New Paradigm for Modeling Growth
Modeling growth is, naturally, of great interest to biomedical engineers. After all, most of the materials that we deal with in the body are tissues that are constantly growing and shrinking. Currently, growth is usually modeled by one-to-one mapping, with each 'point' in the body of interest referenced to its original location. This works well for stretching and shrinking; however, a major difficulty arises when the body adds or loses material of interest. For instance, as an ice cube freezes, water is added to the solid. This growth of the ice cube cannot be modeled by one-to-one mapping because a point did not necessarily exist at time = 0.
To remedy this, many researchers have included correction factors; often that only make the model match a particular application. However, there appears to be a shift in modeling growth (in which our own Dr. Humphrey is involved) in which growth is tracked spatially. In other words, the point is referenced to its current position, and the body is seen as a mixture of the states that can be present. To go back to our ice cube example, the entire area/volume will be seen as having present both ice and water. A variable density gradient can then be used at each point that will tell you what is present -- if a point is ice, it's density gradient will equal the density gradient for ice, if it is water, it will equal water, while if it is in the process of freezing, it will be somewhere between the two.
This new approach to modeling growth is generally called 'mixture theory' because it views a tissue as a 'mixture' of the solid and liquid states. For more information (at a much deeper level!), visit the links below. The first one goes through the mathematics for this approach, and is from a professor at Columbia University. The second one is co-authored by Dr. Humphrey, and applies this type of theory to a biomechanical problem.
http://www.columbia.edu/~ga29/pdf/AteshianBMMB07b.pdf
http://content.karger.com/ProdukteDB/produkte.asp?Doi=80699 (must access via A&M subscription)
To remedy this, many researchers have included correction factors; often that only make the model match a particular application. However, there appears to be a shift in modeling growth (in which our own Dr. Humphrey is involved) in which growth is tracked spatially. In other words, the point is referenced to its current position, and the body is seen as a mixture of the states that can be present. To go back to our ice cube example, the entire area/volume will be seen as having present both ice and water. A variable density gradient can then be used at each point that will tell you what is present -- if a point is ice, it's density gradient will equal the density gradient for ice, if it is water, it will equal water, while if it is in the process of freezing, it will be somewhere between the two.
This new approach to modeling growth is generally called 'mixture theory' because it views a tissue as a 'mixture' of the solid and liquid states. For more information (at a much deeper level!), visit the links below. The first one goes through the mathematics for this approach, and is from a professor at Columbia University. The second one is co-authored by Dr. Humphrey, and applies this type of theory to a biomechanical problem.
http://www.columbia.edu/~ga29/pdf/AteshianBMMB07b.pdf
http://content.karger.com/ProdukteDB/produkte.asp?Doi=80699 (must access via A&M subscription)
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