An important function of calculus is solving problems in which you know the initial values of variables and the rates of change of those variables, called the derivatives, but you really want to know the final values. Equations involving functions of the variables and their derivatives are called differential equations, and solving them often involves integrals, which are a kind of sum. There are two main classes of differential equations: ordinary differential equations, for when there are two variables, and partial differential equations, for when there are more. Partial differential equations can model systems that evolve in both space and time. Another important use of calculus is in finding optimal configurations of a system. Searching for the best configuration is called optimization.
Bioengineers design materials and devices used in biomedical applications, such as artificial organs. The mathematical models describing how objects made of particular materials respond to stress and strain use partial differential equations. Sometimes, solutions to simple versions of these equations can be used as guides in the early parts of design. Eventually, the full model is solved on a computer using methods based on calculus, and engineers may use calculus again later to optimize the function or design.
Biomedical devices frequently involve electronic controls, and calculus is the main tool for analyzing the electronic circuits and signals in these controls. Signal processing, in particular, uses integrals called the Laplace transform and Fourier transform. Bioengineers are concerned with how electric signals propagate in space and time; for example, studying how electrical signals propagate through heart tissue can help predict, prevent and treat heart attacks. Models of the electrical signals in the heart are based on partial differential equations such as Maxwell’s equations.
Chemical engineering is an important component of bioengineering. A bioengineer may be developing a new chemical process for producing a material to help heal wounds, while another might be analyzing a chemical pathway in a cell colony to better understand how to control its growth. Predicting and understanding evolution of such complex systems of reactions involves applying chemical kinetics equations based on reaction rates. These are ordinary differential equations, which can be solved analytically (with pencil and paper) or numerically on a computer. In a cell colony, chemical reactions interact with each other at different rates at different points in space. Diffusion carries chemicals from one point to another, changing their concentration. Partial differential equations are used to model these systems.
Biology is full of systems that flow. Fluid flow is governed by a set of partial differential equations called the Navier-Stokes equations. Tissue engineering seeks to reproduce tissue, such as arteries and veins using synthetic materials or even living cells. Solving the Navier-Stokes equations helps determine pressure, flow rate and other characteristics. Also, biomedical devices often involve fluid flow. For instance, a device may be designed to precisely move individual cells or nanoscopic packages of chemicals from place to place through networks of tiny channels. This is called microfluidics. Solving the Navier-Stokes equations for different shaped channels helps scientists understand the design requirements of microfluidic systems.
Almost any aspect of bioengineering involves computer simulations to solve the mathematical models. Different ways to solve problems on a computer are called numerical methods, and most of these methods involve calculus in their very design.