Modeling the relative impact of capsular tissue effects on implanted glucose sensor time lag and signal attenuation

Abstract

Little is known mechanistically about why implanted glucose sensors lag behind blood glucose levels in both the time to peak sensor response and the magnitude of peak sensor response. A mathematical model of glucose transport from capillaries through surrounding tissue to the sensor surface was constructed to address how different aspects of the tissue affect glucose transport to an implanted sensor. Physiologically relevant values of capsule diffusion coefficient, capsule porosity, cellular glucose consumption, capsule thickness, and subcutaneous vessel density were used as inputs to create simulated sensor traces that mimic experimental instances of time lag and concentration attenuation relative to a given blood glucose profile. Using logarithmic sensitivity analysis, each parameter was analyzed to study the effect of these variables on both lag and attenuation. Results identify capsule thickness as the strongest determinant of sensor time lag, while subcutaneous vessel density and capsule porosity had the largest effects on attenuation of glucose that reaches the sensor surface. These findings provide mechanistic insight for the rational design of sensor modifications that may alleviate the deleterious consequences of tissue effects on implanted sensor performance.

Fig. 1

a Histological image of a foreign body capsule that has formed around an implanted glucose sensor at 14 days post-implantation. H&E staining demonstrates lack of vasculature (red) relative to cells (blue) within capsule. Image borrowed from [12]. b Masson’s trichrome stain of a cross-section of fibrous tissue 3 weeks post-implantation shows presence of extracellular matrix (blue) aswellascells (red) within capsule tissue. Provides basis for modeling capsule as dense, avascular network composed of inflammatory cells and extracellular matrix. Image borrowed from [13]

Fig. 2

Schematic representation of model for diffusion of glucose through native tissue and the fibrous capsule

Fig. 3

Simulated recreation of experimental glucose sensor traces. Raw blood glucose data (filled circle) was fit to a continuous curve, C_{bg,experimental},(dash-dot line) and then input into the model to give a simulated glucose sensor trace, C_s,model, (dash line). This output was compared in shape to raw sensor data (filled square), which is fit by a solid line, C_{bg,experimental}. The raw sensor output data was converted from nanoamperes to micromolar using an experimentally determined standard curve relating the two, C_{sexperimental} = 276i_{sexperimental} – 1130 (R²=0.9993, data not shown). The simulated trace produced a lag time (Δt_est) and attenuation (ΔC_est) and a shape similar to data in Koschwanez et al. [13]. The lag time and attenuation from [13] are represented in the figure by Δt_exp and ΔC_exp. All parameters used for this study were the same except for vessel density and capsule porosity. Vessel density was increased to 100 cm²/cm³ as a representation of how density has decreased from its subcutaneous value. Capsule porosity was calculated from a Darcy’s Law constant for fibrin matrices to be 0.925, thus representing the loosely organized matrix early in inflammation (data not shown)

Fig. 4

Effect of a capsule diffusion coefficient, b capsule porosity, c vascular density in surrounding native tissue, d cellular uptake of glucose in the capsule, and e the capsule thickness on sensor lag times (solid line) and attenuation (broken line). Results have been normalized to baseline values in Table 2

Fig. 5

a Box and whisker plots of lag sensitivity to five different model parameters. b Box and whisker plots of attenuation sensitivity to five different model parameters. Sensitivities are represented as unitless values to allow for comparison across parameters

Fig. 6

Scatter plot of sensor time lag vs. glucose attenuation for all of the conditions simulated with the transport model. The three variables that showed the greatest sensitivity—capsule thickness, vessel density, and capsule porosity—are represented by the red, green, and blue circles, respectively. The benchmark value of time lag and attenuation calculated using the variables in Table 2 is labeled in magenta. This data point is common to all simulation trials. The open circles clustered around the benchmark value are the results obtained for the less sensitive variables of capsule diffusion coefficient and cellular glucose uptake