Use principles of electroneutrality and electrochemical equilibrium to relate known bath concentrations of a positively charged ion (sodium, Na +) and a negatively charged ion (chloride, Cl-, or gadolinium, Gd -2) to a measured tissue concentration of the negative species.

Governing Equations

1) Electroneutrality

No net charge in the tissue. (The sum of all fixed and mobile charges in a tissue is zero; a tissue is electrically neutral only after fully equilibrating with its bathing solution.)

electro1

Note: Under normal physiologic conditions, the concentrations of sodium and chloride in the tissue are much greater than the concentrations of other mobile ions.

  • other ions have a negligible effect on net tissue electroneutrality
  • [Gd -2] t and [other] t can be ignored in the description of tissue electroneutrality
electro2

Therefore, in tissue:

2) Ideal Donnan Theory

Description of charged species distribution across an interface

Assumptions:

  • the electrical environments on each side of the interface are spatially homogeneous
  • electrical inhomogeneities at the interface are confined to a small region:

fcd1

  • the interface region is small enough to be ignored

Donnan Electrochemical Equilibrium Relation

fcd2

For a physiologic saline or media solution containing the charged contrast agent Gd -2, the Donnan relation can be written as:

fcd3

Solve for concentrations sodium and chloride in tissue in terms of the assumed tissue concentration of contrast agent and known bath concentrations of other ions.

fcd4fdc5

Combine Electroneutrality and Donnan relations:

fcd6
fdc7fdc8

Note: bath concentration of sodium is equal to the bath concentration of chloride

fcd9fdc10

Experimental results have shown that this expression underpredicts cartilage tissue FCD by a factor of 2. Therefore, in order to match true cartilage FCD, this expression is scaled by an empirical factor of 2.

fdc11

Definition of Variables

c i concentration of charged species i, c t = in tissue, c b = in bath
z i valence of charged species i
FCD t fixed charged density of tissue
[Na +] t concentration of sodium in tissue
[Na +] b conc. of sodium in bath z Na =+1
[Cl -] t concentration of chloride in tissue
[Cl -] b conc. of chloride in bath, z Cl = -1
[Gd -2] t concentration of Gd-DTPA -2 in tissue
[Gd -2] b conc. of Gd in bath z Gd = -2