To address the given optimization problem, we first need to translate the natural language description into a symbolic representation. This involves defining variables, an objective function, and constraints based on the provided information.

Given:
- Variables: `milligrams of vitamin B4` and `milligrams of vitamin D`
- Objective Function: Minimize \(2 \times (\text{milligrams of vitamin B4})^2 + 5 \times \text{milligrams of vitamin B4}\)
- Constraints:
  1. Immune support index for milligrams of vitamin B4 is 1.
  2. Immune support index for milligrams of vitamin D is 8.
  3. Total combined immune support index from both vitamins squared must be at least 16.
  4. Total combined immune support index from both vitamins must be at least 16.
  5. \(10 \times \text{milligrams of vitamin B4} - 10 \times \text{milligrams of vitamin D} \geq 0\)
  6. Total combined immune support index from both vitamins must not exceed 19.

Let's assign symbolic variables:
- Let `x1` represent `milligrams of vitamin B4`.
- Let `x2` represent `milligrams of vitamin D`.

The objective function in terms of these symbolic variables is: \(2x_1^2 + 5x_1\)

Constraints:
1. \(x_1\) has an immune support index of 1, which doesn't directly translate into a constraint without more context but implies its coefficient in any immune-related calculation is 1.
2. \(x_2\) has an immune support index of 8.
3. \((x_1)^2 + (x_2)^2 \geq 16\)
4. \(x_1 + x_2 \geq 16\), considering the immune support indices are 1 and 8 respectively, this becomes \(x_1 + 8x_2 \geq 16\).
5. \(10x_1 - 10x_2 \geq 0\)
6. \(x_1 + 8x_2 \leq 19\)

Symbolic representation:
```json
{
  'sym_variables': [('x1', 'milligrams of vitamin B4'), ('x2', 'milligrams of vitamin D')],
  'objective_function': '2*x1**2 + 5*x1',
  'constraints': [
    '(x1)**2 + (x2)**2 >= 16',
    'x1 + 8*x2 >= 16',
    '10*x1 - 10*x2 >= 0',
    'x1 + 8*x2 <= 19'
  ]
}
```

Now, let's implement this in Gurobi:

```python
from gurobipy import *

# Create a new model
m = Model("Vitamin_Optimization")

# Add variables
x1 = m.addVar(name="milligrams_of_vitamin_B4", lb=-GRB.INFINITY, ub=GRB.INFINITY)
x2 = m.addVar(name="milligrams_of_vitamin_D", lb=-GRB.INFINITY, ub=GRB.INFINITY)

# Set the objective function
m.setObjective(2*x1**2 + 5*x1, GRB.MINIMIZE)

# Add constraints
m.addConstr(x1**2 + x2**2 >= 16, name="total_combinedimmune_support_index_squared")
m.addConstr(x1 + 8*x2 >= 16, name="total_combined_immune_support_index_min")
m.addConstr(10*x1 - 10*x2 >= 0, name="vitamin_b4_over_vitamin_d")
m.addConstr(x1 + 8*x2 <= 19, name="total_combined_immune_support_index_max")

# Optimize the model
m.optimize()

# Print solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Milligrams of Vitamin B4: {x1.x}")
    print(f"Milligrams of Vitamin D: {x2.x}")
else:
    print("No optimal solution found")
```