Advanced Biomechanics – Mixture Theory and Biology

Here we take a look at how to apply mixture theory to biological problems. There are a range of possible applications. Some of the classic ones are the mechanics of cartilage, bone, and ligaments, but one might also consider drug delivery, pressure ulcers and traumatic brain injury. Similar methods also apply to chemical contamination of the waterways and oil seepage into groundwater supplies. We are assuming that you have already taken the introductory continuum course or had a similar experience.

Here is the syllabus (Syllabus_Advanced_Cell_Tissue_Mechanics_2019). We may make edits as we go, but this is a good starting point.

This is the presentation from day 1 (Biomechanics_Overview_Small). The goal was to provide a background for why mixture theory is important in biomechanics. In addition to the fact that fluids and solids interact in most tissues, there are a number of transport problems that require it – drug delivery, nutrient transport in the disc, and blood flow in the lamina cribrosa to name a few. It also provides a framework for better understanding materials with microstructure. That allows us to consider the effects of tissue damage, cell signaling in bone, and the electro-mechanical coupling within cardiac tissues.

Day 2: The Fundamentals. We jumped right into the derivation for the Balance of Mass. It required us talking about some basic definitions and some asides where we reviewed some results from Continuum Mechanics. https://youtu.be/Hm98VAFnlkQ

Day 3: The Conflict. Here we looked at the derivation of the balance of linear momentum for a fluid and solid. We did not include chemical reactions (one component changing into the other) but we did introduce momentum transfer between the constituents. https://youtu.be/0OdBKs8Vg4Q

Day 4: Ligaments and Tendons – Day 1. Here we are starting to develop a process for modeling tendons and ligaments. This is our first attempt at properly defining the constituents of the mixture, how they interact, and what kind of results we can obtain by using mixture theory. We started by including collagen fibers, ground substance and plasma/interstitial fluid. This may not be the most useful approach, but it is a good starting point. https://youtu.be/MpAfAm2umqw

Day 5: Ligaments and Tendons – Day 2. Finishing up the general derivation and looking at what kinds of questions you can ask with mixture theory models. https://youtu.be/cN3nEzDzNPI

Day 6: Finishing Ligaments and Tendons and Beginning Liquid Armorhttps://youtu.be/acHrH2uvQcc

This lecture did not go well. Technical difficulties. Similar issues with Day 7. Will come back to that. 

Day 8: Introduction to Fighting Cancer powerpoint. Research_Cancer_v5_small

Here are the classic papers – first author Mary Schuff! Schuff_Mixtures_1_JMB_2012 and Schuff_Mixtures_2_JMB_2012

Day 9: Fighting Cancer https://youtu.be/CEYMsreNYYw and Day 10: https://youtu.be/Wq7kWBdPD-o   More Fighting Cancer. Going over papers by Mary Schuff. Remarkable stuff! This is where we introduce models of diffusion, Taylor dispersion, and derive the real diffusion equation!

 

 

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