### Fast and Accurate Numerical Solution of a Cell Population Balance Model Using a Spectral Coefficient Method

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Krystian Ganko, Massachusetts Institute of Technology
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Partial integro-differential equations (PIDEs) in the form of population balance models (PBMs) appear in a variety of physical contexts that describe distributions of particles changing in time (e.g., atmospheric aerosols, crystallization, cell culture growth). However, their efficient numerical solution is often complicated by arbitrarily complex integral operator kernels and non-constant growth rate functions. In particular, numerical solution of integral operators suffers from the curse of dimensionality, whereas non-constant growth rate functions may contain destabilizing discontinuities and/or cause moment closure issues. In this work, we developed a spectral coefficient-based numerical method in Julia for quickly and accurately solving a mass-distributed, mitotically-dividing cell PBM containing time-invariant partitioning, fission rate, and constant/linear mass growth rate functions. The implementation was further extended to feature physically-motivated, time-variant growth rates due to suspension cell culture contact inhibition. The implementation, performance, and error of the numerical method on this first order hyperbolic PIDE solver are all discussed, with focus on error control when varying the number of spectral modes used in expanding the integral and non-constant coefficient terms. The method is benchmarked against a lightweight Finite Difference (FD) numerical scheme on a similar number of grid points, showing that >50 spectral modes are required in the integral and non-constant coefficient terms to achieve comparable accuracies to the FD solver. Although the coefficient-based method is slower than the benchmark method used, future improvements to the speed of the spectral scheme through the use of low-rank integral kernel approximations and other dimensionality-reducing techniques are discussed.

**Abstract Author(s): **Krystian Ganko, Richard D. Braatz