Two-Compartment Model of Radioimmunotherapy delivered through Cerebrospinal Fluid

European Journal of Nuclear Medicine and Molecular Imaging

Ping He, Kim Kramer, Peter Smith Jones, Pat Zanzonico, John Humm, Steven M. Larson, Nai-Kong V. Cheung

Corresponding Author:

Dr. Nai-Kong V. Cheung

Department of Pediatrics,

Memorial Sloan-Kettering Cancer Center

New York, NY 10065

Email:

Supplemental Tables

Table 1: Nomenclature for Parameters and Variables

Nomenclature
Parameters
Dose(mg) / Dose administered
kAR(M-1s-1) and k-AR(s-1) / Association and dissociation constants of specific binding, respectively
kNAR(M-1s-1) and k-NAR(s-1) / Association and dissociation constants of nonspecific binding, respectively
R0 / Initial concentration of receptors
NA / Avogadro’s Number
V(mL) / Volume of the CSF space
VV(mL) and VS(mL) / Volume of ventricles and volume of subarachnoid space
CL(mL/hour) / Clearance rate of CSF
Inf(mg/hour) / Infusion rate (Continuous Infusion)
Immunoreactivity / Percent immunoreactivity
TumorVent / Percent of total tumor load in ventricles
Nonspecificity / Percent of nonspecific binding compared to specific binding
MW(Da) / Molecular weight of antibody
DT(μm) / Diameter of tumor cells
S(cm2) / Surface area of CSF space
CI(MBq/mg) / Specific activity of the isotope
t1/2 (hour) / Half-life of isotope
Variables
n / Layers of tumor cells
CSpecies(M) / Concentration of the Species
CISpecies(kBq/mL) / Radioactivity of the Species
BV and BS / Free antibody withimmunoreactivity in ventricles and subarachnoid space, respectively
NV and NS / Free antibody withoutimmunoreactivity in ventricles and subarachnoid space, respectively
RV and RS / Free specific binding receptors in ventricles and subarachnoid space, respectively
NRV and NRS / Free nonspecific binding receptors in ventricles and subarachnoid space, respectively
ARV and ARS / Bound specific antibody-receptor complex in ventricles and subarachnoid space, respectively
NARV and NARS / Bound nonspecific antibody-receptor complex in ventricles and subarachnoid space, respectively
AUC[CISpecies](min*kBq/mL) / Area under the radioactivity vs. time curve for the particular species

* For different species present, “B” stands for binding, “N” stands for nonbinding, “R” stands for receptor, “AR” stands for bound antibody-receptor complex, “V” stands for ventricle, and “S” stands for subarachnoid space.

Table 2: Parameter values

Parameter Values
Parameter / Value used during Optimization / Literature Values / Source
Dose / 2 mg or 0.2 mg / 2 mg / [1]
MW / 150,000 Da / 150 kD
CI0 / 185 MBq/mg (5 mCi/mg) / 185MBq/mg (5 mCi/mg)
Immunoreactivity / 50% / 50% / [1, 2]
Nonspecificity / 1% / 1-10% / [3]
TumorVent / 0% / 0-5% / Assumption
V / 140 mL / 140 mL / [4, 5]
CL / 20 mL/hour / 20 mL/hour
Vv / 30 mL / 30 mL / [6]
Vs / 110 mL / 110 mL (V - Vv)
kAR / 3x104 M-1s-1 / 3x103-3x105 M-1s-1 / [7]
k-AR / 3x10-4 s-1 / 3x10-5-3x10-3 s-1
kNAR / 3x102 M-1s-1 / PercentNBx kAR / [7]
k-NAR / 3x10-4 s-1 / k-AR / [7]
S / 1800 cm2 / ≥1800 cm2 / [8]
t1/2 / 193 hour / 193 hour / [1]
DT / 10 μm / 10 μm / [2]
R0 / 2.865x1014 antigen/ml / 5.73x1011-5.73x1014antigen/ml

Supplement Fig. 1 Comparison of model prediction to experimental data obtained from CSF sampling.

a. Comparison of model predictions to patient data. The patient received 2 mg dose at specific activity of 185 MBq/mg (5 mCi/mg). The antigen density was fixed at 2.865 x 1014 antigens/ml. The ventricular volume and the clearance rate were fitted to be 25 ml and 18ml/hour, respectively. a1 : Therapeutic ratio and a2: AUC(CIAR)

b. Comparison of data fitting between 1-compartment model and 2-compartment model. The dose administered was 3.54 mg at specific activity of 185 MBq/mg (5 mCi/mg). The antigen density was 2.865 x 1014 antigens/ml. The ventricular volume was 40 ml and the clearance rate 20 ml/hour. b1 : Therapeutic ratio and b2: AUC(CIAR)

Model Specifics

The pharmacokinetics model of the CSF space is divided into the ventricular compartment and the subarachnoid compartment. The free antibody is further divided into two subdivision, those with immunoreactivity (BV or BS) and those that have lost immunoreactivity during isotope labeling (NV or NS). Only those with immunoreactivity will show any binding behavior. The specific binding of 131I-3F8 antibody to the GD2 receptors is modeled using the same binding kinetics as presented by Lv et al., which takes into account that binding only occurs on the surface of the CSF space. It is represented as the following,

Here kAR and k-AR are the association and dissociation constant, respectively. The ratio of k-AR to kAR is Kd, which is a measurement of binding affinity. Lowering the Kd is equivalent to raising the affinity. Since tumor cells have little or no presence in the brain ventricles, most of the specific binding is in the subarachnoid space. Only a small percentage of the total tumor load in the ventricles (TumorVent) will express specific binding.

Nonspecific binding occurs in both the ventricular compartment and the subarachnoid compartment. However, for an antibody such as 3F8, the amount of nonspecific binding is small comparing to the amount of specific binding. Therefore, the amount of nonspecific binding can be represented as a percentage of the specific binding (Nonspecificity). Nonspecific binding will have smaller receptor number and smaller affinity as well, so its kinetics will be determined by the following set of equations.

Here kNAR and k-NAR are the association and dissociation constants of nonspecific binding. The affinity for nonspecific binding is equal to TumorVent times the affinity of specific binding. The kinetics for nonspecific binding is applied to both the ventricular compartment and subarachnoid compartment, but only to the antibody that has immunoreactivity.

Finally, we combine the binding kinetics with the movement of antibody by convection of CSF, and we get a system of 12 differential equations.

The ventricular compartment is modeled by the following 6 differential equations.

The subarachnoid compartment is modeled by the following 6 corresponding equations.

The system of equations is then evaluated using MATLAB, a computer software package. We used the differential equation solver, ode45, which is based on the 4th order Runge-Kutta method. (See the reference at the end of the section for more information on ode45 and Runge-Kutta method.) The following table summarizes the initial values we used for our simulation.

Initial Values(in M)
CBV0 /
CNV0 /
CRV0 /

CARV0 / 0
CNRV0 /

CNARV0 / 0
CBS0 / 0
CNS0 / 0
CRS0 /

CARS0 / 0
CNRS0 /

CNARS0 / 0

The physical decay is taken into account when calculating the radioactivity of each species. The kinetics used in this model can be summarized as

and

WhereCI is the specific activity of the isotope, and t1/2 is its half-life. To calculate the radioactivity of specific species, we used the following equation.

CSpecies is the concentration of the species (kBq/g), MW is the molecular weight (Da),CISpecies is its radioactivity (kBq/g), and is the density of CSF or tumor, which are approximated to be 1g/ml.

The area under the curve of the radioactivity of a species (AUC[CISpecies]) is calculated by estimating the following integration with trapezoidal method.

The therapeutic ratio is calculated as the following,

The radiation absorbance is calculated as

Reference for ode45 and Runge-Kutta:

Dunn SM, Constantinides A, and Moghe PV.Numerical Methods in Biomedical Engineering.Academic Press, 2006, pg. 224.

References:

1.Kramer K, Humm JL, Souweidane MM, Zanzonico PB, Dunkel IJ, Gerald WL, et al. Phase I study of targeted radioimmunotherapy for leptomeningeal cancers using intra-Ommaya 131-I-3F8. J Clin Oncol. 2007;25:5465-70.

2.Lv Y, Cheung NK, Fu BM. A pharmacokinetic model for radioimmunotherapy delivered through cerebrospinal fluid for the treatment of leptomeningeal metastases. J Nucl Med. 2009;50:1324-31.

3.Xu H, Cheung IY, Guo HF, Cheung NK. MicroRNA miR-29 modulates expression of immunoinhibitory molecule B7-H3: potential implications for immune based therapy of human solid tumors. Cancer Res. 2009;69:6275-81. doi:0008-5472.CAN-08-4517 [pii]

10.1158/0008-5472.CAN-08-4517.

4.Guyton A, Hall J. Textbook of Medical Physiology. Philadelphia, PA: Saunders; 2000.

5.Silverberg GD, Heit G, Huhn S, Jaffe RA, Chang SD, Bronte-Stewart H, et al. The cerebrospinal fluid production rate is reduced in dementia of the Alzheimer's type. Neurology. 2001;57:1763-6.

6.Brassow F, Baumann K. Volume of brain ventricles in man determined by computer tomography. Neuroradiology. 1978;16:187-9.

7.Xu H, Hu J, Cheung NK. Induction of tumor cell death by anti-GD2 monoclonal antibody (MoAb): Requirement of antibody Fc and a long residence time (slow Koff). Journal of Clinical Oncology, 2007 ASCO Annual Meeting Proceedings Part I. 2007;Vol 25:Abstract #13507.

8.Blasberg RG, Patlak CS, Shapiro WR. Distribution of methotrexate in the cerebrospinal fluid and brain after intraventricular administration. Cancer treatment reports. 1977;61:633-41.

1