Skip to main content
Log in

Numerical simulation of skin transport using Parareal

  • Published:
Computing and Visualization in Science

Abstract

In silico investigation of skin permeation is an important but also computationally demanding problem. To resolve all scales involved in full detail will not only require exascale computing capacities but also suitable parallel algorithms. This article investigates the applicability of the time-parallel Parareal algorithm to a brick and mortar setup, a precursory problem to skin permeation. The C++ library Lib4PrM implementing Parareal is combined with the UG4 simulation framework, which provides the spatial discretization and parallelization. The combination’s performance is studied with respect to convergence and speedup. It is confirmed that anisotropies in the domain and jumps in diffusion coefficients only have a minor impact on Parareal’s convergence. The influence of load imbalances in time due to differences in number of iterations required by the spatial solver as well as spatio-temporal weak scaling is discussed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

Notes

  1. The coarse method is often represented by \(\mathcal {G}\), probably because of the French word “gros” for coarse.

  2. http://user.cscs.ch/computing_systems/piz_dora/.

  3. http://www.top500.org/list/2014/11.

  4. https://gcc.gnu.org.

References

  1. Anissimov, Y.G., Roberts, M.S.: Diffusion modeling of percutaneous absorption kinetics: 3. Variable diffusion and partition coefficients, consequences for stratum corneum depth profiles and desorption kinetics. J. Pharm. Sci. 93(2), 470–487 (2004). doi:10.1002/jps.10567

    Article  Google Scholar 

  2. Anissimov, Y.G., Roberts, M.S.: Diffusion modelling of percutaneous absorption kinetics: 4. Effects of slow equilibration process within stratum corneum on absorbtion and desorption kinetics. J. Pharm. Sci. 98, 772–781 (2009). doi:10.1002/jps.21461

    Article  Google Scholar 

  3. Arteaga, A., Ruprecht, D., Krause, R.: A stencil-based implementation of Parareal in the C\(++\) domain specific embedded language STELLA. Appl. Math. Comput. (2015). doi:10.1016/j.amc.2014.12.055

  4. Aubanel, E.: Scheduling of tasks in the Parareal algorithm. Parallel Comput. 37, 172–182 (2011). doi:10.1016/j.parco.2010.10.004

    Article  MathSciNet  MATH  Google Scholar 

  5. Bal, G.: On the convergence and the stability of the Parareal algorithm to solve partial differential equations. In: Kornhuber, R., et al. (eds.) Domain Decomposition Methods in Science and Engineering, Lecture Notes in Computational Science and Engineering, vol. 40, pp. 426–432. Springer, Berlin (2005). doi:10.1007/3-540-26825-1_43

    Google Scholar 

  6. Bylaska, E.J., Weare, J.Q., Weare, J.H.: Extending molecular simulation time scales: parallel in time integrations for high-level quantum chemistry and complex force representations. J. Chem. Phys. 139(7), 074114 (2013). doi:10.1063/1.4818328

    Article  Google Scholar 

  7. Celledoni, E., Kvamsdal, T.: Parallelization in time for thermo-viscoplastic problems in extrusion of aluminium. Int. J. Numer. Methods Eng. 79(5), 576–598 (2009). doi:10.1002/nme.2585

    Article  MathSciNet  MATH  Google Scholar 

  8. Demmel, J.W., Eisenstat, S.C., Gilbert, J.R., Li, X.S., Liu, J.W.H.: A supernodal approach to sparse partial pivoting. SIAM J. Matrix Anal. Appl. 20(3), 720–755 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  9. Dick, B., Vogel, A., Khabi, D., Rupp, M., Küster, U., Wittum, G.: Utilization of empirically determined energy-optimal CPU-frequencies in a numerical simulation code. Comput. Vis. Sci. (2015). doi:10.1007/s00791-015-0249-8

  10. Dongarra, J., et al.: Applied Mathematics Research for Exascale Computing. Technical Report LLNL-TR-651000, Lawrence Livermore National Laboratory (2014). http://science.energy.gov/~/media/ascr/pdf/research/am/docs/EMWGreport.pdf

  11. Elwasif, W.R., Foley, S.S., Bernholdt, D.E., Berry, L.A., Samaddar, D., Newman, D.E., Snchez, R.S.: A dependency-driven formulation of parareal: parallel-in-time solution of PDEs as a many-task application. In: Proceedings of the 2011 ACM International Workshop on Many Task Computing on Grids and Supercomputers, p. 1524 (2011). doi:10.1145/2132876.2132883

  12. Emmett, M., Minion, M.L.: Toward an efficient parallel in time method for partial differential equations. Commun. Appl. Math. Comput. Sci. 7, 105132 (2012). doi:10.2140/camcos.2012.7.105

    Article  MathSciNet  Google Scholar 

  13. Falgout, R.D., Friedhoff, S., Kolev, T.V., MacLachlan, S.P., Schroder, J.B.: Parallel time integration with multigrid. SIAM J. Sci. Comput. 36, C635C661 (2014)

    Google Scholar 

  14. Farhat, C., Chandesris, M.: Time-decomposed parallel time-integrators: theory and feasibility studies for fluid, structure, and fluid-structure applications. Int. J. Numer. Methods Eng. 58(9), 13971434 (2003). doi:10.1002/nme.860

    Article  MathSciNet  Google Scholar 

  15. Gander, M.J., Vandewalle, S.: On the superlinear and linear convergence of the Parareal algorithm. In: Widlund, O.B., Keyes, D.E. (eds.) Domain Decomposition Methods in Science and Engineering, Lecture Notes in Computational Science and Engineering, vol. 55, pp. 291–298. Springer, Berlin (2007). doi:10.1007/978-3-540-34469-8_34

  16. Hansen, S., Lehr, C.M., Schaefer, U.F.: Modeling the human skin barrier—towards a better understanding of dermal absorption. Adv. Drug Deliv. Rev. (2013). doi:10.1016/j.addr.2012.12.002

  17. Kreienbuehl, A., Benedusi, P., Ruprecht, D., Krause, R.: Time parallel gravitational collapse simulation (2015) (in preparation)

  18. Li, X., Demmel, J., Gilbert, J., iL. Grigori, Shao, M., Yamazaki, I.: SuperLU Users’ Guide. Technical Report LBNL-44289, Lawrence Berkeley National Laboratory (1999). http://crd.lbl.gov/~xiaoye/SuperLU/. Last update: August 2011

  19. Lions, J.L., Maday, Y., Turinici, G.: A “parareal” in time discretization of PDE’s. C. R. l’Acad. Sci. Ser. I Math. 332, 661668 (2001). doi:10.1016/S0764-4442(00)01793-6

    MathSciNet  Google Scholar 

  20. Minion, M.L., Speck, R., Bolten, M., Emmett, M., Ruprecht, D.: Interweaving PFASST and parallel multigrid. SIAM J. Sci. Comput. (2015). arxiv:1407.6486

  21. Minion, M.L.: A hybrid Parareal spectral deferred corrections method. Commun. Appl. Math. Comput. Sci. 5(2), 265301 (2010). doi:10.2140/camcos.2010.5.265

    Article  MathSciNet  Google Scholar 

  22. Mula, O.: Some contributions towards the parallel simulation of time dependent neutron transport and the integration of observed data in real time. Ph.D. Thesis, Université Pierre et Marie Curie - Paris VI (2014). https://tel.archives-ouvertes.fr/tel-01081601

  23. Naegel, A., Heisig, M., Wittum, G.: A comparison of two- and three-dimensional models for the simulation of the permeability of human stratum corneum. Eur. J. Pharm. Biopharm. 72(2), 332–338 (2009). doi:10.1016/j.ejpb.2008.11.009. http://www.sciencedirect.com/science/article/B6T6C-4V1KMMP-1/2/b906a3a90140385ba35b48bed48fdef7

  24. Querleux, B. (ed.): Computational Biophysics of the Skin. Pan Stanford Publishing, Singapore (2014)

    Google Scholar 

  25. Randles, A., Kaxiras, E.: Parallel in time approximation of the lattice Boltzmann method for laminar flows. J. Comput. Phys. 270, 577586 (2014). doi:10.1016/j.jcp.2014.04.006

    Article  MathSciNet  Google Scholar 

  26. Reiter, S., Vogel, A., Heppner, I., Rupp, M., Wittum, G.: A massively parallel geometric multigrid solver on hierarchically distributed grids. Comput. Vis. Sci. 16(4), 151–164 (2013). doi:10.1007/s00791-014-0231-x

    Article  Google Scholar 

  27. Rim, J.E., Pinsky, P.M., van Osdol, W.W.: Using the method of homogenization to calculate the effective diffusivity of the stratum corneum with permeable corneocytes. J. Biomech. 41(4), 788–796 (2008). doi:10.1016/j.jbiomech.2007.11.011. http://www.sciencedirect.com/science/article/B6T82-4RWHXFR-2/2/bfe8e93f74d145a105071a106d6d227c

  28. Rim, J.E., Pinsky, P.M., van Osdol, W.W.: Multiscale modeling framework of transdermal drug delivery. Ann. Biomed. Eng. 37(6), 1217–1229 (2009)

    Article  Google Scholar 

  29. Ruprecht, D., Speck, R., Emmett, M., Bolten, M., Krause, R.: Poster: Extreme-scale space–time parallelism. In: Proceedings of the 2013 Conference on High Performance Computing Networking, Storage and Analysis Companion, SC’13 Companion (2013). http://sc13.supercomputing.org/sites/default/files/PostersArchive/tech_posters/post148s2-file3.pdf

  30. Ruprecht, D., Speck, R., Krause, R.: Parareal for diffusion problems with space- and time-dependent coefficients. In: Domain Decomposition Methods in Science and Engineering XXII, Lecture Notes in Computational Science and Engineering, vol. 104, pp. 3–10. Springer, Switzerland (2015). doi:10.1007/978-3-319-18827-0_1

  31. Ruprecht, D.: Convergence of Parareal with spatial coarsening. PAMM 14(1), 1031–1034 (2014). doi:10.1002/pamm.201410490

    Article  Google Scholar 

  32. Samaddar, D., Newman, D.E., Snchez, R.S.: Parallelization in time of numerical simulations of fully-developed plasma turbulence using the Parareal algorithm. J. Comput. Phys. 229, 65586573 (2010). doi:10.1016/j.jcp.2010.05.012

    Article  Google Scholar 

  33. Speck, R., Ruprecht, D., Krause, R., Emmett, M., Minion, M.L., Winkel, M., Gibbon, P.: A massively space–time parallel N-body solver. In: Proceedings of the International Conference on High Performance Computing, Networking, Storage and Analysis, SC’12, p. 92:1–92:11. IEEE Computer Society Press, Los Alamitos, CA, USA (2012). doi:10.1109/SC.2012.6

  34. Vogel, A., Calotoiu, A., Strube, A., Reiter, S., Nägel, A., Wolf, F., Wittum, G.: 10,000 performance models per minute-scalability of the UG4 simulation framework. In: Träff, J.L., Hunold, S., Versaci, F. (eds.) Euro-Par 2015: parallel processing, pp. 519–531. Springer, Berlin (2015)

  35. Vogel, A., Reiter, S., Rupp, M., Nägel, A., Wittum, G.: UG4: A novel flexible software system for simulating pde based models on high performance computers. Comput. Vis. Sci. 16(4), 165–179 (2013). doi:10.1007/s00791-014-0232-9

    Article  Google Scholar 

  36. Wang, T.F., Kasting, G.B., Nitsche, J.M.: A multiphase microscopic diffusion model for stratum corneum permeability. I. Formulation, solution, and illustrative results for representative compounds. J. Pharm. Sci. 95(3), 620–648 (2006). doi:10.1002/jps.20509

    Article  Google Scholar 

  37. Wang, T.F., Kasting, G.B., Nitsche, J.M.: A multiphase microscopic diffusion model for stratum corneum permeability. II. Estimation of physicochemical parameters, and application to a large permeability database. J. Pharm. Sci. 96(11), 3024–3051 (2007). doi:10.1002/jps.20883

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Andreas Kreienbuehl.

Additional information

Communicated by Volker Schulz.

This research is funded by the Deutsche Forschungsgemeinschaft (DFG) as part of the “Exasolvers” project in the Priority Programme 1648 ”Software for Exascale Computing” (SPPEXA) and by the Swiss National Science Foundation (SNSF) under the lead agency agreement as grant SNF-145271. The research of A.K., D.R., and R.K. is also funded through the FUtuRe SwIss Electrical InfraStructure (FURIES) project of the Swiss Competence Centers for Energy Research (SCCER).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kreienbuehl, A., Naegel, A., Ruprecht, D. et al. Numerical simulation of skin transport using Parareal. Comput. Visual Sci. 17, 99–108 (2015). https://doi.org/10.1007/s00791-015-0246-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00791-015-0246-y

Keywords

Navigation