Mathematical modeling and the quantification of brain dynamics

Neuroimaging greatly expanded the fundamental understanding of brain functions, and it has revealed novel treatment options in disciplines such as neurology, neurosurgery, and neuropsychiatry. The last 30 years have witnessed a flourish of approaches that include novel opportunities to image not only structure in ever-increasing resolution but also, and perhaps more importantly, the basic mechanisms of brain work that include the roles of regional cerebral blood flow and energy metabolism, neuronal network and neurotransmitter system activity, and most recently the abnormal deposition of amyloid-beta in brain tissue and the abnormalities of second messenger cascades that likely underlie important neuropathology. The quantification of brain images is vital to the appropriate understanding and interpretation of these experimental and clinical findings. While many brain imaging agents, such as markers of amyloid-beta in dementia, are used with the ultimate goal of application to clinical prognostication and differential diagnosis, others will be fundamental research tools for understanding new drugs, such as antipsychotics, antidepressants, and anxiolytics, as well as drugs for relief of devastating neurological disorders such as multiple sclerosis, stroke, and dementia. This chapter provides a brief introduction to some of the quantitative methods of understanding brain work and brain functions that neuroscientists developed in the last 30 years, and it highlights their importance to future tests of treatment. Here, an overall description of the basic elements of quantification, and, in particular, mathematical modeling of dynamic brain images, is presented both to justify the role of such modeling in initial study development, and to validate specifications for use in clinical settings. Quantification and kinetic modeling are just as important as image reconstruction and structural identification of regions of interest, and they are fundamental components of all new brain imaging tools. The quantitative methods presented in this brief introduction continue to underpin the routine approaches and hence matter to most clinicians and clinician scientists involved in brain imaging.