13 Introduction
Motivation
Chronic wounds persist when the tissue in the wound bed cannot sustain the coordinated cell activity required for repair (Marston 2004). In diabetic foot ulcers, wounds can remain open for weeks or months under standard care and can progress to infection and limb-threatening complications (Marston 2004). A dermagraft is a living dermal substitute in which human fibroblasts are seeded onto a bioabsorbable scaffold and delivered to the wound as an organized tissue construct rather than as isolated cells (Marston 2004). In a multicenter randomized trial, Dermagraft improved complete wound closure at 12 weeks relative to standard care alone (Marston et al. 2003).
The dermis is a vascularized connective tissue layer populated by fibroblasts and supported by extracellular matrix, whereas the engineered scaffold used to culture a substitute such as Dermagraft is initially avascular (Betts et al. 2022; Marston 2004). Oxygen must therefore reach scaffold-seeded cells by diffusion through fluid-filled pores (Phillips et al. 2012; Milo et al. 2010). A representative oxygen diffusion coefficient in water is on the order of \(2 \times 10^{-9}\ \mathrm{m^2/s}\) (Milo et al. 2010). That is large enough to sustain transport across cellular distances but small enough that millimeter-scale geometry becomes a design constraint. The tissue-engineering question for this module asks: given a cell oxygen-consumption rate and a scaffold diffusion coefficient, what is the maximum scaffold thickness that keeps oxygen high enough for cells to remain viable everywhere in the construct?
Learning Objectives
- Relate microscopic random motion to a macroscopic diffusion coefficient and Fick’s laws of transport.
- Apply a mass balance to a solute to model transport.
- Use a reaction-diffusion model to predict oxygen profiles and the maximum viable scaffold thickness.
Before You Start
This section assumes students have experience with derivatives and integrals. If this isn’t the case, don’t stress. You’ll be building calculus fluency in your math classes this year, and the calculus appendix covers the derivatives and integrals used here.
What You’ll Learn
By the end of the module, a student should be able to do the following.
- Understand the concept of flux and apply Fick’s first law to compute the steady flux through a slab.
- Estimate how long a graft takes to respond to a change in its surroundings using the characteristic diffusion time \(\tau = L^2/(2D)\).
- Explain steady state and zero-order consumption, and choose the boundary conditions that fit a graft on a sealed base.
- Simplify the reaction-diffusion equation to obtain a steady-state one-dimensional oxygen profile in a scaffold.
- Compute the maximum scaffold thickness consistent with a minimum viable oxygen concentration.