Mathematics as the Invisible Architect: Bridging Natural Phenomena and Practical Applications

Mathematics: The Invisible Driver of Civilization

Mathematics, alongside philosophy, has systematically shaped human cognitive abilities, driving the progress of human civilization. Over history, it has evolved to address societal needs while simultaneously advancing as an "invisible culture" through individual and collective intellectual efforts. Fundamental concepts such as distance and space have been refined with mathematical tools, enabling simplified representations of complex phenomena and fostering systematic understanding.

Mathematical tools, including Maxwell's equations for electromagnetism, the Navier-Stokes equations for fluid dynamics, elasticity equations for material properties, and the heat conduction equation, are grounded in conservation laws. They describe relationships between physical quantities over time and space and have broad applications, such as fingerprint recognition, voice analysis, data compression, medical imaging, cryptography, animation, and financial modeling.

The Role of Mathematical Modeling

Mathematical modeling simplifies complex phenomena by distilling essential features and causal relationships into well-posed equations that ensure uniqueness, stability, and existence. A classic example is Newton’s discovery of F=ma  (force equals mass times acceleration), which demonstrates how careful analysis of relevant variables can yield fundamental laws of nature.

Fingerprint recognition illustrates modeling in practice. The dermal layer beneath the skin remains unchanged throughout life, making it unique to each individual. By defining a "distance" metric to measure similarity or dissimilarity between fingerprints, invariant features are extracted to ensure reliable comparison. These metrics are mapped to coordinate systems, akin to assigning an address to each fingerprint.

Since 2015, remarkable advancements in deep learning have driven significant shifts in mathematical applications. While this blog does not address deep learning in detail, my separate post, Paradigm Shift in Mathematics Through Deep Learning, delves into this topic.

Mathematics in Medical Imaging: Advances and Limitations in Imaging Tissue Properties

In the 20th century, mathematics became integral to medical imaging, driving innovations such as X-ray, CT, MRI, ultrasound, and PET. By the 1990s, efforts to visualize electrical, mechanical, and optical properties of tissues—beyond the scope of traditional imaging—sparked critical research in functional imaging, offering unprecedented diagnostic insights.

Visualizing these tissue properties involves applying physical stimuli (e.g., electrical currents, magnetic fields, or mechanical vibrations) and measuring the body's responses using techniques such as electrodes, coils, MRI, ultrasound, or optics. These responses are interpreted through mathematical models based on partial differential equations, including Maxwell's and elasticity equations.

Unlike conventional imaging methods, this process is an inverse problem with nonlinear characteristics, making it particularly challenging. To become viable diagnostic tools, new imaging methods must address critical requirements such as repeatability, robustness, spatial resolution, real-time functionality, convenience, non-invasiveness, and uniqueness.

Electrical and mechanical tissue properties, such as conductivity and elasticity, vary significantly with disease progression. Direct measurement of these properties within the body is infeasible, requiring researchers to solve complex inverse problems using mathematical models. This advanced field of medical imaging demands multidisciplinary expertise in mathematics, engineering, and medicine, alongside strong analytical and programming skills.

Interdisciplinary collaboration is equally essential, as effective communication between researchers and clinicians helps bridge the gap between theoretical advancements and practical applications. Novel imaging methods targeting biological tissue properties have transformative potential, enabling applications such as early tumor detection, direct neural activity visualization, optimization of electromagnetic therapies, and improved bioelectromagnetic source imaging.

Commercialization Challenges in Imaging Tissue Properties

Imaging technologies focused on tissue properties face significant entry barriers due to their complexity and the need for interdisciplinary integration. Success requires foundational knowledge in mathematical modeling, partial differential equations, and scientific computing, combined with biomedical expertise in areas such as bioelectromagnetics, MR physics, and advanced image processing. Even prominent scientists have struggled to integrate these diverse disciplines, leading to impractical theories or unrealistic assumptions. To overcome these challenges, foundational theories must be continually refined through rigorous analysis, simulations, and experimental validation. This iterative process ensures that imaging technologies remain both accurate and clinically relevant.

Efforts to commercialize imaging techniques for visualizing tissue properties have often failed due to poor reproducibility and limited understanding of clinical environments. Many of these failures originate from academic, lab-driven research that lacks market-oriented innovation, compounded by misaligned R&D funding strategies influenced by the early successes of CT and MRI. These cases underscore the importance of aligning technological innovation with market needs to avoid counterproductive outcomes, even with significant investments.