Mathematical Modeling and the Quantification of Brain Dynamics

  • Albert GjeddeEmail author
  • Dean F. Wong
Part of the Neuromethods book series (NM, volume 71)


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.

Key words

Binding potential Clearance Kinetics Neuroimaging Quantitative analysis Neuroreceptor mapping 



Global Excellence Award 2010, Capital Region, Denmark (Gjedde). NIH-NIDA midcareer award K24 DA000412 (Wong). Special thanks for technical assistance to Ayon Nandi, MS; and Rebecca Mellinger-Pilgram, BS, Johns Hopkins University.


  1. 1.
    Gjedde A, Bauer WR, Wong DF (2011) Neurokinetics: The dynamics of neurobiology in vivo. Springer, New YorkGoogle Scholar
  2. 2.
    Kuikka JT et al (1991) Mathematical modelling in nuclear medicine. Eur J Nucl Med 18(5):351–362PubMedCrossRefGoogle Scholar
  3. 3.
    Sheppard CW (1948) The theory of the study of transfers within a multi-compartment system. J Appl Phys 19(70)Google Scholar
  4. 4.
    Rescigno A, Beck J (1972) Compartments. In: Rosen R (ed) Foundations of mathematical biology, 1st edn. Academic, New York, pp 255–322Google Scholar
  5. 5.
    Rescigno A, Beck JS (1987) The use and abuse of models. J Pharmacokinet Biopharm 15(3):327–344PubMedGoogle Scholar
  6. 6.
    Gjedde A (1980) Rapid steady-state analysis of blood-brain glucose transfer in rat. Acta Physiol Scand 108(4):331–339PubMedCrossRefGoogle Scholar
  7. 7.
    Gjedde A (2008) Functional brain imaging celebrates 30th anniversary. Acta Neurol Scand 117(4):219–223PubMedCrossRefGoogle Scholar
  8. 8.
    Kety SS, Schmidt CF (1948) The nitrous oxide method for the quantitative determination of cerebral blood flow in man; theory, procedure and normal values. J Clin Invest 27(4):476–483CrossRefGoogle Scholar
  9. 9.
    Raichle ME et al (1983) Brain blood flow measured with intravenous H2(15)O. II. Implementation and validation. J Nucl Med 24(9):790–798PubMedGoogle Scholar
  10. 10.
    Ohta S et al (1996) Cerebral [15O]water clearance in humans determined by PET: I. Theory and normal values. J Cereb Blood Flow Metab 16(5):765–780PubMedCrossRefGoogle Scholar
  11. 11.
    Gjedde A (1981) High- and low-affinity transport of d-glucose from blood to brain. J Neurochem 36(4):1463–1471PubMedCrossRefGoogle Scholar
  12. 12.
    Ter-Pogossian MM et al (1970) The measure in vivo of regional cerebral oxygen utilization by means of oxyhemoglobin labeled with radioactive oxygen-15. J Clin Invest 49(2):381–391PubMedCrossRefGoogle Scholar
  13. 13.
    Ohta S et al (1992) Oxygen consumption of the living human brain measured after a single inhalation of positron emitting oxygen. J Cereb Blood Flow Metab 12(2):179–192PubMedCrossRefGoogle Scholar
  14. 14.
    Sokoloff L et al (1977) The [14C]deoxyglucose method for the measurement of local cerebral glucose utilization: theory, procedure, and normal values in the conscious and anesthetized albino rat. J Neurochem 28(5):897–916PubMedCrossRefGoogle Scholar
  15. 15.
    Gjedde A (1982) Calculation of cerebral glucose phosphorylation from brain uptake of glucose analogs in vivo: a re-examination. Brain Res 257(2):237–274PubMedGoogle Scholar
  16. 16.
    Gjedde A et al (1985) Comparative regional analysis of 2-fluorodeoxyglucose and methylglucose uptake in brain of four stroke patients. With special reference to the regional estimation of the lumped constant. J Cereb Blood Flow Metab 5(2):163–178PubMedCrossRefGoogle Scholar
  17. 17.
    Reivich M et al (1979) The [18F]fluorodeoxyglucose method for the measurement of local cerebral glucose utilization in man. Circ Res 44(1):127–137PubMedCrossRefGoogle Scholar
  18. 18.
    Bass L et al (2011) Analogue tracers and lumped constant in capillary beds. J Theor Biol 285(1):177–181PubMedCrossRefGoogle Scholar
  19. 19.
    Hasselbalch SG et al (2001) The [18F]fluorodeoxyglucose lumped constant determined in human brain from extraction fractions of [18F]F-fluorodeoxyglucose and glucose. J Cereb Blood Flow Metab 21(8):995–1002PubMedCrossRefGoogle Scholar
  20. 20.
    Kuwabara H, Evans AC, Gjedde A (1990) Michaelis-Menten constraints improved cerebral glucose metabolism and regional lumped constant measurements with [18F]fluorodeoxyglucose. J Cereb Blood Flow Metab 10(2):180–189PubMedCrossRefGoogle Scholar
  21. 21.
    Garnett ES, Firnau G, Nahmias C (1983) Dopamine visualized in the basal ganglia of living man. Nature 305(5930):137–138PubMedCrossRefGoogle Scholar
  22. 22.
    Gjedde A et al (1991) Dopa decarboxylase activity of the living human brain. Proc Natl Acad Sci U S A 88(7):2721–2725PubMedCrossRefGoogle Scholar
  23. 23.
    Kumakura Y et al (2005) PET studies of net blood-brain clearance of FDOPA to human brain: age-dependent decline of [18F]fluorodopamine storage capacity. J Cereb Blood Flow Metab 25(7):807–819PubMedCrossRefGoogle Scholar
  24. 24.
    Wagner HN Jr et al (1983) Imaging dopamine receptors in the human brain by positron tomography. Science 221(4617):1264–1266PubMedCrossRefGoogle Scholar
  25. 25.
    Wong DF et al (1984) Effects of age on dopamine and serotonin receptors measured by positron tomography in the living human brain. Science 226(4681):1393–1396PubMedCrossRefGoogle Scholar
  26. 26.
    Wong DF et al (1997) Quantification of neuroreceptors in the living human brain: III. D2-like dopamine receptors: theory, validation, and changes during normal aging. J Cereb Blood Flow Metab 17(3):316–330PubMedCrossRefGoogle Scholar
  27. 27.
    Wong DF, Gjedde A, Wagner HN Jr (1986) Quantification of neuroreceptors in the living human brain. I. Irreversible binding of ligands. J Cereb Blood Flow Metab 6(2):137–146PubMedCrossRefGoogle Scholar
  28. 28.
    Wong DF et al (1986) Quantification of Neuroreceptors in the living human brain. II. Inhibition studies of receptor density and affinity. J Cereb Blood Flow Metab 6(2):147–153PubMedCrossRefGoogle Scholar
  29. 29.
    Gjedde A, Wong DF (2001) Quantification of neuroreceptors in living human brain. v. endogenous neurotransmitter inhibition of haloperidol binding in psychosis. J Cereb Blood Flow Metab 21(8):982–994PubMedCrossRefGoogle Scholar
  30. 30.
    Mintun MA et al (1984) A quantitative model for the in vivo assessment of drug binding sites with positron emission tomography. Ann Neurol 15(3):217–227PubMedCrossRefGoogle Scholar
  31. 31.
    Farde L et al (1986) Quantitative analysis of D2 dopamine receptor binding in the living human brain by PET. Science 231(4735):258–261PubMedCrossRefGoogle Scholar
  32. 32.
    Gjedde A et al (2005) Mapping neuroreceptors at work: on the definition and interpretation of binding potentials after 20 years of progress. Int Rev Neurobiol 63(1):1–20PubMedCrossRefGoogle Scholar
  33. 33.
    Kuhar MJ et al (1978) Dopamine receptor binding in vivo: the feasibility of autoradiographic studies. Life Sci 22(2):203–210PubMedCrossRefGoogle Scholar
  34. 34.
    Innis RB et al (2007) Consensus nomenclature for in vivo imaging of reversibly binding radioligands. J Cereb Blood Flow Metab 27(9):1533–1539PubMedCrossRefGoogle Scholar
  35. 35.
    Wong DF et al (1997) Quantification of neuroreceptors in the living human brain: IV Effect of aging and elevations of D2-like receptors in schizophrenia and bipolar illness. J Cereb Blood Flow Metab 17(3):331–342PubMedCrossRefGoogle Scholar
  36. 36.
    Wong DF et al (1998) Quantification of extracellular dopamine release in schizophrenia and cocaine use by means of TREMBLE. In: Carson RE, Herscovitch P, Daube-Witherspoon ME (eds) Quantitative functional brain imaging with positron emission tomography, 1st edn. Academic, San Diego, pp 463–468CrossRefGoogle Scholar
  37. 37.
    Gjedde A et al (2010) Inverted-U-shaped correlation between dopamine receptor availability in striatum and sensation seeking. Proc Natl Acad Sci U S A 107(8):3870–3875PubMedCrossRefGoogle Scholar
  38. 38.
    Koepp MJ et al (1998) Evidence for striatal dopamine release during a video game. Nature 393(6682):266–268PubMedCrossRefGoogle Scholar
  39. 39.
    Wong DF et al (2006) Increased occupancy of dopamine receptors in human striatum during cue-elicited cocaine craving. Neuropsychopharmacology 31(12):2716–2727PubMedCrossRefGoogle Scholar
  40. 40.
    Wong DF et al (2008) Mechanisms of dopaminergic and serotonergic neurotransmission in Tourette syndrome: clues from an in vivo neurochemistry study with PET. Neuropsychopharmacology 33(6):1239–1251PubMedCrossRefGoogle Scholar
  41. 41.
    Laruelle M, Abi-Dargham A (1999) Dopamine as the wind of the psychotic fire: new evidence from brain imaging studies. J Psychopharmacol 13(4):358–371PubMedCrossRefGoogle Scholar
  42. 42.
    McConathy J, Kilts CD, Goodman MM (2001) Radioligands for PET and SPECT imaging of the central noradrenergic system. CNS Spectr 6(8):704–709PubMedGoogle Scholar
  43. 43.
    Scott DJ et al (2007) Time-course of change in [11C]carfentanil and [11C]raclopride binding potential after a nonpharmacological challenge. Synapse 61(9):707–714PubMedCrossRefGoogle Scholar
  44. 44.
    Maarrawi J et al (2007) Motor cortex stimulation for pain control induces changes in the endogenous opioid system. Neurology 69(9):827–834PubMedCrossRefGoogle Scholar
  45. 45.
    Laruelle M et al (1997) Imaging D2 receptor occupancy by endogenous dopamine in humans. Neuropsychopharmacology 17(3):162–174PubMedCrossRefGoogle Scholar
  46. 46.
    Yokoi F et al (2002) Dopamine D2 and D3 receptor occupancy in normal humans treated with the antipsychotic drug aripiprazole (OPC 14597): a study using positron emission tomography and [11C]raclopride. Neuropsychopharmacology 27(2):248–259PubMedCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  1. 1.Department of Neuroscience and PharmacologyUniversity of CopenhagenCopenhagenDenmark
  2. 2.Departments of Radiology, Psychiatry, Neuroscience, and Environmental Health Sciences, Carey Business SchoolJohns Hopkins UniversityBaltimoreUSA
  3. 3.NeuroimagingUniversity of CopenhagenCopenhagenDenmark

Personalised recommendations