cancer mouse model. Experimental data demonstrated a 3-fold increase in response using nano-vectored drug <i>vs</i>. free drug delivery, in excellent quantitative agreement with the model predictions. Our model results implicate that therapeutically targeting blood volume fraction, e.g., through vascular normalization, would achieve a better outcome due to enhanced drug delivery.</p><p>Author Summary</p><p>Cancer treatment efficacy can be significantly enhanced through the elution of drug from nano-carriers that can temporarily stay in the tumor vasculature. Here we present a relatively simple yet powerful mathematical model that accounts for both spatial and temporal heterogeneities of drug dosing to help explain, examine, and prove this concept. We find that the delivery of systemic chemotherapy through a certain form of nano-carriers would have enhanced tumor kill by a factor of 2 to 4 over the standard therapy that the patients actually received. We also find that targeting blood volume fraction (a parameter of the model) through vascular normalization can achieve more effective drug delivery and tumor kill. More importantly, this model only requires a limited number of parameters which can all be readily assessed from standard clinical diagnostic measurements (e.g., histopathology and CT). This addresses an important challenge in current translational research and justifies further development of the model towards clinical translation.</p></div

Parameter calibration from patient data demonstrates model predictivity.

<p>(<i>A</i>) Nonlinear regression analysis of Eq. S2 (coefficient of determination <i>R</i>² = 0.86) to the measurements of kill fraction and blood volume fraction BVF from histopathology images of 21 patients with CRC metastatic to liver (standard deviations reflect variability of measured values across 20 slides per patient). Inset: parameter values obtained from fit. (<i>B</i>) Linear regression analysis of Hounsfield Unit measurements from pre-treatment arterial-phase contrast-enhanced CT data from 18 patients and blood volume fraction (BVF) measurements from histopathology leads to calibration of BVF parameter (inset). (<i>C</i>) Side-by-side boxplots of <i>f</i>_kill values measured from histopathology and predicted by mathematical model Eq. S2 based on calibration in <i>A</i> and <i>B</i> (18 data points in each set, symbols). In each boxplot, the thick horizontal line is the median; the box is defined by the 25th and 75th percentiles (lower and upper quartile); the diamond is the mean. A paired t-test at the 0.05 significance level resulted in <i>P</i> = 0.44, indicating that the observed difference between the two data sets is not significant. (<i>D</i>) Predictions of Eq. S2 (open circles, average relative error ≈ 24%) compared, for each patient, to the direct measurements from histopathology post-treatment and resection (filled circles, with standard deviation of multiple measurements per patient).</p

Numerical simulations of the general integro-differential model (Eqs 6 and 7) in a cylindrically symmetric domain.

<p>As cell kill ensues over several cell cycles, (<i>A</i>) successive cell layers next to the blood vessel (<i>r</i> = <i>r</i>_b) die out, i.e., tumor volume fraction <i>φ</i> decreases; (<i>B</i>) local drug concentration <i>σ</i> increases due to an enhancement of drug penetration; and (<i>C</i>) cell kill accelerates further from the vessel and deep into the tumor. Input parameters: <i>r</i>_b / <i>L</i> = 0.102 and BVF = 0.01. The duration of the entire simulation was 10 (<i>λ</i>_<i>k</i><i>λ</i>_<i>u</i><i>φ</i>₀<i>σ</i>₀)^−1/2, where time unit is a characteristic cell apoptosis time. Drug concentration and tumor volume fraction were normalized by their initial values, and radial distance by the diffusion penetration distance <i>L</i>. The fraction of tumor kill <i>f</i>_kill is calculated from <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004969#pcbi.1004969.e012" target="_blank">Eq 12</a> (<b>Methods</b>).</p

Testing the efficacy of drug-loaded nano-carriers in mice.

<p>Comparison of fraction of tumor killed measured across three different treatment BALB/c mice groups (n = 10 per group) over a period of 17 days (from day 14 to day 31 after 4T1 tumor cell inoculation, see <b><a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004969#sec003" target="_blank">Methods</a></b>) showing a roughly 3-fold increase in kill from nano-vectored drug vs. free drug. At each time point, tumor volume measurements from the three drug treatment groups were first normalized to the measurement from the control (PBS) group (no drug treatment), and then to the initial tumor volume for each group; <i>f</i>_kill was then calculated as (1 –normalized tumor volume).</p