# Lagrangian Modeling of Marine Microplastics Fate and Transport: The State of the Science
## 4 Numerical Lagrangian Models
### TrackMPD  
Amongst the three-dimensional numerical models, TrackMPD is an open-source model that has been released in recent years [61] based on the framework of the Particle Tracking and Analysis Toolbox (PaTATO) [122]. TrackMPD is compatible with various ocean models in order to address the fate and transport of marine microplastic debris in coastal areas. TrackMPD is capable of simulating microplastics' behavior in three dimensions considering different physical processes, including beaching and washing-off, biofouling and degradation, advection, windage, turbulent dispersion, and sinking, as well as different physical properties such as density (Equations (7) and (8)), size (Equation (10)), and shape (Equations (2)-(4), (7) and (8)). The physical properties of microplastics can be parametrized in TrackMPD, particularly in the calculation of the settling velocity. TrackMPD utilizes the empirical equations of [99] and [123] in estimating the settling velocity of microplasticswith different shapes as a function of particle density and radius. Accordingly, the settling velocity (\(v_{s}\)) of spherical particles (ms\({}^{-1}\)) is obtained based on a calibrated formulation provided by [123], as shown in Equations (2) and (3), while this is different, as determined in Equation (4), for cylindrical fragments (mms\({}^{-1}\)) [99]. The effects of biofouling on the fate and transport of microplastics can be investigated by TrackMPD. In other words, TrackMPD uses some empirical equations in order to approximate the thickness of the biofilm layer. In this respect, the thickness of biofilm layers of spherical and cylindrical particles can be estimated based on Equations (7) and (8), respectively [96]. Accordingly, biofilm thickness can be given either a constant or variable value over time. The case of constant biofilm thickness is referred to as stationary biofouling, while the case of variable biofilm thickness is considered as non-stationary biofouling. TrackMPD simply predicts mechanical degradation by using an empirical relation (Equation (10)). In this respect, the size of plastic particles (their diameter or length) decreases based on a constant rate and time. While TrackMPD can simulate biofouling and degradation, it suffers from incapability in simulating some other transformational processes such as hetero- and homo-aggregation. To simulate the horizontal and vertical movements of plastic particles due to advection, TrackMPD employs the Runge-Kutta method [61]. In this respect, the velocity of the flow is calculated iteratively in three directions at the location of a particle to consider velocity values (ms\({}^{-1}\)) at past and future times. Then, to calculate particle displacements (m) and obtain their new location in each direction, the previous location of microplastics is added to the multiplication of the internal time interval (\(\Delta t_{j}\)) and the flow velocity's components (\(U=(u,v,w)\)), as shown in Equation (19). Moreover, the horizontal (in \(x\) or \(y\) directions) and vertical diffusion of particles is predicted in TrackMPD based on a random-walk model, as demonstrated in Equation (20) and (21), respectively [61]. As with D-WAQ PART, TrackMPD can consider the windage effect in the calculation of particle velocity. Accordingly, the horizontal wind-induced drift of microplastics is considered using Equations (22)-(25) [61]. TrackMPD estimates the probability of microplastics' washing-off based on a Monte Carlo method, as shown in Equation (16) [61]. The probability (\(P\)) of being washed off in TrackMPD is an exponential function of the time step (\(t\)), and the half-life of plastic litter (\(T\)) is equal to the time that microplastics remain on the beach [87]. If \(P\) is higher than a random number, which varies from 0 to 1, beached microplastics can be transported (i.e., washed off) seaward at extreme tides. D-WAQ PART uses a basically similar approach in the calculation of the probability of wash-off.