Application of state equations approach to solve the classical Kubelka-Munk differential equations in turbid environment

The Kubelka-Munk (KM) model is frequently used to describe the optical properties (absorption and scattering) of inhomogeneous media in terms of the measured reflection and transmittance. The classical KM is modeled by two fluxes which counterpropagate in the media and can be described by system of differential equations. This system, on the other hand, can also be rewritten in the form of state equations, widely used in many areas of engineering such as electrical circuits, control systems and more. Here, we describe the simple mathematical expression of light reflectance and transmittance in turbid media based on the KM light propagation model via a system of state equations. To this end, we demonstrate the use of the state equations to solve KM differential equations in a simple and straightforward manner. To validate the solution of our model against the KM solution, different turbid media including tissue phantoms, milk with varying concentrations of sugar, human hands, and mouse brain were tested. We used a spatial frequency domain imaging system to recover the absorption and scattering properties of each of the media. These measured properties were then introduced into both models to obtain diffuse light reflectance and transmittance information. Experimental results demonstrate significant concurrence between the two approaches within a minute difference in all tested cases. The results presented in this study demonstrate the validity of our alternative strategy to the traditional KM solution.