Usage of Applicable Mathematics in Biomedical Engineering (Macro and Micro Biomechanics)

Biomedical engineering is an emerging discipline which links medicine, biology, and technology in order to tackle medical problems using an engineering approach. As such, it is heavily influenced by engineering principles and uses applicable mathematics in nearly all of its aspects. It is important to note that applied mathematics (vector algebra, calculus, numerical approaches to solving functions, etc.) is a language, which evolved to serve the needs of science and particularly those of classical (Newtonian) mechanics. Thus, it describes and explores natural phenomena using mathematical functions and relations. As biomedical engineering incorporates many engineering spheres, it employs applicable mathematics in numerous situations. The following is an overview of the applications of mathematics in solving macro-biomechanical, and micro and nano-scale challenges in biomedical engineering. Biomechanics is a field of biomedical engineering which analyses the behaviour of structures and materials in biological or medical applications [1]. It is closely related to the field of prosthetics and uses mathematical analysis to design and test better devices. Dynamic analysis of systems involves constructing free-body diagrams and resolving forces acting on the components. Using partial derivatives and integration can vastly simplify biomechanical problems by switching between the domains of position, velocity, and acceleration. The position data is usually obtained from sensors on the bodies of people. It is then fed into a computer program, which mathematically resolves the velocity and acceleration vectors [2] . Such techniques are frequently used to analyse the natural movements of limbs (for example in gait analysis), body posture, skin tension [2]. Many challenges exist in biomechanics, which require precise mathematical analysis and lead to sometimes quite complicated mathematical models of movement. One of these is that each muscle has a moment arm and exerts forces in specific directions, is coupled with another muscle, and usually acts on a bone (or a tendon) which is a anatomically complex structure (has a different physical properties and modulus in different directions) which is never a straight beam, but rather a bended and rotated structure. Furthermore, it has been established that each person has a slightly different 'way' of motion, which varies between individuals, race, and age. The differential equations of motion produced by the analysis of biological systems involve the usage of applied mathematics to solve them. It is also evident that constructing engineering models for medical applications (for example a prosthetic limb) is thus quite complex and requires much expenditure in research. Mathematical models are frequently used to establish relationships between various biological processes and, in such a way, eliminate the need for constant experimental analysis. An article in Medical and Biological Engineering and Computing, for example, shows a study of fracture healing rates in mice [3]. In it, a numerical method was established for the healing of a semi-stabilised mouse tibia, which was then experimentally tested [3]. Again, the significance of applied mathematics in biomedical testing in evident. It should be noted that factors such as mass-conservation and the non-negativity of values are reflected in the mathematical model of a biological system and that, as a language, mathematics is made to work and describe the behaviour of biomedical systems. Mathematical exploration of static systems is also important in biomechanics. Testing of biomaterials for stiffness, elasticity, strength, etc. produces graphs (usually stress-strain in most standardised tests), which can be further mathematically interpreted. Using relations between forces, common for applied statics, also employs applicable mathematics in order to convert the experimentally-produced graphs into other types of graphs. On the nanoscale, mathematical algorithms are used to create miniature patterns on the surfaces of materials [4]. It has been established that cell migration and adhesion to surfaces is vastly governed by its micro- or nano-topology [4]. Thus, by creating a mathematical algorithm which produces a pattern on a surface (either by laser-emission or thermochemical processes), one can manipulate a material's biocompatibility and affect the way that cells populate it and migrate on it [4]. The newly employed technology of 3D printing is also an example of the employment of applicable mathematics in biomechanics. Researchers from MIT have developed a technology by which a mixture of a hydrogel and hepatocytes (living liver cells) is used to 3D print, layer by layer, a functional liver. The problem of vascularisation and the high metabolic demand of liver cells is tackled by constructing a 3D function of a capillary network. It is then used as a framework and printed out of slowly dissolvable sugars , which promote vascularisation of the organ [7]. In this biomechanical application, mathematics is used as a tool to construct a micro-architecture in the organ. Mathematical approaches are also being used to study the biomechanical movement of cells with respect to chemotaxical factors [5]. Numerical techniques are employed to ensure that the data obtained from experiment (which is usually vastly non-linear) will decay to a linear, engineering-usable solution [5]. Again, we see how applicable mathematics is used to study a biomedical phenomenon and convert it in a workable form, which can be used to create a model. Finally, mathematical algorithms are used to monitor and predict the biomechanical movement of cells. This is extremely useful as it can help us understand the behaviour and spread of cancer cells (among other things) [6]. It has been established that cells generally move by two techniques - blebs (semicircular pressure-driven extensions) and 'pods' (longer, stretch-like protrusions). A mathematical model can be created to recognize (via a computer) the functions of these techniques and thus monitor cell motility or even predict where the cell would move to, depending on its shape[6]. The exploration of such simple mathematical functions can therefore be used to study the micro biomechanics of cells. Applicable mathematics is used in almost any field of biomedical engineering and namely in biomechanics. The hitherto stated examples serve to briefly portray a section of mathematical applications in a fraction of biomedical research.

[1] Tanaka, Masao. Computational Biomechanics. Vol. 3. N.p.: n.p., 2012. Print.
[2] "Everyone's a Winner." PE: Professional Engineering: 30-31. Print.
[3] Geris, Liesbet. "Mathematical modeling of fracture healing in mice: comparison between experimental data and numerical simulation results." Medical & Biological Imaging & Computing April (2006): n. pag. Print.
[4] King, William. "Thermochemical Method Developed for Nanopatterning." Beckman Institute. N.p., Dec. 2009. Web. 18 Feb. 2013. <http://beckman.illinois.edu/news/2009/12/kingafm>.
[5] Payne, L. E. "Decay for a Keller–Segel Chemotaxis Model." Studies in Applied Mathematics Nov (2009): n. pag. Print.
[6] Kim, Jongrae. "Position and velocity estimation of cell boundary." International Conference on Systems Biology (2009): n. pag. University of Glasgow. Web. 18 Feb. 2013.
[7] Additional resource : IBioSeminars