Aims

The aim was to characterize the population pharmacokinetics of BYL719 in cancer patients and assess the time course of tumour response in relation to drug exposure and dosing schedule.



Methods

Plasma samples and longitudinal tumour size measurements were collected from 60 patients with advanced solid malignancies who received oral BYL719 once daily (30–450 mg) or twice daily at 120 mg or 200 mg. Non-linear mixed effect modelling was employed to develop the population pharmacokinetic and pharmacodynamic model.



Results

The pharmacokinetics were best described by a one compartment disposition model and transit compartments accounting for the lag time in absorption. The typical population oral clearance and volume of distribution estimates with their between-subject variability (BSV) were 10 l h−1 (BSV 26%) and 108 l (BSV 28%), respectively. The estimated optimal number of transit compartments was 8.1, with a mean transit time to the absorption compartment of 1.28 h (BSV 32%). The between-occasion variability in the rate and extent of absorption was 46% and 26%, respectively. Tumour growth was modelled using a turnover model characterized by a zero order growth rate of 0.581 cm week1 and a first order death rate of 0.0123 week−1. BYL719 inhibited tumour growth with an IC50 of 100 ng ml−1 (BSV 154%). Model-based predictions showed potential for additional anti-tumour activity of twice daily dosing at total daily dose below 400 mg, but a loss of efficacy if administered less frequently than once daily.



Conclusions

The proposed model provides a valuable approach for planning future clinical studies and for designing optimized dosing regimens with BYL719.

https://bpspubs.onlinelibrary.wiley.com/doi/pdf/10.1111/bcp.12378

Population pharmacokinetics and pharmacodynamics of BYL719, a phosphoinositide 3-kinase antagonist, in adult patients with advanced solid malignancies

The McKendrick/Von Foerster equation is a transport equation with a non-local boundary condition that appears
frequently in structured population models. A variant of this equation with a size structure has been
proposed as a metastatic growth model by Iwata et al. &nbsp
Here we will show how a family of metastatic models with 1D or 2D structuring variables, based on the Iwata model, can be
reformulated into an integral equation counterpart, a Volterra equation of convolution type, for which a rich numerical and
analytical theory exists. Furthermore, we will point out the potential of this reformulation by addressing questions coming up in
the modelling of metastatic tumour growth. We will show how this approach permits to reduce the computational cost
of the numerical resolution and to prove structural identifiability.

https://hal.archives-ouvertes.fr/hal-00935233/document

Efficient resolution of metastatic tumor growth models by reformulation into integral equations

In this thesis we develop minimal models of the relationship between motility, growth, and evolution of cancer cells. We utilise simple simulations of a population of individual cells in space to examine how changes in mechanical properties of invasive cells and their surroundings can affect the speed of cell migration. We also find that the growth rate of large lesions depends weakly on the migration speed of escaping cells, and has stronger and more complex dependencies on the rates of other stochastic processes in the model, namely the rate at which cells transition to being motile and the reverse rate at which cells cease to be motile. 
To examine how the rates of growth and evolution of an ensemble of cancerous lesions depends on their geometry and underlying fitness landscape, we develop an analytical framework in which the spatial structure is coarse grained and the cancer treated as a continuously growing system with stochastic migration events. Both approaches conclude that the whole ensemble can undergo migration-driven exponential growth regardless of the dependence of size on time of individual lesions, and that the relationship between growth rate and rate of migration is determined by the geometrical constraints of individual lesions. We also find that linear fitness landscapes result in faster-than-exponential growth of the ensemble, and we can determine the expected number of driver mutations present in several important cases of the model. 
Finally, we study data from a clinical study of the effectiveness of a new low-dose combined chemotherapy. This enables us to test some important hypotheses about the growth rate of pancreatic cancers and the speed with which evolution occurs in reality. Despite this, we find that the frequency of resistant mutants is far too high to be explained without resorting to novel mechanisms of cross-resistance to multiple drugs.

Minimal models of invasion and clonal selection in cancer

This paper attempts to present and employ Runge-Kutta Method of fifth-order (RK5) and New Iterative Method for the numerical solution of metastatic cancer model which occur in two compartments of cancer environment. These methods have been proved to be powerful mathematical tools for various phenomena in biomathematics and it is extremely effective for linear and non-linear systems of differential equations. Our numerical experiments illustrate the effect of parameters β!, β" and β# on cancer models which are responsible for the spread or reduction of cancer cells through the boundary of an organ tissue. The results obtained are compared with analytical solutions and show that (RK5) and NIM are powerful numerical techniques to solve systems of differential equations. Finally, all computations and algorithms are implemented using MAPLE 18 software version.

Numerical comparison of runge-kutta (rk5) and new iterative method (nim) for solving metastatic cancer model

We develop a heuristic model embedding Kleiber and Murray laws to describe mass growth, metastasis and vascularization in cancer. We analyze the relevant dynamics using different evolution equations (Verhulst, Gompertz and others). Their extension to reaction diffusion equation of the Fisher type is then used to describe the relevant metastatic spreading in space. Regarding this last point, we suggest that cancer diffusion may be regulated by Levy flights mechanisms and discuss the possibility that the associated reaction diffusion equations are of the fractional type, with the fractional coefficient being determined by the fractal nature of the capillary evolution.

Capillary Network, Cancer and Kleiber Law

Analyzing the effectiveness of different treatments for cancer typically involves collecting data from large clinical studies or animal testing. Such testing is always needed as a final validation for establishing treatment safety, but can be very costly and labor-intensive. Developing alternative testing approaches to be used in preliminary stages of evaluating new treatment strategies would be a great aid in speeding up research and development. We consider the use of mathematical models to describe the progression of cancer and how the influence of anti-cancer drugs can be incorporated into these models. There are many different forms of cancer, but several types share similar mechanisms for how they start and spread. The basic understanding of metastatic cancer consists of the following general stages:

http://eaton.math.rpi.edu/Faculty/Schwendeman/Workshop/MPIReports/2011/Novartis11.pdf

Dynamic models of metastatic tumor growth

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Dynamic models of metastatic tumor growth