Comprehensive model for simultaneous MRI determination of perfusion and permeability using a blood-pool agent in rats rhabdomyosarcoma

Abstract

To present a new compartmental analysis model developed to simultaneously measure tissue perfusion and capillary permeability in a tumor using MRI and a macromolecular contrast medium. Rhadomyosarcomas were implanted subcutaneously in 20 rats and studied by 1.5-T MRI using a fast gradient echo sequence (2D fast SPGR TR/TE/α 13 ms/1.2 ms/60°) after injection of a macromolecular contrast medium. The left ventricle and tumor signal intensities were converted into concentrations and modeled using compartmental analysis, yielding tumor perfusion F, distribution volume V_distribution, volume transfer constant K^trans, rate constant of influx k_pe, and initial extraction (fraction) E. Tumor perfusion was F=43±29 ml·min⁻¹·100 g⁻¹. The permeability study allowed the measurement of k_pe=0.37±0.12 min⁻¹ and K^trans=0.01±0.0031 min⁻¹. The blood volume could be assimilated to the distribution volume (V_distribution=2.9±1.01%) since the capillary leakage was small. The simultaneous assessment of perfusion and permeability allowed quantification of the initial extraction (fraction) E=2.34±1.05%. Quantification of both tumor perfusion and capillary leakage is feasible using MRI using a macromolecular blood pool agent. The method should improve tumor characterization.

Appendix

The blood flow (F), the fractional plasma volume (ν_p), the volume transfer constant (K^trans), the influx rate constant (k_pe), and the extraction fraction (E) were deduced from k_2,1, k_3,2, and k_0,2, using Fick’s general equation of mass balance, which applied to the open two-compartmental model describes the transport of a contrast medium through the plasma compartment and its leakage into the interstitial space [19]:

Equation (8) can also be written as:

$$\frac{1}{{V_{{voxel}} }} \cdot \frac{{d{\left( {q_{p} {\left( t \right)}} \right)}}}{{dt}} = F \cdot \frac{{M_{{voxel}} }}{{V_{{voxel}} }} \cdot {\left( {C_{a} {\left( t \right)} - C_{V} {\left( t \right)}} \right)} - P \cdot S \cdot \frac{{M_{{voxel}} }}{{V_{{voxel}} }} \cdot {\left( {C_{p} {\left( t \right)} - C_{e} {\left( t \right)}} \right)}$$

(9)

where V_voxel is the volume of a voxel. Using the tissue density of the voxel (ρ=M/V) Eq. (9) becomes:

$$\frac{1} {{V_{{voxel}} }} \cdot \frac{{d{\left( {q_{p} {\left( t \right)}} \right)}}} {{dt}} = F \cdot \rho \cdot {\left( {C_{a} {\left( t \right)} - C_{V} {\left( t \right)}} \right)} - P \cdot S \cdot \rho \cdot {\left( {C_{p} {\left( t \right)} - C_{e} {\left( t \right)}} \right)}$$

(10)

By analogy with Eq. (6) of the model, Eq. (10) can be written:

$$\frac{1} {{V_{{voxel}} }} \cdot \frac{{d{\left( {q_{p} {\left( t \right)}} \right)}}} {{dt}} = F \cdot \rho \cdot C_{a} {\left( t \right)} - P \cdot S \cdot \rho \cdot {\left( {C_{p} {\left( t \right)} - C_{e} {\left( t \right)}} \right)} - F \cdot \rho \cdot C_{v} {\left( t \right)}$$

(11)

C_v(t), the concentration in venous plasma at the exit of the capillary, is assumed to be equal to the concentration in capillary plasma C_p(t) giving:

$$\frac{1} {{V_{{voxel}} }} \cdot \frac{{d{\left( {q_{p} {\left( t \right)}} \right)}}} {{dt}} = F \cdot \rho \cdot C_{a} {\left( t \right)} - P \cdot S \cdot \rho \cdot {\left( {C_{p} {\left( t \right)} - C_{e} {\left( t \right)}} \right)} - F \cdot \rho \cdot C_{p} {\left( t \right)}$$

(12)

All parameters were obtained by analogy between Eq. (6) (derived from our compartmental model used with SAAM II) and Eq. (12) (derived from general equation of mass balance) separating the flow and the permeability components:

For flow, the analogy of the amount of CM received by the capillary compartment yields:

$$V_{{voxel}} \cdot F \cdot \rho \cdot C_{a} {\left( t \right)} \Leftrightarrow k_{{2,1}} \cdot q_{a} {\left( t \right)}$$

(13)

In other words, in Eq. (13), the left side (derived from the general equation of mass balance) is considered equivalent to the right side (derived from our imaging compartmental model).

In MRI the extracellular CM concentration is measured in the whole blood volume (Cb). Therefore a correction of C_b by the hematocrit (Ht approx. 0.45) is necessary:

$$q_{a} {\left( t \right)} = V_{{voxel}} \cdot C_{a} {\left( t \right)} = V_{{voxel}} \cdot \frac{{C_{b} {\left( t \right)}}} {{1 - Ht}}$$

(14)

Using Eq. (14), Eq. (13) becomes:

$$F \cdot \rho \cdot C_{a} {\left( t \right)} \Leftrightarrow k_{{2,1}} \cdot C_{a} {\left( t \right)} = k_{{2,1}} \cdot \frac{{C_{b} {\left( t \right)}}} {{1 - Ht}}$$

(15)

The product F×ρ×C_a(t) is equivalent to the product k_2,1×C_b(t)/(1-Ht). Therefore estimation of k_2,1 from experimental data [C_b(t)] allows deduction of volume blood flow:

$$F \cdot \rho \Leftrightarrow \frac{{k_{{2,1}} }} {{1 - Ht}}$$

(16)

The tissue density (ρ) is assumed to be equal to 1 according to literature values [24, 25].

For permeability, the analogy of exchanges between compartments yields:

$$V_{{voxel}} \cdot P \cdot S \cdot \rho \cdot {\left( {C_{p} {\left( t \right)} - C_{e} {\left( t \right)}} \right)} \Leftrightarrow k_{{3,2}} \cdot q_{p} {\left( t \right)}$$

(17)

Replacing q_p by ν_p×C_p(t)×V_voxel, Eq. (17) can be simplified into:

$$P \cdot S \cdot \rho \cdot {\left( {C_{p} {\left( t \right)} - C_{e} {\left( t \right)}} \right)} \Leftrightarrow k_{{3,2}} \cdot \nu _{p} \cdot C_{p} {\left( t \right)}$$

(18)

When using a macromolecular CM with a low leakage rate, we can assume that the concentration in the extravascular extracellular space is negligible in comparison with the concentration in the capillary at the beginning of the experiment. The rate constant kpe is deduced from:

$$\frac{{P \cdot S \cdot \rho }} {{\nu _{p} }} \Leftrightarrow k_{{3,2}} {\left( t \right)}$$

(19)

P×S×ρ is equal to K^trans_, and the ratio K^trans/ν_p is equal to k_pe(t).

For blood volume, the analogy of the venous output yields:

$$V_{{voxel}} \cdot F \cdot \rho \cdot C_{p} {\left( t \right)} \Leftrightarrow k_{{0,2}} \cdot q_{p} {\left( t \right)}$$

(20)

Replacing q_p(t) by ν_p×C_p(t)×V_voxel Eq. (20) can be simplified into:

$$F \cdot \rho \cdot C_{p} {\left( t \right)} \Leftrightarrow k_{{0,2}} \cdot \nu _{p} \cdot C_{p} {\left( t \right)}$$

(21)

The plasma volume ν_p is deduced as follows:

$$\nu _{p} \Leftrightarrow \frac{{F \cdot \rho }} {{k_{{0,2}} }}$$

(22)

The extraction fraction E is calculated according to the model of Renkin [27] and Crone [28] of capillary permeability such as:

$$E = 1 - \exp {\left( { - \frac{{K^{{trans}} }} {F}} \right)}$$

(23)