How do I bind a WPF DataGrid to a variable number of columns?

My WPF application generates sets of data which may have a different number of columns each time. Included in the output is a description of each column that will be used to apply formatting. A simplified version of the output might be something like:

class Data
{
    IList<ColumnDescription> ColumnDescriptions { get; set; }
    string[][] Rows { get; set; }
}

This class is set as the DataContext on a WPF DataGrid but I actually create the columns programmatically:

for (int i = 0; i < data.ColumnDescriptions.Count; i++)
{
    dataGrid.Columns.Add(new DataGridTextColumn
    {
        Header = data.ColumnDescriptions[i].Name,
        Binding = new Binding(string.Format("[{0}]", i))
    });
}

Is there any way to replace this code with data bindings in the XAML file instead?