Power-law tumour growth

A tumour whose proliferating part scales as a power of its size, dn/dt = r n^p.

• Name: power_law_growth
• Field: biology/oncology
• Formalism: ordinary differential equations
• References: Padmanabhan, Meskin, and Al Moustafa 2021; Benzekry et al. 2014

Factory

@register_model(
    field='biology/oncology',
    tags=('tumour growth', 'ode', 'power law'),
    references=GROWTH_REFERENCES,
    defaults={'until': 40.0, 'dt': 0.2},
)
def power_law_growth(n: float = 10.0, *, r: float = 1.32, p: float = 0.58) -> System:
    """Return a tumour whose proliferating part scales as a power of its size, `dn/dt = r n^p`.

    For `p < 1` the volume follows `(n0^(1-p) + (1-p) r t)^(1/(1-p))`, a growth slower than exponential. The defaults
    are the fit to breast cancer volumes in mm³ and days.

    Args:
        n:
            The initial volume.

        r:
            The growth rate.

        p:
            The exponent.

    Returns:
        system:
            The model, with the volume `tumour.n`.

    Examples:
        >>> from skcomplex.models.biology.oncology import power_law_growth
        >>> power_law_growth(p=2 / 3).interactions[0].p
        0.6666666666666666
    """
    return System(
        elements=[PopulationElement('tumour', n=n)],
        interactions=[PowerLawGrowthInteraction('growth', target='tumour', r=r, p=p)],
    )

Components

class PopulationElement(CompartmentElement):
    """Population of one species, a compartment whose state `n` holds its size.

    Args:
        label:
            The label of the population.

        visual:
            How the element is drawn, or `None` for `default_visual_`.

        n:
            The initial size.

    Examples:
        >>> from skcomplex.models.biology.common.populations import PopulationElement
        >>> PopulationElement('rabbits', n=40).n
        40
    """

    def __init__(
        self,
        label: str,
        visual: VisualSpec | None = None,
        *,
        n: Annotated[float, StateSpec(domain=RealSet.closed_open(0, oo))] = 1.0,
    ) -> None:
        super().__init__(label, visual=visual, n=n)

class PowerLawGrowthInteraction(GrowthInteraction):
    """Growth that scales as a power of the size, `dn/dt = r n ** p`, without harvest.

    Args:
        label:
            The label of the interaction.

        target:
            The path of the element or group whose population grows.

        r:
            The growth rate.

        p:
            The exponent, `2/3` for a growing rim and 1 for exponential growth.

        state:
            The name of the state of the target that holds the population.

    Examples:
        >>> from skcomplex.core.elements import CompartmentElement
        >>> from skcomplex.core.system import System
        >>> from skcomplex.engine.runtime import Simulation
        >>> from skcomplex.models.biology.common.populations import PowerLawGrowthInteraction
        >>> system = System(
        ...     elements=[CompartmentElement('tumour', n=1.0)],
        ...     interactions=[PowerLawGrowthInteraction('growth', target='tumour', r=1.0, p=0.5)],
        ... )
        >>> trajectory = Simulation(system, dt=0.01).check().prepare().run(until=2.0).trajectory_
        >>> round(float(trajectory.values('tumour.n')[-1]), 3)
        4.0
    """

    def __init__(
        self,
        label: str,
        *,
        target: Annotated[str, RoleSpec()],
        r: Annotated[float, ParameterSpec(domain=RealSet.open(0, oo))] = 1.32,
        p: Annotated[float, ParameterSpec(domain=RealSet.open(0, oo))] = 0.58,
        state: str = 'n',
    ) -> None:
        super().__init__(label, target=target, r=r, p=p, form='power', state=state)

    def _get_rate(self, namespace: SimpleNamespace) -> Expression:
        """Return the power-law growth, with no harvest.

        Args:
            namespace:
                The namespace of the symbols the rate can use, with the namespace of the target.

        Returns:
            rate:
                The rate of the population.
        """
        return self._get_growth(namespace)

Equations

tumour.n: diff(n_tumour(t), t) == r_growth*n_tumour(t)^p_growth