To solve the given optimization problem using Gurobi, we first need to understand and possibly simplify the constraints provided. The objective function aims to maximize the value of \(1 \times \text{milligrams of vitamin B4} + 5 \times \text{milligrams of vitamin C}\).

Given constraints:
1. Muscle growth index for milligrams of vitamin B4 is 1, and for milligrams of vitamin C is 6.
2. The total combined muscle growth index must be at least 52.
3. The expression \(-1 \times \text{milligrams of vitamin B4} + 1 \times \text{milligrams of vitamin C} \geq 0\).
4. The total combined muscle growth index should not exceed 111.
5. Milligrams of vitamin B4 must be an integer, while milligrams of vitamin C can be any real number.

Let's denote:
- \(x_0\) as the amount of milligrams of vitamin B4,
- \(x_1\) as the amount of milligrams of vitamin C.

The objective function is: \(\max(1x_0 + 5x_1)\).

From constraint 2 and considering the muscle growth indices, we have \(1x_0 + 6x_1 \geq 52\).

Constraint 3 translates to \(-x_0 + x_1 \geq 0\) or \(x_1 \geq x_0\).

The upper bound constraint from point 4 gives us \(1x_0 + 6x_1 \leq 111\).

Since \(x_0\) must be an integer and \(x_1\) can be any real number, we'll ensure to define these variables accordingly in the Gurobi model.

Here's how you could implement this optimization problem using Gurobi in Python:

```python
from gurobipy import *

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

# Define variables
x0 = m.addVar(vtype=GRB.INTEGER, name="milligrams_of_vitamin_B4")
x1 = m.addVar(vtype=GRB.CONTINUOUS, name="milligrams_of_vitamin_C")

# Set the objective function
m.setObjective(1*x0 + 5*x1, GRB.MAXIMIZE)

# Add constraints
m.addConstr(1*x0 + 6*x1 >= 52, "muscle_growth_index_min")
m.addConstr(-x0 + x1 >= 0, "vitamin_balance")
m.addConstr(1*x0 + 6*x1 <= 111, "muscle_growth_index_max")

# Optimize the model
m.optimize()

# Print the solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Milligrams of Vitamin B4: {x0.x}")
    print(f"Milligrams of Vitamin C: {x1.x}")
    print(f"Objective Function Value: {m.objVal}")
else:
    print("No optimal solution found")
```