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.)


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

Therefore, in tissue:

2) Ideal Donnan Theory

Description of charged species distribution across an interface


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


  • the interface region is small enough to be ignored

Donnan Electrochemical Equilibrium Relation


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


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.


Combine Electroneutrality and Donnan relations:


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


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.


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