## Modeling Must Be At Several Levels

In modeling a circulation and then shrinking it, we must use a model, because a bee is too small, too complex, and time cannot be controlled. Using the rendering capabilities of computer graphics, we can explore more functional qualities and then change these at will, but still the problems and models continue to be too complicated, so ultimately a working rendition must come down to multi-scale modeling.

Approaches employing traditional, mono-scale modeling have proven themselves to be inadequate, even using large supercomputers, because the ranges of scales and the large number of variables involved are computationally prohibitive. Thus, there is a growing need to develop systematic modeling and simulation approaches for multi-scale problems. We have made some progress. For example, we can compare local changes in patterns of blood flow within a web of vessels with changes occurring within larger portions of the circulation as seen globally. Such a heterogeneous model uses Navier-Stokes equations to describe the three-dimensional flow within a single artery. We then couple a model of this flow to a systemic, zero-dimensional model of the whole circulation. Using this geometrical multi-scale strategy, we have joined an initial boundary value problem to an initial-value problem to predict a change wrought upon a circulation by surgical intervention, such as might occur following blood loss and onset of hypovolemic shock. Such involved models suggest we can obtain useful 'meta' information by matching conditions prevailing in two sub-models within a single numerical simulation.

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