To solve the given optimization problem using Gurobi, we first need to define the decision variables and then formulate the objective function and constraints based on the provided information.

The variables are:
- `zinc`: milligrams of zinc (continuous)
- `fat`: grams of fat (integer)
- `iron`: milligrams of iron (integer)
- `vitaminK`: milligrams of vitamin K (integer)
- `vitaminB5`: milligrams of vitamin B5 (integer)

The objective function is to minimize:
\[6 \times \text{zinc} + 1 \times \text{fat} + 5 \times \text{iron} + 8 \times \text{vitaminK} + 8 \times \text{vitaminB5}\]

Constraints are numerous and include both lower and upper bounds on various combinations of the variables based on their support indices.

Below is how you might implement this in Gurobi using Python:

```python
from gurobipy import *

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

# Define decision variables
zinc = model.addVar(lb=0, vtype=GRB.CONTINUOUS, name="milligrams_of_zinc")
fat = model.addVar(lb=0, vtype=GRB.INTEGER, name="grams_of_fat")
iron = model.addVar(lb=0, vtype=GRB.INTEGER, name="milligrams_of_iron")
vitaminK = model.addVar(lb=0, vtype=GRB.INTEGER, name="milligrams_of_vitamin_K")
vitaminB5 = model.addVar(lb=0, vtype=GRB.INTEGER, name="milligrams_of_vitamin_B5")

# Objective function
model.setObjective(6*zinc + 1*fat + 5*iron + 8*vitaminK + 8*vitaminB5, GRB.MINIMIZE)

# Constraints (only a few examples shown due to the large number)
# Immune support index constraints
model.addConstr(0.67*zinc + 0.33*iron >= 95, "immune_support_1")
model.addConstr(0.67*zinc + 0.33*vitaminK <= 133, "immune_support_2")

# Cardiovascular support index constraints
model.addConstr(0.5*fait + 0.5*iron <= 329, "cardiovascular_support_1")
model.addConstr(0.4*zinc + 0.3*vitaminK + 0.3*vitaminB5 <= 240, "cardiovascular_support_2")

# Cognitive support index constraints
model.addConstr(0.6*fait + 0.4*iron <= 140, "cognitive_support_1")
model.addConstr(0.7*zinc + 0.15*vitaminK + 0.15*vitaminB5 <= 295, "cognitive_support_2")

# Energy support index constraints
model.addConstr(0.3*fait + 0.7*vitaminB5 >= 68, "energy_support_1")
model.addConstr(0.4*zinc + 0.6*iron <= 166, "energy_support_2")

# Muscle growth index constraints
model.addConstr(0.6*iron + 0.4*vitaminK >= 40, "muscle_growth_1")
model.addConstr(0.5*zinc + 0.5*vitaminB5 <= 110, "muscle_growth_2")

# Additional constraints (only one example)
model.addConstr(-6*zinc + vitaminK >= 0, "additional_constraint")

# Optimize model
model.optimize()

# Print solution
for v in model.getVars():
    print("%s %f" % (v.varName, v.x))
```