To solve this optimization problem, we first need to define the variables and the objective function symbolically, followed by translating the constraints into a symbolic representation. Finally, we'll express these in Gurobi code for computation.

### Symbolic Representation of the Problem

Let's denote:
- \(x_0\) as the quantity of milligrams of calcium.
- \(x_1\) as the quantity of milligrams of vitamin E.

The objective function to minimize is given by: 
\[4x_0 + 5x_1\]

The constraints can be summarized as follows:

1. Cardiovascular support index constraint:
   - Minimum combined cardiovascular support index: \(23x_0 + x_1 \geq 12\)
   - Maximum combined cardiovascular support index: \(23x_0 + x_1 \leq 38\)

2. Muscle growth index constraints:
   - Combined muscle growth index should be at least 48: \(22x_0 + 7x_1 \geq 48\)
   - Combined muscle growth index should not exceed 66: \(22x_0 + 7x_1 \leq 66\)

3. Cognitive performance index constraints:
   - Combined cognitive performance index must be at least 43: \(x_0 + 21x_1 \geq 43\)
   - Combined cognitive performance index should not exceed 81: \(x_0 + 21x_1 \leq 81\)

4. Immune support index constraints:
   - Combined immune support index must be at least 7: \(26x_0 + 6x_1 \geq 7\)
   - Combined immune support index should not exceed 31: \(26x_0 + 6x_1 \leq 31\)

5. Additional constraint:
   - \(6x_0 - 3x_1 \geq 0\)

Symbolic representation in JSON format:

```json
{
    'sym_variables': [('x0', 'milligrams of calcium'), ('x1', 'milligrams of vitamin E')],
    'objective_function': '4*x0 + 5*x1',
    'constraints': [
        '23*x0 + x1 >= 12',
        '23*x0 + x1 <= 38',
        '22*x0 + 7*x1 >= 48',
        '22*x0 + 7*x1 <= 66',
        'x0 + 21*x1 >= 43',
        'x0 + 21*x1 <= 81',
        '26*x0 + 6*x1 >= 7',
        '26*x0 + 6*x1 <= 31',
        '6*x0 - 3*x1 >= 0'
    ]
}
```

### Gurobi Code

Now, let's implement this optimization problem using Gurobi in Python:

```python
from gurobipy import *

# Create a model
m = Model("Optimization_Problem")

# Define variables
x0 = m.addVar(vtype=GRB.CONTINUOUS, name="milligrams_of_calcium")
x1 = m.addVar(vtype=GRB.CONTINUOUS, name="milligrams_of_vitamin_E")

# Set the objective function
m.setObjective(4*x0 + 5*x1, GRB.MINIMIZE)

# Add constraints
m.addConstr(23*x0 + x1 >= 12, "cardiovascular_support_index_min")
m.addConstr(23*x0 + x1 <= 38, "cardiovascular_support_index_max")
m.addConstr(22*x0 + 7*x1 >= 48, "muscle_growth_index_min")
m.addConstr(22*x0 + 7*x1 <= 66, "muscle_growth_index_max")
m.addConstr(x0 + 21*x1 >= 43, "cognitive_performance_index_min")
m.addConstr(x0 + 21*x1 <= 81, "cognitive_performance_index_max")
m.addConstr(26*x0 + 6*x1 >= 7, "immune_support_index_min")
m.addConstr(26*x0 + 6*x1 <= 31, "immune_support_index_max")
m.addConstr(6*x0 - 3*x1 >= 0, "additional_constraint")

# Optimize the model
m.optimize()

# Print solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Milligrams of Calcium: {x0.x}")
    print(f"Milligrams of Vitamin E: {x1.x}")
    print(f"Objective Function Value: {m.objVal}")
else:
    print("No optimal solution found")
```