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Faculty of Science - Mahidol University

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<strong>Mahidol</strong> <strong>University</strong> Annual Research Abstracts, Vol. 33 297<br />

neuroendocrine ensemble exhibits prominent pulsatile secretory<br />

patterns governed by nonlinear and time-delayed feedforward and<br />

feedback signal interchanges. Formulation and subsequent<br />

bifurcation analysis <strong>of</strong> the model provide a qualitative and<br />

mathematical frame work for a better understanding <strong>of</strong> the delayed<br />

responsive mechanisms as well as the dynamic variations in different<br />

pathological situations.<br />

(Mathematical Medicine and Biology, 22(1)(2005) 15-33.)<br />

MATHEMATICAL MODELS FOR PRESSURE<br />

CONTROLLED VENTILATION OF OLEIC ACID-<br />

INJURED PIGS (NO. 777)<br />

Crooke P.S. 1 , Kongkul K. 2 , Lenbury Y. 2 , Adams A.B. 3 , Carter<br />

C.S. 3 , Marini J.J. 3 , Hotchkiss J.R. 3<br />

1 Department <strong>of</strong> Mathematics, Vanderbilt <strong>University</strong>, Nashville,<br />

TN 37240, USA. E-mail : pscrooke@math.vanderbilt.edu;<br />

2 Department <strong>of</strong> Mathematics, <strong>Faculty</strong> <strong>of</strong> <strong>Science</strong>, <strong>Mahidol</strong><br />

<strong>University</strong>, Bangkok 10400, Thailand. E-mail :<br />

konvika@tsu.ac.th, scylb@mahidol.ac.th; 3 Pulmonary and<br />

Critical Care Medicine, Regions Hospital and <strong>University</strong> <strong>of</strong><br />

Minnesota, St. Paul MN 55101 USA. E-mail :<br />

hotchkissjr@upmc.edu<br />

Key words : oleic acid injury, mathematical model, variable<br />

compliance<br />

One-compartment, mathematical models for pressure<br />

controlled ventilation, incorporating volume dependent<br />

compliances, linear and nonlinear resistances, are constructed and<br />

compared with data obtained from healthy and (oleic acid) lunginjured<br />

pigs. Experimental data are used to find parameters in the<br />

mathematical models and were collected in two forms. Firstly, the<br />

Pe-V curves for healthy and lung injured pigs were constructed;<br />

these dta are used to compute compliance functions for each animal.<br />

Secondly, dynamic data from pressure controlled ventilation for a<br />

variety <strong>of</strong> applied pressures are used to estimate resistance parameters<br />

in the models. The models were then compared against the collected<br />

dynamic data. The best mathematical models are ones with<br />

compliance functions <strong>of</strong> the form C(V) = a + bV where a and b are<br />

constants obtained from the Pe-V curves and the resistive pressures<br />

during inspiration change from a linear relation Pr = RQ to nonlinear<br />

relation Pr = RQ where Q is the flow into the one-compartment lung<br />

and is a positive number. The form <strong>of</strong> the resistance terms in the<br />

mathematical models indicate the possible presence <strong>of</strong> gas-liquid<br />

foams in the experimental data.<br />

(Mathematical Medicine and Biology 22 (2005) 99-112)<br />

DELAY EFFECT IN MODELS OF POPULATION<br />

GROWTH (NO. 778)<br />

Giang D.V. 1 , Lenbury Y. 2 , Seidman T.I. 3<br />

1 Hanoi Insititue <strong>of</strong> Mathematics, 18 Hoang Quoc Viet, 10307<br />

Hanoi, Vietnam, E-mai dvgiang@math.ac.vn ; 2 Department <strong>of</strong><br />

Mathematics, <strong>Faculty</strong> <strong>of</strong> <strong>Science</strong>, <strong>Mahidol</strong> <strong>University</strong>, Bangkok<br />

10400, Thailand. E-mail : scylb@mahidol.ac.th; 3 Department<br />

<strong>of</strong> Mathematics and Statistics, <strong>University</strong> <strong>of</strong> Maryland Baltimore<br />

County, Baltimore, MD 21250 USA. E-mail<br />

Seidman@math.umbc.edu<br />

Key words : Delay differential equations, Comparison theorem Oneparameter<br />

semi-group<br />

Frist we systematize earlier results on the global stability<br />

<strong>of</strong> the model x + μx = f(x(‘ – τ)) <strong>of</strong> population growth. Second, we<br />

investigate the effect <strong>of</strong> delay on the asymptotic behavior when the<br />

nonlinearity f is a unimodal function. Our results can be applied to<br />

several population models [Elements <strong>of</strong> Mathematical Ecology, 2001<br />

[7]; Appl. Anal. 43 (1992) 109-124; Math. Comput. Modelling, in<br />

press; Funkt. Boil. Med. 256 (1982) 156-164; Math. Comput.<br />

Modelling 35 (2002) 719-731; Mat. Stos. 6 (1976) 25-40] because<br />

the function f does not need to be monotone or differentiable.<br />

Specifically, our results generalize earlier result <strong>of</strong> [Delay Differential<br />

Equations with Applications in Population Dynamics 1993], since<br />

our function f may not be differentiable.<br />

(Journal <strong>of</strong> Mathematical Analysis and Applications 305 (2005)<br />

631-643.)<br />

THREE-DIMENSION MAGNETOTELLURIC<br />

INVASION : DATA SPACE METHOD (NO. 779)<br />

Siripunvaraporn W. 1 , Egbert G. 2 , Lenbury Y. 3 , Uyeshima M. 4<br />

1 Department <strong>of</strong> Physics, <strong>Faculty</strong> <strong>of</strong> <strong>Science</strong>, <strong>Mahidol</strong> <strong>University</strong>,<br />

Bangkok 10400, Thailand. E-mail : scwsp@mahidol.ac.th;<br />

2 College <strong>of</strong> Oceanic and Atmospheric <strong>Science</strong>s, Oregon State<br />

<strong>University</strong>, Corvallis OR 97331, USA; 3 Department <strong>of</strong><br />

Mathematics <strong>Faculty</strong> <strong>of</strong> <strong>Science</strong>, <strong>Mahidol</strong> <strong>University</strong>, Bangkok<br />

10400, Thailand. E-mail scylb@mahidol.ac.th; 4 Earthquake<br />

Research Insitute, <strong>University</strong> <strong>of</strong> Tokyo, 1-1-1 Yayoi, Bunkyoku,<br />

Tokyo 113-0032, Japan.<br />

Key words : Magnetotellurics, Data-space method, 3D inversion,<br />

Occam’s inversion<br />

A three-dimensional magnetotelluric (MT) minimum<br />

structure inversion algorithm has been developed based on a dataspace<br />

variant <strong>of</strong> the Occam approach. Computational costs associated<br />

with construction and inversion <strong>of</strong> model-space matrices make a<br />

model-space Occam approach to 3D MT inversion impractical. These<br />

difficulties are overcome with a data-space approach,where matrix<br />

dimensions depend on the size <strong>of</strong> the data set, rather than the number<br />

<strong>of</strong> model parameters. With the transformation to data space it<br />

becomes feasible to invert modest 3D MT data sets on a PC. To<br />

reduce computational time, a relaxed convergence criterion is used<br />

for the iterative forward modeling code used to compute the<br />

sensitivity matrix. This allows reduction in computational time by<br />

more than 70% without affecting the inversion results. Numerical<br />

experiments with synthetic data show that reasonable fits can be<br />

obtained within a small number <strong>of</strong> iterations, with a few additional<br />

iterations required to eliminate unnecessary structure and find the<br />

model with minimum norm.<br />

(Physics <strong>of</strong> the Earth and Planetary Interiors. 150 (2005) 3-14.)<br />

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