Nanoparticle uptake by a semi-permeable, spherical cell from an external planar diffusive field. I. Mathematical model and asymptotic solution

Abstract In this paper, we consider the diffusion of nanoparticles taken up by a semi-permeable spherical cell (of radius a ) placed in the path of a diffusive particle field generated by an external planar source at a distance $$z_0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>z</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> from the cell center. The cell interior and exterior are characterized by different diffusive properties, while the cell is able to accommodate a different saturation level of particles at steady state than is present in the external medium. The situation models the practical problem of biological cells exposed from one direction. The conflict of geometries is handled by the introduction of an effective boundary condition at a virtual spherical boundary at a radius R creating a finite domain problem. A closed-form, large-time asymptotic solution for the local concentration interior to the cell is developed valid for $$a \ll R \le z_0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>a</mml:mi> <mml:mo>≪</mml:mo> <mml:mi>R</mml:mi> <mml:mo>≤</mml:mo> <mml:msub> <mml:mi>z</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:mrow> </mml:math> . From the solution of the finite domain problem, we readily deduce a matched asymptotic expansion solution for the physical, infinite domain problem. Using this matched asymptotic expansion solution, we derive a time asymptotic approximation for the rate of nanoparticle accumulation in the cell. We contrast the resulting time dependence of this experimentally relevant quantity with the rate of accumulation found under strictly spherically symmetric conditions.