## Step 1: Define the symbolic representation of the problem
The problem involves two variables: 'milligrams of vitamin E' and 'milligrams of vitamin B9'. Let's denote 'milligrams of vitamin E' as $x_1$ and 'milligrams of vitamin B9' as $x_2$. The objective function to minimize is $4.45x_1 + 1.29x_2$. The constraints are:
- $x_1$ has a cognitive performance index of 1 and a muscle growth index of 2.
- $x_2$ has a cognitive performance index of 7 and a muscle growth index of 2.
- The total combined cognitive performance index is $x_1 + 7x_2 \geq 21$.
- The total combined muscle growth index is $2x_1 + 2x_2 \geq 20$.
- The constraint $-3x_1 + 6x_2 \geq 0$.
- The total combined cognitive performance index is $x_1 + 7x_2 \leq 47$.
- The total combined muscle growth index is $2x_1 + 2x_2 \leq 28$.
- $x_1$ must be an integer.
- $x_2$ can be a non-integer.

## 2: Convert the problem into a symbolic representation
The symbolic representation of the variables is:
- $x_1$ represents 'milligrams of vitamin E'.
- $x_2$ represents 'milligrams of vitamin B9'.

The objective function is: $4.45x_1 + 1.29x_2$.

The constraints are:
- $x_1 + 7x_2 \geq 21$.
- $2x_1 + 2x_2 \geq 20$.
- $-3x_1 + 6x_2 \geq 0$.
- $x_1 + 7x_2 \leq 47$.
- $2x_1 + 2x_2 \leq 28$.
- $x_1 \geq 0$ and is an integer.
- $x_2 \geq 0$.

## 3: Provide the symbolic problem description in JSON format
```json
{
'sym_variables': [('x1', 'milligrams of vitamin E'), ('x2', 'milligrams of vitamin B9')],
'objective_function': '4.45*x1 + 1.29*x2',
'constraints': [
    'x1 + 7*x2 >= 21',
    '2*x1 + 2*x2 >= 20',
    '-3*x1 + 6*x2 >= 0',
    'x1 + 7*x2 <= 47',
    '2*x1 + 2*x2 <= 28',
    'x1 is an integer',
    'x1 >= 0',
    'x2 >= 0'
]
}
```

## 4: Implement the problem using Gurobi
```python
import gurobi

def solve_optimization_problem():
    # Create a new Gurobi model
    model = gurobi.Model()

    # Define the variables
    x1 = model.addVar(name="x1", vtype=gurobi.GRB.INTEGER, lb=0)  # milligrams of vitamin E
    x2 = model.addVar(name="x2", vtype=gurobi.GRB.CONTINUOUS, lb=0)  # milligrams of vitamin B9

    # Define the objective function
    model.setObjective(4.45 * x1 + 1.29 * x2, gurobi.GRB.MINIMIZE)

    # Add constraints
    model.addConstr(x1 + 7 * x2 >= 21, name="cognitive_performance_index")
    model.addConstr(2 * x1 + 2 * x2 >= 20, name="muscle_growth_index_min")
    model.addConstr(-3 * x1 + 6 * x2 >= 0, name="vitamin_constraint")
    model.addConstr(x1 + 7 * x2 <= 47, name="cognitive_performance_index_max")
    model.addConstr(2 * x1 + 2 * x2 <= 28, name="muscle_growth_index_max")

    # Optimize the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Optimal solution found.")
        print(f"Milligrams of vitamin E: {x1.varValue}")
        print(f"Milligrams of vitamin B9: {x2.varValue}")
        print(f"Objective function value: {model.objVal}")
    else:
        print("No optimal solution found.")

solve_optimization_problem()
```