if my wife was rear-ended and has had a back injury, is there a formula to determine a dollar amount for pain and suffering?

There isn't any mathematical formula for determining compensation for pain and suffering.  The amount depends on the facts of the case.  If someone has residual complaints such as pain or other problems after completing their medical treatment, this would mean additional compensation for pain and suffering when compared to someone not having residucal complaints after completion of medical treatment.  If the medical reports state that future treatment is necessary, the estimated cost of that future treatment discounted to present value should be included in the settlement.

When your wife completes her medical treatment and is released by the doctor or is declared by the doctor to be permanent and stationary which means having reached a point in her medical treatment where no further improvement is anticipated, she should obtain her medical bills, medical reports and documentation of any wage loss.  Her personal injury claim filed with the at-fault party's insurance carrier should include these items.  Compensation for the medical bills is straight reimbursement.  Compensation for wage loss is straight reimbursement.  The medical reports will document the nature and extent of her injury and will be used to determine compensation for pain and suffering. 

I would usually ask for quadruple the medical bills to compensate for pain and suffering, but NOT expecting to get that.  This would be a starting point in negotiations with the insurance company.  The insurance company will respond with a much lower offer and your wife can continue negotiations to try to get the insurance company to increase its offer.  If the case is settled with the at-fault party's insurance carrier, NO lawsuit is filed.  If your wife is dissatisfied with settlement offers from the at-fault party's insurance carrier, she should reject the settlement offers and file a lawsuit for negligence against the at-fault party/registered owner of the vehicle (if someone other than the at-fault driver).  If the case is NOT settled with the at-fault party's insurance carrier, your wife will need to file her lawsuit for negligence against the at-fault party prior to the expiration of the applicable statute of limitations or she will lose her rights forever in the matter.