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This is summary of pharmacokinetic two compartment model.

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- 1. Two Compartment Model
- 2. Agenda <ul><li>Tofts Model </li></ul><ul><li>EPIC Macros </li></ul>
- 3. Pharmacokineticc Behavior of Contrast Agents Review S : capillary surface area P : capillary permeability F : capillary blood flow
- 4. Summary of variables Materials and Methods m/min Permeability constant P m 2 /kg Surface area per unit mass S L/kg ·min Isodirectional transfer rate per unit volume between the plasma and the tumor compartments K p↔t L/kg ·min Transfer rate per unit volume from the tumor to the plasma compartment K t p L/kg ·min Transfer rate per unit volume from the plasma to the tumor compartment K p t L/kg ·min Transfer rate per unit volume from the plasma compartment to the kidneys K p k L/kg ·min Isodirectional transfer rate per unit volume between the plasma and the EES compartment K p↔e L/kg ·min Transfer rate per unit volume from the EES to the plasma compartment K e p L/kg ·min Transfer rate per unit volume from the plasma to the EES compartment K p e mM Concentration in the tumor compartment Ct [CA t (t)] mM Concentration in the extravascular extracellular space(EES) compartment Ce [CA e (t)] mM Concentration in the plasma compartment Cp [CA p (t)] Units Meaning Standardized quantities Variable
- 5. Summary of variables Materials and Methods Units Meaning Standardized quantities Variable mM Contrast agent concentration [CA] Normalized signal intensity S/S 0 l/min Excretion rate constant β l/min Distribution rate constant α kg/L Normalized concentration amplitudes for unit dose a 1,2 mmole/kg Dose D Plasma volume fraction of the tumor volume v p EES volume fraction of the tumor volume v e L/kg Tumor volume (per unit mass of tumor) or inverse tumor density V T L/kg Tumor EES volume (per unit mass of tumor) V t L/kg Plasma volume (per unit mass of body) V p L/kg EES volume (per unit mass of body) V e m 3 /kg ·min Permeability-surface area product per unit mass, isodirectional transfer between the plasma and the tumor compartments P p↔t S pt m 3 /kg ·min Permeability-surface area product per unit mass, transfer from the tumor to the plasma compartment P t p S pt m 3 /kg ·min Permeability-surface area product per unit mass, transfer from the plasma to the tumor compartment P p t S pt
- 6. Tofts model Review Kidneys Tumor Compartment Extravascular Extracellular Space Compartment (Whole body) Plasma Compartment (Whole body) Excretion Tow-compartment model with parameters defined in the text Distribution
- 7. Pharmacokinetic behavior of contrast agents Materials and Methods [CA p (t)] (mM) : The constast agent concentration in the plasma compartment [CA e (t)] (mM) : The contrast agent concentration in the EES compartment V p (L/kg) : The plasma volume of the whole animal (per unit mass) K e p (L/kg·min) : The trasnfer rate per unit volume from the volume of distribution (EES) to the plasma compartment K p e ( L/kg·min) : The transfer rate per unit volume from the plasma to the EES compartment K p k ( L/kg·min) : The excretion transfer rate per unit volume caused by the unidirectional transit of the contrast agent from the plasma compartment to the kidneys K i j = P i j S ij K j i = P j i S ji = P i j S ij K i↔j = P i ↔j S ij P i j (m/minute) : The permeability constant between compartments i and j S ij (m 2 /kg) : The surface area per unit mass of the capillaries (S ij = S ji ) (1) (2)
- 8. Pharmacokinetic behavior of contrast agents Materials and Methods V e (L/kg) : The volume of distribution or EES volume per unit mass Important to note : V p and V e are more accurately described as the volume of the compartment permeated by the contrast agent per unit mass Contrast agents of differing size will express different compartment volumes in the same animal Assume : Simple diffusive trasport : (3)
- 9. Pharmacokinetic behavior of contrast agents Materials and Methods D (mmole/kg) : The dose K p↔e (L/kg·min) : Represents the iso-directional transfer rate per unit volume between the plasma and EES compartments Distribution rate constant (1/minute) : Excretion rate constant (1/minute) : (4) (5) (6)
- 10. Pharmacokinetic behavior of contrast agents Materials and Methods The normalized concentration amplitudes for unit dose, a 1,2 (kg/liter), are given by : Essentially, the tumor compartment is composed of fractions of the plasma and EES compartments [CA t (t)] (mM) : The constrast agent concentration in the tumor compartment v p : The tumor’s plasma volume fraction v e : The tumor’s EES volume fraction permeated by the agent (7) (8) (9)
- 11. Pharmacokinetic behavior of contrast agents Materials and Methods (10) <ul><li>Contrast agent concentration for the plasma and EES compartment </li></ul><ul><li>Eq.(4) , Eq.(5) Eq.(8) </li></ul><ul><li>nothing that the transfer rates per unit concentration of the EES of the tumor are different from those averaged for the whole body </li></ul><ul><li>the transfer rates incorporated in a 1,2 , α and β are averages for the entire body and different from those in the tumor </li></ul>Assume : The plasma volume fraction of the tumor is much less than the EES volume fraction of the tumor is much less than the EES volume fraction of the tumor Simple diffusive transport
- 12. Signal Intensity to Contrast Agent Concentration Calibration Materials and Methods <ul><li>The two-compartment model used interactions of contrast agent concentrations in different tissues to elucidate pharmacokinetic parameters </li></ul><ul><li>The concentration calibration curve was used to relate normalized signal intensity, S/S 0 </li></ul>C 1 - TR : repetition time , T 10 : longitudianl relaxation time in the absence of contrast agent C 2 - the product of TR and the logitudinal relaxivity of the agent r 1 (1/mM·sec) C 1 - should be approximately the same for any agent under the same viscosity, pH and temperature conditions C 2 - linearly proportional to the longitudinal relaxivity and is therefore agent dependent
- 13. Contrast Agent Materials and Methods <ul><li>PAMAM-TU-DTPA </li></ul><ul><li>Gd-DTPA </li></ul>MRI : Animals <ul><li>Female Sprague-Dawley rats had mammary tumors induced with 180 mg/kg (avg dose) N-ethyl-N-nitrosourea(ENU) by intraperitoneal injection at 30 days of age </li></ul><ul><li>Tumor size : 2 to 5 cm ( 2-5 months postinduction) </li></ul><ul><li>Imaging parameter - sequence : FLASH ( θ = 80° , TR = 70 ms , TE = 6ms , FOV = 300 mm , 256 x 256 , slice thickness = 3.0 mm , # of slices = 5 , acquisition time = 51 s , total time = 120 min. - coil : human head coil - system : 1.5 T Siemens ( at the St. Francis Medical Center. Peoria, IL) </li></ul>
- 15. Mathematics of the Two-Compartment Model
- 22. Relating Signal Intensities to Contrast Agent Concentrations

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