Our recently described nonlinear dynamical model of cell injury is here applied to the problems of brain ischemia and neuroprotection. We discuss measurement of global brain ischemia injury dynamics by time course analysis. Solutions to proposed experiments are simulated using hypothetical values for the model parameters. The solutions solve the global brain ischemia problem in terms of “master bifurcation diagrams” that show all possible outcomes for arbitrary durations of all lethal cerebral blood flow (CBF) decrements. The global ischemia master bifurcation diagrams: (1) can map to a single focal ischemia insult, and (2) reveal all CBF decrements susceptible to neuroprotection. We simulate measuring a neuroprotectant by time course analysis, which revealed emergent nonlinear effects that set dynamical limits on neuroprotection. Using over-simplified stroke geometry, we calculate a theoretical maximum protection of approximately 50% recovery. We also calculate what is likely to be obtained in practice and obtain 38% recovery; a number close to that often reported in the literature. The hypothetical examples studied here illustrate the use of the nonlinear cell injury model as a fresh avenue of approach that has the potential, not only to solve the brain ischemia problem, but also to advance the technology of neuroprotection.
References
[1]
Leibniz, G.W. Note H to Bayle’s article ‘Rorarius’ (1697). In Philosophical Texts; Woolhouse, R., Francks, R., Eds.; Oxford University Press: New York, NY, USA, 1998.
Seidl, S.E.; Potashkin, J.A. The promise of neuroprotective agents in Parkinson’s disease. Front. Neurol. 2011, 2, 68.
[4]
O’Collins, V.E.; Macleod, M.R.; Donnan, G.A.; Horky, L.L.; van der Worp, B.H.; Howells, D.W. 1,026 experimental treatments in acute stroke. Ann. Neurol. 2006, 59, 467–477, doi:10.1002/ana.20741.
[5]
Zivin, J.A. Acute stroke therapy with tissue plasminogen activator (tPA) since it was approved by the U.S. Food and Drug Administration (FDA). Ann. Neurol. 2009, 66, 6–10, doi:10.1002/ana.21750.
[6]
Macleod, M.R.; Petersson, J.; Norrving, B.; Hacke, W.; Dirnagl, U.; Wagner, M.; Schwab, S. European Hypothermia Stroke Research Workshop. Hypothermia for Stroke: Call to action 2010. Int. J. Stroke 2010, 5, 489–492, doi:10.1111/j.1747-4949.2010.00520.x.
[7]
Cheng, Y.D.; Al-Khoury, L.; Zivin, J.A. Neuroprotection for ischemic stroke: Two decades of success and failure. NeuroRx 2004, 1, 36–45, doi:10.1602/neurorx.1.1.36.
[8]
Van der Worp, H.B.; de Haan, P.; Morrema, E.; Kalkman, C.J. Methodological quality of animal studies on neuroprotection in focal cerebral ischaemia. J. Neurol. 2005, 252, 1108–1114, doi:10.1007/s00415-005-0802-3.
[9]
Savitz, S.I.; Fisher, M. Future of neuroprotection for acute stroke: In the aftermath of the SAINT trials. Ann. Neurol. 2007, 61, 396–402.
[10]
O’Collins, V.E.; Macleod, M.R.; Cox, S.F.; van Raay, L.; Aleksoska, E.; Donnan, G.A.; Howells, D.W. Preclinical drug evaluation for combination therapy in acute stroke using systematic review, meta-analysis, and subsequent experimental testing. J. Cereb. Blood Flow Metab. 2011, 31, 962–975, doi:10.1038/jcbfm.2010.184.
[11]
Macrae, I.M. Preclinical stroke research—advantages and disadvantages of the most common rodent models of focal ischaemia. Br. J. Pharmacol. 2011, 164, 1062–1078, doi:10.1111/j.1476-5381.2011.01398.x.
[12]
DeGracia, D.J.; Huang, Z.F.; Huang, S. A nonlinear dynamical theory of cell injury. J. Cereb. Blood Flow Metab. 2012, 32, 1000–1013, doi:10.1038/jcbfm.2012.10.
[13]
DeGracia, D.J. Towards a dynamical network view of brain ischemia and reperfusion. Part I: Background and preliminaries. J. Exp. Stroke Transl. Med. 2010, 3, 59–71, doi:10.6030/1939-067X-3.1.59.
[14]
DeGracia, D.J. Towards a dynamical network view of brain ischemia and reperfusion. Part II: A post-ischemic neuronal state space. J. Exp. Stroke Transl. Med. 2010, 3, 72–89, doi:10.6030/1939-067X-3.1.72.
[15]
DeGracia, D.J. Towards a dynamical network view of brain ischemia and reperfusion. Part III: Therapeutic implications. J. Exp. Stroke Transl. Med. 2010, 3, 90–103, doi:10.6030/1939-067X-3.1.90.
[16]
DeGracia, D.J. Towards a dynamical network view of brain ischemia and reperfusion. Part IV: Additional considerations. J. Exp. Stroke Transl. Med. 2010, 3, 104–114, doi:10.6030/1939-067X-3.1.104.
Kaplan, B.; Brint, S.; Tanabe, J.; Jacewicz, M.; Wang, X.J.; Pulsinelli, W. Temporal thresholds for neocortical infarction in rats subjected to reversible focal cerebral ischemia. Stroke 1991, 22, 1032–1039, doi:10.1161/01.STR.22.8.1032.
[21]
Heiss, W.D. Experimental evidence of ischemic thresholds and functional recovery. Stroke 1992, 23, 1668–1672, doi:10.1161/01.STR.23.11.1668.
[22]
Heiss, W.D. Flow thresholds of functional and morphological damage of brain tissue. Stroke 1983, 14, 329–331.
[23]
Rossolini, G.; Piantanelli, L. Mathematical modeling of the aging processes and the mechanisms of mortality: Paramount role of heterogeneity. Exp. Gerontol. 2001, 36, 1277–1288, doi:10.1016/S0531-5565(01)00092-4.
[24]
Kirino, T. Delayed neuronal death. Neuropathology 2000, 20, S95–S97, doi:10.1046/j.1440-1789.2000.00306.x.
[25]
Bodine-Fowler, S. Skeletal muscle regeneration after injury: An overview. J. Voice 1994, 8, 53–62.
[26]
Hertz, L.; Dienel, G.A. Energy metabolism in the brain. Int. Rev. Neurobiol. 2002, 51, 1–102.
[27]
Mainzer, K. Thinking in Complexity: The Computational Dynamics of Matter, Mind, and Mankind, 5th ed.; Springer: Berlin, Germany, 2007.
[28]
Glass, L.; Kaplan, D. Time series analysis of complex dynamics in physiology and medicine. Med. Prog. Technol. 1993, 19, 115–128.
[29]
Gourévitch, B.; Bouquin-Jeannès, R.L.; Faucon, G. Linear and nonlinear causality between signals: Methods, examples and neurophysiological applications. Biol. Cybern. 2006, 95, 349–369.
[30]
Kaplan, D.; Glass, L. Understanding Nonlinear Dynamics; Springer: New York, NY, USA, 1995.
[31]
Pereda, E.; Quiroga, R.Q.; Bhattacharya, J. Nonlinear multivariate analysis of neurophysiological signals. Prog. Neurobiol. 2005, 77, 1–37, doi:10.1016/j.pneurobio.2005.10.003.
[32]
Chou, I.C.; Voit, E.O. Recent developments in parameter estimation and structure identification of biochemical and genomic systems. Math. Biosci. 2009, 219, 57–83.
[33]
Smith, M.L.; Bendek, G.; Dahlgren, N.; Rosen, I.; Wieloch, T.; Siesjo, B.K. Models for studying long-term recovery following forebrain ischemia in the rat. 2. A 2-vessel occlusion model. Acta Neurol. Scand. 1984, 69, 385–401.
[34]
Levenberg, K. A Method for the solution of certain non-linear problems in least squares. Q. Appl. Math. 1944, 2, 164–168.
[35]
Hu, B. Personal communication. University of Maryland: Baltimore, MD, USA, 2012.
[36]
Dienel, G. Personal communication. University of Arkansas: Little Rock, AR, USA, 2012.
[37]
VanGilder, R.L.; Huber, J.D.; Rosen, C.L.; Barr, T.L. The transcriptome of cerebral ischemia. Brain Res. Bull. 2012, 88, 313–319, doi:10.1016/j.brainresbull.2012.02.002.
[38]
Mehra, A.; Lee, K.H.; Hatzimanikatis, V. Insights into the relation between mRNA and protein expression patterns: I. Theoretical considerations. Biotechnol. Bioeng. 2003, 84, 822–833, doi:10.1002/bit.10860.
[39]
Sonenberg, N.; Hershey, J.W.B.; Mathews, M.B. Translational Control of Gene Expression; Cold Spring Harbor Laboratory Press: Cold Spring Harbor, NY, USA, 2000.
DeGracia, D.J.; Jamison, J.T.; Szymanski, J.J.; Lewis, M.K. Translation arrest and ribonomics in post-ischemic brain: Layers and layers of players. J. Neurochem. 2008, 106, 2288–2301, doi:10.1111/j.1471-4159.2008.05561.x.
[42]
Huang, S.; Guo, Y.P.; May, G.; Enver, T. Bifurcation dynamics in lineage-commitment in bipotent progenitor cells. Dev. Biol. 2007, 305, 695–713, doi:10.1016/j.ydbio.2007.02.036.
[43]
Nedergaard, M.; Dirnagl, U. Role of glial cells in cerebral ischemia. Glia 2005, 50, 281–286.
[44]
Del Zoppo, G.J. Virchow’s triad: The vascular basis of cerebral injury. Rev. Neurol. Dis. 2008, 5 (Suppl. 1), S12–S21.
[45]
Lipton, S.A. Similarity of neuronal cell injury and death in AIDS dementia and focal cerebral ischemia: Potential treatment with NMDA open-channel blockers and nitric oxide-related species. Brain Pathol. 1996, 6, 507–517.
[46]
Baron, J.C. How healthy is the acutely reperfused ischemic penumbra? Cerebrovasc. Dis. 2005, 20 (Suppl. 2), 25–31, doi:10.1159/000089354.
Sommer, C. Neuronal plasticity after ischemic preconditioning and TIA-like preconditioning ischemic periods. Acta Neuropathol. 2009, 117, 511–523.
[49]
Hadjiev, D.I.; Mineva, P.P. Transient ischemic attack may present a target for normobaric hyperoxia treatment. Med. Hypotheses 2010, 75, 128–130, doi:10.1016/j.mehy.2010.02.008.
[50]
Powers, W.J.; Zazulia, A.R. The use of positron emission tomography in cerebrovascular disease. Neuroimaging Clin. N. Am. 2003, 13, 741–758.
[51]
Carey, L.M.; Seitz, R.J. Functional neuroimaging in stroke recovery and neurorehabilitation: Conceptual issues and perspectives. Int. J. Stroke 2007, 2, 245–264, doi:10.1111/j.1747-4949.2007.00164.x.
[52]
Kitagawa, K. Ischemic tolerance in the brain: Endogenous adaptive machinery against ischemic stress. J. Neurosci. Res. 2012, 90, 1043–1054, doi:10.1002/jnr.23005.
[53]
Aronowski, J.; Ostrow, P.; Samways, E.; Strong, R.; Zivin, J.A.; Grotta, J.C. Graded bioassay for demonstration of brain rescue from experimental acute ischemia in rats. Stroke 1994, 25, 2235–2240, doi:10.1161/01.STR.25.11.2235.
[54]
Motulsky, H.J.; Ransnas, L.A. Fitting curves to data using nonlinear regression: A practical and nonmathematical review. FASEB J. 1987, 1, 365–374.
