Brain-on-a-chip: Design and modeling

The aim of this paper is to lay the foundations of the basic science that will guide the design and making of an implantable neuro-glial-vascular unit optimized to perform certain fundamental processes that could facilitate monitoring and supporting the proper functionality of the brain. Such an engineered unit is called brain-on-a-chip. We first provide a possible structure of a brain-on-a-chip and then focus on the mathematical modeling of the coupled mechano-electrochemistry of a neuron and its membrane. We propose a constrained Lagrangian formulation that links the Hodgkin-Huxley model of the electronic membrane and the motion and diffusion processes of a triphasic porous medium that fills the inside of the neuron. The three phases of the triphasic medium are: solid, fluid and ionic. Lastly, a simplified Lagrangian formulation more suitable for practical applications is given whose corresponding Euler-Lagrange equations are obtained from the non-conservative form of Hamilton’s principle.