Although each of the rates described in are individually important for describing antibody distribution in tumors, it is the relative values of these rates that ultimately determine the extent of antibody penetration. A simple and qualitative approach to considering which processes dominate and which are limiting in a particular situation is to define dimensionless ratios of the rates [24
] (). Three such dimensionless numbers are useful in understanding the tumor penetration problem - two analogous to previously defined terms (the Thiele modulus and the Biot number) and one unique to pharmacokinetic modeling (the clearance modulus) [17
Dimensionless numbers predicting antibody penetration into tumorsa
The Biot number is a ratio between the rates of antibody extravasation from the capillary versus diffusion through the tumor tissue. For typical parameter values, the Biot number is small (Bi << 1), indicating that extravasation is a more significant barrier to targeting than subsequent diffusion. When the Biot number is small, the antibody concentration at the surface of the tumor [Ab]surf can be approximated as Bi [Ab]plasma. With typical Biot numbers for antibodies of the order of 0.001, a plasma concentration of 1 μM is reduced to only 1 nM at the capillary surface in contact with the tumor tissue. The low Biot number highlights the essential difficulty in antibody delivery to vascularized tumors in contrast to micrometastases.
The Thiele and clearance moduli compare the rates of passive antibody transport due to capillary extravasation and diffusion to the rates of antibody loss due to cellular catabolism or plasma clearance, respectively. Both values can be calculated for specific cases and used to predict whether antibody will reach the target distance R. If both the Thiele and clearance moduli are <1, penetration will be faster than clearance and the antibody will reach the target distance [17
]. If either modulus is >1, clearance will be faster than penetration and the moving antibody front will stop before reaching the target distance. IgGs generally satisfy the clearance modulus criterion (Γ < 1) owing to their long serum half-lifes; however, antibody fragments (scFvs and Fabs) often fail this test because they are rapidly cleared through the kidney. Both IgG and fragments can exhibit internalization-mediated limitations to penetration by the Thiele modulus criterion.
The Thiele and clearance moduli can also be used to predict the approximate penetration depth for an antibody by finding the distance R for which the Thiele or clearance modulus is ~1 (). Because penetration depth is predicted by this analysis to be proportional to the square root of antibody concentration [Ab
, this relationship predicts a requirement for excessive dosages to obtain antibody penetration to a depth of 100 μm for many antibodies. These predictions are consistent with early observations using radiolabeled antibody, where doses of 800 μg to 1 mg per mouse were required to achieve relatively homogeneous targeting of xenografts [25
]. In a similar manner, the dose necessary to achieve saturation can be determined by setting the Thiele and clearance numbers to 1 and solving for [Ab]plasma
In the cited theoretical analyses [14
], high-affinity antibodies were considered. The more general moduli defined here include the possibility of lower affinity interactions (i.e. Kd
). Such antibodies are typically able to penetrate farther into the tumor tissue owing to their ability to dissociate from antigen following binding and continue diffusing through the interstitial space. It is crucial, however, to distinguish penetration from saturation - in the high-affinity limit, all antigen sites are saturated, layer after layer, as antibody penetrates into the tissue. In the low-affinity case, antibody penetrates farther but targets a smaller fraction of antigen on the cell surface owing to its weaker interaction strength. In general, with a sustained plasma concentration the fractional antigen saturation of targeted cells will be approximately: