MATHEMATICAL AND STATISTICAL MODELS
FOR RESEARCH

SAMUEL LITWIN, Ph.D., Member

Interpretation of clinical or biological data often requires a mathematical or statistical model. Such models can increase the power of experiments to decide among clinical treatments or to evaluate hypotheses, especially when their ramifications are explored through computation. An example illustrates the work.

Woodchuck hepatitis virus (WHV) is being used to model human hepatitis B virus (HBV) infection. In humans, HBV causes cirrhosis and, ultimately, may cause hepatocellular carcinoma (HCC). We report a preliminary study of viral clearance in woodchucks. This model lays groundwork for analysis of viral clearance data, estimation of rates of virus killing, and viral rebound after, for example, treatment with lamivudine.

We divide the liver into several pools (Figure 1) including infected, I, and uninfected, U, cells. Time dependencies of all quantities are modeled with differential equations. I(t) and U(t) are the pool sizes as functions of time.

All cells are initially infected, thus I(0) = C where C is the nominal liver size. Pool I is viewed here as not replicating. When cells are killed by cytotoxic T lymphocytes (CTLs) other liver cells start replication. Such cells may go into pool X, a state committed to replication. Cells in this pool, still infected, are killed at the same rate as those in the infected pool. Replicating cells from pool X form two cells in pool U, that is, replication is modeled to disinfect both daughters. A final pool of cells, Y, consists of replicating uninfected cells. Replicating infected cells, X, are killed at the same rate as are quiescent infected cells, I. Cells in pools U and Y are killed at the slow liver turnover rate.

New free virus particles, pool V, are produced from infected cells. These can reinfect cells U or Y. Free virus is cleared at a fixed rate per virion. When an uninfected cell is attacked by virus, it shifts to the infected pool of the same relative state, i.e., a U cell becomes an I cell, Y becomes X. Pool size I is determined by:

where k₁ is the death rate of infected cells, r₁ is the rate at which cells enter the replication cycle, and, k_v determines the rate of infection of quiescent uninfected cells U by free virus V. The quantity in brackets approximates the deficit in total cells, and power p controls the rate at which total cell count approaches its nominal level, C. Recruitment into replication phase becomes more intense as total cell count falls. The size of replicating infected pool X is determined by:

where the first term represents recruitment from the quiescent infected pool; the second term, k₁X, death while in pool X; r₂X, leaving this pool to enter pool U; and, k_vVY, recruitment of newly infected replicating cells. Cells enter pool X at rate r₁ but leave it at rate r₂, the rate of cell reproduction. Rate parameters r₂ and k₁ are derived from others below. The equation for Y uses k₂ << k₁, since uninfected cells have a much longer lifetime than infected ones. C' is slightly larger than C to account for normal liver cell turnover. New uninfected cells enter pool U at rate:

where the "2"s indicate that two uninfected cells are created. k₂ << k₁ is the death rate of uninfected cells. Uninfected replicating cells Y are produced at rate:

New free virions are generated by infected cells at rate k_g, cleared by a general clearing process with rate k_-v, and further reduced by uptake to all cells.

To limit liver growth to C cells, the factor controlling cell recruitment out of quiescence and into the cell cycle, r₁U, or r₁I, is multiplied by a function of total liver deficit. This function dampens liver growth beyond C cells, while p controls the asymptotic behavior of U as it approaches C. C' will be slightly greater than C to allow for normal cell death.

Setting I = X = dU/dt = dY/dt = V =0, the equilibrium conditions for liver size and complete elimination of virus yields:

These three equations yield the steady state solution:

Behavior of the model is illustrated in Figure 2. We make the assumption that the total liver pool size, (I + X + U + Y), must not dip below 20% of its nominal size to avoid death from liver failure.

PUBLICATIONS

BOUCK, J., LITWIN, S., SKALKA, A.M., KATZ,R.A. In vivo selection for intronic splicing signals from a randomized pool. Nucleic Acids Res. 26:4516-4523, 1998.

Feder, M.M., Siegfried, J.M., Balshem, A., Litwin, S., Keller, S.M., Liu, Z., Testa, J.R. Clinical relevance of chromosome abnormalities in non- small cell lung cancer. Cancer Genet. Cytogenet. 102:25-31, 1998.

MASON, W.S., ZHOU, T., NUNES, F., CONDREAY, L.D., LITWIN, S., SUMMERS, J. Antiviral therapy for chronic hepadnavirus infections. In Therapies for Viral Hepatitis, edited by R.F. Schinazi, J-P. Sommadossi, H.C. Thomas. International Medical Press, London, pp. 177-184, 1998.

PEHRSON, J.R., LITWIN S., MYERS, C.B., Cohen, L.H. Pyrimidine dimer formation as a probe of nucleosome core and linker structure in situ. Methods (in press).

SAUTER, E.R., NESBIT, M., WATSON, J.C., LITWIN, S., KLEIN-SZANTO, A., HERLYN, M. Vascular endothelial growth factor is a marker of tumor invasion and metastasis in squamous cell carcinomas of the head and neck. Clin. Cancer Res. (in press).

^§   Fox Chase researcher

Illustrations or unpublished data in these reports should not be used without permission of the author.

Fox Chase Cancer Center

Scientific Report 1998