Foundations of modeling in cryobiology-II: Heat and mass transport in bulk and at cell membrane and ice-liquid interfaces

doi:10.1016/j.cryobiol.2019.09.014

Review

. 2019 Dec:91:3-17.

doi: 10.1016/j.cryobiol.2019.09.014. Epub 2019 Oct 4.

Foundations of modeling in cryobiology-II: Heat and mass transport in bulk and at cell membrane and ice-liquid interfaces

Daniel M Anderson¹, James D Benson², Anthony J Kearsley³

Affiliations

¹ Applied and Computational Mathematics Division, National Institute of Standards and Technology, Gaithersburg, MD, 20899-8910, USA; Department of Mathematical Sciences, George Mason University, Fairfax, VA, 22030, USA. Electronic address: danders1@gmu.edu.
² Applied and Computational Mathematics Division, National Institute of Standards and Technology, Gaithersburg, MD, 20899-8910, USA; Department of Biology, University of Saskatchewan, Saskatoon, SK, S7N 5E2, Canada. Electronic address: james.benson@usask.ca.
³ Applied and Computational Mathematics Division, National Institute of Standards and Technology, Gaithersburg, MD, 20899-8910, USA. Electronic address: anthony.kearsley@nist.gov.

PMID: 31589832
PMCID: PMC7098062
DOI: 10.1016/j.cryobiol.2019.09.014

Review

Foundations of modeling in cryobiology-II: Heat and mass transport in bulk and at cell membrane and ice-liquid interfaces

Daniel M Anderson et al. Cryobiology. 2019 Dec.

. 2019 Dec:91:3-17.

doi: 10.1016/j.cryobiol.2019.09.014. Epub 2019 Oct 4.

Authors

Daniel M Anderson¹, James D Benson², Anthony J Kearsley³

Affiliations

¹ Applied and Computational Mathematics Division, National Institute of Standards and Technology, Gaithersburg, MD, 20899-8910, USA; Department of Mathematical Sciences, George Mason University, Fairfax, VA, 22030, USA. Electronic address: danders1@gmu.edu.
² Applied and Computational Mathematics Division, National Institute of Standards and Technology, Gaithersburg, MD, 20899-8910, USA; Department of Biology, University of Saskatchewan, Saskatoon, SK, S7N 5E2, Canada. Electronic address: james.benson@usask.ca.
³ Applied and Computational Mathematics Division, National Institute of Standards and Technology, Gaithersburg, MD, 20899-8910, USA. Electronic address: anthony.kearsley@nist.gov.

PMID: 31589832
PMCID: PMC7098062
DOI: 10.1016/j.cryobiol.2019.09.014

Abstract

Modeling coupled heat and mass transport in biological systems is critical to the understanding of cryobiology. In Part I of this series we derived the transport equation and presented a general thermodynamic derivation of the critical components needed to use the transport equation in cryobiology. Here we refine to more cryobiologically relevant instances of a double free-boundary problem with multiple species. In particular, we present the derivation of appropriate mass and heat transport constitutive equations for a system consisting of a cell or tissue with a free external boundary, surrounded by liquid media with an encroaching free solidification front. This model consists of two parts-namely, transport in the "bulk phases" away from boundaries, and interfacial transport. Here we derive the bulk and interfacial mass, energy, and momentum balance equations and present a simplification of transport within membranes to jump conditions across them. We establish the governing equations for this cell/liquid/solid system whose solution in the case of a ternary mixture is explored in Part III of this series.

Keywords: Cryobiology; Interfacial conditions; Membrane boundary conditions; Transport processes.

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Figures

**Figure 1:**
Regions under consideration. Solid lines indicate interfaces. Bulk transport occurs away from solid lines. Here we make no assumptions about region geometry.

**Figure 2:**
Schematic of a typical pillbox balance equation. We assume the total width dx⁺ + dx⁻ and cross sectional area A are constant from time t to t + dt. We have that $d x^{-} = d x_{o}^{-} + V_{n}$ and $d x^{+} = d x_{o}^{+} - V_{n} d t$ . The volume in the left region at time t, for example, is given by $A d x_{o}^{-}$ , and if, for example, [ρ] =mass/volume, then the mass at time t is $ρ^{-} A d x_{o}^{-} + ρ^{+} A d x_{o}^{+} = (ρ^{-} d x_{o}^{-} + ρ^{+} d x_{o}^{+}) A$ and the mass at time t + dt is $(ρ^{-} d x^{-} + ρ^{+} d x^{+}) A = ρ^{-} (d x_{o}^{-} + V_{n} d t) A + ρ^{+} (d x_{o}^{+} - V_{n} d t) A$ . The change of mass from t to t + dt is then (ρ⁻ − ρ⁺)V_nAdt. On the other hand, note that if the velocity into the left region is v⁻ and the velocity out of the right region is v⁺, then the normal velocity is $v \cdot \hat{n}$ and the mass flux over time dt is $(ρ^{-} v^{-} - ρ^{+} v^{+}) \cdot \hat{n} A d t$ .

**Figure 3:**
Schematic of cell membrane. The interior radius is a and the exterior is b and we assume a ≫ b – a.

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References

1. Abazari A, Elliott JAW, Law GK, McGann LE, and Jomha NM. A biomechanics triphasic approach to the transport of non dilute solutions in articular cartilage. Biophys. Journal, 97:3054–3064, 2009. - PMC - PubMed
1. Abazari A, Thompson RB, Elliott JAW, and McGann LE. Transport phenomena in articular cartilage cryopreservation as predicted by the modified triphasic model and the effect of natural inhomogeneities. Biophysical Journal, 102(6):1284–1293, 2012. - PMC - PubMed
1. Acker JP, Elliott JAW, and McGann LE. Intercellular ice propagation: experimental evidence for ice growth through membrane pores. Biophys J, 81(3):1389–97, September 2001. - PMC - PubMed
1. Aitta A, Huppert HE, and Worster MG. Diffusion-controlled solidification of a ternary melt from a cooled boundary. J. Fluid Mech, 432: 201–217, 2001.
1. Ambrosi D and Preziosi L. Modeling injection molding processes with deformable porous preforms. SIAM J. Appl. Math, 61:22–42, 2000.

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[1] Abazari A, Elliott JAW, Law GK, McGann LE, and Jomha NM. A biomechanics triphasic approach to the transport of non dilute solutions in articular cartilage. Biophys. Journal, 97:3054–3064, 2009. - PMC - PubMed

[2] Abazari A, Elliott JAW, Law GK, McGann LE, and Jomha NM. A biomechanics triphasic approach to the transport of non dilute solutions in articular cartilage. Biophys. Journal, 97:3054–3064, 2009. - PMC - PubMed

[3] Abazari A, Thompson RB, Elliott JAW, and McGann LE. Transport phenomena in articular cartilage cryopreservation as predicted by the modified triphasic model and the effect of natural inhomogeneities. Biophysical Journal, 102(6):1284–1293, 2012. - PMC - PubMed

[4] Abazari A, Thompson RB, Elliott JAW, and McGann LE. Transport phenomena in articular cartilage cryopreservation as predicted by the modified triphasic model and the effect of natural inhomogeneities. Biophysical Journal, 102(6):1284–1293, 2012. - PMC - PubMed

[5] Acker JP, Elliott JAW, and McGann LE. Intercellular ice propagation: experimental evidence for ice growth through membrane pores. Biophys J, 81(3):1389–97, September 2001. - PMC - PubMed

[6] Acker JP, Elliott JAW, and McGann LE. Intercellular ice propagation: experimental evidence for ice growth through membrane pores. Biophys J, 81(3):1389–97, September 2001. - PMC - PubMed

[7] Aitta A, Huppert HE, and Worster MG. Diffusion-controlled solidification of a ternary melt from a cooled boundary. J. Fluid Mech, 432: 201–217, 2001.

[8] Aitta A, Huppert HE, and Worster MG. Diffusion-controlled solidification of a ternary melt from a cooled boundary. J. Fluid Mech, 432: 201–217, 2001.

[9] Ambrosi D and Preziosi L. Modeling injection molding processes with deformable porous preforms. SIAM J. Appl. Math, 61:22–42, 2000.

[10] Ambrosi D and Preziosi L. Modeling injection molding processes with deformable porous preforms. SIAM J. Appl. Math, 61:22–42, 2000.

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Foundations of modeling in cryobiology-II: Heat and mass transport in bulk and at cell membrane and ice-liquid interfaces

Affiliations

Foundations of modeling in cryobiology-II: Heat and mass transport in bulk and at cell membrane and ice-liquid interfaces

Authors

Affiliations

Abstract

Figures

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References

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Grants and funding

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