Research interests
PDE analysis and computations, complex fluids, numerical analysis, mathematical modeling in physics and engineering, mathematical modeling in biology and medicine, bioinformatics, fluid dynamics, finite difference schemes.
PDE analysis and computations, complex fluids, numerical analysis, mathematical modeling in physics and engineering, mathematical modeling in biology and medicine, bioinformatics, fluid dynamics, finite difference schemes.
My research centers on the energetic variational approach (EnVarA), a framework that derives thermodynamically consistent PDE models for complex fluids and multiphase systems. I study how energy laws, dissipation mechanisms, and microstructural interactions shape the dynamics of multi‑component flows, including diffuse‑interface (phase‑field) models and systems with moving contact lines. My work has clarified the structure of free energies in ternary mixtures, established equivalence between different modeling approaches, and developed energy‑stable numerical schemes that preserve dissipation at the discrete level. These contributions provide a unified mathematical foundation for modeling mixtures, interfaces, and complex materials.
Key references:
M.-H. Giga, A. Kirshtein, C. Liu, Variational Modeling and Complex Fluids, Springer (2017).
Kirshtein, J. Brannick, C. Liu, Energy Dissipation Laws in Multi‑Component Phase‑Field Models, CMS (2020).
Brannick, A. Kirshtein, C. Liu, Diffusive Interface Methods with Energetic Variational Approaches, Elsevier (2016).
I develop thermodynamically consistent phase‑field and phase‑field–mechanics models for solid‑state sintering, focusing on microstructure evolution driven by surface energy reduction. My work resolves long‑standing issues in earlier models, such as artificial void generation and inconsistent mass transport, by deriving governing equations directly from energetic variational principles. These models capture grain boundary motion, rigid‑body rotation, and densification dynamics, and have been validated through numerical simulations. This research advances predictive modeling of ceramic and metallic materials, enabling improved control of microstructure and material properties.
Key references:
X. Dai , B. Qian, A. Kirshtein, Q. Yang, A Phase-Field-Micromechanics Study on the Microstructural Evolution during Viscous Sintering,// Powder Technology, 456 (2025): 120823 (url) (arXiv)
Q. Yang and A. Kirshtein, A thermodynamically consistent phase-field-micromechanics model of sintering with coupled diffusion and grain motion, // Journal of the American Ceramic Society, 2025, 108 (3), e20279 (url) (arXiv)
Q. Yang, Y. Gao, A. Kirshtein, Q. Zhen, C. Liu, A free-energy-based and interfacially consistent phase-field model for solid-state sintering without artificial void generation, //Computational Materials Science. 2023, 229: 112387. (url)
Q. Yang, A. Kirshtein, Y. Ji, C. Liu, J. Shen, L.-Q. Chen, A thermodynamically consistent Phase-Field model for viscous sintering, //Journal of American Ceramic Society. 2018;00:1–12. (url) (PDF)
I work on mechanistic and data‑driven models of cancer progression, focusing on the interaction networks between cancer cells, immune cells, cytokines, and therapeutic agents. Using ODE‑based and reaction–diffusion frameworks, my research identifies patient‑specific tumor behaviors, classifies immune microenvironments, and predicts treatment responses. I have developed models for colon cancer progression, immune‑mediated tumor control, and the effects of FOLFIRI chemotherapy, integrating clinical and gene‑expression data to estimate parameters and validate predictions. These models provide insight into tumor heterogeneity and support personalized treatment strategies.
Key references:
A. Budithi, S. Su, A. Kirshtein, L. Shahriyari, Data driven mathematical model of FOLFIRI treatment for colon cancer, //Cancers. 2021, 13(11), 2632. (url) (PDF)
T. Le, S. Su, A. Kirshtein, L Shahriyari, Data driven mathematical model of osteosarcoma, //Cancers. 2021; 13(10):2367. (url) (PDF)
A. Kirshtein, S. Akbarinejad, W. Hao, T. Le, S. Su, R.A. Aronow, L. Shahriyari, Data Driven Mathematical Model of Colon Cancer Progression, //J. Clin. Med. 2020, 9, 3947. (url) (PDF)
T. Le, R. Aronow, A. Kirshtein, L. Shahriyari, A review of digital cytometry methods: estimating the relative abundance of cell types in a bulk of cells, //Briefing in Bioinformatics, 2020. (url)
In collaboration with Shahriyari Lab
I study mathematical models of opinion formation, polarization, and collective decision‑making in social systems. My work develops differential‑integral and particle‑based formulations of Hegselmann–Krause–type models, establishing convergence properties, concentration inequalities, and numerical methods with provable accuracy. I have also contributed to models of candidate–voter interactions and the effects of mandatory voting policies, showing how social influence structures shape long‑term opinion distributions. These models provide quantitative insight into consensus formation, polarization, and the impact of institutional interventions.
Key references:
C. Börgers, N. Dragovic, A. Haensch, A. Kirshtein, A particle method for continuous Hegselmann-Krause opinion dynamics, // In: Cherifi, H., Rocha, L.M., Cherifi, C., Donduran, M. (eds) Complex Networks & Their Applications XII. COMPLEX NETWORKS 2023. Studies in Computational Intelligence, vol 1142, pp 457–469. Springer, Cham. (url) (arXiv)
C. Börgers, N. Dragovic, A. Haensch, A. Kirshtein, L. Orr, ODEs and Mandatory Voting, //CODEE Journal: Vol. 17, Article 11. (url) (arXiv)
C. Börgers, N. Dragovic, A. Kirshtein, Candidate voter dynamics, //Physica A: Vol. 682, 131176 (url)