Migrating pages that inherit from a class that in turn inherits from System.Web.UI.Page

I'm trying to migrate an existing .NET site/app to Umbraco.  So far, I've migrated our master pages to Umbraco templates.

Now I'm trying to migrate the actual .aspx pages.  Many of these .aspx pages inherit from a custom base class page which in turn inherits from System.Web.UI.Page.

So, when I try to preview a page in Umbraco, the master has a user control that calls a method in the custom base class.  Meanwhile, the page that is passed to the method does not inherit from our custom base class, rather it's an Umbraco page, and I get the error message: Unable to cast object of type 'ASP.default_aspx' to type 'Com.MyCompany.MobileComponent.MobileComponentPage'.

Any ideas how to make this work whilst minimizing the changes to that base class?