Tubular bioreactor models that include Onsager-Curie scalar cross-phenomena to describe stress-dependent rates of cell proliferation.

The theory of heterogeneous catalysis in chemical reactors is employed to simulate laminar flow through tubes at large mass transfer Peclet numbers in which anchorage-dependent cells (i) adhere to a protein coating on the inner surface at r=R(wall), (ii) receive nutrients and oxygen from an aqueous medium via transverse diffusion toward the active wall, and (iii) proliferate in the presence of viscous shear at the cell/aqueous-medium interface. This process is modeled as convective diffusion in cylindrical coordinates with chemical reaction at the boundary, where chemical reaction describes the rate of nutrient consumption. The formalism of irreversible thermodynamics is employed to describe an unusual coupling between viscous shear, or velocity gradients at the cell/aqueous-medium interface, and rates of nutrient consumption. Linear transport laws in chemically reactive systems that obey Curie's theorem predict the existence of cross-phenomena between fluxes (i.e., scalar reaction rates) and driving forces (i.e., 2nd-rank velocity gradient tensor) whose tensorial ranks differ by an even integer-in this case, two. This methodology for stress-dependent chemical reactions yields an additional zeroth-order contribution, via the magnitude of the velocity gradient tensor, to heterogeneous kinetic rate expressions because nutrient consumption and cell proliferation are stress-sensitive. Computer simulations of nutrient consumption suggest that bioreactor designs should consider stress-sensitive reactions when the shear-rate-based Damköhler number (i.e., defined for the first time in this study as the stress-dependent zeroth-order rate of nutrient consumption relative to the rate of nutrient diffusion toward active cells adhered to the tube wall) is greater than 10-20% of the stress-free Damköhler number. Models of bioreactor performance are presented for simple 1st-order, simple 2nd-order, and complex chemical kinetic rate expressions, where the latter considers adsorption/desorption equilibria via the Fowler-Guggenheim modification of the Langmuir isotherm for cell-protein docking on active sites, accompanied by cell-cell attraction. Stress sensitivity is magnified in physically realistic cell-based tubular bioreactors with complex stress-free kinetic rate expressions relative to simulations with simple 1st- and 2nd-order kinetics.

Hepatic sequestration of chlordecone and hexafluoroacetone evaluated by pharmacokinetic modeling.

Chlordecone (CD) and mirex (M) differ by a single carbonyl group in CD in place of two chlorines in M. Although both compounds are lipophilic, their tissue distributions differ markedly: CD concentrations are highest in liver; M concentrations are highest in fat. We used tissue time course data in rats from our laboratory for CD and M and literature data from monkeys to develop PBPK models to study differences in liver and fat partitioning. The PK model for M had partitioning in tissue without specific hepatic binding. The CD model had partitioning similar to M, and also included liver binding: the maximal binding (B(max)) and binding affinity constant (Kd) required to describe the rat data were 370 nmol/g liver and 100 nM, respectively. To see if other ketones with electron withdrawing constituents at the alpha carbon were also preferentially distributed to liver, we developed a PBPK description for tissue distribution of hexafluoroacetone (HFA). Compared to acetone, HFA is known to be preferentially sequestered in liver and more slowly excreted unchanged from the body. Acetone is more equally distributed to tissues. HFA distribution was evaluated with a PBPK model that included hepatic binding. B(max) and Kd were 1.58 micromol/g liver and 301 microM. In summary, liver sequestration of CD and HFA most likely represents relatively high-affinity but reversible binding of activated carbonyls in these compounds (activated by the presence of electron withdrawing substituents on the alpha-carbons) with glutathione and glutathione transferases, that are present at much higher concentrations in liver than in other tissues. Strong, but reversible hemithioketal formation with active sulfhydryls may also be associated with the toxic responses to CD and HFA.

Electric-field-enhanced nutrient consumption in dielectric biomaterials that contain anchorage-dependent cells.

This research contribution addresses electric-field stimulation of intra-tissue mass transfer and cell proliferation in viscoelastic biomaterials. The unsteady state reaction-diffusion equation is solved according to the von Kármán-Pohlhausen integral method of boundary layer analysis when nutrient consumption and tissue regeneration occur in response to harmonic electric potential differences across a parallel-plate capacitor in a dielectric-sandwich configuration. The partial differential mass balance with diffusion and electro-kinetic consumption contains the Damköhler (Λ(2)) and Deborah (De) numbers. Zero-field and electric-field-sensitive Damköhler numbers affect nutrient boundary layer growth. Diagonal elements of the 2nd-rank diffusion tensor are enhanced in the presence of weak electric fields, in agreement with the formalism of equilibrium and nonequilibrium thermodynamics. Induced dipole polarization density within viscoelastic biomaterials is calculated via the real and imaginary components of the complex dielectric constant, according to the Debye equation, to quantify electro-kinetic stimulation. Rates of nutrient consumption under zero-field conditions are described by third-order kinetics that include local mass densities of nutrients, oxygen, and attached cells. Thinner nutrient boundary layers are stabilized at shorter dimensionless diffusion times when the zero-field intra-tissue Damköhler number increases above its initial-condition-sensitive critical value [i.e., {Λ(2)(zero-field)}(critical)≥53, see Eq. (23)], such that the biomaterial core is starved of essential ingredients required for successful proliferation. When tissue regeneration occurs above the critical electric-field-sensitive intra-tissue Damköhler number, the electro-kinetic contribution to nutrient consumption cannot be neglected. The critical electric-field-sensitive intra-tissue Damköhler number is proportional to the Deborah number.

Stress-sensitive tissue regeneration in viscoelastic biomaterials subjected to modulated tensile strain.

This research contribution addresses the mechanochemistry of intra-tissue mass transfer for nutrients, oxygen, growth factors, and other essential ingredients that anchorage-dependent cells require for successful proliferation on biocompatible surfaces. The unsteady state reaction-diffusion equation (i.e., modified diffusion equation) is solved according to the von Kármán-Pohlhausen integral method of boundary layer analysis when nutrient consumption and tissue regeneration are stimulated by harmonically imposed stress. The mass balance with diffusion and stress-sensitive kinetics represents a rare example where the Damköhler and Deborah numbers appear together in an effort to simulate the development of mass transfer boundary layers in porous viscoelastic biomaterials. The Boltzmann superposition integral is employed to calculate time-dependent strain in terms of the real and imaginary components of dynamic compliance for viscoelastic solids that transmit harmonic excitation to anchorage-dependent cells. Rates of nutrient consumption under stress-free conditions are described by third-order kinetics which include local mass densities of nutrients, oxygen, and attached cells that maintain dynamic equilibrium with active protein sites in the porous matrix. Thinner nutrient mass transfer boundary layers are stabilized at shorter dimensionless diffusion times when the stress-free intra-tissue Damköhler number increases above its initial-condition-sensitive critical value. The critical stress-sensitive intra-tissue Damköhler number, above which it is necessary to consider the effect of harmonic strain on nutrient consumption and tissue regeneration, is proportional to the Deborah number and corresponds to a larger fraction of the stress-free intra-tissue Damköhler number in rigid biomaterials.

Nutrient diffusion and simple nth-order consumption in regenerative tissue and biocatalytic sensors.

This contribution addresses intra-tissue molar density profiles for nutrients, oxygen, growth factors, and other essential ingredients that anchorage-dependent cells require for successful proliferation on biocompatible surfaces. One-dimensional transient and steady state models of the reaction-diffusion equation are solved to correct a few deficiencies in the first illustrative example of diffusion and zeroth-order rates of consumption in tissues with rectangular geometry, as discussed in Ref. [(Griffith and Swartz, 2006) 1]. The functional form of the molar density profile for each species depends on geometry and the magnitude of the species-specific intra-tissue Damköhler number. The tissue's central core is reactant starved at high consumption rates and low rates of intra-tissue diffusion when the Damköhler number exceeds its geometry-sensitive critical value. Ideal tissue engineering designs avoid the diffusion-limited regime such that attached cells are exposed to all of the ingredients required for proliferation everywhere within a regenerative matrix. Analytical and numerical molar density profiles that satisfy the unsteady state modified diffusion equation with pseudo-homogeneous n(th)-order rates of intra-tissue consumption (i.e., n=0,1,2) allow one to (i) predict von Kármán-Pohlhausen mass transfer boundary layer thicknesses, measured inward from the external biomaterial surface toward its central core, and, most importantly, (ii) estimate the time required to achieve steady state conditions for regenerative tissue growth and biocatalytic sensing.

Pressure-sensitive nutrient consumption via dynamic normal stress in rotational bioreactors.

Pressure-sensitive biological response is simulated in "rotating-cup" bioreactors with unidirectional modulations in compressive stress at the cylindrical wall that stimulate bone-tissue growth. Anchorage-dependent mammalian cells (i) adhere to a protein coating, (ii) receive nutrients and oxygen from an aqueous medium via radial diffusion toward the active surface, and (iii) respond to physiological modulations in centrifual-force-induced fluid pressure at the cell/aqueous-medium interface. This process is modeled by the classic diffusion equation (i.e., Fick's second law), with a time-dependent reaction/diffusion boundary condition at the wall. Non-reversing angular velocity modulations resemble pulsations at physiological frequencies. Computer simulations of nutrient consumption profiles suggest that rotational bioreactor designs should consider the effects of normal stress when the pressure-sensitive Damköhler number (i.e., ratio of the pressure-dependent zeroth-order rate of nutrient consumption relative to the rate of nutrient diffusion toward active cells adhered to the cylindrical wall), evaluated under steady rotation, is greater than approximately 10-20% of the stress-free Damköhler number (i.e., beta(0,1st-order)=0.025) for simple 1st-order stress-free kinetics, and approximately 1% of the stress-free Damköhler number (i.e., beta(0,2nd-order)=0.40) for complex 2nd-order stress-free nutrient consumption. When the peak-to-peak amplitude of angular velocity modulations of the cylindrical wall is the same as or larger than the angular velocity for steady rotation, the effect of non-reversing centrifugal-force-induced dynamic normal stress in rotational bioreactors, superimposed on steady rotation, can be significant when one is below the critical value of the pressure-sensitive Damköhler number that has been identified under steady rotation.

Stress-sensitive nutrient consumption via steady and non-reversing dynamic shear in continuous-flow rotational bioreactors.

Stress-sensitive biological response is simulated in a modified parallel-disk viscometer that implements steady and unidirectional dynamic shear under physiological conditions. Anchorage-dependent mammalian cells adhere to a protein coating on the surface of the rotating plate, receiving nutrients and oxygen from an aqueous medium that flows radially and tangentially, accompanied by transverse diffusion in the z-direction toward the active surface. This process is modeled as radial convection and axial diffusion with angular symmetry in cylindrical coordinates. The reaction/diffusion boundary condition on the surface of the rotating plate includes position-dependent stress-sensitive nutrient consumption via the zr- and zTheta-elements of the velocity gradient tensor at the cell/aqueous-medium interface. Linear transport laws in chemically reactive systems that obey Curie's theorem predict the existence of cross-phenomena between scalar reaction rates and the magnitude of the second-rank velocity gradient tensor, selecting only those elements of nabla v experienced by anchorage-dependent cells that are bound to protein-active sites. Stress sensitivity via the formalism of irreversible thermodynamics introduces a zeroth-order contribution to heterogeneous reaction rates that must be quenched when nutrients, oxygen, chemically anchored cells, or vacant active protein sites are not present on the surface of the rotating plate. Computer simulations of nutrient consumption profiles via simple nth-order kinetics (i.e., n=1,2) suggest that rotational bioreactor designs should consider stress-sensitivity when the shear-rate-based Damköhler number (i.e., ratio of the stress-dependent zeroth-order rate of nutrient consumption relative to the rate of nutrient diffusion toward active cells adhered to the rotating plate) is greater than approximately 25% of the stress-free Damköhler number. Rotational bioreactor simulations are presented for simple 1st-order, simple 2nd-order, and complex stress-free kinetics, where the latter includes a 4th-order rate expression that considers adsorption/desorption equilibria via the Fowler-Guggenheim modification of the Langmuir isotherm for receptor-mediated cell-protein binding, accompanied by the formation of receptor complexes. Dimensionless parameters are identified to obtain equivalent stress-free nutrient consumption in the exit streams of 2-dimensional creeping-flow rotational bioreactors and 1-dimensional laminar-flow tubular bioreactors. Modulated rotation of the active plate at physiological frequencies mimics pulsatile cardiovascular flow and demonstrates that these rotational bioreactors must operate above the critical stress-sensitive Damköhler number, identified under steady shear conditions, before dynamic shear has a distinguishable effect on bioreactor performance.