## Step 1: Define the symbolic representation of the variables
The variables are defined as follows: 
- $x_0$ represents 'milligrams of calcium'
- $x_1$ represents 'milligrams of vitamin B4'
- $x_2$ represents 'milligrams of vitamin B1'

## Step 2: Convert the objective function into a symbolic representation
The objective function to maximize is $4.36x_0^2 + 4.01x_0x_1 + 9.16x_0 + 3.03x_1 + 6.23x_2$.

## 3: Define the constraints in symbolic notation
The constraints are:
- $1.38x_0 \leq 108$ (cognitive performance index for calcium)
- $0.27x_0 \leq 57$ (muscle growth index for calcium)
- $1.11x_1 \leq 108$ (cognitive performance index for vitamin B4)
- $1.31x_1 \leq 57$ (muscle growth index for vitamin B4)
- $1.33x_2 \leq 108$ (cognitive performance index for vitamin B1)
- $1.8x_2 \leq 57$ (muscle growth index for vitamin B1)
- $1.38x_0 + 1.33x_2 \geq 17$ (total cognitive performance index from calcium and vitamin B1)
- $0.27x_0^2 + 1.31x_1^2 \geq 13$ (total muscle growth index from calcium squared and vitamin B4 squared)
- $0.27x_0 + 1.31x_1 + 1.8x_2 \geq 12$ (total muscle growth index from calcium, vitamin B4, and vitamin B1)
- $1.38^2x_0^2 + 1.33^2x_2^2 \leq 70$ (total cognitive performance index from calcium squared and vitamin B1 squared)
- $1.11x_1 + 1.33x_2 \leq 108$ (total cognitive performance index from vitamin B4 and vitamin B1)
- $1.38^2x_0^2 + 1.11^2x_1^2 \leq 51$ (total cognitive performance index from calcium squared and vitamin B4 squared)
- $1.38x_0 + 1.11x_1 + 1.33x_2 \leq 51$ (total cognitive performance index from calcium, vitamin B4, and vitamin B1)
- $1.31x_1 + 1.8x_2 \leq 35$ (total muscle growth index from vitamin B4 and vitamin B1)
- $0.27x_0 + 1.8x_2 \leq 42$ (total muscle growth index from calcium and vitamin B1)
- $0.27x_0 + 1.31x_1 + 1.8x_2 \leq 42$ (total muscle growth index from calcium, vitamin B4, and vitamin B1)

## 4: Provide the symbolic representation of the problem
```json
{
    'sym_variables': [
        ('x0', 'milligrams of calcium'), 
        ('x1', 'milligrams of vitamin B4'), 
        ('x2', 'milligrams of vitamin B1')
    ], 
    'objective_function': '4.36x0^2 + 4.01x0*x1 + 9.16x0 + 3.03x1 + 6.23x2', 
    'constraints': [
        '1.38x0 <= 108',
        '0.27x0 <= 57',
        '1.11x1 <= 108',
        '1.31x1 <= 57',
        '1.33x2 <= 108',
        '1.8x2 <= 57',
        '1.38x0 + 1.33x2 >= 17',
        '0.27x0^2 + 1.31x1^2 >= 13',
        '0.27x0 + 1.31x1 + 1.8x2 >= 12',
        '1.9044x0^2 + 1.7689x2^2 <= 70',
        '1.11x1 + 1.33x2 <= 108',
        '1.9044x0^2 + 1.2321x1^2 <= 51',
        '1.38x0 + 1.11x1 + 1.33x2 <= 51',
        '1.31x1 + 1.8x2 <= 35',
        '0.27x0 + 1.8x2 <= 42',
        '0.27x0 + 1.31x1 + 1.8x2 <= 42'
    ]
}
```

## Step 5: Implement the optimization problem using Gurobi
```python
import gurobi

def optimize():
    model = gurobi.Model()

    # Define variables
    x0 = model.addVar(name="milligrams of calcium", lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY)
    x1 = model.addVar(name="milligrams of vitamin B4", lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY)
    x2 = model.addVar(name="milligrams of vitamin B1", lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY)

    # Objective function
    model.setObjective(4.36 * x0**2 + 4.01 * x0 * x1 + 9.16 * x0 + 3.03 * x1 + 6.23 * x2, gurobi.GRB.MAXIMIZE)

    # Constraints
    model.addConstr(1.38 * x0 <= 108)
    model.addConstr(0.27 * x0 <= 57)
    model.addConstr(1.11 * x1 <= 108)
    model.addConstr(1.31 * x1 <= 57)
    model.addConstr(1.33 * x2 <= 108)
    model.addConstr(1.8 * x2 <= 57)
    model.addConstr(1.38 * x0 + 1.33 * x2 >= 17)
    model.addConstr(0.27 * x0**2 + 1.31 * x1**2 >= 13)
    model.addConstr(0.27 * x0 + 1.31 * x1 + 1.8 * x2 >= 12)
    model.addConstr(1.9044 * x0**2 + 1.7689 * x2**2 <= 70)
    model.addConstr(1.11 * x1 + 1.33 * x2 <= 108)
    model.addConstr(1.9044 * x0**2 + 1.2321 * x1**2 <= 51)
    model.addConstr(1.38 * x0 + 1.11 * x1 + 1.33 * x2 <= 51)
    model.addConstr(1.31 * x1 + 1.8 * x2 <= 35)
    model.addConstr(0.27 * x0 + 1.8 * x2 <= 42)
    model.addConstr(0.27 * x0 + 1.31 * x1 + 1.8 * x2 <= 42)

    # Solve the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Objective: ", model.objVal)
        print("Milligrams of calcium: ", x0.varValue)
        print("Milligrams of vitamin B4: ", x1.varValue)
        print("Milligrams of vitamin B1: ", x2.varValue)
    else:
        print("The model is infeasible")

optimize()
```