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 in vivo 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 vs. 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.Author SummaryCancer 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.
We consider stabilization for a linear ordinary differential equation system with input dynamics governed by a heat equation, subject to boundary control matched disturbance. The active disturbance rejection control (ADRC) approach is applied to estimate, in real time, the disturbance with both constant high gain and timevarying high gain. The disturbance is canceled in the feedback loop. The close loop systems with constant high gain and time-varying high gain are shown respectively, to be practically stable and asymptotically stable.
Implementations of the heat balance integral method are discussed in which exponential functions are used in place of the familiar polynomial approximants. The rationale is based upon that of least-squares in that the use of 'appropriate' basis functions can enhance solution accuracy. Whilst this is true in principle it is shown that considerable skill must be exercised when deviating from polynomial approximants. The discussions are illustrated by application to a familiar single-phase Stefan problem that is typical of heat transfer problems exhibiting decay-like spatial solution profiles.
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.
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