Keywords
AI-based mathematical modeling
Differential equations
Philosophy of science
Fractional calculus
Fractals
Applied Mathematics
Computational fractalbased methodologies in medicine and biology
Engineering applications in medicine and biology
Biomedical decisionmaking
Probabilistic clinical reasoning
Stochastic processes
Brownian motion
Data analytics-based models
Data distribution
Dynamic precision medicine
Public health challenges
Predictive bias
Complexity
Computational complexity
Complex systems
Algorithmic thinking and processes
How to Cite
Abstract
The ultimate reason for the ubiquity of mathematics in modern science entails the essence of mathematical thinking and processes so that complex phenomena including those emerging in medical and biological systems can be understood, and thus, scientific models at their crux can be generated. The consequent complexities and uncertainties require the applications of stochastic processes in mathematical modeling with Artificial Intelligence (AI) techniques used in realms of medicine and biology. Within these conditions, clinical evaluation evidence and model explainability are considered to ensure accountable, effective and safe uses of AI in clinical settings, along with robust, reliable as well as accurate understanding of various complex processes that manifest huge numbers of heterogeneous temporospatial scales. The role of philosophy within science can be expounded by its juxtaposition with models and empirical data explicated by philosophy whose pillars are driven into semantic, pragmatic and syntactic structures of scientific theory that also make up the foundational tenets of algorithmic thinking and patterns. While philosophy of science examines and reflects on the concepts, theories, arguments and methods of science, it should also be borne in mind that scientific theory, by its definition, relates to applications validated by its predictions as units of analyses. Concerning mathematical models, their structure and behavior in target systems are also to be addressed so that explicit common patterns that are implicit in scientific practice can be included in this complex influx. On the other hand, critical functions of mathematical modeling from the pragmatic aspect include the unification of models and data, model fitting to the data, identification of mechanisms depending on observations as well as predictions of future observations. Given these, philosophy of science in medical and biological fields is stated to prompt a comprehensive understanding to construct holistic mathematical models particularly in complex sciences including different attributes of complexity, evolution and adaptation. Regarding the position of AI, its algorithms and mathematical modeling, the methods of neural networks, statistics, operations research, fractional calculus, fractals, and so forth can be employed with AI being capable of uncovering hidden insights embedded in big data concerning medical and biological issues in view of contemporary scientific thinking and processes. In addition, the treatment and handling of uncertainty in clinical medicine and biological problems over their processes may disclose compelling challenges due the fact that uncertainties are one of the intrinsic features of nearly all mathematical models which are formed based on three basic types of uncertainty: interval, Bayesian and stochastic. Accordingly, the current overview aims at providing answers built on sophisticated models considering the explanation and interpretation of design and formulation considering that the extant research literature could have some fragmented points in terms of original and application-oriented works. To these ends, the opportunities, challenges, limitations and conjunctures with respect to mathematical modeling in medicine and biology are addressed while role of philosophy of science is discussed within the context of mathematical modeling and applications in medicine and biology. In addition to these points, the delineation of forecasting, prediction, estimation and approximation concerning different mathematical modeling with the integration of AI in medicine and biology is explained. Thereby, an overview is inclusively presented by comprising the principles underpinning the medical and biological systems within a framework in relation to the diagnostic and disease-related treatment processes and follow-up, which can provide new directions in novel formulations, designs and interpretations based on mathematical modeling processes to be constructed and solved through practicality as well as to-the-point specific means.
References
Abirami, A., P. Prakash, et al., 2023 Fractional diffusion equation for medical image denoising using adi scheme. Journal of Population Therapeutics and Clinical Pharmacology 30: 52–63.
Amigó, J. M. and M. Small, 2017 Mathematical methods in medicine: neuroscience, cardiology and pathology. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375: 20170016.
Arthur, J. L., 1985 An introduction to stochastic modeling (howard m. taylor and samuel karlin). SIAM Review 27: 581. Baleanu, D., Y. Karaca, L. Vázquez, and J. E. Macías-Díaz, 2023.
Advanced fractional calculus, differential equations and neural networks: Analysis, modeling and numerical computations. Physica Scripta 98: 110201.
Behinfaraz, R., A. A. Ghavifekr, R. De Fazio, and P. Visconti, 2023 Utilizing fractional artificial neural networks for modeling cancer cell behavior. Electronics 12: 4245.
Brugnach, M., C. Pahl-Wostl, K.-E. Lindenschmidt, J. Janssen, T. Filatova, et al., 2008 Chapter four complexity and uncertainty: Rethinking the modelling activity. Developments in Integrated Environmental Assessment 3: 49–68.
Chambers, R. B., 2000 The role of mathematical modeling in medical research:“research without patients?”. Ochsner Journal 2: 218–223.
Chang, Y.-C., 2024 Deep-learning estimators for the hurst exponent of two-dimensional fractional brownian motion. Fractal and Fractional 8: 50.
Chen, L., P. Ji, Y. Ma, Y. Rong, and J. Ren, 2023 Custom machine learning algorithm for large-scale disease screening-taking heart disease data as an example. Artificial Intelligence in Medicine 146: 102688.
Chin-Yee, B., 2017 The new medical model: why medicine needs philosophy. Canadian Medical Association. Journal 189: E896.
Coscia, V., 2011 On the mathematical theory of living systems, i: Complexity analysis and representation. Mathematical and computer modelling 54: 1919–1929.
Daun, S., J. Rubin, Y. Vodovotz, and G. Clermont, 2008 Equationbased models of dynamic biological systems. Journal of critical care 23: 585–594.
Evans, L. C., 2022 Partial differential equations, volume 19. American Mathematical Society. Fortier, P. J. and H. Michel, 2003 Computer systems performance evaluation and prediction. Digital Press.
Ge,W., C. Lueck, H. Suominen, and D. Apthorp, 2023 Has machine learning over-promised in healthcare?: A critical analysis and a proposal for improved evaluation, with evidence from parkinson’s disease. Artificial Intelligence in Medicine 139: 102524.
Grizzi, F., A. Castello, D. Qehajaj, C. Russo, and E. Lopci, 2019 The complexity and fractal geometry of nuclear medicine images. Molecular Imaging and Biology 21: 401–409.
Havlin, S., S. Buldyrev, A. Goldberger, R. Mantegna, S. Ossadnik, et al., 1995 Fractals in biology and medicine. Chaos, Solitons & Fractals 6: 171–201.
Heid, E., C. J. McGill, F. H. Vermeire, and W. H. Green, 2023 Characterizing uncertainty in machine learning for chemistry. Journal of Chemical Information and Modeling 63: 4012–4029.
Jin, W., M. Fatehi, R. Guo, and G. Hamarneh, 2024 Evaluating the clinical utility of artificial intelligence assistance and its explanation on the glioma grading task. Artificial Intelligence in Medicine 148: 102751.
Joshi, M., S. Bhosale, and V. A. Vyawahare, 2023 A survey of fractional calculus applications in artificial neural networks. Artificial Intelligence Review 56: 13897–13950.
Karaca, Y., 2022a Multi-chaos, fractal and multi-fractional AI in different complex systems. In Multi-Chaos, Fractal and Multi-Fractional Artificial Intelligence of Different Complex Systems, pp. 21–54, Elsevier.
Karaca, Y., 2022b Theory of complexity, origin and complex systems. In Multi-Chaos, Fractal and Multi-fractional Artificial Intelligence of Different Complex Systems, pp. 9–20, Elsevier.
Karaca, Y., 2023a Computational complexity-based fractional-order neural network models for the diagnostic treatments and predictive transdifferentiability of heterogeneous cancer cell propensity. Chaos Theory and Applications 5: 34–51.
Karaca, Y., 2023b Fractional calculus operators-bloch partial differential equation-artificial neural networks-computational complexity modeling of the micro-macrostructural brain tissues with diffusion mri signal processing and neuronal multi-components. Fractals 31: 2340204–547.
Karaca, Y. and D. Baleanu, 2022 Computational fractional-order calculus and classical calculus ai for comparative differentiability prediction analyses of complex-systems-grounded paradigm. In Multi-Chaos, Fractal and Multi-fractional Artificial Intelligence of Different Complex Systems, pp. 149–168, Elsevier.
Karaca, Y., D. Baleanu, and R. Karabudak, 2022a Hidden markov model and multifractal method-based predictive quantization complexity models vis-á-vis the differential prognosis and differentiation of multiple sclerosis’ subgroups. Knowledge-Based Systems 246: 108694.
Karaca, Y., D. Baleanu, M. Moonis, and Y.-D. Zhang, 2020a Theory, analyses and predictions of multifractal formalism and multifractal modelling for stroke subtypes’ classification. In Computational Science and Its Applications–ICCSA 2020: 20th International Conference, Cagliari, Italy, July 1–4, 2020, Proceedings, Part II 20, pp. 410–425, Springer.
Karaca, Y., D. Baleanu, M. Moonis, Y. D. Zhang, and O. Gervasi, 2023a Editorial special issue: Part IV-III-II-I series: Fractals fractional AI-based analyses and applications to complex systems.
Karaca, Y., D. Baleanu, Y. D. Zhang, O. Gervasi, and M. Moonis, 2022b Multi-Chaos, Fractal and Multi-Fractional Artificial Intelligence of Different Complex Systems. Academic Press.
Karaca, Y. and C. Cattani, 2018 Naive bayesian classifier. Comput. Methods Data Anal pp. 229–250.
Karaca, Y. and M. Moonis, 2022 Shannon entropy-based complexity quantification of nonlinear stochastic process: diagnostic and predictive spatiotemporal uncertainty of multiple sclerosis subgroups. In Multi-Chaos, Fractal and Multi-Fractional Artificial Intelligence of Different Complex Systems, pp. 231–245, Elsevier.
Karaca, Y., M. Moonis, and D. Baleanu, 2020b Fractal and multifractional-based predictive optimization model for stroke subtypes’ classification. Chaos, Solitons & Fractals 136: 109820.
Karaca, Y., M. Moonis, Y.-D. Zhang, and C. Gezgez, 2019 Mobile cloud computing based stroke healthcare system. International Journal of Information Management 45: 250–261.
Karaca, Y., M. Ur Rahman, M. A. El-Shorbagy, and D. Baleanu, 2023b Multiple solitons, bifurcations, chaotic patterns and fission/ fusion, rogue waves solutions of two-component extended (2+ 1)-d itô calculus system. Fractals 31: 2350135.
Karlsson Faronius, H., 2023 Solving partial differential equations with neural networks.
Karpov, A. V., 2009 Mathematical modeling in medicine. Mathematical Models of Life Support Systems 2: 312.
Koen-Alonso, M. and P. Yodzis, 2005 7.3| dealing with model uncertainty in trophodynamic models: A patagonian example. DYNAMIC FOOD WEBS: MULTISPECIES p. 381.
Li, C. and D. Zhao, 2024 A non-convex fractional-order differential equation for medical image restoration. Symmetry 16: 258.
Magin, R. L., 2010 Fractional calculus models of complex dynamics in biological tissues. Computers & Mathematics with Applications 59: 1586–1593.
Mohamadou, Y., A. Halidou, and P. T. Kapen, 2020 A review of mathematical modeling, artificial intelligence and datasets used in the study, prediction and management of covid-19. Applied Intelligence 50: 3913–3925.
Mubayi, A., C. Kribs, V. Arunachalam, and C. Castillo-Chavez, 2019 Studying complexity and risk through stochastic population dynamics: Persistence, resonance, and extinction in ecosystems.
In Handbook of Statistics, volume 40, pp. 157–193, Elsevier. Ning, X., L. Jia, Y.Wei, X.-A. Li, and F. Chen, 2023 Epi-dnns: Epidemiological priors informed deep neural networks for modeling covid-19 dynamics. Computers in biology and medicine 158: 106693.
Oberguggenberger, M., 2005 The mathematics of uncertainty: models, methods and interpretations. In Analyzing uncertainty in civil engineering, pp. 51–72, Springer.
Pinchover, Y. and J. Rubinstein, 2005 An introduction to partial differential equations, volume 10. Cambridge university press.
Saaty, T. L. and J. M. Alexander, 1981 Thinking with models: mathematical models in the physical, biological, and social sciences. RWS Publications.
Takale, K., U. Kharde, and S. Gaikwad, 2024 Investigation of fractional order tumor cell concentration equation using finite difference method. Baghdad Science Journal .
Torres, N. V. and G. Santos, 2015 The (mathematical) modeling process in biosciences. Frontiers in genetics 6: 169934. Vosika, Z. B., G. M. Lazovic, G. N. Misevic, and J. B. Simic-Krstic, 2013 Fractional calculus model of electrical impedance applied to human skin. PloS one 8: e59483.
Wang, S.-H., Y. Karaca, X. Zhang, and Y.-D. Zhang, 2022 Secondary pulmonary tuberculosis recognition by rotation angle vector grid-based fractional fourier entropy. Fractals 30: 2240047.
Wang, W., M. Luo, P. Guo, Y. Wei, Y. Tan, et al., 2023 Artificial intelligence-assisted diagnosis of hematologic diseases based on bone marrow smears using deep neural networks. Computer Methods and Programs in Biomedicine 231: 107343.
Winther, R. G., 2012 Mathematical modeling in biology: philosophy and pragmatics. Frontiers in Plant Science 3: 102.
Yang, Q., D. Chen, T. Zhao, and Y. Chen, 2016 Fractional calculus in image processing: a review. Fractional Calculus and Applied Analysis 19: 1222–1249.
Yang, Z., Z. Hu, H. Ji, K. Lafata, E. Vaios, et al., 2023 A neural ordinary differential equation model for visualizing deep neural network behaviors in multi-parametric mri-based glioma segmentation. Medical Physics 50: 4825–4838.
Zimmermann, H.-J., 2000 An application-oriented view of modeling uncertainty. European Journal of operational research 122: 190–198.
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.