Construct a decision tree and find the expected value for the surgery using the provided Scenario's and Decision tree template picture attached to this request. Scenario $1$: Ankle Fracture Decision Tree DecisionTree

In this scenario, we are deciding between performing an immediate Below$-$Knee Amputation $($BKA$)$ or attempting surgical debridement and antibiotics to save the ankle.

$-$The outcomes for both options have associated probabilities and utility values. Below is the action plan i would create that provides that best solution.

Step $1$: Decision Tree Construction

Initial Decision Node: Start with the ankle fracture.

Two Branches:

Option $1$: Perform immediate BKA $($Below$-$Knee Amputation$).$

Option $2$: Perform Surgical Debridement with antibiotics.

Step $2$: List the Outcomes for Each Option

Option $1($BKA$)$:

Outcome $1\mathrm{.}1$: Successful amputation with no further complications $($probability: P$1\mathrm{.}1,$ utility: U$1\mathrm{.}1).$

Outcome $1\mathrm{.}2$: Complications $($probability: P$1\mathrm{.}2,$ utility: U$1\mathrm{.}2).$

Option $2($Surgical Debridement$)$:

Outcome $2\mathrm{.}1$: Complete recovery with no long$-$term effects $($probability: P$2\mathrm{.}1,$ utility: U$2\mathrm{.}1).$

Outcome $2\mathrm{.}2$: Minor long$-$term disability $($probability: P$2\mathrm{.}2,$ utility: U$2\mathrm{.}2).$

Outcome $2\mathrm{.}3$: Infection spreads, requiring BKA $($probability: P$2\mathrm{.}3,$ utility: U$2\mathrm{.}3).$

Outcome $2\mathrm{.}4$: Infection spreads, requiring AKA $($Above$-$Knee Amputation$)($probability: P$2\mathrm{.}4,$ utility: U$2\mathrm{.}4).$

Outcome $2\mathrm{.}5$: Death due to septic complications $($probability: P$2\mathrm{.}5,$ utility: U$2\mathrm{.}5=0).$

Step $3$: Fill in Probabilities and Utilities

We will now assign probabilities and utilities for each outcome based on the literature or patient preferences. For simplicity, let$$s assume hypothetical values:

Immediate BKA:

P$1\mathrm{.}1=0\mathrm{.}95,$ U$1\mathrm{.}1=0\mathrm{.}8$

P$1\mathrm{.}2=0\mathrm{.}05,$ U$1\mathrm{.}2=0\mathrm{.}5$

Surgical Debridement:

P$2\mathrm{.}1=0\mathrm{.}6,$ U$2\mathrm{.}1=1\mathrm{.}0$

P$2\mathrm{.}2=0\mathrm{.}2,$ U$2\mathrm{.}2=0\mathrm{.}7$

P$2\mathrm{.}3=0\mathrm{.}1,$ U$2\mathrm{.}3=0\mathrm{.}8$

P$2\mathrm{.}4=0\mathrm{.}05,$ U$2\mathrm{.}4=0\mathrm{.}3$

P$2\mathrm{.}5=0\mathrm{.}05,$ U$2\mathrm{.}5=0$

Step $4$: Calculate Expected Values for Each Option

Option $1($BKA$)$:

Expected value for BKA:

EV$1=$P$1\mathrm{.}1\backslash$times U$1\mathrm{.}1+$P$1\mathrm{.}2\backslash$times U$1\mathrm{.}2$ EV$1=(0\mathrm{.}95\backslash$times $0\mathrm{.}8)+(0\mathrm{.}05\backslash$times $0\mathrm{.}5)=0\mathrm{.}76+0\mathrm{.}025=0\mathrm{.}785$

Option $2($Surgical Debridement$)$:

Expected value for debridement:

EV$2=$P$2\mathrm{.}1\backslash$times U$2\mathrm{.}1+$P$2\mathrm{.}2\backslash$times U$2\mathrm{.}2+$P$2\mathrm{.}3\backslash$times U$2\mathrm{.}3+$P$2\mathrm{.}4\backslash$times U$2\mathrm{.}4+$P$2\mathrm{.}5\backslash$times U$2\mathrm{.}5$ EV$2=(0\mathrm{.}6\backslash$times $1\mathrm{.}0)+(0\mathrm{.}2\backslash$times $0\mathrm{.}7)+(0\mathrm{.}1\backslash$times $0\mathrm{.}8)+(0\mathrm{.}05\backslash$times $0\mathrm{.}3)+(0\mathrm{.}05\backslash$times $0)$EV$2=0\mathrm{.}6+0\mathrm{.}14+0\mathrm{.}08+0\mathrm{.}015=0\mathrm{.}835$

Step $5$: Compare the Expected Values

Option $1($BKA$)$: Expected value $=0\mathrm{.}785$

Option $2($Surgical Debridement$)$: Expected value $=0\mathrm{.}835$