A new treatment for breast cancer using a combination of two drugs: AZD9496 and palbociclib

In this study, we examined a mathematical model of breast cancer (BC) treatment that combines an oral oestrogen receptor inhibitor, AZD9496 with Palbociclib, a selective inhibitor of cyclin- dependent kinases CDK4 and CDK6. Treatment is described by analytical functions that enable us to control the dosage and time interval of the treatment, thus personalising the treatment for each patient. Initially, we investigated the effect of each treatment separately, and finally, we investigated the combination of both treatments. By applying numerical simulations, we confirmed that the combination of AZD9496 with palbociclib was the optimal treatment for BC. The dosage of AZD9496 increased and decreased throughout the treatment period, while the intervals were constant between treatments. Palbociclib changed almost cyclically, whereas the time intervals remained constant. To investigate the mathematical model, we applied the singularly perturbed homotopy analysis method, which is a numerical algorithm. The significant advantage of this method is that the mathematical model does not have to contain a small parameter (as is standard in perturbation theory). However, it is possible to artificially introduce a small parameter into the system of equations, making it possible to study the model using asymptotic methods.

A mathematical model for treatment using chemo-immunotherapy

In this study, we investigated a mathematical model for chemoimmunotherapy (a combination of chemotherapy and immunotherapy) for brain cancer. In most cases, the standard protocol for cancer treatment is fixed in terms of treatment time intervals and dosages. We offer a wide range of non-fixed protocols, which essentially vary in terms of time intervals and dosages (i.e., personalised medicine). The functions that describe this treatment are explicit and analytical. Hence, the parameters of the function can be easily changed and a new protocol can be obtained. We compared different protocols and obtained an optimal solution. In addition, we applied the singular perturbed vector field (SPVF) method to determine the hierarchy of the system of equations, which enabled us to identify the equilibrium points of the mathematical model and investigate their stability.

Combination of singularly perturbed vector field method and method of directly defining the inverse mapping applied to complex ODE system prostate cancer model

We propose a new method to solve a system of complex ordinary differential equations (ODEs) with hidden hierarchy. Given a complex system of the ODE, the hierarchy of the system is generally hidden. Once we reveal the hierarchy of the system, the system can be reduced into subsystems called slow and fast subsystems. This division of slow and fast subsystems reduces the analysis and hence reduces the computation time, which can be expensive. In our new method, we first apply the singularly perturbed vector field method that is the global quasi-linearization method. This method exposes the hierarchy of a given complex system. Subsequently, we apply a version of the homotopy analysis method called the method of directly defining the inverse mapping. We applied our new method to the immunotherapy of advanced prostate cancer.

Method of directly defining the inverse mapping applied to prostate cancer immunotherapy — Mathematical model

In this paper, we apply the method of directly defining the inverse mapping introduced by Liao and Zhao [On the method of directly defining inverse mapping for nonlinear differential equations, Numer. Algorithms 72(4) (2016) 989–1020] to the problem of prostate cancer immunotherapy. We extend this method in two directions: first, we apply the method to a system of nonlinear ordinary differential equation, and second, we propose a new technique for finding the base functions in the considered algorithm.

Israeli surrogacy law in practice

This article examines surrogacy arrangements in Israel. The Law of Surrogacy was legislated in March 1996. Surrogacy agreements are strictly limited by the criteria set forth in the law. An authorizations committee was set up by the Ministry of Health to examine each application for approval of a surrogacy agreement. Prior to filing the request, the intended parents must find an appropriate surrogate mother. Psychological evaluation of the parties is imperative in matching the parents with a surrogate mother and both sides undergo these evaluations. The surrogate mother is entitled to psychological counseling throughout the period of her attempts to conceive, the pregnancy, and for a period of 6 months after the birth. Surrogacy is a dynamic and rapidly evolving area, in both law and psychology. The constant advancements in medical technology indicate that future trends--indirectly related to surrogacy--include posthumous reproduction and reproductive cloning. Surrogacy in Israel is restricted to couples with a serious infertility problem.