.. _elimination:

Elimination, Clearance and Volume of Distribution
#####################################################
Pharmacokinetics is the study of "what the body does to the drug." By and large, the body removes it either by :term:`metabolism` (primarily in the liver) or by direct :term:`elimination` of the unchanged drug (primarily in the kidneys).

.. note:: See the :ref:`sum_link_elim_ke_example_tut` for the :ref:`tut_script` used to generate results in this section. 

Consider, as a first example, a bolus dose of 100 |nbsp| mg administered intravenously (IV) at time *t0=1*. Under first order kinetics, the rate of elimination is directly proportional to the amount of drug, $S(t)$, in the body at time *t* and the constant of proportionality is known as the elimination rate constant, *KE*:

.. math::

    \frac{dS}{dt} = -KE \cdot S(t)
    
This :term:`ordinary differential equation<ordinary differential equations>` (ODE) has a closed form solution:

.. math::

    S(t) = S(t0) \cdot \exp\{-KE \cdot (t-t0)\}
    
|ie| an exponential decay curve, where the initial conditions are

.. math::

    S(t0) = \text{100~mg}.

We can specify this directly in the |predictions| section of the control script,

.. literalinclude:: /_autogen/indiv_examples/elimination/elim_ke_example/tut_sections/PREDICTIONS.pyml
    :language: pyml
    
where, for now, we treat *KE* as if it were a constant measurable quantity, :pyml:`c[KE]`, defined in the |data_file|. From this closed form solution, we can predict the concentration at different time points and synthesize observations (:numref:`fig_elim_ke_example`).
  
.. figure:: /_autogen/indiv_examples/elimination/elim_ke_example/images/gen_dense/000001.*
    :name: fig_elim_ke_example
    :align: center
    :width: 50%
    
    |AvT| curve for an intravenously administered 100 |nbsp| mg bolus dose at time *t0=1* with first order elimination. (Observations are noiseless in this example.)
    

.. _elimination_vol:
    
Volume of Distribution
=====================================
In practice, however, we cannot measure the *amount* of drug in the body from a blood plasma sample, only its *concentration* (|ie| amount of drug per unit volume).

.. note:: 
  
   See the :ref:`sum_link_elim_v_example_tut` for the :ref:`tut_script` used to generate results in this section. 

This measured concentration will clearly be influenced by the physiological volume of blood plasma in an individual's body. Less obvious, however, is that the concentration will also be influenced by the physiochemical properties of the drug that determine how the drug is distributed throughout the body; some drugs are distributed mostly in the blood plasma (which is directly observed) whereas others get distributed to all tissues (which are not). The scaling factor -- a combination of physiological and physiochemical properties -- that relates amounts to concentrations is known as the :term:`volume of distribution`, *V*, which varies greatly between drugs and, to a lesser extent, between individuals.

We model the observed concentration by including the volume of distribution parameter (with a value of 20 |nbsp| L in this example) in the |predictions| section of the control script:

.. literalinclude:: /_autogen/indiv_examples/elimination/elim_v_example/tut_sections/PREDICTIONS.pyml
    :language: pyml

which has the effect of scaling the observations (:numref:`fig_elim_v_example`; note the scale of the *y*-axis).

.. figure:: /_autogen/indiv_examples/elimination/elim_v_example/images/gen_dense/000001.*
    :name: fig_elim_v_example
    :align: center
    :width: 50%
    
    |CvT| curve for an intravenously administered 100 |nbsp| mg bolus dose with first order elimination


.. _elimination_cl:
    
Clearance
===============

.. note:: 
  
   See the :ref:`sum_link_elim_cl_example_tut` for the :ref:`tut_script` used to generate results in this section. 


Because we can only measure concentrations, it makes sense to define the rate of elimination also in terms of concentrations:

.. math::

    \frac{dS}{dt} &= -KE \cdot V \cdot \frac{S(t)}{V} \\
                  &= -KE \cdot V \cdot C(t) \\
                  &= -CL \cdot C(t) \\

where

.. math::

    C(t) = \frac{S(t)}{V}
    
is the concentration at time *t* and

.. math::

    CL = KE \cdot V
    
is a constant of proportionality known as the :term:`clearance`, *CL*, that relates elimination to concentration. Again, this permits a closed form solution,

.. math::

    S(t) = S(0) \cdot \exp\{-(CL/V) \cdot (t-t0)\}

where we have substituted *CL/V* for the rate constant of elimination, *KE*:

.. literalinclude:: /_autogen/indiv_examples/elimination/elim_cl_example/tut_sections/PREDICTIONS.pyml
    :language: pyml
    
Because this is a mathematical equivalence, the |CvT| curve (:numref:`fig_elim_v_example`) is unchanged.
