INTRODUCTION:

In the fields of medicine, biotechnology and pharmacology, drug discovery is the process by which new candidate medications are discovered. Drug discovery is a complex undertaking facing many challenges, not the least of which is a high attrition rate as many promising candidates prove ineffective or toxic in the clinic owing to a poor understanding of the diseases, and thus the biological systems, they target. Therefore, it is broadly agreed that to increase agreed that to increase the productivity of drug discovery one needs a far deeper understanding of the molecular mechanisms of diseases, taking into account the full biological context of the drug target and moving beyond individual genes and proteins [1]. Mathematical methods are increasingly being used in drug discovery to enquire into biological systems, with a view to understanding the behavior in a more holistic way.

Present difficulties in drug development include an increase in cost and duration of drug development, and only few new medical entities reach approval. It takes from 10 to 15 years to bring a new drug to market — at a cost of more than $1 billion. Many new potential drugs fail because researchers lack reliable information about their behavior. That leads to problems for both pharma industry and public health. Moreover, one can observe some lack of interest of drug pharma for some disease areas due to high potential costs of research. Mathematical model based approaches also been suggested to expand the use of simulations in support of clinical drug development for predicting outcomes of planned trials [2].

PRELIMINARIES:

Drugs:

A drug is any substance (other than food that provides nutritional support) that, when inhaled, injected, smoked, consumed, absorbed via a patch on the skin, or dissolved under the tongue causes a physiological change in the body [3].

Drug Discovery:

Drug discovery is the process through which potential new medicines are identified. It involves a wide range of scientific disciplines, including biology, chemistry, and pharmacology [4]. Drug discovery cycle is shown in Snapshot – 1 [5].

Snapshot – 1

Drug Development:

Drug development is the process of bringing a new pharmaceutical drug to the market once a lead compound has been identified through the process of drug discovery [ 6 ].

Mathematical model:

A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling [ 7 ].

SURVEY:

MATHEMATICAL MODELS IN DRUG DISCOVERY AND DEVELOPMENT:

Princeton researchers, Christodoulos Floudas and Meghan Bellows Peterson, have developed a way to use mathematical models to take some of the guesswork out of discovering new drugs. Using the technique, they have identified several potential new drugs for fighting HIV. Snapshot – 2 shows a graphic of their drug candidate (red) attached to HIV (blue).

Snapshot – 2

Using mathematical concepts, they have developed a method of discovering new drugs for a range of diseases by calculating which physical properties of biological molecules may predict their effectiveness as medicines [8].

Mathematical models of the dynamics of a drug within the host are now frequently used to guide drug development. In [ 9 ], Hannah C. Slater et al argued that integrating within–host pharmacokinetic and pharmacodynamic models with mathematical models for the population– level transmission of malaria is key to guiding optimal drug design to aid malaria elimination. Mathematical models are frequently employed to guide product development. These models incorporate two factors. The first is the pharmacokinetics of the compound: how the drug concentration increases and decays over time as determined by its absorption, distribution, metabolism, and excretion. This is typically described by a set of differential equations, broadly representing the physical compartments where these different effects take place. The second is the pharmacodynamics, describing the relationship between the drug concentration and its killing efficacy. This is summarized as a ‘dose response curve’ showing the efficacy as a function of the measured concentration in the blood. By combining these two models, predictions can be made of the likely efficacy and its decay over time for different dosing schedules. Such approaches are increasingly being evaluated to inform anti malarial drug development, including optimizing the dosing schedule and to explore the impact of combinations of therapies. In [10], L.G. De Pillis et al have presented a competition model of cancer tumor growth that includes both the immune system response and drug therapy. This is a four–population model that includes tumor cells, host cells, immune cells, and drug interaction. Also they analyzed the stability of the drug–free equilibria with respect to the immune response in order to look for target basins of attraction. Using optimal control theory with constraints and numerical simulations, they obtained new therapy protocols that they then compare with traditional pulsed periodic treatment. The optimal control generated therapies produce larger oscillations in the tumor population over time. However, by the end of the treatment period, total tumor size is smaller than that achieved through traditional pulsed therapy, and the normal cell population suffers nearly no oscillations.

The design of a mathematical model of a biological system is governed by the need to distill the essential behavior of the system and the need to answer specific questions about that system. The model they have developed, which is built from combining some of the most useful aspects of previously existing models, does in fact exhibit the qualitative behavior we wished to reproduce, including jeffs phenomenon and tumor dormancy.

In [2], Yulia Balykina has presented modern approaches to the research and development of new drugs based on pharmacodynamics and pharmacokinetics modeling. The basic concepts are considered. Software tools for such kind of modeling under large amounts of data are described.

In [11], Christy Chuang – Steinet et al have covered some traditional statistical support to show how statistics has been used in many aspects of drug development. They also highlighted some areas where a statistician’s contribution will be crucial in moving forward, in view of the FDA’s (Food and Drug Administration) Critical Path initiative and the pharmaceutical industry’s collective effort to take advantage of the FDA’s call for innovation.

In [12] Orit Lavia et al have reviewed the aspects of multidrug resistance ( MDR ) that have been mathematically studied, and explains how, from a methodological perspective, mathematics can be used to study drug resistance. They also discussed the quantitative approaches of mathematical analysis, and demonstrated how mathematics can be used in combination with other experimental and clinical tools. They emphasized the potential benefits of integrating analytical and mathematical methods into future clinical and experimental studies of drug resistance.

In [13], David Orrell et al have developed the Mathematical Models to optimize the scheduling of anti – cancer drugs. An advantage of this approach is that we can quickly simulate thousands of possible schedules for combinations of different drugs. This allows our partners to prioritize the most effective drug combinations and the best schedules for validation in vivo.

MATHEMATICAL MODELING IN VARIOUS DISEASE THERAPIES:

In [14], Jean Clairambault has presented the mathematical models that have been designed in the frame of continuous deterministic cell population dynamics that aim at optimization of cancer treatments using chronotherapeutics.

In [15] Ivana Ilic has presented some new possibilities of using mathematics in medical sciences and medical practice. The so – called mathematical way of thinking is presented, which is, basically, the starting point of usual way of thinking, and also the essence of scientific and technological literacy, necessary for the development of theoretical and practical modern medicine.

In [16], Urszula Ledzewicz et al have formulated the problem of how to transfer the system from an initial condition in the malignant region of the state space through therapy into a benign region as an optimal control problem based on stepanova’s mathematical model of immunological activity during cancer growth. Also they were considered the combinations of a chemotherapeutic agent and an immune boost and both qualitative information about the structure of optimal controls and some quantitative illustrations of these solutions were given.

In [17], Jean Clairambault has listed the most necessary mathematical methods, enforcing already existing methods, should be further developed towards designing and applying optimized individualized treatments of cancer in the clinic.

Ordinary differential equation based models are useful in cancer biology to study how biological systems change over time. Ordinary differential equations (ODE), and mathematical models in general, can support experimental findings and lead to new avenues of scientific discovery.

In [ 18 ], Margaret P. Chapman et al have provided an overview of the ODE modeling framework, and presented examples of how ODEs can be used to address problems in cancer biology.

In [19], Mihai Ilea et al proposed an original mathematical model with small parameter for the interactions between these two cancer cell sub – populations and the mathematical model of a vascular tumor. The graphical output for a mathematical model of a vascular tumor shows the differences in the evolution of the tumor populations of proliferating, quiescent and necrotic cells.

In [20], De Pillis LG et al presented a new mathematical model that describes tumor –immune interactions, focusing on the role of natural killer and CD8+ T cells in tumor surveillance, with the goal of understanding the dynamics of immune – mediated tumor rejection. The model describes tumor – immune cell interactions using a system of differential equations. The functions describing tumor – immune growth, response, and interaction rates, as well as associated variables, are developed using a least–squares method combined with a numerical differential equations solver. In [21], A. G .Lopez et al presented a dynamical model of cancer growth including three interacting cell populations of tumor cells, healthy host cells and immune effectors cells. The tumor–immune and the tumor – host interactions are characterized to reproduce experimental results. A thorough dynamical analysis of the model is carried out, showing its capability to explain theoretical and empirical knowledge about tumor development. A chemotherapy treatment reproducing different experiments is also introduced. This simple model can serve as a foundation for the development of more complicated and specific cancer models.

In [22], Carlo Bianca et al presented an improved version of a mathematical model related to the competition between immune system cells and mammary carcinoma cells under the action of a vaccine (Triplex). The model describes in detail both the humoral and cellular response of the immune system to the tumor associate antigen and the recognition process between B cells, T cells and antigen presenting cells.

In [23], Mark Chaplain presented a review of a variety of mathematical models which have been used to describe the formation of apillary networks and then focus on a specific recent model which uses novel mathematical modelling techniques to generate both 2 and 3 dimensional vascular structures. The modelling focusses on key events of angiogenesis such as the migratory response of endothelial cells to exogenous cytokines (tumour angiogenic factors, TAF) secreted by a solid tumour, endothelial cell proliferation, endothelial cell interactions with extracellular matrix macromolecules such as fibronectin, matrix degradation, capillary sprout branching and anastomosis. Numerical simulations of the model, using parameter values based on experimental data, are presented and the theoretical structures generated by the model are compared with the morphology of actual capillary networks observed in vivo experiments. At last they discussed the use of the mathematical model as a possible angiogenesis assay and implications for chemotherapy regimes.

In [24], Benjamin Ribba et al proposed a multiscale model of cancer growth based on the genetic and molecular features of the evolution of colorectal cancer. They investigated the role of gene – dependent cell cycle regulation in the response of tumors to therapeutic irradiation protocols.

In [25],Bachman JW et al have combined radiation treatment with differentiation therapy. During differentiation therapy, a differentiation promoting agent is supplied (e.g., TGF – beta) such that cancer stem cells differentiate and become more radiosensitive. Then radiation can be used to control them. For this they have considered three types of cancer: head and neck cancer, brain cancers (primary tumors and metastatic brain cancers), and breast cancer and they used mathematical modeling to show that combination therapy of the above type can have a large beneficial effect for the patient; increasing treatment success and reducing side effects. Cancer development is a stepwise process through which normal somatic cells acquire mutations which enable them to escape their normal function in the tissue and become self–sufficient in survival. The number of mutations depends on the patient's age, genetic susceptibility and on the exposure of the patient to carcinogens throughout their life.

In [26], Heiko Enderling et al have presented a simplified model of this mutation and expansion process, in which they assume that the loss of two tumour suppressor genes (TSGs) is sufficient to give rise to a cancer. Their mathematical model of the stepwise development of breast cancer verifies the idea that the normal mutation rate in genes is only sufficient to give rise to a tumour within a clinically observable time if a high number of breast stem cells and TSGs exist or genetic instability is involved as a driving force of the mutation pathway. Furthermore, this model shows that if a mutation occurred in stem cells pre–puberty, and formed a field of cells with this mutation through clonal formation of the breast, it is most likely that a tumour will arise from within this area. They apply different treatment strategies, namely surgery and adjuvant external beam radiotherapy and targeted intra operative radiotherapy (TARGIT) and use the model to identify different sources of local recurrence and analyse their prevention.

In [27], Marcello Delitala et al have presented a mathematical model for immune response against cancer aimed at reproducing emerging phenomena arising from the interactions between tumor and immune cells. The model is stated in terms of integro – differential equations and described the dynamics of tumor cells, characterized by heterogeneous antigenic expressions, antigen – presenting cells and T – cells.

The presented model seems able to mimic the recognition, learning and memory aspects of immune response and highlights how the immune system might act as an additional selective pressure leading, eventually, to the selection for the most resistant cancer clones.

In [28], Philipp M. Altrock et al have examined recent topics of importance to basic and clinical cancer research, including methodology to describe cancer at various scales. They began with models that describe clonal evolution in tumour development and determined the temporal sequence of mutational events. They then discussed mathematical models that describe cancer across multiple scales, such as hybrid models that combine cellular dynamics and micro environmental factors, followed by modeling of metastasis dynamics and immunotherapy and completed with an outlook on open problems that require quantitative investigation. In [29], R. P. Araujo et al have presented a concise history of the study of solid tumour growth, illustrating the development of mathematical approaches from the early decades of the twentieth century to the present time. Most importantly these mathematical investigations are interwoven with the associated experimental work, showing the crucial relationship between experimental and theoretical approaches, which together have moulded our understanding of tumour growth and contributed to current anti – cancer treatments.

CONCLUSION:

In the course of this survey, it is thrilling and unexpected to see the traces that mathematics and mathematical models have left behind to pharmacology. Infinite contributions that the models could provide in analyzing, determining, deciding and paving way for a new area of research. In this short survey we could manage to provide a very small glimpse of the contribution of mathematical models to drug discovery, development and several therapies.

REFERENCES:

1. http://chekhov.cs.vt.edu/PAPERS/sys_biol_drug_disc.pdf

2. Yulia Balykina. Modern Approaches to Drugs Research and Development using Mathematical Modeling. Cloud of Science. 1; 2014 :566 – 578.

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7. https://en.wikipedia.org/wiki/Mathematical_model

8. http://www.princeton.edu/engineering/news/archive/?id=4492

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11. https://www.sas.com/storefront/aux/en/sppharmastat/60622_excerpt.pdf

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Received on 26.07.2017 Modified on 19.08.2017

Research J. Pharm. and Tech 2017; 10(12): 4397-4401.

DOI: 10.5958/0974-360X.2017.00810.1