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Project 2 Beginning with the seven-compartment model developed by Lakin et. al. [1], the following improvements were made: The cerebrospinal fluid compartment was split into two compartments: ventricular CSF, and subarachnoid CSF. The capillary compartment was split into two compartments: the capillary bed and the choroid plexus, which is the site of CSF production. The subclavicular portion of the body is represented as four compartments: the spinal theca, the extracranial veins and the extracranial arteries (with a heart pump driving the blood flow), and a "rest of body" compartment. CSF flow is allowed between the subarachnoid CSF and the spinal theca. This breaks with the "Kellie-Monroe doctrine" (intracranial volume remains constant) and allows for small but important volume adjustments between the intra- and extracranial compartments. This yields a system of thirteen ordinary differential equations in the thirteen compartmental pressures. Autoregulation of the type developed in [2] is then incorporated into the model, so as to maintain constant blood flow into the capillary bed and the choroid plexus. Also, constant production of CSF is built into the model, which corresponds well with available data. At this point, the system has the form: C dP/dt + Z P = Q where C and Z are 13 by 13 matrices of compliances and fluidities respectively; P and Q are 13 component vectors of pressures and flows. The system, is in fact, nonlinear since the compliances are pressure-dependent. The rank of C is 8, and the rank of Z is 12, so simplification proceeds as follows. Row operations are devised to reduce C to as close to upper triangular form as possible. . Then, making use of these row operations, the whole system is reduced to a 12 by 12 system . However, 4 of the 12 equations are algebraic, and one can solve for four of the pressures in terms of the other eight. By replacing these four variables, and moving certain pressure terms to the right hand side of the equation to act as forcing, one reduces the system to an 8 by 8 system of the same form as above. Based on the respiratory mechanics involved, it appears appropriate to use Po , the pressure in the extracranial venous compartment, as a forcing term, since the expansion and contraction of the thoracic space in respiration causes pressure changes at this site. In fact, clinical evidence suggests that these pressure changes also affect the rate at which blood is pumped by the heart, since the ability of the heart to fill is affected by the venous pressure. Both of these effects will introduce forcing with the respiratory frequency, which is incommensurate with the cardiac frequency; this is expected to result in a class of solutions known as "almost periodic" which correspond to clinically observed data. Work that remains in order to implement this model is calibration of the scale values of the resistances and compliances, determination of the precise functional forms to be used for the pressure-dependent compliances as well as the extracranial venous pressure forcing. This will proceed in consultation with a neurologist and cardiologist participating in the project. Implementation of the model on Maple or Matlab can then proceed. The further goal is to develop the model to the point at which it is sufficiently realistic to allow the study of the effects of microgravity on cerebrovascular function, and to establish links with researchers at the NASA facility at Ames to facilitate such studies. Dr. Kadas, the Principal Investigator on this project, is Professor of Mathematics at Saint Michael's College in Colchester, Vermont. Her areas of expertise include biomathematics and modeling. [1] Lakin, W.D., Yu, J., Penar, P. (1996) Mathematical modeling of human intracranial Pressure Dynamics. Nova J. Mathematics, Algebra, Game Theory 5:103-130 [2] Kadas, Z.M., Lakin, W.D., Yu, J., Penar, P. (1997) A mathematical model of the intracranial system including autoregulation. Neurological Research 19 441-450 |