# Analytical and Finite Element Modeling of Nanomembranes for Miniaturized, Continuous Hemodialysis

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Experimental Section

#### 2.1. Silicon Nanomembranes

#### 2.2. Analytical Model

**Figure 1.**Diagram of system considered by the analytical models. The system is initially at some constant concentration (depicted as red shading) and becomes less concentrated in a gradient over time. Mathematically, the system is one-dimensional; however, it is shown here with a second dimension to illustrate the relationship between the temporal and spatial evolution of the concentration profile.

#### 2.3. Experimental System and the Finite Element Model

**Figure 2.**Photos, schematic, and COMSOL finite element model of the experimental system. (

**a**) Photo of the experimental system taken from the access side. The dialysate channel passes over the membrane (in the center of the chip) while the blood channel passes underneath it; (

**b**) Photo of the same system taken from the other side. The four squares in the corners of the chip in the photographs are additional regions of membrane, but are not in use during this particular experiment; (

**c**) Schematic of active section of experimental system. The membrane spans 70% of the blood channel’s width, and the dialysate and blood are in counterflow. The dashed line represents the membrane; (

**d**) Snapshot of the finite element model. It is colored with an example heat map representing concentration of a solute as it moves through the system, where the highest concentration is bright red while the lowest is dark blue.

#### 2.4. Prediction of Dialyzer Adequacy

#### 2.5. Design Selection

## 3. Results and Discussion

#### 3.1. Validation of Models

**Figure 3.**A comparison between the analytical model and a simple finite element model incorporating ideal geometry (schematic inset). Though the analytical model is one-dimensional and the finite element model is three-dimensional, they are in near-exact agreement regardless of blood channel height. These data for a solute of hydrodynamic radius 1.5 nanometers and a blood flow rate of 10 millimeters per minute.

**Figure 4.**Comparison between the analytical model, the complex geometry finite element model, and experimental results. Experiments were performed with n = 3 or 4, and error bars represent standard error of the mean. Experiments varying blood channel height used solute with diffusion coefficient $1.6\times {10}^{-6}$ and experiments varying diffusion coefficient used blood channel height of 0.3 mm. Residence time of blood channel fluid near the membrane was 10 s in all cases. The finite element model was highly predictive of experimental results, while the analytical model captured trends but systematically over-predicted clearance.

#### 3.2. Proof-of-Principle Design

**Table 1.**Generation rates $\dot{G}$, physiological concentrations, target clearance rates, and predicted steady-state concentrations during continuous nanomembrane dialysis for selected uremic toxins as well as albumin. "Physiological concentrations" refer to reference concentrations expected of patients in good health. Values may vary from patient-to-patient and serve here only as approximations of averages for a mean body weight of 70 kg. An example target clearance calculation is supplied in Equation (15). Steady-state concentrations are predicted using Equation (7) for $\beta <0.2$ (see Equation (9)) and (11) otherwise.

Plasma Solute | $\dot{G}$ (mmol/h) | Physiological Concentration (mM) | Target Clearance (L/h) | Steady-State Concentration (mM) |
---|---|---|---|---|

$4.6$ [18] | $2.3$ | $4.6$ | ||

Creatinine | $0.58$ [19] | $0.11$ [18] | $5.3$ | $0.29$ |

${\beta}_{2}$-microglobulin | $7.7\times {10}^{-4}$ [20] | $1.5\times {10}^{-4}$ [21] | $5.1$ | $1.5\times {10}^{-2}$ |

Albumin | $1.0\times {10}^{-2}$ [22] | $0.65$ [18] | $1.5\times {10}^{-2}$ | $0.65$ |

**Figure 5.**Dialyzer optimization process. Volume per time values in plots (

**a**–

**c**) are products of the volumetric flow rate of blood and the fractional clearance of urea, albumin, and ${\beta}_{2}$-microglobulin, respectively. Curves shown are polynomial fits used to express all values as a function of channel height only. (

**a**) Isolines of urea clearance with changing blood channel height and blood residence time; line B (green) is the isoline where urea clearance is at the desired value; (

**b**) Isocurves of albumin clearance with changing channel height and pore size, with residence time calculated as a function of channel height in order to maintain constant urea clearance. Curve D (green) is the desired curve in this plot; (

**c**) ${\beta}_{2}$-microglobulin clearance as a function of channel height with constant urea and albumin clearances. The shape of this relationship is complex owing to the simultaneous change of three separate variables represented by changes in blood channel height (channel height itself, in addition to blood residence time and pore size); (

**d**) Final proof-of-principle design selection. While a channel height of 0.05 millimeters does not correspond to the global maximum of ${\beta}_{2}$-microglobulin clearance, it is a good compromise between clearance and practicality.

#### 3.3. Finite Element Models of Practical Designs

**Figure 6.**Depictions of the practical designs referenced in Table 2. (

**a**) The trench chip design, consisting of a series of triangular channels etched under a continuous membrane. Sixty-one individual channels are fed by a single inlet and empty into a single outlet, and 11 such chips would be required for a total active membrane area of 9 square centimeters; (

**b**) The trapezoidal chip design, incorporating 13 regularly spaced membranes instead of a single continuous membrane. Fourteen chips are needed to reach 9 square centimeters of membrane area. While in the original design used by Johnson et al. the trapezoidal windows were the blood channels with dialysate on the other side, this was reversed here due to the need for such thin blood channels; (

**c**) The lift-off design. This free-standing membrane can be fabricated up to approximately 9 square centimeters in a single sheet and would need to be held up by periodically placed supports; however, these would not affect the clearance characteristics.

**Table 2.**Comparison between target physiological concentrations of various toxins and steady-state concentration results from three-dimensional finite element models of practical geometries depicted in Figure 6. Values are steady state concentrations in mM. While the trapezoidal and lift-off designs have too much clearance of albumin and not enough of smaller solutes, the trench design achieves a more desirable distribution of steady state concentrations.

Plasma Solute | Physiological | Trapezoidal | Trench | Lift-off |
---|---|---|---|---|

Urea | $4.6$ | 24 | $2.1$ | $5.7$ |

Creatinine | $0.11$ | $1.8$ | $0.14$ | $3.9$ |

${\beta}_{2}$-microglobulin | $1.5\times {10}^{-4}$ | $1.3\times {10}^{-2}$ | $6.0\times {10}^{-4}$ | $2.3\times {10}^{-3}$ |

Albumin | $0.65$ | $7.0$ | $2.0\times {10}^{-2}$ | $0.10$ |

#### 3.4. Discussion

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- United States Renal Data System 2014 Annual Data Report: End Stage Renal Disease. In National Institutes of Health, National Institute of Diabetes and Digestive and Kidney Diseases; Bethesda: Rockville, MD, USA, 2014.
- Fleming, G. Renal replacement therapy review: Past, present and future. Organogenesis
**2011**, 7, 2–12. [Google Scholar] [CrossRef] [PubMed] - Himmelfarb, J.; Ikizler, T. Hemodialysis. N. Engl. J. Med.
**2010**, 363, 1833–1845. [Google Scholar] [CrossRef] [PubMed] - Dhondt, A.; Vanholder, R.; van Biesen, W.; Lameire, N. The removal of uremic toxins. Kidney Int. Suppl.
**2000**, 76, S47–S59. [Google Scholar] [CrossRef] [PubMed] - Bleyer, A.J.; Russell, G.B.; Satko, S.G. Sudden and cardiac death rates in hemodialysis patients. Kidney Int.
**1999**, 55, 1553–1559. [Google Scholar] [CrossRef] [PubMed] - Foley, R.N.; Gilbertson, D.T.; Murray, T.; Collins, A.J. Long interdialytic interval and mortality among patients receiving hemodialysis. N. Engl. J. Med.
**2011**, 365, 1099–1107. [Google Scholar] [CrossRef] [PubMed] - FHN Trial Group; Chertow, G.M.; Levin, N.W.; Beck, G.J.; Depner, T.A.; Eggers, P.W.; Gassman, J.J.; Gorodetskaya, I.; Greene, T.; James, S.; et al. In-center hemodialysis six times per week versus three times per week. N. Engl. J. Med.
**2010**, 363, 2287–2300. [Google Scholar] - Weinhandl, E.D.; Liu, J.; Gilbertson, D.T.; Arneson, T.J.; Collins, A.J. Survival in daily home hemodialysis and matched thrice-weekly in-center hemodialysis patients. J. Am. Soc. Nephrol.
**2012**, 23, 895–904. [Google Scholar] [CrossRef] [PubMed] - United States Renal Data System 2013 Annual Data Report: End Stage Renal Disease, Chapter 3: Hospitalization. In National Institutes of Health, National Institute of Diabetes and Digestive and Kidney Diseases; Bethesda: Rockville, MD, USA, 2013.
- Armignacco, P.; Lorenzin, A.; Neri, M.; Nalesso, F.; Garzotto, F.; Ronco, C. Wearable devices for blood purification: Principles, miniaturization, and technical challenges. Semin. Dial.
**2015**, 28, 125–130. [Google Scholar] [CrossRef] [PubMed] - Johnson, D.G.; Khire, T.S.; Lyubarskaya, Y.L.; Smith, K.J.; Desormeaux, J.P.; Taylor, J.G.; Gaborski, T.R.; Shestopalov, A.A.; Striemer, C.C.; McGrath, J.L. Ultrathin silicon membranes for wearable dialysis. Adv. Chronic Kidney Dis.
**2013**, 20, 508–515. [Google Scholar] [CrossRef] [PubMed] - Striemer, C.C.; Gaborski, T.R.; McGrath, J.L.; Fauchet, P.M. Charge- and size-based separation of macromolecules using ultrathin silicon membranes. Nature
**2007**, 445, 749–753. [Google Scholar] [CrossRef] [PubMed] - Snyder, J.; Clark, A.J.; Fang, D.; Gaborski, T.; Striemer, C.; Fauchet, P.; McGrath, J. An experimental and theoretical analysis of molecular separations by diffusion through ultrathin nanoporous membranes. J. Membr. Sci.
**2011**, 369, 119–129. [Google Scholar] [CrossRef] [PubMed] - DesOrmeaux, J.; Winans, J.; Wayson, S.; Gaborski, T.; Khire, T.; Striemer, C.; McGrath, J. Nanoporous silicon nitride membranes fabricated from porous nanocrystalline silicon templates. Nanoscale
**2014**, 6, 10798–10805. [Google Scholar] [CrossRef] [PubMed] - Chung, H.; Chan, C.; Khire, T.; Marsh, G.; Clark, A.J.; Waugh, R.; McGrath, J. Highly permeable silicon membranes for shear free chemotaxis and rapid cell labeling. Lab Chip
**2014**, 14, 2456–2468. [Google Scholar] [CrossRef] [PubMed] - Miller, J.; Carter, R.; McNabb, K.; DesOrmeaux, J.; Striemer, C.; Winans, J.; Gaborski, T. Lift-off of large-scale ultrathin nanomembranes. J. Micromech. Microeng.
**2015**, 25. [Google Scholar] [CrossRef] - Lopot, F.; Kotyk, P.; Bláha, J.; Válek, A. Analysis of the urea generation rate and the protein catabolic rate in hemodialyzed patients. Artif. Organs
**1995**, 19, 832–836. [Google Scholar] [CrossRef] [PubMed] - The Master Surgeon Reference Tables. Available online: http://www.webcitation.org/6dVDRW1Ss (accessed on 3 December 2015).
- Medscape: Creatinine. Available online: http://www.webcitation.org/6dVDhenk2 (accessed on 3 December 2015).
- Karlsson, F.A.; Wibell, L.; Evrin, P.E. beta 2-Microglobulin in clinical medicine. Scand. J. Clin. Lab. Investig. Suppl.
**1980**, 154, 27–37. [Google Scholar] - Johnson, H.J.; Flye, M.; Javadpour, N. Serum beta 2 microglobulin levels in patients with testicular cancer. Urology
**1980**, 16, 522–524. [Google Scholar] [CrossRef] - Medscape: Hypoalbuminemia. Available online: http://www.webcitation.org/6dVDpwKJ7 (accessed on 3 December 2015).
- Liabeuf, S.; Drueke, T.B.; Massy, Z.A. Protein-bound uremic toxins: New insight from clinical studies. Toxins
**2011**, 3, 911–919. [Google Scholar] [CrossRef] [PubMed] - Mumtaz, A.; Anees, M.; Bilal, M.; Ibrahim, M. Beta-2 microglobulin levels in hemodialysis patients. Saudi J. Kidney Dis. Transpl.
**2010**, 21, 701–706. [Google Scholar] [PubMed] - Liabeuf, S.; Barreto, D.V.; Barreto, F.C.; Meert, N.; Glorieux, G.; Schepers, E.; Temmar, M.; Choukroun, G.; Vanholder, R.; Massy, Z.A.; European Uraemic Toxin Work Group. Free p-cresylsulphate is a predictor of mortality in patients at different stages of chronic kidney disease. Nephrol. Dial. Transpl.
**2010**, 25, 1183–1191. [Google Scholar] [CrossRef] [PubMed] - MacRae, J.M.; Pandeya, S.; Humen, D.P.; Krivitski, N.; Lindsay, R.M. Arteriovenous fistula-associated high-output cardiac failure: A review of mechanisms. Am. J. Kidney Dis.
**2004**, 43, e17–e22. [Google Scholar] [CrossRef] [PubMed] - Mortensen, N.A.; Okkels, F.; Bruus, H. Reexamination of Hagen-Poiseuille flow: Shape dependence of the hydraulic resistance in microchannels. Phys. Rev. E Stat. Nonlinear Soft Matter Phys.
**2005**, 71. [Google Scholar] [CrossRef] [PubMed] - Krivitski, N.M.; Kislukhin, V.V.; Snyder, J.W.; MacGibbon, D.R.; Kuznetsova, O.A.; Reasons, A.M.; Depner, T.A. In vivo measurement of hemodialyzer fiber bundle volume: Theory and validation. Kidney Int.
**1998**, 54, 1751–1758. [Google Scholar] [CrossRef] [PubMed]

© 2015 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons by Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Burgin, T.; Johnson, D.; Chung, H.; Clark, A.; McGrath, J.
Analytical and Finite Element Modeling of Nanomembranes for Miniaturized, Continuous Hemodialysis. *Membranes* **2016**, *6*, 6.
https://doi.org/10.3390/membranes6010006

**AMA Style**

Burgin T, Johnson D, Chung H, Clark A, McGrath J.
Analytical and Finite Element Modeling of Nanomembranes for Miniaturized, Continuous Hemodialysis. *Membranes*. 2016; 6(1):6.
https://doi.org/10.3390/membranes6010006

**Chicago/Turabian Style**

Burgin, Tucker, Dean Johnson, Henry Chung, Alfred Clark, and James McGrath.
2016. "Analytical and Finite Element Modeling of Nanomembranes for Miniaturized, Continuous Hemodialysis" *Membranes* 6, no. 1: 6.
https://doi.org/10.3390/membranes6010006