.. _bp_rf:

Blood Pressure
---------------

Data for blood pressure is obtained from NCD RISC :cite:p:`NCDRiskFactorCollaborationNCD-RisC2017` and represents years 1975-2014. Most countries across the world are covered in the dataset, 
which provides information on mean systolic and diastolic blood pressure, as well as the proportion of 
people with a systolic blood pressure (SBP) > 140 by age group (18-19, 20-24, …, 80-84, 85+) and gender.

Modelling principles for systolic blood pressure
``````````````````````````````````````````````````

Systolic blood pressue (SBP) is modelled with a log-normal distribution:

.. math::
   SBP(n,y) \leadsto LogNormal(\hat\sigma(n,y),\hat\mu(n,y))

The parameters of the log-normal distribution are fitted using the average systolic blood pressure, 
:math:`\overline{SBP}`
and proportion of people with high blood pressure, :math:`\mathbb{P}_{BP>140}`.
For each year, :math:`y`, the following equation system is solved:

.. math::
    \begin{eqnarray}
        &\min_{\lambda(y)}\sum_n ( \overline{SBP} - \displaystyle e^{\hat\mu(n,y) - \frac{1}{2} \hat{σ}^2(n,y)} )^2 \\
        &\hat{σ}(n,y) = σ_0+λ(y)*(n-n_0)\\
        &\hat\mu(n,y) = \ln(140)-\Phi^{-1} (1- \mathbb{P}_{BP>140})
    \end{eqnarray}



Where, :math:`n_0` is the first available age (here 18) and :math:`σ_0` is computed to match the data :math:`\overline{SBP}, \mathbb{P}_{BP>140}` (for the first available age group (here 18-19).

During the calibration procedure a higher weight was used to fit the percentage of people with a high blood pressure 
compared to those with an average SBP, in order to model the impact of high blood pressure more accurately.  
:numref:`BP_RF_GammaDistribution` and :numref:`SBP_RF` illustrate the methodology and its outputs for France. 

As shown in :numref:`SBP_RF`, blood pressure rises as a function of age. For example, the blood pressure of an 
individual in the 50th percentile increases from 107 at 20 years of age to 134 at 65 years. 

.. _BP_RF_GammaDistribution:
.. figure:: images/cdfBPcalibration.png
   :alt: BP_gamma
   :width: 600px
   :align: center
   :figclass: align-center

   Fitting gamma distribution on average blood pressure and share of people with raised blood pressure 
   (Women - France - 2015). 


.. _SBP_RF:
.. figure:: images/cdfBP.png
   :width: 600px
   :align: center
   :figclass: align-center

   Systolic blood pressure: fitted gamma distribution for different age 
   (Women - France - 2015). 


Relative risks and baseline risks
````````````````````````````````````
The link between SBP and diseases are modelled through relative risks. Relative risk 
(by age group and by gender) are provided by IHME (:cite:p:`GBD2016RiskFactorsCollaborators2017`) through a 
linear function (see :numref:`RR_stroke`). 

.. _RR_stroke:
.. figure:: images/rrBP.png
   :width: 600px
   :align: center
   :figclass: align-center

   Relative risk for Stroke and Hemorrhagic Stroke based on blood pressure level, men and women aged 50 
   
For each disease, the baseline risk (:math:`BR`) is computed as follows:

.. math::
   BR = \frac{1}{\int_{80}^{250} RR(x)dBP(x)}
   
Where: 

 * :math:`RR(x)` = Relative risk based on blood pressure 
 * :math:`dBP`   = Change in blood pressure (SBP modelled with a log-normal distribution)
 * :math:`UR`    = Underlying Risk.