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Researchers have observed that response of tumor cells to treatment varies depending on whether the cells are grown in monolayer, as

Cancer therapies are tested thoroughly on monolayer layers to identify not only their effectiveness but also the specific manner in which they impede cell division or induce apoptosis. It is understood that the effectiveness of treatment in monolayer does not predict equivalent effectiveness

As part of a study on metronomic therapy of breast cancer, Klement et al. [

The results of this paper raise a few questions which may be approached through simulations. The first is whether it is likely that the results in Klement et al. [

The spontaneous cessation of tumor spheroid growth was conjectured to be due to the inability of nutrients to penetrate to the core of the spheroid, which subsequently undergoes necrosis [

To make sense of how model simulations can reflect therapies, it is necessary to tune general models with good qualitative behavior to the specifics of a particular cell line and therapy. This paper considers treatment of SK-N-SH neuroblastoma cells with 15-deoxy-

The data published in these two papers [

The nonlinear dynamic model developed here includes five compartments,

A cell cycle model (compartments

Monolayer layers exhibit exponential growth, at least in the short run, and the cell cycle for such cultures is modeled by a system of ordinary differential equations given below.

The rate of change of

The rate of change of

The rate of change of

Writing the system as

Here

The equation

Using the Routh-Hurwitz Criterion, we determined that the monolayer dies out when

Table

Summary of cell cycle parameters for SK-N-SH monolayer culture (default parameters). Note that values for

Cell line | SK-N-SH |

Source | Kim et al. (2003) |

Observed | 62.87% |

Observed | 26.93% |

Observed | 10.20% |

Observed | n/a |

Observed | Taken as 0 |

| |

Source 2 | Multiple |

Observed | |

| |

Calculated | 0.40 |

Calculated | 0.85 |

Calculated | 1.59 |

Calculated | 3.85 |

The experiments described in Kim et al. [

Summary of treatment parameters for SK-N-SH monolayer culture. Initial conditions have total cells at 100, divided into percents given by

Cell line | SK-N-SH |

Source | Kim et al. (2003) |

Treatment | 15-Deoxy- |

Parameter(s) | |

Computed final total control | 147.83 |

Initial conditions for all runs | |

| |

Treatment intensity | 2 |

Fitted parameter | |

Observed at 24 hrs | 57.9%, 25.5%, 16.6%, 8.04% |

Computed: % | 56.33%, 27.28%, 16.39% |

Computed total | 119.34 |

Computed mortality | 8% |

| |

Treatment intensity | 4 |

Fitted parameter | |

Observed at 24 hrs | 51.9%, 26.4%, 21.7%, 10% |

Computed: % | 51.34%, 26.78%, 21.88% |

Computed total | 109.90 |

Computed mortality | 10.09% |

| |

Treatment intensity | 8 |

Fitted parameter | |

Observed at 24 hrs | 48.2%, 23.1%, 28.7%, 17% |

Computed: % | 47.73%, 27.54%, 24.72% |

Computed total | 97.37 |

Computed mortality | 17.03% |

| |

Treatment intensity | 12 |

Fitted parameter | |

Observed at 24 hrs. | 37.6%, 20%, 42.4%, 36% |

Computed: % | 36.45%, 29.31%, 34.24% |

Computed total | 73.83 |

Computed mortality | 26.25% |

Wallace and Guo [

The path to enter the quiescent state, ^{6} ^{3}.

The rate of change of

The rate of change of

The rate of change of

The rate of change of

The rate of change of

When

The passage of

The return from the quiescent state to the proliferating state would occur for some fraction of cells when nutrients cease to be blocked by the nonquiescent cells,

A class of functions that behaves this way is given by

Both functions ^{6} ^{3} the effect near zero is easily avoided for small

Finally, the fraction of cells at the

These equations were tuned to spheroid culture data in Carlsson et al. [

Cell cycle parameters

Summary of spheroid growth data estimated from Carlsson et al. [

Type, cell line | (Day, reported diameter in mm), | Thickness of viable cell rim at end of trial (day, |
---|---|---|

Neuroblastoma, SK-N-SH | (2, 0.19), (7, 0.21), (11, 0.27), (15, 0.32), (19, 0.35), (25, 0.39), (31, 0.41) | (31, 50–150 |

Summary of default parameters for spheroid model. Spheroid volumes and viable cell volume estimated from Table

Cell line | SK-N-SH |

(Time, computed volume) | (2, |

(Time, computed volume) | (7, |

(Time, computed volume) | (11, |

(Time, computed volume) | (15, |

(Time, computed volume) | (19, |

(Time, computed volume) | (25, |

(Time, computed volume) | (31, |

(Time, computed volume of necrosis ( | (31, (15.6–4.85) |

(Time, computed volume of live cells ( | (31, (20.5–31.25) |

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Chosen | |

Fitted | |

Chosen | |

Fitted | |

Fitted | 0 |

Fitted | |

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The two most difficult parameters to identify from data are

Four parameters remain to be identified. Of these, two (

Numerical exploration indicates that the two parameters that appear in this equation,

Two parameters remain,

In addition to matching data, overall long-term behavior of the model was compared to qualitative observations and was found to behave well. Initial conditions for all data matching and treatment simulations were set by the volume of the first data point in [

With all parameters specified for the model of an untreated SK-N-SH spheroid, it only remains to alter those parameters corresponding to treatment with 15-deoxy-

To determine the impact of treatment intensity on spheroid volume and persistence of the spheroid, we performed a bifurcation analysis of the spheroid model. In particular, we calculated numerically (using the MATLAB function “lsqnonlin”) the volume of proliferating, quiescent, and necrotic cells at equilibrium as a function of

Rapidly growing tumor monolayers exhibit characteristic doubling times, death rates, and cell cycle proportions that are enough to determine a linear model completely, leading to simple algebraic expressions for all parameters. Data for treated monolayers are another matter. In the example studied here, two parameters,

Simulated monolayer treatment of SK-N-SH neuroblastoma cells with 15-deoxy-

Model simulation of cell cycle analysis after 24 hours of treatment, versus measured values. Data is taken from [

It is possible to extend the linear model of the cell cycle in monolayer growth to a nonlinear model of spheroid growth, as illustrated in Figure

Model predictions for spheroid growth to 100 days. (a) Relative sizes of proliferating (

The SK-N-SH neuroblastoma spheroid model with default (control) parameters given in Tables

Response of SK-N-SH neuroblastoma spheroid and monolayer models to increasing 15-deoxy-

The purpose of constructing these models was to simulate the different response to treatment between monolayer and spheroid cultures. Figure

Bifurcation diagram for equilibrium of necrotic cells in spheroid model. Each line represents the equilibrium volume of cells at a different treatment intensity ^{6} ^{3}. Filled circles denote the equilibria using the fitted pair of parameters

Long-term behavior of SK-N-SH neuroblastoma spheroid, under hypothetical continued treatment, at values of

The behavior of the monolayer model under treatment conditions is consistent with the conclusions in [

Fitting a linear model to cell cycle information for a monolayer can be done statistically. In (

Figure

Our results give a spheroid model that not only has the correct qualitative growth behavior [

The ability of the model to approximate spheroid growth over time, while including cell cycle dynamics, should make it useful to the experimenter who wishes to predict the results of specific therapeutic actions on preangiogenic tumors. Its utility as a predictor fits the proposed workflow model in McGuire et al. [

Figure

An accurate model for growth of

The authors declare that they have no competing interests.

The authors wish to thank Dartmouth College for its generous support of undergraduate interns.