ARDS Mathematical Equations

 

ARDS Markov equations

2024-04-29

By Samuel Y Huang and Alexander Huang

Modeling Acute Respiratory Distress Syndrome (ARDS) with a stochastic modeling process that utilizes transition diagrams and first step analyses

\[ \{X_0, X_1, X_2, \ldots\}, \text{ where } X_t \text{ is the state at time } t. \]

In terms of ARDS this means \[ \{X_{ARDSStage(t=1)} , X_{ARDSStage(t=2)}, X_{ARDSStage(t=3)}, \ldots\}, \text{ where } X_{ARDSStage(t=t)} \text{ is the state at time } t. \]

Where ARDS is defined via stages as below \[ \text{ARDS stage} = \begin{cases} \text{Mild} & \text{if } PaO2/FiO2 < 100, \\ \text{Moderate} & \text{if } 100 \leq PaO2/FiO2 \leq 200, \\ \text{Severe} & \text{if } 200 < PaO2/FiO2 \leq 300. \end{cases} \] The Markov Chains have a crucial property

\[ X_{t+1} \text{ depends only on } X_t. \]

Questions of Interest - Starting from Mild Acute Respiratory Distress Syndrome, what is the probability of decompensating to moderate or severe or death, what is the probability of downgrading to the floors

• Starting from Moderate Acute Respiratory Distress Syndrome, what is the probability of decompensating to severe or death, what is the probability of downgrading to the floors or improving to mild ARDS

• Starting from Severe Acute Respiratory Distress Syndrome, what is the probability of decompensating to death, what is the probability of downgrading to the floors or improving to mild or moderate ARDS

• For patients on the floor, what is the probability of decompensating to mild, moderate, or severe ARDS or death

• For patients that are dead it is a closed loop

Definition: The state of a Markov chain at time T is the value of

\[ X_{ARDSStage(t=t)} \]

if

\[ X_{ARDSStage(t=6)} = mild \] we say that at time 6 they are in Mild ARDS

Definition: The state space of a Markov Chain S, is the set of values that each \[ X_{ARDSStage(t)} \]

can take. For example, \[ S = \left \{ mild, moderate, severe, floors, death \right \} \] Let S have size N = 5

Definition: A trajectory of a Markov chain is a particular set of values for

\[ \{X_0, X_1, X_2, \ldots\} \] For example if in our ARDS case \[ \{X_{ARDSStage(t=1)} = Floors, X_{ARDSStage(t=2)} = Mild, X_{ARDSStage(t=3)} = Moderate, X_{ARDSStage(t=4)} = Severe, X_{ARDSStage(t=5)} = Death, \ldots\} \] Then the trajectory up to time t = 5 is Floors -> Mild -> Moderate -> Severe -> Death

Trajectory just means Path

Markov Property

The basic property of a Markov Chain is that the only the most recent point in the trajectory affects that happens next. \[ P(X_{t+1} = s \mid X_t = s_t, X_{t-1} = s_{t-1}, \ldots, X_0 = s_0) = P(X_{t+1} = s \mid X_t = s_t), \]

P(Xt+1) depends on P(Xt) but does not depend on Xt-1, …X2, X1, X0.

The Transition Matrix

\[ \left(\begin{array}{c|cccc} \text{State} & \text{Mild} & \text{Moderate} & \text{Severe} & \text{Floors} & \text{Death} \\ \hline Mild & P_{mild,mild} & P_{mild,moderate} & P_{mild,severe} & P_{mild,floors} & P_{mild,death} \\ Moderate & P_{moderate,mild} & P_{moderate,moderate} & P_{moderate,severe} & P_{moderate,floors} & P_{moderate,death} \\ Severe & P_{severe,mild} & P_{severe,moderate} & P_{severe,severe} & P_{severe,floors} & P_{severe,death} \\ Floors & P_{floors,mild} & P_{floors,moderate} & P_{floors,severe} & P_{floors,floors} & P_{floors,death} \\ Death & P_{death,mild} & P_{death,moderate} & P_{death,severe} & P_{death,floors} & P_{death,death} \end{array}\right) \]

The row represents Now from or from X(t) The columns represent Next or To (Xt+1) Entry (i,j) is the conditional probability that next = j given that now = i: the probability of going from state i to state j

\[ p_{ij} = P(X_{t+1} = j \mid X_t = i). \]

\[ \sum_{j=1}^Np_{ij} = \sum_{j=1}^N P(X_{t+1} = j \mid X_t = i) = \sum_{j=1}^N P\{X_t = i\}(X_{t+1} = j) = 1. \]

Tne transition matrix P list all possible states in the state space S. P is a square matrix (NxN), because Xt+1 and Xt both take values in the same state space S (of size N) The rows of P each sum to 1

Let’s define the distribution of X at a certain time

\[ \{X_{ARDSStage(t=1)} , X_{ARDSStage(t=2)}, X_{ARDSStage(t=3)}, \ldots\}, \text{ where } X_{ARDSStage(t=t)} \text{ is the state at time } t. \]

Where the state space is as below: \[ S = \left \{ mild, moderate, severe, floors, death \right \} \] What we need to determine the probability distribution of each time point. Let’s say X0, this will be the chance that in individual starts off from time 0 at each end point from being admitted from the emergency department. Pi is an Nx1 vector denoting the probability distribution of X0

\[ \pi = \begin{pmatrix} \pi_1 \\ \pi_2 \\ \vdots \\ \pi_N \end{pmatrix} = \begin{pmatrix} P(X_0 = 1) \\ P(X_0 = 2) \\ \vdots \\ P(X_0 = N) \end{pmatrix} \] We will write \[ X_0 \sim \pi^T \] to denote the row vector of probabilities is given by the row vector pi row transposition

How do we calculate the probability of the current state? at the current time? using the partition rule below \[ \text{Use the Partition Rule, conditioning on } X_0: \\ P(X_1 = j) = \sum_{i=1}^N P(X_1 = j \mid X_0 = i) P(X_0 = i) \\ = \sum_{i=1}^N p_{ij} \pi_i \quad \text{by definitions} \\ = \sum_{i=1}^N \pi_i p_{ij} \\ = (\pi^T P)_{j}. \\ \text{(pre-multiplication by a vector from Section 8.5).} \\ \text{This shows that } P(X_1 = j) = (\pi^T P)_{j} \text{ for all } j. \\ \text{The row vector } \pi^T P \text{ is therefore the probability distribution of } X_1: \\ X_0 \sim \pi^T \\ X_1 \sim \pi^T P. \]

Clinical Correlate Let’s say we are on our second timepoint t = 1 let’s say moderate ARDS. What is the probability they had the specific trajectory mild - > moderate ARDS.

Now we want to describe the probability distribution after time 0. How do we get the states at time T?

Trajectory Probability Can characterize the trajectory of ARDS by probabilities.

• For example Transition of the trajectory from Severe -> Moderate -> Mild -> Floors

• Of great interest will be the probability of regressing from ARDS within one day or 7 days

• Do most patients deescalate in a predictable pattern? or is this more of a random disease where there will be a uniform distribution of trajectories?

Recall a trajectory is the sequence of values.

Because of the Markov Property (Memoryless) we can find the proability of any trajectory by multiplying together the starting proability and all subsequent single step properties

Let’s envision a 7 day course (trajectory) of ARDS.

What is the probability of Mild -> Severe -> Moderate -> Mild -> Floors -> Severe -> Death?

Where the starting probability for Mild is as follows \[ X_0 \sim \pi^T \]

\[ \ T \sim \left(\begin{array}{c|cccc} \text{State} & \text{Mild} & \text{Moderate} & \text{Severe} & \text{Floors} & \text{Death} \\ \hline Mild & P_{mild,mild} & P_{mild,moderate} & P_{mild,severe} & P_{mild,floors} & P_{mild,death} \\ Moderate & P_{moderate,mild} & P_{moderate,moderate} & P_{moderate,severe} & P_{moderate,floors} & P_{moderate,death} \\ Severe & P_{severe,mild} & P_{severe,moderate} & P_{severe,severe} & P_{severe,floors} & P_{severe,death} \\ Floors & P_{floors,mild} & P_{floors,moderate} & P_{floors,severe} & P_{floors,floors} & P_{floors,death} \\ Death & P_{death,mild} & P_{death,moderate} & P_{death,severe} & P_{death,floors} & P_{death,death} \end{array}\right) \]

Let’s define the probability of each of the different states \[ \text{Let } X_0 \sim (\text{Mild}). \text{ What is the probability of the trajectory Mild } \rightarrow \text{ Severe } \rightarrow \text{ Moderate } \rightarrow \text{ Mild } \rightarrow \text{ Floors } \rightarrow \text{ Severe } \rightarrow \text{ Death?} \\ P(\text{Mild, Severe, Moderate, Mild, Floors, Severe, Death}) = P(X_0 = \text{Mild}) \times p_{\text{mild, severe}} \times p_{\text{severe, moderate}} \times p_{\text{moderate, mild}} \times p_{\text{mild, floors}} \times p_{\text{floors, severe}} \times p_{\text{severe, death}} \]

This is theoretical.

A Markov Chain maps the changes of the states in the aggregate population.