Different ODE models of tumor growth can deliver similar results

Abstract

Background: Simeoni and colleagues introduced a compartmental model for tumor growth that has proved quite successful in modeling experimental therapeutic regimens in oncology. The model is based on a system of ordinary differential equations (ODEs), and accommodates a lag in therapeutic action through delay compartments. There is some ambiguity in the appropriate number of delay compartments, which we examine in this note.

Methods: We devised an explicit delay differential equation model that reflects the main features of the Simeoni ODE model. We evaluated the original Simeoni model and this adaptation with a sample data set of mammary tumor growth in the FVB/N-Tg(MMTVneu)202Mul/J mouse model.

Results: The experimental data evinced tumor growth heterogeneity and inter-individual diversity in response, which could be accommodated statistically through mixed models. We found little difference in goodness of fit between the original Simeoni model and the delay differential equation model relative to the sample data set.

Conclusions: One should exercise caution if asserting a particular mathematical model uniquely characterizes tumor growth curve data. The Simeoni ODE model of tumor growth is not unique in that alternative models can provide equivalent representations of tumor growth.

Keywords: Cancer chemotherapy; Delay differential equations; Mathematical model; Ordinary differential equations; Tumor growth.

Fig. 1

Schematic representation of the Simeoni tumor growth model. The tumor resides in compartment Z₁, with growth described by a tumor growth function. c(t) denotes the plasma concentration of an anticancer agent if present. The drug elicits its effect decreasing the tumor growth rate by a factor proportional to c(t)*Z₁(t) through the constant parameter k₁. Tumor cells cycle successively through transit compartments Z₂, Z₃, Z₄ before cell death. k₂ is a first-order rate constant of transit. The number of transit compartments is arbitrary. The system of ordinary differential equations describing this model is given in the text

Fig. 2

Schematic representation of the Simeoni tumor growth model, with transit compartments replaced by a single compartment in which tumor cell death is delayed relative to drug treatment. The delay is explicitly incorporated into the system of ordinary differential equations describing this model

Fig. 3

a, b Time course of tumor growth in 21 untreated tumor-bearing mice over the course of the experiment. The X-axis (time) denotes days, and the Y-axis (volume) denotes mm³

Fig. 4

Observed tumor sizes and fitted values of the 21 untreated tumor-bearing mice over the course of the experiment. The Simeoni tumor growth function was fit to the tumor size data from the entire cohort of animals, and individual fits were then derived from the mixed model analysis undertaken in Monolix

Fig. 5

a, b Time course of tumor growth in 19 treated tumorbearing mice over the course of the experiment. Treatment consisted of a single dose of cisplatin (5 mg/kg) on day 0. The X-axis (time) denotes days, and the Y-axis (volume) denotes mm³

Fig. 6

Observed tumor sizes and fitted values of the 19 treated tumor-bearing mice over the course of the experiment. A system of delay differential equations incorporating the Simeoni tumor growth function was fit to the tumor size data from the entire cohort of animals, and individual fits were then derived from the mixed model analysis undertaken in Monolix